handbook of
statistics 14
Statistical Methods
in Finance
Edited by
i ;.s
C.R.kau
l

ELSEVIER SCIENCE B.V.
Sara Burgerhartstraal 25
P.O. Box 211, 1000 AE Amsterdam, The Netherlands
ISBN: 0-444-81964-9
© 1996 Elsevier Science B.V. All rights reserved.
No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form
or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior
written permission of the publisher, Elsevier Science B.V . Copyright & Permissions Department. P.O.
Box 521, 1000 AM Amsterdam. The Netherlands.
Special regulations for readers in the U.S.A.-This publication has been registered wilh the Copyright
Clearance Center Inc. (CCC). 222 Rosewood Drive. Danvers, MA 0192.1. Information can be obtained
from the CCC about conditions under which photocopies of parts oi' this publication may be made in
the U.S.A. All other copyright questions, including photocopying outside the U.S.A., should be
referred to the Publishers unless otherwise specified.
No responsibility is assumed by the publisher for any injury and or damage to persons or property as a
matter of products liability, negligence or otherwise, or from any use of operation of any methods,
products, instructions or ideas contained in the material herein.
This book is printed on acid-free paper.
Printed in The Netherlands.

Table of contents
Preface v
Contributors xv
PART I. ASSET PRICING
Ch. I. Econometric Evaluation of Asset Pricing Models 1
W. E. Person and R. Jagannathan
1. Introduction 1
2. Cross-sectionaI regression methods for testing beta pricing models 3
3. Asset pricing models and stochastic discount factors 10
4. The generalized method of moments 15
5. Model diagnostics 23
6. Conclusions 28
Appendix 29
References 30
Ch. 2. Instrumental Variables Estimation of Conditional Beta
Pricing Models 35
C. R. Harvey and C. M. Kirby
1. Introduction 35
2. Single beta models 37
3. Models with multiple betas 44
4. Latent variables models 46
5. Generalized method of moments estimation 47
6. Closing remarks 58
References 58

VIII
Table of contents
Ch. 3. Semiparametric Mclhods for Asset Pricing Models 61
B. N. Lehmann
1. Introduction 61
2. Some relevant aspects of the generalized method of moments (GMM) 62
3. Asset pricing relations and their econometric implications 68
4. Efficiency gains within alternative beta pricing formulations 74
5. Concluding remarks 87
References 88
PART II. TERM STRUCTURES OF INTEREST RATES
Ch. 4. Modeling the Term Structure 91
A. R. Pagan, A. D. Hall, and V. Martin
1. Introduction 91
2. Characteristics of term structure data 92
3. Models of the term structure 104
4. Conclusion 116
References 116
PART III. VOLATILITY
Ch. 5. Stochastic Volatility 119
E. Ghysels, A. C. Harvey and E. Renault
1. Introduction 119
2. Volatility in financial markets 120
3. Discrete lime models 139
4. Continuous time models 153
5. Statistical inference 167
6. Conclusions 182
References 183
Ch. 6. Slock Price Volatility 193
£ E. LeRoy
1. Introduction 193
2. Statistical issues 195
3. Dividend-smoothing and non stationarity 198
4. Bubbles 201
5. Time-varying discount rates 203

Table of contents
ix
6. Interpretation 204
7. Conclusion 206
References 207
Ch. 7. GARCH Models of Volatility 209
1. Introduction 209
2. GARCH models 210
3. Statistical inference 224
4. Statistical properties 229
5. Conclusions 234
References 235
PART IV. PREDICTION
Ch. 8. Forecast Evaluation and Combination 241
F. X. Diebold and J. A. Lopez
1. Evaluating a single forecast 242
2. Comparing the accuracy of multiple forecasts 247
3. Combining forecasts 252
4. Special topics in evaluating economic and financial forecasts 256
5. Concluding remarks 264
References 265
Ch. 9. Predictable Components in Stock Returns 269
G. Kaul
I Introduction 269
2. Why predictability 270
3. Predictability of stock returns: The methodology 273
4. Power comparisons 287
5. Conclusion 291
References 292
Ch. 10. Interest Rate Spreads as Predictors of Business Cycles 297
K. Lahiri and 7. G. Wang
1. Introduction 297
2. Hamilton's non-linear filter 299
3. Empirical results 301
4. Implications for the monetary transmission mechanism 308

X
Table of contents
5. Conclusion 311
Acknowledgement 313
References 313
PART V. ALTERNATIVE PROBABILISTIC MODELS
Ch. 11. Nonlinear Time Series, Complexity Theory, and Finance 317
W. A. Brock and P. J. F. de Lima
1. Introduction 317
2. Nonlinearity in stock returns 326
3. Long memory in stock returns 337
4. Asymmetric information structural models and stylized features of stock returns 349
5. Concluding remarks 353
References 353
Ch. 12. Count Data Models for Financial Data 363
A. C. Cameron and P. K. Trivedi
1. Introduction 363
2. Stochastic process models for count and duration data 366
3. Econometric models of counts 371
4. Concluding remarks 388
Acknowledgement 389
References 390
Ch. 13. Financial Applications of Stable Distributions 393
/. //. McCulloch
1. Introduction 393
2. Basic properties of stable distributions 394
3. Stable portfolio theory 401
4. Log-stable option pricing 404
5. Parameter estimation and empirical issues 415
Appendix 420
Acknowledgements 421
References 421
Ch. 14. Probability Distributions for Financial Models 427
./. B. McDonald
1. Introduction 427
2. Alternative models 428

Table of contents
xi
Applicalions in finance 437
Appendix A: Special functions 454
Appendix B: Data 456
Acknowledgement 458
References 45H
PART VI. APPLICATIONS OF SPECIALIZED STATISTICAL METHODS
Ch. 15. Bootstrap Based Tests in Financial Models 463
(7. S, Maddala and H. Li
I. Introduction 463
2
3
4
5
6
7
8
A review of different bootstrap methods 464
Issues in the generation of bootstrap samples and the test statistics 466
A critique of the application of bootstrap methods in financial models 469
Bootstrap methods for model selection using trading rules 476
Bootstrap methods in long-horizon regressions 478
Impulse response analysis in nonlinear models 483
Conclusions 484
References 485
Ch 16. Principal Components and Factor Analyses 489
C, R. Rao
1. Introduction 489
2. Principal components 490
3. Model based principal components 496
4. Factor analysis 498
5. Conclusions 503
References 5(M
Ch. 17. Errors-in-Variables Problems in Financial Models 507
G. S. Maddala and M'. Nimalendran
1. Introduction 507
2. Grouping methods 508
3. Alternatives to the two-pass estimation method 513
4. Direct and reverse regression methods 514
5. Latent variables / structural equation models with measurement 515
6. Artificial neural networks (ANN) as alternatives to MIMIC models 522
7. Signal extraction methods and tests for rationality 523
8. Qualitative and limited dependent variable models 523
9. Factor analysis with measurement errors 524
10. Conclusion 525
References 525

Ml
Table of contents
Ch. 18. Financial Applications of Artificial Neural Networks 529
M. Qi
\. Introduction 529
2. Artificial Neural Networks 529
3. Relationship between ANN and interpretational statistical models 533
4. ANN implementation and interpretation 537
5. Financial applications 544
6. Conclusions 547
Acknowledgement 548
References 548
Ch. 19. Applications of Limited Dependent Variable Models in Finance 553
G. S. Maddala
1. Introduction 553
2. Studies on loan discrimination and default 553
3. Studies on bond ratings and bond yields 555
4. Event studies 557
5. Savings and loan and bank failures 559
6. Miscellaneous other applications 562
7. Suggestions for future research 564
References 565
PART VII. MISCELLANEOUS OTHER PROBLEMS
Ch. 20. Testing Option Pricing Models 567
D. S. Bates
1. Introduction 367
2. Option pricing fundamentals 568
3. Time series-based tests of option pricing models 574
4. Implicit parameter estimation 587
5. Implicit parameter tests of alternate distributional hypotheses 600
6. Summary and conclusions 604
References 605
Ch. 21. Peso Problems: Their Theoretical and Empirical Implications 613
M. D. D. Evans
1. Introduction 613
2. Peso problems and forecast errors 615
3. Peso problems, asset prices and fundamentals 626
4. Risk aversion and peso problems 634

Table of contents
5. Econometric issues 641
6. Conclusion 644
References 645
Ch. 22. Modeling Market Microstructure Time Series 647
J. Ilasbrouck
1. Introduction 647
2. Simple univariate models of prices 651
3. Simple bivariate models of prices and trades 657
4. General specifications 667
5. Time 673
6. Discreteness 677
7. Nonlinearity 679
8. Multiple mechanisms and markets 680
9. Summary and directions for further work 685
References 687
Ch. 23. Statistical Methods in Tests of Portfolio Efficiency: A Synthesis
J. Shanken
1. Introduction 693
2. Testing efficiency with a riskless asset 695
3. Testing efficiency without a riskless asset 701
4. Related work 708
References 709
Subject Index 713
Contents of Previous Volumes 719

Preface
As with the earlier volumes in this series, the main purpose of this volume of
the Handbook of Statistics is to serve as a source reference and teaching
supplement to courses in empirical finance. Many graduate students and researchers in
the finance area today use sophisticated statistical methods but there is as yet no
comprehensive reference volume on this subject. The present volume is intended
to fill this gap.
The first part of the volume covers the area of asset pricing. In the first paper,
Ferson and Jagannathan present a comprehensive survey of the literature on
econometric evaluation of asset pricing models. The next paper by Harvey and
Kirby discusses the problems of instrumental variable estimation in latent
variable models of asset pricing. The next paper by Lehman reviews semi-parametric
methods in asset pricing models. Chapter 23 by Shanken also falls in the category
of asset pricing.
Part II of the volume on term structure of interest rates consists of only one
paper by Pagan, Hall and Martin. The paper surveys both the econometric and
finance literature in this area, and shows some similarities and divergences
between the two approaches. The paper also documents several stylized facts in the
data that prove useful in assessing the adequacy of the different models.
Part III of the volume deals with different aspects of volatility. The first paper
by Ghysels, Harvey and Renault present a comprehensive survey on the
important topic of stochastic volatility models. These models have their roots both
in mathematical finance and financial econometrics and are an attractive
alternative to the popular ARCH models. The next paper by LeRoy presents a critical
review of the literature on variance-bounds tests for market efficiency. The third
paper by Palm on GARCH models of stock price volatility, surveys some more
recent developments in this area. Several surveys on the ARCH models have
appeared in the literature and these are cited in the paper. The paper surveys
developments since the appearance of these surveys.
Part IV of the volume deals with prediction problems. The first paper by Diebold
and Lopez deals with the statistical methods of evaluation of forecasts. The second
paper by Kaul, reviews the literature on the predictability of stock returns. This area
has always fascinated those involved in making money in financial markets as well
as academics who presumably are interested in studying whether one can, in fact,
make money in the financial markets. The third paper by Lahiri reviews statistical

VI
Preface
evidence on interest rate spreads as predictors of business cycles. Since there is not
much of a literature to survey in this area, Lahiri presents some new results.
Part V of the volume deals with alternative probabilistic models in finance. The
first paper by Brock and deLima surveys several areas subsumed under the rubic
"complexity theory." This includes chaos theory, nonlinear time series models,
long memory models and models with asymmetric information. The next paper
by Cameron and Trivedi surveys the area of count data models in finance. In
some financial studies, the dependent variable is a count, taking non-negative
integer values. The next paper by McCulloch surveys the literature on stable
distributions. This area was very active in finance in the early 60's due to the work
by Mandelbrot but since then has not received much attention until recently when
interest in stable distributions has revived. The last paper by McDonald reviews
the variety of probability distributions which have been and can be used in the
statistical analysis of financial data.
Part VI deals with application of specialized statistical methods in finance. This
part covers important statistical methods that are of general applicability (to all
the models considered in the previous sections) and not covered adequately in the
other chapters. The first paper by Maddala and Li covers the area of bootstrap
methods. The second paper by Rao covers the area of principal component and
factor analyses which has, during recent years, been widely used in financial
research particularly in arbitrage pricing theory (APT). The third paper by
Maddala and Nimalendran reviews the area of errors in variables models as
applied to finance. Almost all variables in finance suffer from the errors in
variables problems. The fourth paper by Qi surveys the applications of artificial
neutral networks in financial research. These are general nonparametric nonlinear
models. The final paper by Maddala reviews the applications of limited dependent
variable models in financial research.
Part VII of the volume contains surveys of miscellaneous other problems. The
first paper by Bates surveys the literature on testing option pricing models. The
next paper by Evans discusses what are known in the financial literature as "peso
problems." The next paper by Hasbrouck covers market microstructure, which is
an active area of research in finance. The paper discusses the fime series work in
this area. The final paper by Shanken gives a comprehensive survey of tests of
portfolio efficiency.
One important area left out has been the use of Bayesian methods in finance.
In principle, all the problems discussed in the several chapters of the volume can
be analyzed from the Bayesian point of view. Much of this work remains to be
done.
Finally, we would like to thank Ms. Jo Ducey for her invaluable help at several
stages in the preparation of this volume and patient assistance in seeing the
manuscript through to publication.
G. S. Maddala
C. R. Rao

Contributors
D. S. Bates, Department of Finance, Wharton School, University of Pennsylvania,
Philadelphia, PA 19104, USA (Ch. 20)
W. A. Brock, Department of Economics, University of Wisconsin, Madison, WI
53706, USA (Ch. 11)
A. C. Cameron, Department of Economics, University of California at Davis,
Davis, CA 95616-8578, USA (Ch. 12)
P. J. F. de Lima, Department of Economics, The Johns Hopkins University,
Baltimore, MD 21218, USA (Ch. 11)
F. X. Diebold, Department of Economics, University of Pennsylvania,
Philadelphia, PA 19104, USA (Ch. 8)
M. D. D. Evans, Department of Economics, Georgetown University, Washington
DC 20057-1045, USA (Ch. 21)
W. E. Ferson, Department of Finance, University of Washington, Seattle, WA
98195, USA (Ch. 1)
E. Ghysels, Department of Economics, The Pennsylvania State University,
University Park, PA 16802 and CIRANO (Centre interuniversitaire de
recherche en analyse des organisations), Universite de Montreal, Montreal,
Quebec, Canada H3A2A5 (Ch. 5)
A. D. Hall, School of Business, Bond University, Gold Coast, QLD 4229, Australia
(Ch. 4)
A. C. Harvey, Department of Statistics, London School of Economics, Houghton
Street, London WC2A 2AE, UK (Ch. 5)
C. R. Harvey, Department of Finance, Fuqua School of Business, Box 90120, Duke
University, Durham, NC 27708-0120, USA (Ch. 2)
J. Hasbrouck, Department of Finance, Stern School of Business, 44 West 4th
Street, New York, NY 10012-1126, USA (Ch. 22)
R. Jagannathan, Finance Department, School of Business and Management, The
Hong Kong University of Science and Technology, Clear Water Bay, Kowloon,
Hong Kong (Ch. 1)
G. Kaul, University of Michigan Business School, Ann Harbor, MZ 48109-
1234 (Ch. 9)
C. M. Kirby, Department of Finance, College of Business & Mgm., University of
Maryland, College Park, MD 20742, USA (Ch. 2)
K. Lahiri, Department of Economics, State University of New York at Albany,
Albany, NY 12222 USA (Ch. 10)
XV

XVI
Contributors
B. N. Lehmann, Graduate School of International Relations, University of
California at San Diego, 9500 Gilman Drive, LaJolla, CA 92093-0519, USA
(Ch. 3)
S. F. LeRoy, Department of Economics, University of California at Santa
Barbara, Santa Barbara, CA 93106-9210 (Ch. 6)
H. Li, Department of Management Science, The Chinese University of Hongkong,
302 Leung Kau Kui Building, Shatin, NT, Hong Kong (Ch. 15)
J. A. Lopez, Department of Economics, University of Pennsylvania, Philadelphia,
PA 19104, USA (Ch. 8)
G. S. Maddala, Department of Economics, Ohio State University, 1945 N. High
Street, Columbus, OH 43210-1172, USA (Chs. 15, 17, 19)
V. Martin, Department of Economics, University of Melbourne, Parkville, VIC
3052, Australia (Ch. 4)
J. H. McCulloch, Department of Economics and Finance, 410 Arps Hall, 1945 N.
High Street, Columbus, OH 43210-1172, USA (Ch. 13)
J. B. McDonald, Department of Economics, Brigham Young University, Provo, UT
84602, USA (Ch. 14)
M. Nimalendran, Department of Finance, College of Business, University of
Florida, Gainesville, FL 32611, USA (Ch. 17)
A. R. Pagan, Economics Program, RSSS, Australian National University,
Canberra, ACT 0200, Australia (Ch. 4)
F. C. Palm, Department of Quantitative Economics, University of Limburg, P.O.
Box 616, 6200 MD Maastricht, The Netherlands (Ch. 7)
M. Qi, Department of Economics, College of Business Administration, Kent State
University, P.O. Box 5190, Kent, OH 44242 (Ch. 18)
C. R. Rao, The Pennsylvania State University, Center for Multivariate Analysis,
Department of Statistics, 325 Classroom Bldg., University park, PA 16802-
6105, USA (Ch. 16)
E. Renault, Institut D'Economie Industrielle, Universite des Sciences Sociales,
Place Anatole France, F-31042 Toulouse Cedex, France (Ch. 5)
J. Shanken, Department of Finance, Simon School of Business, University of
Rochester, Rochester, NY 14627, USA (Ch. 23)
P. K. Trivedi, Department of Economics, Indiana University, Bloomington, IN
47405-6620, USA (Ch. 12)
J. G. Wang, AT&T, Rm. N460-WOS, 412 Mt. Kemble Avenue, Morristown, NJ
07960, USA (Ch. 10)

G. S. Maddala and C. R. Rao, eds., Handbook of Statistics, Vol. 14
© 1996 Elsevier Science B.V. All rights reserved.
Econometric Evaluation of Asset Pricing Models
Wayne E. Ferson and Ravi Jagannathan
We provide a brief review of the techniques that are based on the generalized
method of moments (GMM) and used for evaluating capital asset pricing models.
We first develop the CAPM and multi-beta models and discuss the classical two-
stage regression method originally used to evaluate them. We then describe the
pricing kernel representation of a generic asset pricing model; this representation
facilitates use of the GMM in a natural way for evaluating the conditional and
unconditional versions of most asset pricing models. We also discuss diagnostic
methods that provide additional insights.
1. Introduction
A major part of the research effort in finance is directed toward understanding
why we observe a variety of financial assets with different expected rates of return.
For example, the U.S. stock market as a whole earned an average annual return
of 11.94% during the period from January of 1926 to the end of 1991. U.S.
Treasury bills, in contrast, earned only 3.64%. The inflation rate during the same
period was 3.11% (see Ibbotson Associates 1992).
To appreciate the magnitude of these differences, note that in 1926 a nice
dinner for two in New York would have cost about $10. If the same $10 had been
invested in Treasury bills, by the end of 1991 it would have grown to $110, still
enough for a nice dinner for two. Yet $10 invested in stocks would have grown to
$6,756. The point is that the average return differentials among financial assets
are both substantial and economically important.
A variety of asset pricing models have been proposed to explain this
phenomenon. Asset pricing models describe how the price of a claim to a future
payoff is determined in securities markets. Alternatively, we may view asset pri-
* Ferson acknowledges financial support from the Pigott-PACCAR Professorship at the
University of Washington. Jagannathan acknowledges financial support from the National Science
Foundation, grant SBR-9409824. The views expressed herein are those of the authors and not
necessarily those of the Federal Reserve Bank of Minneapolis or the Federal Reserve System.
1

2
W. E. Ferson and R. Jagannathan
cing models as describing the expected rates of return on financial assets, such as
stocks, bonds, futures, options, and other securities. Differences among the
various asset pricing models arise from differences in their assumptions that restrict
investors' preferences, endowments, production, and information sets; the
stochastic process governing the arrival of news in the financial markets; and the
type of frictions allowed in the markets for real and financial assets.
While there are differences among asset pricing models, there are also
important commonalities. All asset pricing models are based on one or more of three
central concepts. The first is the law of one price, according to which the prices of
any two claims which promise the same future payoff must be the same. The law
of one price arises as an implication of the second concept, the no-arbitrage
principle. The no-arbitrage principle states that market forces tend to align the
prices of financial assets to eliminate arbitrage opportunities. Arbitrage
opportunities arise when assets can be combined, by buying and selling, to form
portfolios that have zero net cost, no chance of producing a loss, and a positive
probability of gain. Arbitrage opportunities tend to be eliminated by trading in
financial markets, because prices adjust as investors attempt to exploit them. For
example, if there is an arbitrage opportunity because the price of security A is too
low, then traders' efforts to purchase security A will tend to drive up its price. The
law of one price follows from the no-arbitrage principle, when it is possible to buy
or sell two claims to the same future payoff. If the two claims do not have the
same price, and if transaction costs are smaller than the difference between their
prices, then an arbitrage opportunity is created. The arbitrage pricing theory
(APT, Ross 1976) is one of the most well-known asset pricing model based on
arbitrage principles.
The third central concept behind asset pricing models is financial market
equilibrium. Investors' desired holdings of financial assets are derived from an
optimization problem. A necessary condition for financial market equilibrium in
a market with no frictions is that the first-order conditions of the investors'
optimization problem be satisfied. This requires that investors be indifferent at the
margin to small changes in their asset holdings. Equilibrium asset pricing models
follow from the first-order conditions for the investors' portfolio choice problem
and from a market-clearing condition. The market-clearing condition states that
the aggregate of investors' desired asset holdings must equal the aggregate
"market portfolio" of securities in supply.
The earliest of the equilibrium asset pricing models is the Sharpe-Lintner-
Mossin-Black capital asset pricing model (CAPM), developed in the early
1960s. The CAPM states that expected asset returns are given by a linear function
of the assets' betas, which are their regression coefficients against the market
portfolio. Merton (1973) extended the CAPM, which is a single-period model, to
an economic environment where investors make consumption, savings, and
investment decisions repetitively over time. Econometrically, Merton's model
generalizes the CAPM from a model with a single beta to one with multiple betas.
A multiple-beta model states that assets' expected returns are linear functions of a
number of betas. The APT of Ross (1976) is another example of a multiple-beta

Econometric evaluation of asset pricing models
3
asset pricing model, although in the APT the expected returns are only
approximately a linear function of the relevant betas.
In this paper we emphasize (but not exclusively) the econometric evaluation of
asset pricing models using the generalized method of moments (GMM, Hansen
1982). We focus on the GMM because, in our opinion, it is the most important
innovation in empirical methods in finance within the past fifteen years. The
approach is simple, flexible, valid under general statistical assumptions, and often
powerful in financial applications. One reason the GMM is "general" is that
many empirical methods used in finance and other areas can be viewed as special
cases of the GMM.
The rest of this paper is organized as follows. In Section 2 we develop the
CAPM and multiple-beta models and discuss the classical two-stage regression
procedure that was originally used to evaluate these models. This material
provides an introduction to the various statistical issues involved in the empirical
study of the models; it also motivates the need for multivariate estimation
methods. In Section 3 we describe an alternative representation of the asset
pricing models which facilitates the use of the GMM. We show that most asset
pricing models can be represented in this stochastic discount factor form. In
Section 4 we describe the GMM procedure and illustrate how to use it to estimate
and test conditional and unconditional versions of asset pricing models. In
Section 5 we discuss model diagnostics that provide additional insight into the causes
for statistical rejections and that help assess specification errors in the models. In
order to avoid a proliferation of symbols, we sometimes use the same symbols to
mean different things in different subsections. The definitions should be clear from
the context. We conclude with a summary in Section 6.
2. Cross-sectional regression methods for testing beta pricing models
In this section we first derive the CAPM and generalize its empirical specification
to include multiple-beta models. We then describe the intuitively appealing cross-
sectional regression method that was first employed by Black, Jensen, and Scholes
(1972, abbreviated here as BJS) and discuss its shortcomings.
2.1. The capital asset pricing model
The CAPM was the first equilibrium asset pricing model, and it remains one of
the foundations of financial economics. The model was developed by Sharpe
(1964), Lintner (1965), Mossin (1966), and Black (1972). There are a huge number
of theoretical gapers which refine the necessary assumptions and provide
derivations of the CAPM. HeTewe provide-a-brief-review of the theory.
Let Rit denote one plus the return on asset i during period t, i = 1,2,..., N. Let
Rmt denote the corresponding gross return for the market portfolio of all assets in
the economy. The return on the market portfolio envisioned by the theory is not
observable. In view of this, empirical studies of the CAPM commonly assume

4
W. E. Ferson and R. Jagannathan
that the market return is an exact linear function of the return on an observable
portfolio of common stocks.1 Then, according to the CAPM,
E(Ru) = So + SiPt (2.1)
where
ft = Cov(tf;„tfm,)/Var(tfm,).
According to the CAPM, the market portfolio with return Rmt is on the
minimum-variance frontier of returns. A return is said to be on the minimum-
variance frontier if there is no other portfolio with the same expected return but
lower variance. If investors are risk averse, the CAPM implies that Rmt is on the
positively sloped portion of the minimum-variance frontier, which implies that
the coefficient b\ > 0. In equation (2.1), d0 = E(Ro,), where the return i?0i is
referred to as a zero-beta asset to Rmt because of the condition Cov(Rot,Rmt) = 0.
To derive the CAPM, assume that investors choose asset holdings at each date
t — 1 so as to maximize the following one-period objective function:
V[E(Rpt\I),Var(Rpl\I)} (2.2)
where Rpt denotes the date t return on the optimally chosen portfolio and E(-\I)
and Var(-|7) denote the expectation and variance of return, conditional on the
information set / of the investor as of time t-\. We assume that the function V[- > •]
is increasing and concave in its first argument, decreasing in its second argument,
and time-invariant. For the moment we assume that the information set /includes
only the unconditional moments of asset returns, and we drop the symbol / to
simplify the notation. The first-order conditions for the optimization problem
given above can be manipulated to show that the following must hold:
E(RU) = E(R0t) + pipE(Rpt - R0I) (2.3)
for every asset / = 1, 2, ..., N, where Rpt is the return on the optimally chosen
portfolio, Rot is the return on the asset that has zero covariance with Rpt, and pip
= Cov(Rit,Rpt)IVar(Rpt).
To get from the first-order condition for an investor's optimization problem, as
stated in equation (2.3), to the CAPM, it is useful to understand some of the
properties of the minimum-variance frontier, that is, the set of portfolio returns
with the minimum variance, given their expected returns. It can be readily verified
that the optimally chosen portfolio of the investor is on the minimum-variance
frontier.
One property of the minimum-variance frontier is that it is closed to portfolio
formation. That is, portfolios of frontier portfolios are also on the frontier.
1 When this assumption fails, it introduces market proxy error. This source of error is studied by
Roll (1977), Stambaugh (1982), Kandel (1984), Kandel and Stambaugh (1987), Shanken (1987),
Hansen and Jagannathan (1994), and Jagannathan and Wang (1996), among others. We will ignore
proxy error in our discussion.

Econometric evaluation of asset pricing models
5
Suppose that all investors have the same beliefs. Then every investor's optimally
chosen portfolio will be on the same frontier, and hence the market portfolio of
all assets in the economy - which is a portfolio of every investor's optimally
chosen portfolio - will also be on the frontier. It can be shown (Roll 1977) that
equation (2.3) will hold if Rpt is replaced by the return of any portfolio on the
frontier and RQt is replaced by its corresponding zero-beta return. Hence we can
replace an investor's optimal portfolio in equation (2.3) with the return on the
market portfolio to get the CAPM, as given by equation (2.1).
2.2. Testable implications of the CAPM
Given an interesting collection of assets, and if their expected returns and market-
portfolio betas Pi are known, a natural way to examine the CAPM would be
to estimate the empirical relation between the expected returns and the betas and
see if that relation is linear. However, neither betas nor expected returns are
observed by the econometrician. Both must be estimated. The finance literature
first attacked this problem by using a two-step, time-series, cross-sectional
approach.
Consider the following sample analogue of the population relation given in
(2.1):
Rt = 50 + Sibi + <?,-, i = 1,... ,JV (2.4)
which is a cross-sectional regression of i?; on bt, with regression coefficients equal
to S0 and ^i- IQ equation (2.4), i?, denotes the sample average return of the asset, i,
and bt is the (OLS) slope coefficient estimate from a regression of the return, Rit,
over time on the market index return, Rmt; bt is a constant. Let ut = Ri—E(Rit)
and vt = Pf-bi. Substituting these relations for E(RU) and /?, in (2.1) leads to (2.4)
and specifies the composite error as et = Ut + d^t- This gives rise to a classic
errors-in-variables problem, as the regressor bt in the cross-sectional regression
model (2.4) is measured with error. Using finite time-series samples for the
estimate of bh the regression (2.4) will deliver inconsistent estimates of d0 and di, even
with an infinite cross-sectional sample. However, the cross-sectional regression
will provide consistent estimates of the coefficients as the time-series sample size T
(which is used in the first step to estimate the beta coefficient /},) becomes very
large. This is because the first-step estimate of pt is consistent, so as T becomes
large, the errors-in-variables problem of the second-stage regression vanishes.
The measurement error in beta may be large for individual securities, but it is
smaller for portfolios. In view of this fact, early research focused on creating
portfolios of securities in such a way that the betas of the portfolios could be
estimated precisely. Hence one solution to the errors-in-variables problem is to
work with portfolios instead of individual securities. This creates another
problem. Arbitrarily chosen portfolios tend to exhibit little dispersion in their betas. If
all the portfolios available to the econometrician have the same betas, then
equation (2.1) has no empirical content as a cross-sectional relation. Black,
Jensen, and Scholes (BJS, 1972) came up with an innovative solution to overcome

6
W. E. Ferson and R. Jagannathan
this difficulty. At every point in time for which a cross-sectional regression is run,
they estimate betas on individual securities based on past history, sort the
securities based on the estimated values of beta, and assign individual securities to
beta groups. This results in portfolios with a substantial dispersion in their betas.
Similar portfolio formation techniques have become standard practice in the
empirical finance literature.
Suppose that we can create portfolios in such a way that we can view the
errors-in-variables problem as being of second-order importance. We still have to
determine how to assess whether there is empirical support for the CAPM. A
standard approach in the literature is to consider specific alternative hypotheses
about the variables which determine expected asset returns. According to the
CAPM, the expected return for any asset is a linear function of its beta only.
Therefore, one natural test would be to examine if any other cross-sectional
variable has the ability to explain the deviations from equation (2.1). This is the
strategy that Fama and MacBeth (1973) followed by incorporating the square of
beta and measures of nonmarket (or residual time-series) variance as additional
variables in the cross-sectional regressions. More recent empirical studies have
used the relative size of firms, measured by the market value of their equity, the
ratio of book-to-market-equity, and related variables.2 For example, the
following model may be specified:
E(Ru) = d0 + SiP, + <5size LMEt (2.5)
where LMEt is the natural logarithm of the total market value of the equity
capital of firm i. In what follows we will first show that these ideas extend easily to
the general multiple-beta model. We will then develop a sampling theory for the
cross-sectional regression estimators.
2.3. Multiple-beta pricing models and cross-sectional regression methods
According to the CAPM, the expected return on an asset is a linear function of its
market beta. A multiple-beta model asserts that the expected return is a linear
function of several betas, i.e.,
E(Rit)=50+ J2 W* (2-6)
k=\,...JC
where fiik, k — 1,... ,K, are the multiple regression coefficients of the return of
asset i on K economy-wide pervasive risk factors, fk,k — \,...,K. The coefficient
(5o is the expected return on an asset that has p0k = 0, for k = 1,..., K; i.e., it is the
expected return on a zero- (multiple-) beta asset. The coefficient bk, corresponding
to the £th factor, has the following interpretation: it is the expected return
differential, or premium, for a portfolio that has Pik = 1 and ptJ = 0 for all j =£ k,
2 Fama and French (1992) is a prominent recent example of this approach. Berk (1995) provides a
justification for using relative market value and book-to-price ratios as measures of expected returns.

Econometric evaluation of asset pricing models
7
measured in excess of the zero-beta asset's expected return. In other words, it is
the expected return premium per unit of beta risk for the risk factor, k. Ross
(1976) showed that an approximate version of (2.6) will hold in an arbitrage-free
economy. Connor (1984) provided sufficient conditions for (2.6) to hold exactly in
an economy with an infinite number of assets in general equilibrium. This version
of the multiple-beta model, the exact APT, has received wide attention in the
finance literature. When the factors, /&, are observed by the econometrician, the
cross-sectional regression method can be used to empirically evaluate the
multiple-beta model.3 For example, the alternative hypothesis that the size of the firm
is related to expected returns, given the factor betas, may be examined by using
cross-sectional regressions of returns on the K factor betas and the LMEh similar
to equation (2.5), and by examining whether the coefficient Ssize is different from
zero.
2.4. Sampling distributions for coefficient estimators: The two-stage,
cross-sectional regression method
In this section we follow Shanken (1992) and Jagannathan and Wang (1993, 1996)
in deriving the asymptotic distribution of the coefficients that are estimated using
the cross-sectional regression method. For the purposes of developing the
sampling theory, we will work with the following generalization of equation (2.6):
E(**)=f>,^* + f>2*fl* (2.7)
fc=0 k=\
where {Aik} are observable characteristics of firm /, which are assumed to be
measured without error (the first "characteristic," when k = 0, is the constant
1.0). One of the attributes may be the size variable LMEt. The ft are regression
betas on a set of A^ economic risk factors, which may include the market index
return. Equation (2.7) can be written more compactly using matrix notation as
li=Xy (2.8)
where Rt = [Rlt,... ,Rm], A* = E(Rt),X = [A : ft1, and the definition of the matrices
A and ji and the vector y follow from (2.7).
The cross-sectional method proceeds in two stages. First, ft is estimated by
time-series regressions of Ru on the risk factors and a constant. The estimates are
denoted by b. Let x = [A : b], and let R denote the time-series average of the return
vector Rt. Let g denote the estimator of the coefficient vector obtained from the
following cross-sectional regression:
g = (x'x)~xx!R (2.9)
3 See Chen (1983), Connor and Korajczyk (1986), Lehmann and Modest (1987), and McElroy and
Burmeister (1988) for discussions on estimating and testing the model when the factor realizations are
not observable under some additional auxiliary assumptions.

8
W. E. Ferson and R. Jagannathan
where we assume that x is of rank I + K\ + K2. If b and R converge respectively
to fi and E(Rt) in probability, then g will converge in probability to y.
Black, Jensen, and Scholes (1972) suggest estimating the sampling errors
associated with the estimator, g, as follows. Regress Rt on x at each date / to obtain
gt, where
g, = {x'x)~lx'Rt . (2.10)
The BJS estimate of the covariance matrix of Tl! (g — y) is given by
v = T-1'£(gt-g)(gt-g)' (2.11)
t
which uses the fact that g is the sample mean of the gt's. Substituting the
expression for gt given in (2.10) into the expression for v given in (2.11) gives
v = (x'x)-lx'[T-l^2(Rt - R)(Rt - R)'}x{x'XyX . (2.12)
t
To analyze the BJS covariance matrix estimator, we write the average return
vector, R, as
R=xy + (R-n) -{x-X)y . (2.13)
Substitute this expression for R into the expression for g in (2.9) to obtain
g-y = (x'xylx'[(R - ») - (b - fi)y2] . (2.14)
Assume that b is a consistent estimate of ft and that TXI2{R — fi) —>d u and
Txl2{b — fi) —>,/ h, where u and h are random variables with well-defined
distributions and —></ indicates convergence in distribution. We then have
Tl/2(g - y) -^d (x'x)~lx'u - (x'x)~lx'hy2 . (2.15)
In (2.15) the first term on the right side is that component of the sampling error
that arises from replacing u by the sample average R. The second term is the
component of the sampling error that arises due to replacing fi by its estimate b.
The usual consistent estimate of the asymptotic variance of u is given by
T-lJ2(Rt-R)(Rt-R)' . (2.16)
t
Therefore, a consistent estimate of variance of the first term in (2.15) is given by
(x'x)-lx'[T-1 J2(Rt -R)(Rt -R)']x{x'xYx
t
which is the same as the expression for the BJS estimate for the covariance matrix
of the estimated coefficients v, given in (2.12). Hence if we ignore the sampling
error that arises from using estimated betas, then the BJS covariance estimator

Econometric evaluation of asset pricing models
9
provides a consistent estimate of the variance of the estimator g. However, if the
sampling error associated with the betas is not small, then the BJS covariance
estimator will have a bias. While it is not possible to determine the magnitude of
the bias in general, Shanken (1992) provides a method to assess the bias under
additional assumptions.4
Consider the following univariate time-series regression for the return of asset i
on a constant and the kth economic factor:
Ru = ocik + Pikfkt + eat . (2.17)
We make the following additional assumptions about the error terms in (2.17): (1)
the error £,& is mean zero, conditional on the time series of the economic factors
fk; (2) the conditional covariance of eikt and £/&, given the factors, is a fixed
constant <7y«. We denote the matrix of the {ffywjy by Z«. Finally, we assume that
(3) the sample covariance matrix of the factors exists and converges in probability
to a constant positive definite matrix Q, with the typical element Q«.
Theorem 2.1. (Shanken, 1992/Jagannathan and Wang, 1996)
Txl2(g — y) converges in distribution to a normally distributed random variable
with zero mean and covariance matrix V + W, where V is the probability limit of
the matrix v given in (2.12) and
W= Y. {^x)-xX'{y2ky2l{Qr^Ilkl^)}x{xlx)-x (2.18)
l,k=\,...fa
where JJW is defined in the appendix.
Proof. See the appendix.
Theorem 2.1 shows that in order to obtain a consistent estimate of the co-
variance matrix of the BJS two-step estimator g, we first estimate v (a consistent
estimate of V) by using the BJS method. We then estimate W by its sample
analogue.
Although the cross-sectional regression method is intuitively very appealing,
the above discussion shows that in order to assess the sampling errors associated
with the parameter estimators, we need to make rather strong assumptions. In
addition, the econometrician must take a stand on a particular alternative
hypothesis against which to reject the model. The general approach developed in
Section 4 below has, among its advantages, weaker statistical assumptions and
the ability to handle both unspecified as well as specific alternative hypotheses.
4 Shanken (1992) uses betas computed from multiple regressions. The derivation which follows
uses betas computed from univariate regressions, for simplicity of exposition. The two sets of betas are
related by an invertible linear transformation. Alternatively, the factors may be orthogonalized
without loss of generality.

10
W. E. Ferson and R. Jagannathan
3. Asset pricing models and stochastic discount factors
Virtually all financial asset pricing models imply that any gross asset return
Ri,t+i, multiplied by some market-wide random variable mt+1, has a constant
conditional expectation:
E, {mt+iRitt+i} = l,all i. (3.1)
The notation Et{} will be used to denote the conditional expectation, given a
market-wide information set. Sometimes it will be convenient to refer to
expectations conditional on a subset Z, of the market information, which are
denoted as E(-1 Zt). For example, Zt can represent a vector of instrumental variables
for the public information set which are available to the econometrician. When Zt
is the null information set, the unconditional expectation is denoted as E(-). If we
take the expected values of equation (3.1), it follows that versions of the same
equation must hold for the expectations E(-1Zt) and E().
The random variable mt+\ has various names in the literature. It is known as a
stochastic discount factor, an equivalent martingale measure, a Radon-Nicodym
derivative, or an intertemporal marginal rate of substitution. We will refer to an
mt + i which satisfies (3.1) as a valid stochastic discount factor. The motivation for
use of this term arises from the following observation. Write equation (3.1) as
Pit = E,{m,+iXij+i}
where Xiyt+1 is the payoff of asset i at time t + 1 (the market value plus any cash
payments) and R^t+i = Xif+\jPit. Equation (3.1) says that if we multiply a future
payoff Xtj+i by the stochastic discount factor mt+\ and take the expected value,
we obtain the present value of the future payoff.
The existence of an mt+\ that satisfies (3.1) says that all assets with the same
payoffs have the same price (i.e., the law of one price). With the restriction that
mt+1 is a strictly positive random variable, equation (3.1) becomes equivalent to a
no-arbitrage condition. The condition is that all portfolios of assets with payoffs
that can never be negative, but are positive with positive probability, must have
positive prices.
The no-arbitrage condition does not uniquely identify mt+\ unless markets are
complete, which means that there are as many linearly independent payoffs
available in the securities markets as there are states of nature at date t + 1. To
obtain additional insights about the stochastic discount factor and the
no-arbitrage condition, assume for the moment that the markets are complete. Given
complete markets, positive state prices are required to rule out arbitrage
opportunities.5 Let qts denote the time t price of a security that pays one unit at date
t + 1 if, and only if, the state of nature at t + 1 is s. Then the time t price of a
5 See Debreu (1959) and Arrow (1970) for models of complete markets. See Beja (1971),
Rubinstein (1976), Ross (1977), Harrison and Kreps (1979), and Hansen and Richard (1987) for further
theoretical discussions.

Econometric evaluation of asset pricing models
11
security that promises to pay {XiiS>l+l} units at date t + 1, as a function of the
state of nature s, is given by
2_^1ts^i,s,t+\ = / Kts{<ltslftts)Xi,s,t+\
s s
where nts is the probability, as assessed at time t, that state s occurs at time t + 1.
Comparing this expression with equation (3.1) shows that mStt+\ = qts/nts is the
value of the stochastic discount factor in state s, under the assumption that the
markets are complete. Since the probabilities are positive, the condition that the
random variable defined by {mst+\} is strictly positive is equivalent to the
condition that all state prices are positive.
Equation (3.1) is convenient for developing econometric tests of asset pricing
models. Let Rt+\ denote the vector of gross returns on the N assets on which the
econometrican has observations. Then (3.1) can be written as
E{Rt+lmt+l} - I =0 (3.2)
where 1_ denotes the N vector of ones and 0 denotes the N vector of zeros. The set
of N equations given in (3.2) will form the basis for tests using the generalized
method of moments. It is the specific form of mt+ \ implied by a model that gives
the equation empirical content.
3.1. Stochastic discount factor representations of the CAPM
and multiple-beta asset pricing models
Consider the CAPM, as given by equation (2.1):
E(Rit+l) = d0 + dlpi
where
ft = Cov(Ri!+uRmt+i)/Yar(Rmt+i) .
The CAPM can also be expressed in the form of equation (3.1), with a particular
specification of the stochastic discount factor. To see this, expand the expected
product in (3.1) into the product of the expectations plus the covariance, and then
rearrange to obtain
E(Ri[+l) = l/E(i»r+i) + Cov(Rit+i;-mt+i/E(mt+l)) . (3.3)
Equating terms in equations (2.1) and (3.3) shows that the CAPM of equation
(2.1) is equivalent to a version of equation (3.1), where
E(Rit+iml+i) = 1
where
mt+\ = cq - c\Rmt+\
c0 = [1 +E(Rmt+l)dl/Yar(Rml+l)}/So
(3.4)

12
W. E. Ferson and R. Jagannathan
and
ci = <5i/[<50Var(tfm(+i)].
Equation (3.4) was originally derived by Dybvig and Ingersoll (1982).
Now consider the following multiple-beta model which was given in equation
(2.6):
E(Rlt+i) = S0 + ]T dkpik .
k=\,...,K
It can be readily verified by substitution that this model implies the following
stochastic discount factor representation:
E(Rit+lmit+l) = 1
where
mu+i = Co + ci/h+i + ■ • • + cKfKt+i
with
co = [1 + £{4E(/*)/Var(/t)}]/«5o (3.5)
k
and
cj = - {8j/50Var(fj)}, j=l,...,K .
The preceding results apply to the CAPM and multiple-beta models, interpreted
as statements about the unconditional expected returns of the assets. These
models are also interpreted as statements about conditional expected returns in
some tests where the expectations are conditioned on predetermined, publicly
available information. All of the analysis of this section can be interpreted as
applying to conditional expectations, with the appropriate changes in notation. In
this case, the parameters c0, cu S0, Su etc., will be functions of the time /
information set.
3.2. Other examples of stochastic discount factors
In equilibrium asset pricing models, equation (3.1) arises as a first-order condition
for a consumer-investor's optimization problem. The agent maximizes a lifetime
utility function of consumption (including possibly a bequest to heirs). Denote
this function by V(-). If the allocation of resources to consumption and to
investment assets is optimal, it is not possible to obtain higher utility by changing
the allocation. Suppose that an investor considers reducing consumption at time /
to purchase more of (any) asset. The utility cost at time / of the forgone
consumption is the marginal utility of consumption expenditures Ct, denoted
by (dV jdCt) > 0, multiplied by the price Pu of the asset, measured in the same
units as the consumption expenditures. The expected utility gain of selling the
share and consuming the proceeds at time / + 1 is

Econometric evaluation of asset pricing models
13
Et{{Pif+l+Dtt+l){dVldCt+l)}
where A,*+i is the cash flow or dividend paid at time t+l. If the allocation
maximizes expected utility, the following must hold:
p^Midv/dc,)} = e,{(/Vh +Dit!+l)(dv/dct+l)}.
This intertemporal Euler equation is equivalent to equation (3.1), with
mt+l = (dv/dct+l)/Et{(dv/dct)} . (3.6)
The mt+i in equation (3.6) is the intertemporal marginal rate of substitution
(IMRS) of the representative consumer. The rest of this section shows how many
models in the asset pricing literature are special cases of (3.1), where mt+\ is
defined by equation (3.6).6
If a representative consumer's lifetime utility function V(-) is time-separable,
the marginal utility of consumption at time t, (dV/dCt), depends only on variables
dated at time t. Lucas (1978) and Breeden (1979) derived consumption-based
asset pricing models of the following type, assuming that the preferences are time-
separable and additive:
V = ^P'u(Ct)
t
where ft is a time discount parameter and «(■) is increasing and concave in current
consumption Ct. A convenient specification for w() is
u(C) = [Cl-« - 1]/(1 - a) . (3.7)
In equation (3.7), a > 0 is the concavity parameter of the period utility function.
This function displays constant relative risk aversion equal to a.7 Based on these
assumptions and using aggregate consumption data, a number of empirical
studies test the consumption-based asset pricing model.8
Dunn and Singleton (1986) and Eichenbaum, Hansen, and Singleton (1988),
among others, model consumption expenditures that may be durable in nature.
Durability introduces nonseparability over time, since the flow of consumption
services depends on the consumer's previous expenditures, and the utility is de-
6 Asset pricing models typically focus on the relation of security returns to aggregate quantities. It
is therefore necessary to aggregate the Euler equations of individuals to obtain equilibrium expressions
in terms of aggregate quantities. Theoretical conditions which justify the use of aggregate quantities
are discussed by Gorman (1953), Wilson (1968), Rubinstein (1974), Constantinides (1982), Lewbel
(1989), Luttmer (1993), and Constantinides and Duffle (1994).
7 Relative risk aversion in consumption is defined as -Cw"(C)/w'(C). Absolute risk aversion is
—u"(C)/u'(C), where a prime denotes a derivative. Ferson (1983) studies a consumption-based asset
pricing model with constant absolute risk aversion.
8 Substituting (3.7) into (3.6) shows that m,+1 = /3(C,+i/'C,)~a:. Empirical studies of this model
include Hansen and Singleton (1982, 1983), Ferson (1983), Brown and Gibbons (1985), Jagannathan
(1985), Ferson and Merrick (1987), and Wheatley (1988).

14
W. E. Ferson and R. Jagannathan
fined over the services. Current expenditures increase the consumer's future
utility of services if the expenditures are durable. The consumer optimizes over the
expenditures Ct; thus, durability implies that the marginal utility, (dV/dCt),
depends on variables dated other than date t.
Another form of time-nonseparability arises if the utility function exhibits
habit persistence. Habit persistence means that consumption at two points in time
are complements. For example, the utility of current consumption is evaluated
relative to what was consumed in the past. Such models are derived by Ryder and
Heal (1973), Becker and Murphy (1988), Sundaresan (1989), Constantinides
(1990), Detemple and Zapatero (1991), and Novales (1992), among others.
Ferson and Constantinides (1991) model both the durability of consumption
expenditures and habit persistence in consumption services. They show that the
two combine as opposing effects. In an example where the effect is truncated at a
single lag, the derived utility of expenditures is
V=(l~arlJ2P'(Ct + bCt^0i . (3.8)
t
The marginal utility at time t is
(dv/dct) = p{c, + bc-x)-" + pt+lbEt {(ct+y + bctya} . (3.9)
The coefficient b is positive and measures the rate of depreciation if the good is
durable and there is no habit persistence. If habit persistence is present and the
good is nondurable, this implies that the lagged expenditures enter with a negative
effect (b < 0).
Ferson and Harvey (1992) and Heaton (1995) consider a form of
time-nonseparability which emphasizes seasonality. The utility function is
(l-ay^PiQ + bQ-rf-"
t
where the consumption expenditure decisions are assumed to be quarterly. The
subsistence level (in the case of habit persistence) or the flow of services (in the
case of durability) is assumed to depend only on the consumption expenditure in
the same quarter of the previous year.
Abel (1990) studies a form of habit persistence in which the consumer evaluates
current consumption relative to the aggregate consumption in the previous
period, consumption that he or she takes as exogenous. The utility function is like
equation (3.8), except that the "habit stock," bCt-\, refers to the aggregate
consumption. The idea is that people care about "keeping up with the Joneses."
Campbell and Cochrane (1995) also develop a model in which the habit stock is
taken as exogenous by the consumer. This approach results in a simpler and more
tractable model, since the consumer's optimization does not have to take account
of the effects of current decisions on the future habit stock.
Epstein and Zin (1989, 1991) consider a class of recursive preferences which
can be written as Vt =F(Ct,CEQt{Vt+i)). CEQt{-) is a time t "certainty equiva-

Econometric evaluation of asset pricing models
15
lent" for the future lifetime utility Vt+\. The function F(-,CEQt(-)) generalizes the
usual expected utility function of lifetime consumption and may be time-non-
separable.
Epstein and Zin (1989) study a special case of the recursive preference model in
which the preferences are
V, = [(1 - P)C? + pEt(V?-nP/(l~x)]l/p ■ (3-10)
They show that when p =£ 0 and 1 - a =£ 0, the IMRS for a representative agent
becomes
[P(Ct+ilCt)p-l}(l-*)lp{Rm,t+x}({l-a-p)lp) (3.11)
where Rm>t+\ is the gross market portfolio return. The coefficient of relative risk
aversion for timeless consumption gambles is a, and the elasticity of substitution
for deterministic consumption is (1 — p)~ . If a = 1 — p, the model reduces to the
time-separable, power utility model. If a = 1, the log utility model of Rubinstein
(1976) is obtained.
In summary, many asset pricing models are special cases of the equation (3.1).
Each model specifies that a particular function of the data and the model
parameters is a valid stochastic discount factor. We now turn to the issue of estimating
the models stated in this form.
4. The generalized method of moments
In this section we provide an overview of the generalized method of moments and
a brief review of the associated asymptotic test statistics. We then show how the
GMM is used to estimate and test various specifications of asset pricing models.
4.1. An overview of the generalized method of moments in asset pricing models
Let xt+\ be a vector of observable variables. Given a model which specifies
mt+\ = m(8,xt+\), estimation of the parameters 8 and tests of the model can then
proceed under weak assumptions, using the GMM as developed by Hansen
(1982) and illustrated by Hansen and Singleton (1982) and Brown and Gibbons
(1985). Define the following model error term:
Uij+i =m{e,xt+])Rut+i - 1 . (4.1)
The equation (3.1) implies that Et{uiit+i} = 0 for all i. Given a sample of TV assets
and T time periods, combine the error terms from (4.1) into a T x N matrix u,
with typical row u't+1. By the law of iterated expectations, the model implies that
E(iijj+i \Zt) = 0 for all / and t (for any Zt in the information set at time t), and
therefore E(ut+\Zt) = 0 for all t. The condition E(ut+\Zt) = 0 says that ut+\ is
orthogonal to Zt and is therefore called an orthogonality condition. These or-

16
W. E. Ferson and R. Jagannathan
thogonality conditions are the basis of tests of asset pricing models using the
GMM.
A few points deserve emphasis. First, GMM estimates and tests of asset pricing
models are motivated by the implication that E(«,-]t+i \Zt) = 0, for any Zt in the
information set at time t. However, the weaker condition H(ut+\Zt) = 0, for a
given set of instruments Zt, is actually used in the estimation. Therefore, GMM
tests of asset pricing models have not exploited all of the predictions of the
theories. We believe that further refinements to exploit the implications of the
theories more fully will be useful.
Empirical work on asset pricing models relies on rational expectations,
interpreted as the assumption that the expectation terms in the model are
mathematical conditional expectations. For example, the rational expectations assumption
is used when the expected value in equation (3.1) is treated as a mathematical
conditional expectation to obtain expressions for E(|Z) and E(). Rational
expectations implies that the difference between observed realizations and the
expectations in the model should be unrelated to the information that the
expectations are conditioned on.
Equation (3.1) says that the conditional expectation of the product of mt+\ and
Ritt+i is the constant 1.0. Therefore, the error term 1 — mt+\Riit+\ in equation (4.1)
should not be predictably different from zero when we use any information
available at time t. If there is variation over time in a return Ri^+\ that is
predictable using instruments Z„ the model implies that the predictability is removed
when Ri,t+\ is multiplied by a valid stochastic discount factor, mt+\. This is the
sense in which conditional asset pricing models are asked to "explain" predictable
variation in asset returns. This idea generalizes the "random walk" model of
stock values, which implies that stock returns should be completely unpredictable.
That model is a special case which can be motivated by risk neutrality. Under risk
neutrality the IMRS is a constant. In this case, equation (3.1) implies that the
return Riit+\ should not differ predictably from a constant.
GMM estimation proceeds by defining an NxL matrix of sample mean
orthogonality conditions, G — (u'Z/T), and letting g = vec(G), where Z is a TxL
matrix of observed instruments with typical row Z/, a subset of the available
information at time t? The vec(-) operator means to partition G into row vectors,
each of length L: (hu h2, ..., hN). Then one stacks the h's into a vector, g, with
length equal to the number of orthogonality conditions, NL. Hansen's (1982)
GMM estimates of 8 are obtained by searching for parameter values that make g
close to zero by minimizing a quadratic form g'Wg, where W is an NLxNL
weighting matrix.
Somewhat more generally, let ut+\{&) denote the random TV vector
Rt+xm(0,x,+\)-\, and define gT(0) = T~%(u,(8) <8>Zt_i). Let 8T denote the
parameter values that minimize the quadratic form c/tAt9t, where AT is any
positive definite NLxNL matrix that may depend on the sample, and let JT
9 This section assumes that the same instruments are used for each of the asset equations. In
general, each asset equation could use a different set of instruments, which complicates the notation.

Econometric evaluation of asset pricing models
17
denote the minimized value of the quadratic form g'TArgT- Jagannathan and
Wang (1993) show that JT will have a weighted chi-square distribution which can
be used for testing the hypothesis that (3.1) holds.
Theorem 4.1. (Jagannathan and Wang, 1993). Suppose that the matrix AT
converges in probability to a constant positive definite matrix A. Assume also that
VTgriOo) -+d N(0,S), where iV(-> •) denotes the multivariate normal distribution,
do are the true parameter values, and S is a positive definite matrix. Let
D = E[dgT/de}\g=9o
and let
Q= (Sl'2)(Al?2)[I - {All2)'D{D'AD)-lD'{All2)\(All2){Sll2)
where A1'2 and S1'2 are the upper triangular matrices from the Cholesky
decompositions of A and S. Then the matrix Q has NL-d\m(Q) nonzero, positive
eigenvalues. Denote these eigenvalues by 1;, i = 1,2,..., NL-dim(6). Then JT
converges to
^lZl + • • • + ^NL-dim(9)XNL-dim(e)
where Xi, i = 1,2,..., NL-dim(6) independent random variables, each with a Chi-
Square distribution with one degree of freedom.
Proof. See Jagannathan and Wang (1993).
Notice that when the matrix A is W = S~l, the matrix Q is idempotent of rank
NL-dim(6). Hence the nonzero eigenvalues of Q are unity. In this case, the
asymptotic distribution reduces to a simple chi-square distribution with NL-
dim(0) degrees of freedom. This is the special case considered by Hansen (1982),
who originally derived the asymptotic distribution of the /r-statistic. The JT-
statistic and its extension, as provided in Theorem 4.1, provide a goodness-of-fit
test for models estimated by the GMM.
Hansen (1982) shows that the estimators of 6 that minimize g'Wg are consistent
and asymptotically normal, for any fixed W. If the weighting matrix Wis chosen
to be the inverse of a consistent estimate of the covariance matrix of the
orthogonality conditions S, the estimators are asymptotically efficient in the class of
estimators that minimize g'Wg for fixed Ws. The asymptotic variance matrix of
this optimal GMM estimator of the parameter vector is given as
Cov(0) = [E(dg/de)'WE(dg/de)]-1 (4.2)
where dg/86 is an NLxdim(6) matrix of derivatives. A consistent estimator for
the asymptotic covariance of the sample mean of the orthogonality conditions is
used in practice. That is, we replace W in (4.2) with Cov(#)-1 and replace
E(dg/d6) with its sample analogue. An example of a consistent estimator for the
optimal weighting matrix is given by Hansen (1982) as

18
W. E. Ferson and R. Jagannathan
Cov(gr) = [(l/r)£ 5>n-i«;+w) ® (Z^)] (4.3)
where <g> denotes the Kronecker product. A special case that often proves useful
arises when the orthogonality conditions are not serially correlated. In that
special case, the optimal weighting matrix is the inverse of the matrix Cov(gr), where
Cov(gr) = [(1/T) 5>,+i«;+1) ® (ZtZ't)} . (4.4)
t
The GMM weighting matrices originally proposed by Hansen (1982) have some
drawbacks. The estimators are not guaranteed to be positive definite, and they
may have poor finite sample properties in some applications. A number of studies
have explored alternative estimators for the GMM weighting matrix. A
prominent example by Newey and West (1987a) suggests weighting the autocovariance
terms in (4.3) with Bartlett weights to achieve a positive semi-definite matrix.
Additional refinements to improve the finite sample properties are proposed by
Andrews (1991), Andrews and Monahan (1992), and Ferson and Foerster (1994).
4.2. Testing hypotheses with the GMM
As we noted above, the /^-statistic provides a goodness-of-fit test for a model
that is estimated by the GMM, when the model is overidentified. Hansen's JT-
statistic is the most commonly used test in the finance literature that has used the
GMM. Other standard statistical tests based on the GMM are also used in the
finance literature for testing asset pricing models. One is a generalization of the
Wald test, and a second is analogous to a likelihood ratio test statistic. Additional
test statistics based on the GMM are reviewed by Newey (1985) and Newey and
West (1987b).
For the Wald test, consider the hypothesis to be tested as expressed in the M-
vector valued function H{9) = 0, where M < dim(0). The GMM estimates of 9 are
asymptotically normal, with mean 9 and variance matrix Cov(0). Given standard
regularity conditions, it follows that the estimates of H are asymptotically
normal, with mean zero and variance matrix HeCov(9)H'e, where subscripts denote
partial derivatives, and that the quadratic form
TH>[HeCov(9)H'e]~lH
is asymptotically chi-square, providing a standard Wald test.
A likelihood ratio type test is described by Newey and West (1987b), Eichen-
baum, Hansen, and Singleton (1988, appendix C), and Gallant (1987). Newey and
West (1987b) call this the D test. Assume that the null hypothesis implies that the
orthogonality conditions E(gr*) — 0 hold, while, under the alternative, a subset
E(gr) — 0 hold. For example, g* = (g, h). When we estimate the model under the
null hypothesis, the quadratic form g*'W*g* is minimized. Let W\x be the upper
left block of W; that is, let it be the estimate of Cov (g)'1 under the null. When we

Econometric evaluation of asset pricing models
19
hold this matrix fixed the model can be estimated under the alternative by
minimizing (/W^ g. The difference of the two quadratic forms
T[g*'Wg* - g'W*ng]
is asymptotically chi-square, with degrees of freedom equal to M if the null
hypothesis is true. Newey and West (1987b) describe additional variations on
these tests.
4.3. Illustrations: Using the GMM to test the conditional CAPM
The CAPM imposes nonlinear overidentifying restrictions on the first and second
moments of asset returns. These restrictions can form a basis for econometric
tests. To see these restrictions more clearly, notice that if an econometrician
knows or can estimate Cov(Rit,Rmt), E(Rmt), Var(i?mr), and E(R0t), it is possible to
compute E(Rit) from the CAPM, using equation (2.1). Given a direct sample
estimate of E(Rit), the expected return is overidentified. It is possible to use the
overidentification to construct a test of the CAPM by asking if the expected
return on the asset is different from the expected return assigned by the model. In
this section we illustrate such tests by using both the traditional, return-beta
formulation and the stochastic discount factor representation of the CAPM.
These examples extend easily to the multiple-beta models.
4.3.1. Static or unconditional CAPMs
If we make the assumption that all the expectation terms in the CAPM refer to
the unconditional expectations, we have an unconditional version of the CAPM. It
is straightforward to estimate and then test an unconditional version of the
CAPM, using equation (3.1) and the stochastic discount factor representation
given in equation (3.4). The stochastic discount factor is
mt+\ = co + c\Rmt+\
where c0 and c\ are fixed parameters. Using only the unconditional expectations,
the model implies that
E{(c0+ci^mr+i)^+i-l} = 0
where Rt+i is the vector of gross asset returns. The vector of sample
orthogonality conditions is
gr = gT(co,ci) = (l/T)^2{(c0 + ciRmt+l)Rt+l - 1} .
t
With assets N > 2, the number of orthogonality conditions is N and the number
of parameters is 2, so the /^-statistic has N - 2 degrees of freedom. Tests of the
unconditional CAPM using the stochastic discount factor representation are
conducted by Carhart et al. (1995) and Jagannathan and Wang (1996), who reject
the model using monthly data for the postwar United States.

20
W. E. Ferson and R. Jagannathan
Tests of the unconditional CAPM may also be conducted using the linear,
return-beta formulation of equation (2.1) and the GMM. Let rt = Rt-.%! be the
vector of excess returns, where i?or is the gross return on some reference asset and
1 is an N vector of ones; also let ut = rt- f5rmt, where /? is the N vector of the betas
of the excess returns, relative to the market, and rmt = Rmt - Rot is the excess
return on the market portfolio. The model implies that
E(ut) = E(utrmt) = 0 .
Let the instruments be Zt = (l,rmt)'. The sample orthogonality condition is then
gT(P) = T-lJ2(rt-Prmt)®Zt .
t
The number of orthogonality conditions is 2N and the number of parameters is
N, so the model is overidentified and may be tested using the /r-statistic.
An alternative approach to testing the model using the return-beta formulation
is to estimate the model under the hypothesis that expected returns depart from
the predictions of the CAPM by a vector of parameters a, which are called
Jensen's alphas. Redefining ut = rt — a — firmt, the model has 2N parameters and
2N orthogonality conditions, so it is exactly identified. It is easy to show that the
GMM estimators of a and /? are the same as the OLS estimators, and equation
(4.4) delivers White's (1980) heteroskedasticity-consistent standard errors. The
CAPM may be tested using a Wald test or the D-statistic, as described above.
Tests of the unconditional CAPM using the linear return-beta formulation are
conducted with the GMM by MacKinlay and Richardson (1991), who reject the
model for monthly U.S. data.
4.3.2. Conditional CAPMs
Empirical studies that rejected the unconditional CAPM, as well as mounting
evidence of predictable variation in the distribution of security rates of return, led
to empirical work on conditional versions of the CAPM starting in the early
1980s. In a conditional asset pricing model it is assumed that the expectation terms
in the model are conditional expectations, given a public information set that is
represented by a vector of predetermined instrumental variables Zt. The multiple-
beta models of Merton (1973) and Cox, Ingersoll, and Ross (1985) are intended to
accommodate conditional expectations. Merton (1973, 1980) and Cox-Ingersoll-
Ross also showed how a conditional version of the CAPM may be derived as a
special case of their intertemporal models. Hansen and Richard (1987) describe
theoretical relations between conditional and unconditional versions of mean-
variance efficiency.
The earliest empirical formulations of conditional asset pricing models were
the latent variable models developed by Hansen and Hodrick (1983) and Gibbons
and Ferson (1985) and later refined by Campbell (1987) and Ferson, Foerster,
and Keim (1993). These models allow time-varying expected returns, but
maintain the assumption that the conditional betas are fixed parameters. Consider the

Econometric evaluation of asset pricing models
21
linear, return-beta representation of the CAPM under these assumptions, writing
E(r(|Z(_i) = /}E(r„,f|Zf_i). The returns are measured in excess of a risk-free asset.
Let r1( be some reference asset with nonzero /?i, so that
E(r1(|Z(_1)=jS1E(rm(|Z(_1) .
Solving this expression for E(rmt\Zt-i) and substituting, we have
E(r,|Zi_i) = CE(n,|£_i)
where C= (j3.//?i) and ./ denotes element-by-element division. With this
substitution, the expected market risk premium is the latent variable in the model,
and C is the N vector of the model parameters. When we form the error term
ut = rt — Cr\t, the model implies E(«f|Zf_i) = 0 and we can estimate and test the
model by using the GMM. Gibbons and Ferson (1985) argued that the latent
variable model is attractive in view of the difficulties in measuring the true market
portfolio, but Wheatley (1989) emphasized that it remains necessary to assume
that ratios of the betas, measured with respect to the unobserved market
portfolio, are constant parameters.
Campbell (1987) and Ferson and Foerster (1995) show that a single-beta latent
variable model is rejected in U.S. data. This finding rejects the hypothesis that
there is a (conditional) minimum-variance portfolio such that the ratios of
conditional betas on this portfolio are fixed parameters. Therefore, the empirical
evidence suggests that conditional asset pricing models should be consistent with
either (1) a time-varying beta or (2) more than one beta for each asset.10
Conditional, multiple-beta models with constant betas are examined
empirically by Ferson and Harvey (1991), Evans (1994), and Ferson and Korajczyk
(1995). They reject such models with the usual statistical tests but find that they
still capture a large fraction of the predictability of stock and bond returns over
time. When allowing for time-varying betas, these studies find that the time-
variation in betas contributes a relatively small amount to the time-variation in
expected asset returns. Intuition for this finding can be obtained by considering
the following approximation. Suppose that time-variation in expected excess
returns is E(r|Z) = A/?, where X is a vector of time-varying expected risk premiums
for the factors and /? is a matrix of time-varying betas. Using a Taylor series, we
can approximate
Var[E(r|Z)] « E(jS)'Var[l]E(jS) + E(l)'Var[jS]E(l) .
The first term in the decomposition reflects the contribution of the time-varying
risk premiums; the second reflects the contribution of time-varying betas. Since
the average beta E(fi) is on the order of 1.0 in monthly data, while the average risk
1 A model with more than one fixed beta, and with time-varying risk premiums, is generally
consistent with a single, time-varying beta for each asset. For example, assume that there are two
factors with constant betas and time-varying risk premiums, where a time-varying combination of the
two factors is a minimum-variance portfolio.

22
W. E. Ferson and R. Jagannathan
premium E(l) is typically less than 0.01, the first term dominates the second term.
This means that time-variation in conditional betas is less important than time-
variation in expected risk premiums, from the perspective of modeling predictable
variation in expected asset returns.
While from the perspective of modeling predictable time-variation in asset
returns, time-variation in conditional betas is not as important as time-variation
in expected risk premiums, this does not imply that beta variation is empirically
unimportant. From the perspective of modeling the cross-sectional variation in
expected asset returns, beta variation over time may be very important. To see
this, consider the unconditional expected excess return vector, obtained from the
model as
E{E(r|Z)} = E{ljS} = E(A)EQS) + Cov(l, J?) .
Viewed as a cross-sectional relation, the term Cov(l, /?) may vary significantly in a
cross section of assets. Therefore, the implications of a conditional version of the
CAPM for the cross section of unconditional expected returns may depend
importantly on common time-variation in betas and expected market risk
premiums. The empirical tests of Jagannathan and Wang (1996) suggest that this is
the case.
Harvey (1989) replaced the constant beta assumption with the assumption that
the ratio of the expected market premium to the conditional market variance is a
fixed parameter, as in
E(rw,|Z,_i)/Var(r,„,|Z,_i) = y .
The conditional expected returns may then be written according to the
conditional CAPM as
E(r,\Z,-i) = yCo\(r„rmt\Z,-i) .
Harvey's version of the conditional CAPM is motivated by Merton's (1980)
model in which the ratio y, called the market price of risk, is equal to the relative
risk aversion of a representative investor in equilibrium. Harvey also assumes that
the conditional expected risk premium on the market (and the conditional market
variance, given fixed y) is a linear function of the instruments, as in
E(rwt|Zt_i) = &'mZ,-\
where bm is a coefficient vector. Define the error terms vt = rmt — 5'mZt-i and
wt = rt(\ - vty). The model implies that the stacked error term ut = (vt,wt)
satisfies E(wt|Zf_i) = 0, so it is straightforward to estimate and then test the model
using the GMM. Harvey (1989) rejects this version of the conditional CAPM for
monthly data in the U.S. In Harvey (1991) the same formulation is rejected when
applied using a world market portfolio and monthly data on the stock markets of
21 developed countries.
The conditional CAPM may be tested using the stochastic discount factor
representation given by equation (3.4): mt+\ — cq( - cuRmt+\. In this case the

Econometric evaluation of asset pricing models
23
coefficients c0, and cu are measurable functions of the information set Zt. To
implement the model empirically it is necessary to specify functional forms for the
c0t and c\t. From the expression (3.4) it can be seen that these coefficients are
nonlinear functions of the conditional expected market return and its conditional
variance. As yet there is no theoretical guidance for specifying the functional
forms. Cochrane (1996) suggests approximating the coefficients using linear
functions, and this approach is followed by Carhart et al. (1995), who reject the
conditional CAPM for monthly U.S. data.
Jagannathan and Wang (1993) show that the conditional CAPM implies an
unconditional two-factor model. They show that
mt+i =a0+ fliE(rm,+i \It) + Rmt+\
(where /, denotes the information set of investors and ao and a.\ are fixed
parameters) is a valid stochastic discount factor in the sense that E(i?^+im,+1) = 1 for
this choice of mt+\. Using a set of observable instruments Z„ and assuming that
E(rmH-i \Zt) is a linear function of Zt, they find that their version of the model
explains the cross section of unconditional expected returns better than does an
unconditional version of the CAPM. Bansal and Viswanathan (1993) develop
conditional versions of the CAPM and multiple-factor models in which the
stochastic discount factor m,+ 1 is a nonlinear function of the market or factor
returns. Using nonparametric methods, they find evidence to support the
nonlinear versions of the models. Bansal, Hsieh, and Viswanathan (1993) compare
the performance of nonlinear models with linear models, using data on
international stocks, bonds, and currency returns, and they find that the nonlinear
models perform better. Additional empirical tests of the conditional CAPM and
multiple-beta models, using stochastic discount factor representations, are
beginning to appear in the literature. We expect that future studies will further refine
the relations among the various empirical specifications.
5. Model diagnostics
We have discussed several examples of stochastic discount factors corresponding
to particular theoretical asset pricing models, and we have shown how to test
whether these models assign the right expected returns to financial assets. The
stochastic discount factors corresponding to these models are particular
parametric functions of the data observed by the econometrician. While empirical
studies based on these parametric approaches have led to interesting insights, the
parametric approach makes strong assumptions about the economic
environment. In this section we discuss some alternative econometric approaches to the
problem of asset pricing models.

24
W. E. Ferson and R. Jagannathan
5.1. Moment inequality restrictions
Hansen and Jagannathan (1991) derive restrictions from asset pricing models
while assuming as little structure as possible. In particular, they assume that the
financial markets obey the law of one price and that there are no arbitrage
opportunities. These assumptions are sufficient to imply that there exists a
stochastic discount factor mt+i (which is almost surely positive, if there is no
arbitrage) such that equation (3.1) is satisfied.
Note that if the stochastic discount factor is a degenerate random variable (i.e.,
a constant), then equation (3.1) implies that all assets must earn the same
expected return. If assets earn different expected returns, then the stochastic
discount factor cannot be a constant. In other words, cross-sectional differences in
expected asset returns carry implications for the variance of any valid stochastic
discount factor, which satisfies equation (3.1). Hansen and Jagannathan make use
of this observation to derive a lower bound on the volatility of stochastic discount
factors. Shiller (1979, 1981), Singleton (1980), and Leroy and Porter (1981) derive
a related volatility bound in specific models, and their empirical work suggests
that the stochastic discount factors implied by these simple models are not volatile
enough to explain expected returns across assets. Hansen and Jagannathan (1991)
show how to use the volatility bound as a general diagnostic device.
In what follows we derive the Hansen and Jagannathan (1991) bound and
discuss their empirical application. To simplify the exposition, we focus on an
unconditional version of the bound using only the unconditional expectations.
We posit a hypothetical, unconditional, risk-free asset with return Rf =
E(mr+i)_1. We take the value of Rf, or equivalently E(wr+ 0, as a parameter to be
varied as we trace out the bound.
The law of one price guarantees the existence of some stochastic discount
factor which satisfies equation (3.1). Consider the following projection of any
such mt+\ on the vector of gross asset returns, Rt+\.
mt+i = R't+lP + et+i (5.1)
where
E(er+1^r+1) = 0
and where /? is the projection coefficient vector. Multiply both sides of equation
(5.1) by Rt+i and take the expected value of both sides of the equation, using
E[Rt+iet+i] = 0, to arrive at an expression which may be solved for /?. Substituting
this expression back into (5.1) gives the "fitted values" of the projection as
<i = K+iP = K+MRt+iK+i)~ll ■ (5-2)
By inspection, the m*+l given by equation (5.2) is a valid stochastic discount
factor, in the sense that equation (3.1) is satisfied when m*+l is used in place of
mt+\. We have therefore constructed a stochastic discount factor m*+l that is also
a payoff on an investment position in the N given assets, where the vector

Econometric evaluation of asset pricing models
25
E(Rt+\R't+l)~l I provides the weights. This payoff is the unique linear least
squares approximation of every admissible stochastic discount factor in the space
of available asset payoffs.
Substituting m*+l for R't+l(3 in equation (5.1) shows that we may write any
stochastic discount factor, mt+\, as
m,+l = m*+l + e,+l
where E{et+\m*t+x) = 0. It follows that Var(w?+i) > Var(/»*+1). This expression is
the basis of the Hansen-Jagannathan bound11 on the variance of mt+1. Since m*+l
depends only on the second moment matrix of the N returns, the lower bound
depends only on the assets available to the econometrician and not on the
particular asset pricing model that is being studied. To obtain an explicit expression
for the variance bound in terms of the underlying asset-return moments,
substitute from the previous expressions to obtain
Var(m,+i) > Var(m,*+1)
= pVai(Rt+l)P (5.3)
= [Cov(m,R')VsLr(Ryl] xVar(R)[Var(R)~lCoy(m,R')}
= [l~ E(m)E(R')]Var(R)~l\l - E(m)E(R)]
where the time subscripts are suppressed to conserve notation and the last line
follows from E(mR) = I = E(m)E(R) + Cov(m,R). As we vary the hypothetical
values of E(m) = Rjl, the equation (5.3) traces out a parabola in E(m), o.(m) space,
where o.(m) is the standard deviation of mt+\. If we place o.(m) on the y axis and
E(m) on the x axis, the Hansen-Jagannathan bounds resemble a cup, and the
implication is that any valid stochastic discount factor mt+\ must have a mean and
standard deviation that place it within the cup.
The lower bound on the volatility of a stochastic discount factor, as given by
equation (5.3), is closely related to the standard mean-variance analysis that has
long been used in the financial economics literature. To see this, recall that if
r = R — Rf is the vector of excess returns, then (3.1) implies that
0 = E(mr) = E(m)E(r) + po(m)o(r) .
Since —1 < p < 1, we have that
ff(m)/E(m) > E(rt)/ff(rt)
for all /. The right side of this expression is the Sharpe ratio for asset /. The Sharpe
ratio is defined as the expected excess return on an asset, divided by the standard
deviation of the excess return (see Sharpe 1994 for a recent discussion of this
ratio). Consider plotting every portfolio that can be formed from the N assets in
the Standard Deviation (x axis) - Mean (y axis) plane. The set of such portfolios
"Related bounds were derived by Kandel and Stambaugh (1987), Mackinlay (1987, 1995), and
Shanken (1987).

26
W. E. Ferson and R. Jagannathan
with the smallest possible standard deviation for a given mean return is the
minimum-variance boundary. Consider the tangent to the minimum-variance
boundary from the point 1/E(m) on the y axis. The tangent point is a portfolio of
the asset returns, and the slope of this tangent line is the maximum Sharpe ratio
that can be attained with a given set of N assets and a given risk-free
rate, Rf = 1/E(m). The slope of this line is also equal to Rf multiplied by the
Hansen-Jagannathan lower bound on a(m) for a given E(m) =R7X- That is, we
have that
ff(m) > E(m)|Max{E(r,-)/ff(r,-)}|
for the given Rf.
The preceding analysis is based on equation (3.1), which is equivalent to the
law of one price. If there are no arbitrage opportunities, it implies that mt+\ is a
strictly positive random variable. Hansen and Jagannathan (1991) show how to
obtain a tighter bound on the standard deviation of mt+\ by making use of the
restriction that there are no arbitrage opportunities. They also show how to
incorporate conditioning variables into the analysis. Snow (1991) extends the
Hansen-Jagannathan analysis to include higher moments of the asset returns. His
extension is based on the Holder inequality, which implies that for given values of
5 and p such that
(l/5) + (l//») = l
it is true that
E(mR) < E(ms)l/dE(Rp)l/p.
Cochrane and Hansen (1992) refine the Hansen-Jagannathan bound to consider
information about the correlation between a given stochastic discount factor and
the vector of asset returns. This provides a tighter set of restrictions than the
original bounds, which only make use of the fact that the correlation must be
between -1 and + 1.
5.2. Statistical inference for moment inequality restrictions
Cochrane and Hansen (1992), Burnside (1994), and Cecchetti, Lam, and Mark
(1994) show how to take sampling errors into account when examining whether a
particular candidate stochastic discount factor satisfies the Hansen-Jagannathan
bound. In what follows we will outline a computation which allows for sampling
errors, following the discussion in Cochrane and Hansen (1992).
Assume that the econometrician has a time series of T observations on a
candidate for the stochastic discount factor, denoted by yt, and the N asset
returns Rt. We also assume that the risk-free asset is not one of the N assets. Hence
v = E(m) = \/RF is an unknown parameter to be estimated. Consider a linear
regression of mt+\ onto the unit vector and the vector of asset returns as
m,+1 = a + R't P + ut+\. We use the regression function in the following system of
population moment conditions:

Econometric evaluation of asset pricing models
27
E(a + R'tp) = v (5.4)
E(*,a + RtR'tp) = \N
E(yt) = v
E[(a + R'tP)2}-E[yt}<0 .
The first equation says that the expected value of mt = a + R'tP = v. The second
equation says that the regression function for mt is a valid stochastic discount
factor. The third equation says that v is the expected value of the particular
candidate discount factor that we wish to test. The fourth equation states that the
Hansen-Jagannathan bound is satisfied by the particular candidate stochastic
discount factor.
We can estimate the parameters v, a, and the N vector /?, using the N + 3
equations in (5.4), by treating the last inequality as an equality and using the
GMM. Treating the last equation as an equality corresponds to the null
hypothesis that the mean and variance of yt place it on the Hansen-Jagannathan
boundary. Under the null hypothesis that the last equation of (5.4) holds as an
equality, the minimized value of the GMM criterion function JT, multiplied by T,
has a chi-square distribution with one degree of freedom. Cochrane and Hansen
(1992) suggest testing the inequality relation using the one-sided test.
5.3. Specification error bounds
The methods we have examined so far are developed, for the most part, under the
null hypothesis that the asset pricing model under consideration by the econo-
metrician assigns the right prices (or expected returns) to all assets. An alternative
is to assume that the model is wrong and examine how wrong the model is. In this
section we will follow Hansen and Jagannathan (1994) and discuss one possible
way to examine what is missing in a model and assign a scalar measure of the
model's misspecification.12
Let yt denote the candidate stochastic discount factor corresponding to a given
asset pricing model, and let m* denote the unique stochastic discount factor that
we constructed earlier, as a combination of asset payoffs. We assume that E[j(if(]
does not equal lN, the N vector of ones; i.e., the model does not correctly price all
of the gross returns. We can project yt on the N asset returns to get yt = R'ta + ut,
and project m* on the vector of asset returns to get m* = R'tP + et. Since the
candidate yt does not correctly price all of the assets, then a. and /? will not be the
same. Define pt = {fi — a)'Rt as the modifying payoff'to the candidate stochastic
12 GMM-based model specification tests are examined in a general setting by Newey (1985). Other
related work includes that by Boudoukh, Richardson, and Smith (1993), who compute approximate
bounds on the probabilities of the test statistics in the presence of inequality restrictions; Chen and
Knez (1992) develop nonparametric measures of market integration by using related methods; and
Hansen, Heaton, and Luttmer (1995) show how to compute specification error and volatility bounds
when there are market frictions such as short-sale constraints and proportional transaction costs.

28
W. E. Ferson and R. Jagannathan
discount factor yt. Clearly, (jt+pt) is a valid stochastic discount factor, satisfying
equation (3.1). Hansen and Jagannathan (1994) derive specification tests based on
the size of the modifying payoff, which measures how far the model's candidate
for a stochastic discount factor yt is from a valid stochastic discount factor.
Hansen and Jagannathan (1994) show that a natural measure of this distance is
5 = E(/?/2), which provides an economic interpretation for the model's mis-
specification. Payoffs that are orthogonal to pt are correctly priced by the
candidate yt, and E(p2) is the maximum amount of mispricing by using yt for any
payoff normalized to have a unit second moment. The modifying payoff pt is also
the minimal modification that is sufficient to make yt a valid stochastic discount
factor.
Hansen and Jagannathan (1994) consider an estimator of the distance measure
5 given as the solution to the following maximization problem:
ST = Maxo.r-1 JTtf - (yt + a%)2 + 2oc'IJV]1/2. (5.5)
t
If a.T is the solution to (5.5), then the estimate of the modifying payoff is tx'TRt. It
can be readily verified that the first-order condition to (5.5) implies that a!TRt
satisfies the sample counterpart to the asset pricing equation (3.1).
To obtain an estimate of the sampling error associated with the estimated value
6t, consider
ut = y2t - (yt + a'TRt)2 + 2a'rlJV •
The sample mean of ut is b\. We can obtain a consistent estimator of the variance
of b\ by the frequency zero spectral density estimators described in Newey and
West (1987a) or Andrews (1991) and applied to the time series {u, - b\}t=lT. Let
sT denote the estimated standard deviation of b\ obtained this way. Then, under
standard assumptions, we have that TXI2(5T — 5)/sT converges to a normal (0,1)
random variable. Hence, using the delta method, we obtain
Tx?2dT/2sT(dT - S) — N(0,1) . (5.6)
6. Conclusions
In this article we have reviewed econometric tests of a wide range of asset pricing
models, where the models are based on the law of one price, the no-arbitrage
principle, and models of market equilibrium with investor optimization. Our
review included the earliest of the equilibrium asset pricing models, the CAPM,
and also considered dynamic multiple-beta and arbitrage pricing models. We
provided some results for the asymptotic distribution of traditional two-pass
estimators for asset pricing models stated in the linear, return-beta formulation.
We emphasized the econometric evaluation of asset pricing models by using

Econometric evaluation of asset pricing models
29
Hansen's (1982) generalized method of moments. Our examples illustrate the
simplicity and flexibility of the GMM approach. We showed that most asset
pricing models can be represented in a stochastic discount factor form, which
makes the application of the GMM straightforward. Finally, we discussed model
diagnostics that provide additional insight into the causes of the statistical
rejections in GMM tests and which help assess the specification errors in these
models.
Appendix
Proof of Theorem 2.1 The proof comes from Jagannathan and Wang (1996).
We first introduce some additional notation. Let IN be the TV-dimensional identity
matrix and lrbe a T-dimensional vector of ones. It follows from equation (2.17)
that
R-V=T-1 (IN ® \'T)'ek, k=l,...,K2
where
e* = (eijti, • • •, eitr, • • •, emi, ■ ■ ■, emr) ■
By the definition of bk, we have that
bk-Pk = [In ® ((fk'fk)-lfk%k
where fk is the vector-demeaned factor realizations, conformable to the vector ek.
In view of the assumption that the conditional covariance of e^ and tjit, given the
time series of the factors (denoted by fk), is a fixed constant aijki, we have that
E[(i4-/J4)(*i-A«i)|/t]
= T~l[IN ® (U'ir'jOlEh^liK/iv ® lr)
= T-%®((g£)-lg)]Xkl(Ilf®lT)
= r-%®[(//i)-1//ir)] = o
where we denote the matrix of the {ffy/u},-,- by S&. The last line follows from the
fact fhat^'lr = 0. Hence we have shown that (bk - /}t) is uncorrelated with
(R - n). Therefore, the terms u and hyi should be uncorrelated, and the
asymptotic variance of Tll2{g — y) in equation (2.15) is given by
(x'xylx'[VsLr(u) + VeLT{hy2)]x{x'x)~l .
Let nt]ti denote the limiting value of Con(\ff fleikl y/Tfltji), as T —> oo. Let the
matrix with nt]ki being its ifh element be denoted by n&. We assume that the
sample covariance matrix of the factors exists and converges in probability to a
constant positive definite matrix Q, with typical element Qw. Since \ff (bik - fiik)
converges in distribution to the random variable Si^y/f' f'k^ik, we have

30
W. E. Person and R. Jagannathan
and
W = {x'x)~xx'y&x(hy2)x(x'x)~x
= J2 (x'x)-lx'{y2ky2l(n^Uk!ai!l)}x(x'xrl
l,k=l,...,k2
where Tl'kl is a matrix whose 1,7th element is the limiting value of Cov(Vffk €,*,
Vffl&ji) as T-^oo.
Q.E.D.
References
Abel, A. (1990). Asset prices under habit formation and catching up with the Jones. Amer. Econom.
Rev. Papers Proc. 80, 38-42.
Andrews, D. W. K. (1991). Heteroskedasticity and autocorrelation consistent covariance matrix
estimation. Econometrica 59, 817-858.
Andrews, D. W. K. and J. C. Monahan (1992). An improved heteroskedasticity and autocorrelation
consistent covariance matrix estimator. Econometrica 60, 953-966.
Arrow, K. J. (1970). Essays in the Theory of Risk-Bearing. Amsterdam: North-Holland.
Bansal, R. and S. Viswanathan (1993). No- arbitrage and arbitrage pricing: A new approach. /. Finance
8, 1231-1262.
Bansal, R., D. A. Hsieh and S. Viswanathan (1993). A new approach to international arbitrage
pricing. /. Finance 48, 1719-1747.
Becker, G. S. and K. M. Murphy (1988). A theory of rational addiction. /. Politic. Econom. 96, 675-
700.
Beja, A. (1971). The structure of the cost of capital under uncertainty. Rev. Econom. Stud. 38(8), 359-
368.
Berk, J. B. (1995). A critique of size-related anomalies. Rev. Financ. Stud. 8, 275-286.
Black, F. (1972). Capital market equilibrium with restricted borrowing. /. Business 45, 444-455.
Black, F., M. C. Jensen and M. Scholes (1972). The capital asset pricing model: Some empirical tests.
In: Studies in the Theory of Capital Markets, M. C. Jensen, ed., New York: Praeger, 79-121.
Boudoukh, J., M. Richardson and T. Smith (1993). Is the ex ante risk premium always positive? A new
approach to testing conditional asset pricing models. /. Financ. Econom. 34, 387-408.
Breeden, D. T. (1979). An intertemporal asset pricing model with stochastic consumption and
investment opportunities. J. Financ. Econom. 7, 265-296.
Brown, D. P. and M. R. Gibbons (1985). A simple econometric approach for utility-based asset pricing
models. /. Financed, 359-381.
Burnside, C. (1994). Hansen-Jagannathan bounds as classical tests of asset-pricing models. /. Business
Econom. Statist. 12, 57-79.
Campbell, J. Y. (1987). Stock returns and the term structure. /. Financ. Econom. 18, 373-399.
Campbell, J. Y. and J. Cochrane (1995). By force of habit. Manuscript, Harvard Institute of Economic
Research, Harvard University.
Carhart, M., K. Welch, R. Stevens and R. Krail (1995). Testing the conditional CAPM. Working
Paper, University of Chicago.
Cecchetti, S. G., P. Lam and N. C. Mark (1994). Testing volatility restrictions on intertemporal
marginal rates of substitution implied by Euler equations and asset returns. /. Finance 49, 123-152.

Econometric evaluation of asset pricing models
31
Chen, N. (1983). Some empirical tests of the theory of arbitrage pricing. J. Finance 38, 1393-1414.
Chen, Z. and P. Knez (1992). A measurement framework of arbitrage and market integration.
Working Paper, University of Wisconsin.
Cochrane, J. H. (1996). A cross-sectional test of a production based asset pricing model. Working
Paper, University of Chicago.
Cochrane, J. H. and L. P. Hansen (1992). Asset pricing explorations for macroeconomics. In: NBER
Macroeconomics Annual 1992, O. J. Blanchard and S. Fischer, eds., Cambridge, Mass.: MIT Press.
Connor, G. (1984). A unified beta pricing theory. /. Econom. Theory 34, 13-31.
Connor, G. and R. A. Korajczyk (1986). Performance measurement with the arbitrage pricing theory:
A new framework for analysis. /. Financ. Econom. 15, 373-394.
Constantinides, G. M. (1982). Intertemporal asset pricing with heterogeneous consumers and without
demand aggregation. J. Business 55, 253-267.
Constantinides, G. M. (1990). Habit formation: A resolution of the equity premium puzzle. /. Politic.
Econom. 98, 519-543.
Constantinides, G. M. and D. Duffle (1994). Asset pricing with heterogeneous consumers. Working
Paper, University of Chicago and Stanford University.
Cox, J. C, J. E. Ingersoll, Jr. and S. A. Ross (1985). A theory of the term structure of interest rates.
Econometrica 53, 385-407.
Debreu, G. (1959). Theory of Value: An Axiomatic Analysis of Economic Equilibrium. New York:
Wiley.
Detemple, J. B. and F. Zapatero (1991). Asset prices in an exchange economy with habit formation.
Econometrica 59, 1633-1657.
Dunn, K. B. and K. J. Singleton (1986). Modeling the term structure of interest rates under non-
separable utility and durability of goods. /. Financ. Econom. 17, 27-55.
Dybvig, P. H. and J. E. Ingersoll, Jr., (1982). Mean-variance theory in complete markets. /. Business
55, 233-251.
Eichenbaum, M. S., L. P. Hansen and K. J. Singleton (1988). A time series analysis of representative
agent models of consumption and leisure choice under uncertainty. Quart. J. Econom. 103, 51-78.
Epstein, L. G. and S. E. Zin (1989). Substitution, risk aversion and the temporal behavior of
consumption and asset returns: A theoretical framework. Econometrica 57, 937-969.
Epstein, L. G. and S. E. Zin (1991). Substitution, risk aversion and the temporal behavior of
consumption and asset returns. /. Politic. Econom. 99, 263-286.
Evans, M. D. D. (1994). Expected returns, time-varying risk, and risk premia. /. Finance 49, 655-679.
Fama, E. F. and K. R. French. (1992). The cross-section of expected stock returns. /. Finance 47, 427-
465.
Fama, E. F. and J. D. MacBeth (1973). Risk, return, and equilibrium: Empirical tests. /. Politic.
Econom. 81, 607-636.
Ferson, W. E. (1983). Expectations of real interest rates and aggregate consumption: Empirical tests.
/. Financ. Quant. Anal. 18, 477-497.
Ferson, W. E. and G. M. Constantinides (1991). Habit persistence and durability in aggregate
consumption: Empirical tests. /. Financ. Econom. 29, 199-240.
Ferson, W. E. and S. R. Foerster (1994). Finite sample properties of the generalized method of
moments tests of conditional asset pricing models. /. Financ. Econom. 36, 29-55.
Ferson, W. E. and S. R. Foerster (1995). Further results on the small-sample properties of the
generalized method of moments: Tests of latent variable models. In: Res. Financ, Vol. 13.
Greenwich, Conn.: JAI Press, pp. 91-114.
Ferson, W. E., S. R. Foerster and D. B. Keim (1993). General tests of latent variable models and
mean-variance spanning. /. Finance 48, 131-156.
Ferson, W. E. and C. R. Harvey (1991). The variation of economic risk premiums. /. Politic. Econom.
99, 385-415.
Ferson, W. E. and C. R. Harvey (1992). Seasonality and consumption-based asset pricing. /. Finance
47,511-552.

32
W. E. Ferson and R. Jagannathan
Ferson, W. E. and R. A. Korajczyk (1995). Do arbitrage pricing models explain the predictability of
stock returns? J. Business 68, 309-349.
Ferson, W. E. and J. J. Merrick, Jr. (1987). Non-stationarity and stage-of-the-business-cycle effects in
consumption-based asset pricing relations. J. Financ. Econom. 18, 127-146.
Gallant, R. (1987). Nonlinear Statistical Models. New York: Wiley.
Gibbons, M. R. and W. Ferson (1985). Testing asset pricing models with changing expectations and an
unobservable market portfolio. J. Financ. Econom. 14, 217-236.
Gorman, W. M. (1953). Community preference fields. Econometrica 21, 63-80.
Hansen, L. P. (1982). Large sample properties of generalized method of moments estimators.
Econometrica 50, 1029-1054.
Hansen, L. P., J. Heaton and E. G. J. Luttmer (1995). Econometric evaluation of asset pricing models.
Rev. Financ. Stud. 8, 237-274.
Hansen, L. P. and R. Hodrick (1983). Risk averse speculation in the forward foreign exchange market:
An econometric analysis of linear models. In: Exchange Rates and International Macroeconomics,
J. A. Frenkel, ed., Chicago: University of Chicago Press.
Hansen, L. P. and R. Jagannathan (1991). Implications of security market data for models of dynamic
economies. J. Politic. Econom. 99, 225-262.
Hansen, L. P. and R. Jagannathan (1994). Assessing specification errors in stochastic discount factor
models. NBER Technical Working Paper No. 153.
Hansen, L. P. and S. F. Richard (1987). The role of conditioning information in deducing testable
restrictions implied by dynamic asset pricing models. Econometrica 55, 587-613.
Hansen, L. P. and K. J. Singleton (1982). Generalized instrumental variables estimation of nonlinear
rational expectations models. Econometrica 50, 1269-1286.
Hansen, L. P. and K. J. Singleton (1983). Stochastic consumption, risk aversion, and the temporal
behavior of asset returns. J. Politic. Econom. 91, 249-265.
Harrison, M. and D. Kreps (1979). Martingales and arbitrage in multi-period securities markets.
/. Econom. Theory 20, 381-408.
Harvey, C. R. (1989). Time-varying conditional covariances in tests of asset pricing models. J. Financ.
Econom. 24, 289-317.
Harvey, C. R. (1991). The world price of covariance risk. J. Finance 46, 111-157.
Heaton, J. (1995). An empirical investigation of asset pricing with temporally dependent preference
specifications. Econometrica 63, 681-717.
Ibbotson Associates. (1992). Stocks, bonds, bills, and inflation. 1992 Yearbook. Chicago: Ibbotson
Associates.
Jagannathan, R. (1985). An investigation of commodity futures prices using the consumption-based
intertemporal capital asset pricing model. /. Finance 40, 175-191.
Jagannathan R. and Z. Wang (1993). The CAPM is alive and well. Federal Reserve Bank of
Minneapolis Research Department Staff Report 165.
Jagannathan, R. and Z. Wang (1996). The conditional-CAPM and the cross-section of expected
returns. J. Finance 51, 3-53.
Kandel, S. (1984). On the exclusion of assets from tests of the mean-variance efficiency of the market
portfolio. J. Finance 39, 63-75.
Kandel, S. and R. F. Stambaugh (1987). On correlations and inferences about mean-variance
efficiency. /. Financ. Econom. 18, 61-90.
Lehmann, B. N. and D. M. Modest (1987). Mutual fund performance evaluation: A comparison of
benchmarks and benchmark comparisons. J. Finance 42, 233-265.
Leroy, S. F. and R. D. Porter (1981). The present value relation: Tests based on implied variance
bounds. Econometrica 49, 555-574.
Lewbel, A. (1989). Exact aggregation and a representative consumer. Quart. J. Econom. 104, 621-633.
Lintner, J. (1965). The valuation of risk assets and the selection of risky investments in stock portfolios
and capital budgets. Rev. Econom. Statist. 47, 13-37.
Lucas, R. E. Jr. (1978). Asset prices in an exchange economy. Econometrica 46, 1429-1445.
Luttmer, E. (1993). Asset pricing in economies with frictions. Working Paper, Northwestern University.

Econometric evaluation of asset pricing models
33
McElroy, M. B. and E. Burmeister (1988). Arbitrage pricing theory as a restricted nonlinear
multivariate regression model. /. Business Econom. Statist. 6, 29-42.
MacKinlay, A. C. (1987). On multivariate tests of the CAPM. /. Financ. Econom. 18, 341-371.
MacKinlay, A. C. and M. P. Richardson (1991). Using generalized method of moments to test mean-
variance efficiency. /. Finance 46, 511-527.
MacKinlay, A. C. (1995). Mulifactor models do not explain deviations from the CAPM. /. Financ.
Econom. 38, 3-28.
Merton, R. C. (1973). An intertemporal capital asset pricing model. Econometrica 41, 867-887.
Merton, R. C. (1980). On estimating the expected return on the market: An exploratory investigation.
/. Financ. Econom. 8, 323-361.
Mossin, J. (1966). Equilibrium in a capital asset market. Econometrica 34, 768-783.
Newey, W. (1985). Generalized method of moments specification testing. /. Econometrics 29, 229-256.
Newey, W. K. and K. D. West (1987a). A simple, positive semi-definite, heteroskedasticity and
autocorrelation consistent covariance matrix. Econometrica 55, 703-708.
Newey, W. K. and K. D. West. (1987b). Hypothesis testing with efficient method of moments
estimation. Internat. Econom. Rev. 28, 777-787.
Novales, A. (1992). Equilibrium interest-rate determination under adjustment costs. /. Econom.
Dynamic Control 16, 1-25.
Roll, R. (1977). A critique of the asset pricing theory's tests: Part 1: On past and potential testability of
the theory. J. Financ. Econom. 4, 129-176.
Ross, S. A. (1976). The arbitrage pricing theory of capital asset pricing. J. Econom. Theory 13, 341-
360.
Ross, S. (1977). Risk, return and arbitrage. In: Risk and Return in Finance, I. Friend and J. L. Bicksler,
eds. Cambridge, Mass.: Ballinger.
Rubinstein, M. (1974). An aggregation theorem for securities markets. /. Financ. Econom. 1, 225-244.
Rubinstein, M. (1976). The valuation of uncertain income streams and the pricing of options. Bell J.
Econom. Mgmt. Sci. 7, 407-425.
Ryder H. E., Jr. and G. M. Heal (1973). Optimum growth with intertemporally dependent preferences.
Rev. Econom. Stud. 40, 1-33.
Shanken, J. (1987). Multivariate proxies and asset pricing relations: Living with the roll critique.
/. Financ. Econom. 18, 91-110.
Shanken, J. (1992). On the estimation of beta-pricing models. Rev. Financ. Stud. 5, 1-33.
Sharpe, W. F. (1964). Capital asset prices: A theory of market equilibrium under conditions of risk.
/. Finance 19, 425-442.
Sharpe, W. F. (1994). The Sharpe ratio. /. Port. Mgmt. 21, 49-58.
Shiller, R. J. (1979). The volatility of long-term interest rates and expectations models of the term
structure. /. Politic. Econom. 87, 1190-1219.
Shiller, R. J. (1981). Do stock prices move too much to be justified by subsequent changes in
dividends? Amer. Econom. Rev. 71, 421-436.
Singleton, K. J. (1980). Expectations models of the term structure and implied variance bounds.
J. Politic. Econom. 88, 1159-1176.
Snow, K. N. (1991). Diagnosing asset pricing models using the distribution of asset returns. /. Finance
46, 955-983.
Stambaugh, R. F. (1982). On the exclusion of assets from tests of the two-parameter model: A
sensitivity analysis. /. Financ. Econom. 10, 237-268.
Sundaresan, S. M. (1989). Intertemporally dependent preferences and the volatility of consumption
and wealth. Rev. Financ. Stud. 2, 73-89.
Wheatley, S. (1988). Some tests of international equity integration. /. Financ. Econom. 21, 177-212.
Wheatley, S. M. (1989). A critique of latent variable tests of asset pricing models. /. Financ. Econom.
23, 325-338.
White, H. (1980). A heteroskedasticity-consistent covariance matrix estimator and a direct test for
heteroskedasticity. Econometrica 48, 817-838.
Wilson, R. (1968). The theory of syndicates. Econometrica 36, 119-132.

G. S. Maddala and C. R. Rao, eds., Handbook of Statistics, Vol. 14
© 1996 Elsevier Science B.V. All rights reserved.
2
Instrumental Variables Estimation of Conditional
Beta Pricing Models
Campbell R. Harvey and Chris Kirby
A number of well-known asset pricing models imply that the expected return on
an asset can be written as a linear function of one or more beta coefficients that
measure the asset's sensitivity to sources of undiversifiable risk. This paper
provides an overview of the econometric evaluation of such models using the method
of instrumental variables. We present numerous examples that cover both single-
beta and multi-beta models. These examples are designed to illustrate the various
options available to researchers for estimating and testing beta pricing models.
We also examine the implications of a variety of different assumptions concerning
the time-series behavior of conditional betas, covariances, and reward-to-risk
ratios. The techniques discussed in this paper have applications in other areas of
asset pricing as well.
1. Introduction
Asset pricing models often imply that the expected return on an asset can be
written as a linear combination of market-wide risk premia, where each risk
premium is multiplied by a beta coefficient that measures the sensitivity of the
return on the asset to a source of undiverifiable risk in the economy. Indeed, this
type of tradeoff between risk and expected return is implied by some of the most
famous models in financial economics. The Sharpe (1964) - Lintner (1965) capital
asset pricing model (CAPM), the Black (1972) CAPM, the Merton (1973)
intertemporal CAPM, the arbitrage pricing theory (APT) of Ross (1976), and the
Breeden (1979) consumption CAPM can all be classified under the general
heading of beta pricing models. Although these models differ in terms of
underlying structural assumptions, each implies a pricing relation that is linear in
one or more betas.
The fundamental difference between conditional and unconditional beta pricing
models is the specification of the information environment that investors use to
form expectations. Unconditional models imply that investors set prices based on
an unconditional assessment of the joint probability distribution of future
returns. Under such a scenario we can construct an estimate of an investor's
35

36
C. R. Harvey and C. Kirby
expected return on an asset by taking an average of past returns. Conditional
models, on the other hand, imply that investors have time-varying expectations
concerning the joint probability distribution of future returns. In order to
construct an estimate of an investor's conditional expected return on an asset we
have to use the information available to the investor at time t — 1 to forecast the
return for time t.
Both conditional and unconditional models attempt to explain the cross-
sectional variation in expected returns. Unconditional models imply that
differences in average risk across assets determine differences in average returns.
There are no time-series predictions other than expected returns are constant.
Conditional models have similar cross-sectional implications: differences in
conditional risk determine differences in conditional expected returns. But
conditional models have implications concerning the time-series properties of
expected returns as well. Conditional expected returns vary with changes in
conditional risk and fluctuations in market-wide risk premiums. In theory, we can
test a conditional beta pricing model using a single asset.
Empirical tests of beta pricing models can be interpreted within the familiar
framework of mean-variance analysis. Unconditional tests seek to determine
whether a certain portfolio is on the efficient portion of the unconditional mean-
variance frontier. The unconditional frontier is determined by the unconditional
means, variances and covariances of the asset returns. Conditional tests of beta
pricing models are designed to answer a similar question: does a certain portfolio
lie on the efficient portion of the mean-variance frontier at each point in time? In
conditional tests, however, the mean-variance frontier is determined by the
conditional means, conditional variances, and conditional covariances of asset
returns.
As a general rule, the rejection of unconditional efficiency does not imply a
rejection of conditional mean-variance efficiency. This is easily demonstrated
using an example given by Dybvig and Ross (1985) and Hansen and Richard
(1987). Suppose we are testing whether the 30-day Treasury bill is unconditionally
efficient using monthly data. Unconditionally, the 30-day bill does not lie on the
efficient frontier. It is a single risky asset (albeit low risk) whose return has
non-zero variance. Thus it is surely dominated by an appropriately chosen
portfolio. At the conditional level, however, the conclusion is much different.
Conditionally, the 30-day bill is nominally risk free. At the end of each month we
know precisely what the return will be over the next month. Because the
conditional variance of the return on the T-bill is zero, it must be conditionally
efficient.
A number of different methods have been proposed for testing beta pricing
models. This paper focuses on one in particular: the method of instrumental
variables. Instrumental variables are a set of data, specified by the econome-
trician, that proxy for the information that investors use to form expectations.
The primary advantage of the instrumental variables approach is that it provides
a highly tractable way of characterizing tinie-varying risk and expected returns.
Our discussion of the instrumental variables methodology is organized along the

Instrumental variables estimation of conditional beta pricing models 37
following lines. Section 2 uses the conditional version of the Sharpe (1964) -
Lintner (1965) CAPM to illustrate how the instrumental variables approach can
be employed to estimate and test single beta models. Section 3 extends the
analysis to multi-beta models. Section 4 introduces the technique of latent
variables. Section 5 provides an overview of the estimation methodology. The
final section offers some brief closing remarks.
2. Single beta models
A. The conditional CAPM
The conditional version of the Sharpe (1964) - Lintner (1965) CAPM is
undoubtedly one of the most widely studied conditional beta pricing models. We can
express the pricing relation associated with this model as:
cr lrt , Cov [rjt,rmt | *Vi]m ,n i m
E[r*|0-l] = Var[r„|fl^] E[r-|fl'-l] ' (1)
where rjt is the return on portfolio j from time t — 1 to time t measured in excess of
the risk free rate, rmt is the excess return on the market portfolio, and ilt~\
represents the information set that investors use to form expectations. The ratio
of the conditional covariance between the return on portfolio j and the return on
the market, Co\[rjt,rmt\Qt-\\, to the variance of the return on the market,
Var[r„rt|ftf_i], is the conditional beta of portfolio j with respect to the market.
Any cross-sectional variation in expected returns can be attributed solely to
differences in conditional beta coefficients.
As it stands the pricing relation shown in (1) is untestable. To make it testable
we have to impose additional structure on the model. In particular, we have to
specify a model for conditional expectations. Thus any test of (1) will be a joint
test of the conditional CAPM and the assumed specification for conditional
expectations. In theory any functional form could be used. Let f(Zt-\) denote the
statistical model that generates conditional expectations where Z is a set of
instrumental variables. The function /(•) could be a linear regression model, a
Fourier flexible form [Gallant (1982)], a nonparametric kernel estimator
[Silverman (1986), Harvey (1991), and Beneish and Harvey (1995)], a seminon-
parametric density [Gallant and Tauchen (1989)], a neural net [Gallant and White
(1990)], an entropy encoder [Glodjo and Harvey (1995)], or a polynomial series
expansion [Harvey and Kirby (1995)].
Once we take a stand on the functional form of the conditional expectations
operator it is straightforward to construct a test of the conditional CAPM. First
we use /(•) to obtain fitted values for the conditional mean of rp. This nails down
the left-hand side of the pricing relation in (1). Then we apply /(•) again to get
fitted values for the three components on the right-hand side of (1). Combining
the fitted values for the conditional mean of rmt, those for the conditional
covariance between rjt and rmt, and those for the conditional variance of rmt yields

38
C. R. Harvey and C. Kirby
fitted values for the right-hand side of (1). If the conditional CAPM is valid then
the pricing errors - the difference between the fitted values for the left-hand and
right-hand sides of (1) - should be small and unpredictable. This is the basic
intuition behind all tests of conditional beta pricing models.
In the presentation that follows we focus on one particular specification for
conditional expectations: the linear model. This model, though very simple, has
distinct advantages over the many nonlinear alternatives. The linear model is
exceedingly easy to implement, and Harvey (1991) shows that it performs well
against nonlinear alternatives in out-of-sample forecasting of the market return.
In addition, the linear specification is actually more general than it may seem.
Recent work has shown that many nonlinear models can be consistently
approximated via an expanding sequencing of finite-dimensional linear models.
Harvey and Kirby (1995) exploit this fact to develop a simple procedure for
constructing analytic tests of both single beta and multi-beta pricing models.
B. Linear conditional expectations
The easiest way to motivate the linear specification for conditional expectations is
to assume that the joint distribution of the asset returns and instrumental
variables is spherically invariant. This class of distributions is analyzed in Vershik
(1964), who shows that it is sufficient for linear conditional expectations, and
applied to tests of the conditional CAPM in Harvey (1991). Vershik (1964)
provides the following characterization. Consider a set of random variables,
{xi,... ,xn}, that have finite second moments. Let H denote a linear manifold
spanned by this set. If all random variables in the linear manifold H that have the
same variance have the same distribution then: (i) H is a spherically invariant
space; (ii) {xi,... ,x„} is spherically invariant; and (iii) every distribution function
of any variable in H is a spherically invariant distribution. The above requirements
are satisfied, for example, by both the multivariate normal and multivariate t
distributions.
A potential disadvantage of Vershik's (1964) definition is that it does not
encompass processes like Cauchy for which the variance is undefined. Blake and
Thomas (1968) and Chu (1973) propose a definition for an elliptical class of
distributions that addresses this shortcoming. A random vector x is said to have
an elliptical distribution if and only if its probability density function p(x) can be
expressed as a function of a quadratic form, p(x) = f(^x'C~:x), where C is
positive definite. When the variance-covariance matrix of x exists it is
proportional to C and the Vershik (1964), Blake and Thomas (1968) and Chu
(1973) definitions are equivalent.2 But the quadratic form of the density also
covers processes like Cauchy that imply linear conditional expectations where the
projection constants depend on the characteristic matrix.
2 Implicit in Chu's (1973) definition is the existence of the density function. Kelker (1970) provides
an alternative approach in terms of the characteristic function. See also Devlin, Gnanadesikan and
Kettenring (1976).

Instrumental variables estimation of conditional beta pricing models
39
C. A general framework for testing the CAPM
A linear specification for conditional expectations implies that the return on
portfolio j can be written as:
rjt = Z,^Sj + Uj, , (2)
where uJt is the error in forecasting the return on portfolio j at time t, Zt_\ is a row
vector of I instrumental variables, and dj is a I x 1 set of time-invariant weights.
Substituting the expression shown in (2) into equation (1) yields the restriction:
Zt-\$j = pr -, | J" , E[M/fKmf|Zf_i] , (3)
h[Umt\Zt-\\
where umt is the error in forecasting the return on the market portfolio. Note that
both the variance term, E[m^|Z,_i], and the covariance term, E[ujtumt\Zt^i], are
conditioned on Zt-\. Therefore, the pricing relation in (3) should be regarded as
an approximation. This is the case because the expectation of the true conditional
covariance is not the covariance conditioned on Zt_\. The two are connected via
the relation: E[Cov(ry/,rIB/|ft/_i)|Z/_i] = Cov(r//,rm/|Z/_i)-Cov(E[r//|fl/_i],E[rIB/
|fl,_i]|Z/_i). An analogous relation holds for the true conditional variance of rmt
and the variance conditioned on Zt-\. There is no way to construct a test of the
original version of pricing restriction given that the true information set Q is
unobservable.
If we multiply both sides of (3) by the conditional variance of the return on the
market portfolio we obtain the restriction:
E[m^i/Z/_i5/|Z/_i] = E[m//mib/Z/_i5ib|Z/_i] . (4)
Notice that the conditional expected return on both the market portfolio and
portfolio j have been moved inside the expectations operator. This can be done
because both of these quantities are known conditional on Z,_i. As a result, we
do not need to specify an explicit model for the conditional variance and co-
variance terms. We simply note that, under the null hypothesis, the disturbance:
eP = umtzt-i8j - UjtUmtZt-\5m , (5)
should have mean zero and be uncorrelated with the instrumental variables. If we
divide ejt by the conditional variance of the market return, then the resulting
quantity can be interpreted as the deviation of the observed return from the
return predicted by the model. Thus ep is essentially just a pricing error. A
negative pricing error implies the model is overpricing while a positive pricing
error indicates that the model is underpricing.
The generalized method of moments (GMM), which is discussed in detail in
Section 5, provides a direct way to test the above restriction. Suppose we have a
total of n assets. We can stack the disturbances in (2) and the pricing errors in (5)
into the (2n + 1) x 1 vector:

40
C. R. Harvey and C. Kirby
£,= («( umt et)'=\ [rmt - Zt-i5m]' J , (6)
where u is the innovation in the lxn vector of conditional means and e is the
1 x n vector of pricing errors. The conditional CAPM implies that s, should be
uncorrected with Z,_i. So if we form the Kronecker product of st with the vector
of instrumental variables:
*k®Z't-\ , (7)
and take unconditional expectations, we obtain the vector of orthogonality
conditions:
E[et ® Z't_x] = 0 . (8)
With n assets there are n + 1 columns of innovations for the conditional means
and n columns of pricing errors. Thus, with I instrumental variables we have
l(2n + 1) orthogonality conditions. Note, however, that there are £(n + 1)
parameters to estimate. This leaves n(. overidentifying restrictions.3
We can obtain consistent estimates of the n£ matrix of coefficients d and the
I x 1 vector of coefficients 5m by minimizing the quadratic objective function:
JT = g'TS^gT , (9)
where:
1 T
t=\
and Sj denotes a consistent estimate of:
oo
So ee J2 e[(*®zufa-j®z;^)'] . (ii)
j=—oo
If the conditional CAPM is true then T times the minimized value of the objective
function converges to a central chi-square random variable with nl degrees of
freedom. Thus we can use this criterion as a measure of the overall goodness-of-fit
of the model.
3 An econometric specification of this form is explored for New York Stock Exchange returns in
Harvey (1989) and Huang (1989), for 17 international equity returns in Harvey (1991), for
international bond returns in Harvey, Solnik and Zhou (1995), and for emerging equity market returns in
Harvey (1995).

Instrumental variables estimation of conditional beta pricing models
41
D. Constant conditional betas
The econometric specification shown in (6) assumes that all of the conditional
moments - the means, variances and covariances - change through time. If some
of these moments are constant then we can construct more powerful tests of the
conditional CAPM by imposing this additional structure. Traditionally, tests of
the CAPM have focused on whether expected returns are proportional to the
expected return on a benchmark portfolio. We can construct the same type of test
within our conditional pricing framework with a specification of the form:
% = (r, -/wfl' , (12)
where /? is a row vector of n beta coefficients. The coefficient /?y represents the ratio
of conditional covariance between the return on portfolio j and the return on the
benchmark to the conditional variance of the benchmark return.
Typically, we think of rmt as a proxy for the market portfolio. It is important to
note, however, that the beta coefficients in (12) are left unrestricted. Thus (12) can
also be interpreted as a test of a single factor latent variables model.4 In the latent
variables framework, /?y represents the ratio of conditional covariance between
the return on portfolio j and an unobserved factor to the conditional covariance
between the return on the benchmark portfolio and this factor. The testable
implication is that E[et|.Zr_i] = 0 where st is the vector of pricing errors associated
with the constant conditional beta model. There are nl orthogonality
conditions and n parameters to estimate so we have l{n — 1) overidentifying
restrictions.
Of course we can easily incorporate the restrictions on the conditional beta
coefficients by changing the specification to:
£; = («; umt b, e, )'=
( [rt - Z^Q' f\
[rmt ~ Zt-\$m]'
l"ltP ~ "*<»<]'
(13)
where b is the disturbance vector associated with the constant conditional beta
assumption. Tests based on this specification may shed additional light on the
plausibility of the assumption of constant conditional betas. With n assets there
are n+\ columns of innovations in the conditional means, n columns in b and n
columns in e. Thus there are £(3n+l) orthogonality conditions, l(n+\)+n
parameters to estimate, and n{2l~ 1) overidentifying restrictions.
E. Constant conditional reward-to-risk ratio
Another formulation of the conditional CAPM assumes that the conditional
reward-to-risk ratio is constant. The conditional reward-to-risk ratio,
4 See, for example, Hansen and Hodrick (1983), Gibbons and Ferson (1985) and Ferson (1990).

42
C. R. Harvey and C. Kirby
E[rm,|f2,_i]/Var[rm,|f2,_i], is simply the price of covariance risk. This version of
the conditional CAPM is examined in Campbell (1987) and Harvey (1989). The
vector of pricing errors for the model becomes:
e, = rt - Xutumt , (14)
where X is the conditional expected return on the market divided by its
conditional variance. To complete the econometric specification we have to include
models for the conditional means. The overall system is:
/ [rt-Zt^g\' \
£, = {ut um e,)'= \ [r„, - Z,_i3m]' \ . (15)
\ [r, - X{umtut)}' j
With n assets there are n + 1 columns of innovations in the conditional means and
n columns in e. Thus with £ instrumental variables there are £(2n + 1)
orthogonality conditions and l + (£(n + l)) parameters. This leaves n£ — 1 over-
identifying restrictions.
One way to simplify the estimation in (15) is to note that E[umtUjt\Zt_i]
= E[umtrjt\Zt-i]. This follows from the fact that:
E[umtuJt\Zt-i] = E[umt{rjt - Zt-idj)\Z,-i]
= ^[umtrjt\Zt-i] - E[umtZt-i5j\Zt_i]
= E[umtrjt\Z,^} - E[umt\Zt_i]Zt_idj
= ^[umtrJt\Zt^} .
As a result, we can drop n of the conditional mean equations. The more
parsimonious system is:
Now we have n + 1 equations and £(n + 1) orthogonality conditions. With £ + 1
parameters there are (n£) — 1 overidentifying restrictions. The specifications
shown in (15) and (16) are asymptotically equivalent. But (16) is more
computationally manageable.
The specifications in (15) and (16) do not restrict X to be the conditional
covariance to variance ratio. We can easily add this restriction:
[rt-Zt-XS( \
[i-, - X(umtr,)]' J
(17)
where m is the disturbance associated with the constant reward-to-risk
assumption. Tests of this specification should shed additional light on the plausibility of
the assumption of a constant price of covariance risk. With n assets there are n
columns in u, one column in um, one column in m and n columns in e. Thus there

Instrumental variables estimation of conditional beta pricing models
43
are l{2n + 2) orthogonality conditions, l(n + 1) + 1 parameters, and n - 1 over-
identifying restrictions.
F. Linear conditional betas
Ferson and Harvey (1994, 1995) explore specifications where the conditional
betas are modelled as a linear functions of the instrumental variables. We could,
for example, specify an econometric system of the form:
Zi.w s?
tll<>i
u2t = rmt — Z t-\ m
uat = [ul^Z'^Ki)' - rmtuUt]' (18)
Zi.w s
t-\°i
uat = (-«/ + Hi) - Z't^lKi(Z^_lSm)'
where the elements of Z''wKt are the fitted conditional betas for portfolio /, nt is
the mean return on portfolio /, and <x,- is the difference between the unrestricted
mean return and the mean return that incorporates the pricing restriction of the
conditional CAPM. Note that (18) uses two sets of instruments. The set used to
estimate the conditional mean return on portfolio i and the conditional beta for
the portfolio, Z''w, includes both asset specific (/) and market-wide (w)
instruments. The conditional mean return on the market is estimated using only the
market-wide instruments. This yields an exactly identified system of equations.5
The intuition behind the system shown in (18) is straightforward. The first two
equations follow from our assumption of linear conditional expectations. They
represent statistical models for expected returns. The third equation follows from
the definition of the conditional beta:
h = {^\u\\Z^_x\)-^\rmmu\Z)\\ . (19)
In (18) the conditional beta is modelled as a linear function of both the asset-
specific and market-wide information. The last two equations deliver the average
pricing error for the conditional CAPM. Note that m is the average fitted return
from the statistical model. Thus a, is the difference between the average fitted
return from our statistical model and the fitted return implied by the pricing
relation of conditional CAPM. It is analogous to the Jensen a. In the current
analysis, however, both the betas and the risk premiums are changing through
time.
Because of the complexity and size of the above system it is difficult to estimate
from more one asset at a time. Thus, in general, not all the cross-sectional
restrictions of conditional CAPM can be imposed, and it is not possible to report
a multivariate test of whether the a, are equal to zero. Note, however, that (18)
5 For analysis of related systems see Ferson (1990), Shanken (1990), Ferson and Harvey (1991),
Ferson and Harvey (1993), Ferson and Korajzcyk (1995), Ferson (1995), Harvey (1995) and Ja-
gannathan and Wang (1996).

44
C. R. Harvey and C. Kirby
does impose one important cross-sectional restriction. Because the system is
exactly identified, the market risk premium, Z^_x8m, will be identical for every
asset examined. There are no overidentifying restrictions, so tests of the model are
based on whether the coefficient a, is significantly different from zero. Additional
insights might be gained by analyzing the time-series properties of the
disturbance:
u6it = rit - Z^MZT-iO)' • (20)
Under the null hypothesis, E[Mfe|Z^j] is equal to zero. Thus diagnostics can be
conducted by regressing u^it on various information variables. We could also
construct tests for time-varying of betas based on the coefficient estimates
associated with Z''wKj.
3. Models with multiple betas
A. The multi-beta conditional CAPM
The conditional CAPM can easily be generalized to a model that has multiple
sources of risk. Consider, for example, a Ar-factor pricing relation of the form:
E[r,|Z,_i] = EK|Z,_i] (E[u'ftuft\Zt-{\yXV[u'ftut\Zt^} (21)
where r is a row vector of n asset returns,/is \ x K vector of factor realizations,
Uf is a vector of innovations in the conditional means of the factors, and u is a
vector of innovations in the conditional means of the returns. The first term on
the right-hand side of (21) represents the conditional expectation of the factor
realizations. It has dimension 1 x k. The second term is the inverse of the k x k
conditional variance-covariance matrix of the factors. The final term measures the
conditional covariance of the asset returns with the factors. Its dimension is k x n.
The multi-beta pricing relation shown in (21) cannot be tested in the same
manner as its single-beta counterpart. Recall that in our analysis of single-beta
models it was possible to take the conditional variance of the market return to the
left-hand side of the pricing relation. As a result, we could move the conditional
means inside the expectations operator. This is not possible with a multi-beta
specification. We can, however, get around this problem by focusing on
specializations of the multi-beta model that parallel those discussed in the
previous section. We begin by considering specifications that restrict the
conditional betas to be linear functions of the instruments.
B. Linear conditional betas
The multi-beta analogue of the linear conditional beta specification shown in (18)
takes the form:

Instrumental variables estimation of conditional beta pricing models
45
Zl.W ff
t-\0i
tHt=f,-Zy_l5f
u3it = W2tu2i(z't-iKi)' -f'i«m]' (22)
Zl.W s
t-\di
M5, = (-a,. + ^-z;i>!.(zr^/)/
where the elements of Z'<wKi are the fitted conditional betas associated with the k
sources of risk and/is a row vector of factor realizations. Note that as before the
system is exactly identified, and the vector of conditional betas:
/?, = (EK«2(|Z«> ]r'E[/>h.(|Z;> ] . (23)
is modelled as a linear function, ZhwKt, of the instruments. This specification can
be tested by assessing the statistical significance of the pricing errors and checking
to see whether the disturbance:
u6u = ru-Zi£Mz?-isf)' - (M)
is orthogonal to instruments. The primary advantage of the above formulation is
that fitted values are obtained for the risk premiums, the expected returns, and the
conditional betas. Thus it is simple to conduct diagnostics that focus on the
performance of the model. Its main disadvantage is that it requires a heavy
parameterization.
C. Constant conditional reward-to-risk ratios
Harvey (1989) suggests an alternative approach for testing multi-beta pricing
relations. His strategy is to assume that the conditional reward-to-risk ratio is
constant for each factor. This results is a multi-beta analogue of the specification
shown in (15):
/ [r,-Z,_,*r\
£, = («( Uft e,)'= I \ft-Z,-i5f]' , (25)
where A is a row vector of k time-invariant reward-to-risk measures. The above
system can be simplified to:
-<■/>*>'=(#:&;$)• w
using the same approach that allowed us to simplify the single-beta specification
discussed earlier.6
6 Kan and Zhang (1995) generalize this formulation by modelling the conditional reward-to-risk
ratios as linear functions of the instrumental variables. Their approach eliminates the need for asset-
specific instruments and permits joint estimation of the pricing relation using multiple portfolios. But
the type of diagnostics that fall out of the linear conditional beta model - fitted expected returns,
betas, etc. - are no longer available.

46
C. R. Harvey and C. Kirby
4. Latent variables models
The latent variables technique introduced by Hansen and Hodrick (1983) and
Gibbons and Ferson (1985) provides a rank restriction on the coefficients of the
linear specifications that are assumed to describe expected returns. Suppose we
assume that ratio formed by taking the conditional beta for one asset and dividing
it by the corresponding conditional beta another asset is constant. Under these
circumstances, the ^-factor conditional beta pricing model implies that all of the
variation in the expected returns is driven by changes in the k conditional risk
premiums. We can still form our estimates of the conditional means by projecting
returns on the ^-dimensional vector of instrumental variables. But if all the
variation in expected returns is being driven changes in the k risk premiums then we
should not need all n£ projection coefficients to characterize the time variation in
the n returns. Thus the basic idea of the latent variables technique is to test
restrictions on the rank of the projection coefficient matrix.
A. Constant conditional beta ratios
First we take the vector of excess returns on our set of portfolios and partition it
as:
»"r=(rir i r2t), (34)
where r\t is a 1 x k vector of returns on the reference assets and j-2r is a 1 x (n — k)
vector of returns on the test assets. Then we partition the matrix of conditional
beta coefficients associated with our multi-factor pricing model accordingly:
P={Pi '■ hi (35)
where fLx is k x k and /J2 is k x (n - k). The pricing relation for the multi-beta
model tells us that:
E[rlr|Zr_!] = yA (36)
and
E[i*|Zr_i] = ytp2 , (37)
where y, is a 1 x k vector of time-varying market-wide risk premiums. We can
manipulate (36) to obtain the relation yt = E[rit\Zt_i}P^1. Substituting this
expression for yt into (37) yields the pricing restriction:
E[i*|Zr_i] = ElmlZ,-!]^ . (38)
This says that the conditional expected returns on the test assets are proportional
to the conditional expected returns on the reference assets. The constants of
proportionality are determined by ratios of conditional betas.

Instrumental variables estimation of conditional beta pricing models
47
The pricing relation in (38) can be tested in much the same manner as the
models discussed earlier.7 The only real difference is that we no longer have to
identify the factors. One possible specification is:
/ [n,-z,_i*,]' \
*, = (*!, "2, et)'= [rx-Z^dj]' , (39)
\[Z,_i*2-Z,_i*itf]7
where «& = /fj"1/^- There are k columns in u\t, n — K columns in u2t and n — k
columns in et. Thus we have l(2n - k) orthogonality conditions and In + k(n — k)
parameters. This leaves {(. — k){n — k) overidentifying restrictions. Note that both
the number of instrumental variables and the total number of assets must be
greater than the number of factors.
B. Linear conditional covariance ratios
An important disadvantage of (39) is that the ratio of conditional betas,
<5 = PllP2, is assumed to be constant. One way to generalize the latent variables
model is to assume the elements of <5 are linear in the instrumental variables.8 This
assumption follows naturally from the previous specifications that imposed the
assumption of linear conditional betas. The resulting latent variables system is:
/ [rlt - Z,_!*i]' \
t,t = (ult u2t £t)'=\ [r* - Z,_i*2]' , (40)
V[z_1^2-z_1^(I®z;_1)4>t/
where i is a k x 1 vector of ones. With the original set of instruments the
dimension of <5* in the final set of moment conditions is l(n — k) and the system is
not identified. Thus the researcher must specify some subset of the original
instruments, Z*, with dimension I * < I to be used in the estimation.
Finally, the parameterization in both (39) and (40) can be reduced by
substituting the third equation block into the second block. For example,
* = («„ et)>=(.^<-Z<-fl) , (41)
\[r2t - Zt^di0] J
In this system, it is not necessary to estimate d2.
5. Generalized method of moments estimation
Contemporary empirical research in financial economics makes frequent use of a
wide variety of econometric techniques. The generalized method of moments has
proven to be particularly valuable, however, especially in the area of estimating
and testing asset pricing models. This section provides an overview of the gen-
7 Harvey, Solnik and Zhou (1995) and Zhou (1995) show to construct analytic tests of latent
variables models.
8 See Ferson and Foerster (1994).

48
C. R. Harvey and C. Kirby
eralized method of moments (GMM) procedure. We begin by illustrating the
intuition behind GMM using a simple example of classical method of moments
estimation. This is followed by brief discussion of the assumptions underlying the
GMM approach to estimation and testing along with a review of some of the key
distributional results. For detailed proofs of the consistency and asymptotic
normality of GMM estimators see Hansen (1982), Gallant and White (1988), and
Potscher and Pracha (1991a,b).
A. The Classical method of moments
The easiest way to illustrate the intuition behind the GMM procedure is to
consider a simple example of classical method of moments (CMM) estimation.
Suppose we observe a random sample %\,X2, ■ ■. ,xj of T observations drawn from
a distribution with probability density function f(x; 0), where 0= [61,62, •■-,0k]
denotes a k x 1 vector of unknown parameters. The CMM approach to
estimation exploits the fact that in general the /h population moment of x about zero:
mj = W] , (42)
can be written as known function of 0. To implement the CMM procedure we first
compute they* sample moment of x about zero:
1 i=\
Then we set the /h sample moment equal to the corresponding population
moment for j = 1,2,..., k:
rh\ = m\{0)
m2 = m2{0)
: : : <44)
mk = mk(0)
This yields a set of k equations in k unknowns that can be solved to obtain an
estimator for the unknown vector 0. Thus the basic idea behind the CMM
procedure is to estimate 0 by replacing population moments with their sample
analogues.
Now let's take a more concrete version of the above example. Suppose that
x\,X2, ■ ■ ■ ,xj is a random sample of size T drawn from a normal distribution with
mean (j. and variance a2. To obtain the classical method of moments estimators of
ix and a2 we note that a2 = rti2 - {m\)2. This implies that the system of moments
equations takes the form:
1 T
TU
±±x2=a2+,2
(45)
T^

Instrumental variables estimation of conditional beta pricing models
49
Consequently, the CMM estimators for the mean and variance are:
1 N
1 i=i
i=i \ i=i
Notice that these are also the maximum likelihood estimators of fj. and a1.
B. The Generalized method of moments
The classical method of moments is just a special case of the generalized method
of moments developed by Hansen (1982). This latter procedure provides a general
framework for estimation and hypothesis testing that can be used to analyze a
wide variety of dynamic economic models. Consider, for example, the class of
models that generate conditional moment restrictions of the form:
ErfBr+x] = 0 , (47)
where E,[ • ] is the expectations operator conditional on the information set at time
t, ut+t = h{Xt+x, #o) is an n x 1 vector of vector of disturbance terms, Xt+t is an
5x1 vector of observable random variables, and 0o is an m x 1 vector of
unknown parameters. The basic idea behind the GMM procedure is to exploit the
moment restrictions in (47) to construct a sample objective function whose
minimizer is a consistent and asymptotically normal estimate of the unknown
vector #o-
In order to construct such an objective function, however, we need to make
some assumptions about the nature of the data generating process. Let Zt denote
the date t realization of an I x 1 vector of observable instrumental variables. We
assume, following Hansen (1982), that the vector process {Xt,Zt]^_oo is strictly
stationary and ergodic. Note that this assumption rules out a number of features
sometimes encountered in economic data such as deterministic trends, unit roots,
and unconditional heteroskedasticity. It accommodates many common forms of
conditional heterogeneity, however, and it does not appear to be overly restrictive
in most applications.9
With suitable restrictions on the data generating process in place we can
proceed to construct the GMM objective function. First we form the Kronecker
product:
jXXt+t,ZuOo) = ut+z®Zt . (48)
Then we note that because Zt is in the information set at time t, the model in (47)
implies that:
9 Although is possible to establish consistency and asymptotic normality of GMM estimators
under weaker assumptions, the associated arguments are too complex for an introductory discussion.
The interested reader can consult Potscher and Prucha (1991a,b) for an overview of recent advances in
the asymptotic theory of dynamic nonlinear econometric models.

50
C. R. Harvey and C. Kirby
Et[f(Xt+t,Zt,00)}=0 . (49)
Applying the law of iterated expectations to equation (49) yields the
unconditional restriction:
E\f(Xt+I,Zt,00)]=0 . (50)
Equation (50) represents a set of n£ population orthogonality conditions. The
sample analogue ofE[f(Xt+r,Zt,8)]:
9A9)=]=Y/{Xt+t,Zue) , (51)
(=1
forms the basis for the GMM objective function. Note that for any given value of
8 the vector gT(0) is just the sample mean of T realizations of the random vector
f[Xt+1,Zt,8). Given that/(-) is continuous and {Xt,Zty^=_ao is strictly stationary
and ergodic we have:
gT(8)^E[f(Xt+1,Zt,8)] (52)
by the law of large numbers. Thus if the economic model is valid the vector gT(6o)
should be close to zero when evaluated for a large number of observations. The
GMM estimator of 8q is obtained by choosing the value of 8 that minimizes the
overall deviation of gT(0) from zero. As long as E\f(Xt+r, Zt, 8)] is continuous in 8
it follows that this estimator is consistent under fairly general regularity conditions.
If the model is exactly identified (m — rd), the GMM estimator is the value of 8
that sets the sample moments equal to zero. For the more common situation
where the model is overidentified (m < rd), finding a vector of parameters that
sets all of the sample moments equal to zero is not feasible. It is possible, however,
to find a value of 8 that sets m linear combinations of the rd sample moment
conditions equal to zero. We simply let AT be an m x n£ matrix such that
ATgT{8) = 0 has a well-defined solution. The value of 8 that solves this system of
equations is the GMM estimator. Although we have considerable leeway in
choosing the weighting matrix At, Hansen (1982) shows that the variance-
covariance matrix of the estimator is minimized by letting At equal D'TS^1 where
DT and St are consistent estimates of:
oo
and So = Y, r«C/') , (53)
J=—oo
with r0(j) = E\f(Xt+r,Zt>Oo)f(Xt+T-j,Zt-j,Oo)']. Before considering how to
derive this result we first have to establish the asymptotic normality of GMM
estimators.
C. Asymptotic normality of GMM estimators
We begin by expressing equation (51) as:
Do
dh(Xt+r,8)
88'

Instrumental variables estimation of conditional beta pricing models
51
y/TgT{6)=±=Y,f{Xt+l,Zt,0)
(54)
The assumption that {Xt, Zty^_ao is stationary and ergodic, along with standard
regularity conditions, implies that a version of the central limit theorem holds. In
particular we have that:
VfgT(e0)^N(0,S0) , (55)
with So given by (53). This result allows us to establish the limiting distribution of
the GMM estimator 6T. First we make the following assumptions:
1. The estimator 8T converges in probability to do-
2. The weighting matrix AT converges in probability to A0 where A0 has
rank m.
3. Define:
1 T
^4e
'dh{Xt+T,0)
T
§ flWt\ A . , HI 1
TfiK d6' ~-i- (56)
t=\ \ Or /
For any 0T such that #7—>#o the matrix DT converges in probability to Z)0 where
Do has rank m.
Then we apply the mean value theorem to obtain:
gT{0T) = gT(00) + D*T(0T - 00) , (57)
where D*T is given by (56) with QT replaced by a vector 8*T that lies somewhere
within the interval whose endpoints are given by 8T and do- Recall that 8T is the
solution to the system of equations ATgT(0) = 0. So if we premultiply equation
(57) by At we have:
ATgr(.Oo) + ATDT{0T - 6>0) = 0 . (58)
Solving (58) for (8T - 00) and multiplying by y/f gives:
Vf(6T - e0) = -[ATDrrlATVfgT(6o) , (59)
and by Slutsky's theorem we have:
Vf(0T-Oo)^-[AoDo}-1Ao
x {the limiting distribution of VfgT(6o)} .
Thus the limiting distribution of the GMM estimator is:
Vf(eT-eo)^N(0,(A0DoylA0SoA'0(AoDo)-1') . (61)
Now that we know the limiting distribution of the generic GMM estimator we
can determine the best choice for the weighting matrix AT- The natural metric by

52
C. R. Harvey and C. Kirby
which to measure our choice is the variance-covariance matrix of the distribution
shown in (61). We want, in other words, to choose the AT that minimizes the
variance-covariance matrix of the limiting distribution of the GMM estimator.
D. The asymptotically efficient weighting matrix
The first step in determining the efficient weighting matrix is to note that S0 is
symmetric and positive definite. Thus S0 can be written as S0 = PP' where P is
nonsingular, and we can express the variance-covariance matrix in (61) as:
V=(A0D0ylA0S0A'0(A0D0yv
= (A0D0r[A0P((A0D0)-[A0P)' (62)
= (h+ (Di0^D0yiD'0(pr1)(H+ MtfDor'wr1)'
where:
H= (A0D0r1A0P-(D'0^D0r1D'0(P)-1 .
At first it may appear a bit odd to define H in this manner, but it simplifies the
problem of finding the efficient choice for At- To see why this is true note that:
HP1 Do = (AoDoy'AoPP-'Do - (D,0S^1Do)-iD'0(Pyip-lDo
= 1-1 (63)
= 0
As a consequence equation (62) reduces to:
V=HH' + (D'0SQ-1Do)-i (64)
Because His an m x n£ matrix with rank m it follows that Hit is positive definite.
Thus (Dr0S0~1 D0)~[ is the lower bound on the asymptotic variance-covariance
matrix of the GMM estimator. It is easily verified by direct substitution that
choosing A0 = D'0So~l achieves this lower bound.
This completes our review of the distribution theory for GMM estimators.
Next we want to consider some of the practical aspects of GMM estimation and
see how we might go about testing the restrictions implied economic models. We
begin with a strategy for implementing the GMM procedure.
E. The estimation procedure
To obtain an estimate for the vector of unknown parameters 60 we have to solve
the system of equations:
ArgT(0) = 0 .
Substituting the optimal choice for the weighting matrix into this expression
yields:

Instrumental variables estimation of conditional beta pricing models
53
D'TSjXgT{e) = 0 , (65)
where St is a consistent estimate of the matrix So- But it is apparent that (65) is
just the first-order condition for the problem:
min JT(B) = gABySr'gAO) . (66)
8
So given a consistent estimate of So we can obtain the GMM estimator for #o by
minimizing the quadratic form shown in equation (66).
In order to estimate #o we need a consistent estimate of So. But, in general, So
is a function of do- The solution to this dilemma is to perform a two-step
estimation procedure. Initially we set St equal to the identify matrix and perform
the minimization to get a first-stage estimate for do- Although this estimate is not
asymptotically efficient it is still consistent. Thus we can use it to construct a
consistent estimate of So. Once we have a consistent estimate of So we obtain the
second-stage estimate for 60 by minimizing the quadratic form shown above.
Let's assume that we have performed the two-step estimation procedure and
obtained the efficient GMM estimate of the vector of parameters do- Typically we
would like to have some way of evaluating how well the model fits the observed
data. One way of obtaining such a goodness-of-fit measure is to construct a test of
the overidentifying restrictions.
F. The test for overidentifying restrictions
Suppose the model under consideration is overidentified (m < nl). Under such
circumstances we can develop a test for the overall goodness-of-fit of the model.
Recall that by the mean value theorem we can express gT(0T) as:
9t(0t) = gT(0o) + D*r(6T - fl0) • (67)
If we multiply equation (67) by \ff and substitute for Vf(6T - Oo) from equation
(59) we obtain:
VfgT(eT) = (I~DT(ATDr)-1AT)VfgT(eo) . (68)
Substituting in the optimal choice for AT yields:
VfgT(eT) = (I~DT(D'rST:1D*rylDlrST-l)VTgT(0o) , (69)
so that by Slutsky's theorem:
VfgT(eT)M'- iMJWA))-1^1) x N(0,So) . (70)
Because So is symmetric and positive definite it can be factored as S0 = W, where
P is nonsingular. Thus (70) can be written as:

54
C. R. Harvey and C. Kirby
Vfp-'gAer^il-p-'Doiiy^Doy'D^Py^xN^r) . (71)
The matrix premultiplying the normal distribution in (71) is idempotent with rank
nl — m. It follows, therefore, that the overidentifying test statistic:
MT = Tgr(0T)'S^lgr(0T) (72)
converges to a central chi-square random variable with nl — m degrees of
freedom. The limiting distribution of Mj remains the same if we use a consistent
estimate ST in place of S0-
Note that in many respects the test for overidentifying restrictions is analogous
to the Lagrange multiplier test in maximum likelihood estimation. The GMM
estimator of Oq is obtained by setting m linear combinations of the nl
orthogonality conditions equal to zero. Thus there are nl — m linearly independent
combinations which have not been set equal to zero. Suppose we took these
nl — m linear combinations of the moment conditions and set them equal to a
(nl — m) x 1 vector of unknown parameters a. The system would then be exactly
identified and Mj would be identically equal to zero. Imposing the restriction that
a = 0 yields the efficient GMM estimator along with a quantity
TgT(6T)'S^.1 gT(6T) that can be viewed as the GMM analogue of the score form
of the Lagrange multiplier test statistic.
The test for overidentifying restrictions is appealing because it provides a
simple way to gauge how well the model fits the data. It would also be convenient,
however, to be able to test restrictions on the vector of parameters for the model.
As we shall see, such tests can be constructed in a straightforward manner.
G. Hypothesis testing in GMM
Suppose that we are interested in testing restrictions on the vector of parameters
of the form:
q(00) = 0 , (73)
where q is a known p x 1 vector of functions. Let the p x m matrix Q0 = dq/dO'
denote the Jacobian of q(6) evaluated at Oq. By assumption Q0 has rank p. We
know that for the efficient choice of the weighting matrix the limiting distribution
of the GMM estimator is:
Vf(eT-e0)^N(o,(iy0sslDo)-1) . (74)
Thus under fairly general regularity conditions the standard large-sample test
criteria are distributed asymptotically as central chi-square random variables with
p degrees of freedom when the restrictions hold.
Let 6ur and ffT denote the unrestricted estimator and the estimator obtained by
minimizing Jt{0) subject to q(0) = 0. The Wald test statistic is based on the
unrestricted estimator. It takes the form:

Instrumental variables estimation of conditional beta pricing models
55
WT = Tq{0T)'{QT{D'TS-TlDT)-lQ'T)-lq{eT) , (75)
where QT, DT and St are consistent estimates of Q0, D0 and So computed using
ffj. The Lagrange multiplier test statistic is constructed using the gradient of Jt(0)
evaluated at restricted estimator. It is given by:
LMT = TgT{erT)%lDT{BfTS^lDT)~lDfTS^gT[»T) » (76)
where DT and St are consistent estimates of Do and So computed from ffj. The
likelihood ratio type test statistic is equal to the difference between the over-
identifying test statistic for the restricted and unrestricted estimations:
LRT = T(gT(tfT)'SjlgT((rT) ~ 9tW)%19t{^t)) ■ (77)
The same estimate St must be used for both estimations.
It should be clear from the foregoing discussion that a consistent estimate of So
is one of the key elements of the GMM approach to estimation and testing. In
practice there are a number of different methods for estimating So, and the
appropriate method often depends on the specific characteristics of the model
under consideration. The discussion below provides an introduction to
heteroskedasticity and autocorrelation consistent estimation of the variance-
covariance matrix. A more detailed treatment can be found in Andrews (1991).
H. Robust estimation of the variance-covariance matrix
The variance-covariance matrix of Vfgr(Oo) is given by:
oo
So = £ r0(j) , (78)
J=-oo
where r0(/) = E[f(Xt+z,Zt,Oo)f(Xt+z-j,Zt-j,Oo)']. Because we have assumed
stationarity, this matrix can also be written as:
oo
So = ro(o) + ^(ro(/) + ro(/)') , (79)
using the relation r0(-y') = r0(/')'. Now we want to consider how we might go
about estimating So consistently. First take the scenario where the vector
J[Xt+z,Zt,Oo) is serially uncorrelated. Under such circumstances the second term
on the right-hand side of equation (79) drops out and
rr(0) = i/r^+tlz(,flr)/(^t,zr,()r)'
t=\
provides a consistent estimate for So-
The case where f[-) exhibits serial correlation is more complicated. Note that
the sum in equation (79) contains an infinite number of terms. It is obviously

56
C. R. Harvey and C. Kirby
impossible to estimate each of these terms. One way to proceed would be to treat
/(•) as if it were serially correlated for a finite number of lags L. Under such
circumstances a natural estimator for S0 would be:
sr = rr(o)+ ]£(/>(/)+ />(/)')
(80)
7=1
where />(/) = \/TY:li+jJ{Xt+x,Zt,er)J{Xt+^j,Zt_j,er)'. As long as the
individual rT(j) in equation (80) are consistent the estimator ST will be consistent
providing that L is allowed to increase at suitable rate as the sample size T
increases. But the estimator of 50 in (80) is not guaranteed to be positive semi-
definite. This can lead to problems in empirical work.
The solution to this difficulty is to calculate St as a weighted sum of the rT{j)
where the weights gradually decline to zero as j increases. If these weights are
chosen appropriately then ST will be both consistent and positive semidefinite.
Suppose we begin by defining the nl(L + 1) x rd(L + 1) partitioned matrix:
CT(L) =
/>(0)
/>(1)
/>(0)
rT{L) rT{L-\)
rT{L)'
rT(L-\)'
/>(o)
(81)
The matrix Ct(L) can always be written in the form Ct(L) = FT where Y is an
(T + L) x n£(L + 1) partitioned matrix. Take L = 2 as an example. The matrix Y
is given by:
Y =
Vf
o o f[xl+1,zx,eT)'
o j[xl+t,zueT)' ■
J[Xi+x,Zi,6T) : J{Xt+%,Zt,Qt)
'■ J[Xt+i,Zt,0t) 0
\j[xT+x,zT,eT)' o o
(82)
From this result it follows that Ct(L) is a positive semidefinite matrix. Next
consider the matrix:
ST(L) = [a07 oliI...ollI]
/>(0) ... rT(L)'
/>(1) ... rT{L-\)'
jT(L) ... rr(0)
a0/'
a]/
(83)
where the a, are scalars. Because St(L) is the partitioned-matrix equivalent of a
quadratic form in a positive semidefinite matrix it must also be positive semi-
definite. Equation (83) can be rearranged to show that:

Instrumental variables estimation of conditional beta pricing models
57
ST(L) = (a20 + --- + a2L)rT(0)
+ E (|! a<a^) (rrW + *>(/)') • (84)
The weighted sum on right-hand side of equation (84) has the general form of an
estimator for the variance-covariance matrix So. Thus if we select the a, so that
the weights in (84) are a decreasing function of L and we allow L to increase with
the sample size at an appropriately slow rate we obtain a consistent positive
semidefinite estimator for S0.
The modified Bartlett weights proposed by Newey and West (1987) have been
used extensively in empirical research. Let wj be the weight placed on rT(j) in the
calculation of the variance-covariance matrix. The weighting function for
modified Bartlett weights takes the form:
Wj={l~l^\ 7 = 0,1,2,...,! ,85j
1 \0 ]>L, X '
where L is the lag truncation parameter. Note that these weights are obtained by
setting at = l/y/T+T for i = 0,1,... ,L. Newey and West (1987) show that if L is
allowed to increase at a rate proportional to T1^ then St based on these weights
will be a consistent estimator of So. Although the weighting scheme proposed by
Newey and West (1987) is popular, recent research has shown that other schemes
may be preferable. Andrews (1991) explores both the theoretical and empirical
performance of a variety of different weighting functions. Based on his results
Parzen weights seem to offer an good combination of analytic tractability and
overall performance. The weighting function for Parzen weights is:
i-£ + ¥ 0<£<i
20 -if i<f<l • (86)
0 i>\
The final question we need to address is how choose the lag truncation
parameter L in (86). The simplest strategy is to follow the suggestions of Gallant
(1987) and set L equal to the integer closest to T1^5. The main advantage of this
plug-in approach is that it is yields an estimator that depends only on the sample
size for the data set in question. An alternative strategy developed by Andrews
(1991), however, may lead to better performance in small samples. He suggests
the following data-dependent approach: use the first-stage estimate of 0O to
construct the sample analogue of J[Xt+t,Zt,6o). Then estimate a first-order
autoregressive model for each element of this vector. The autocorrelation
coefficients along with the residual variances can be used to estimate the value of
L that minimizes the asymptotic truncated mean-squared-error of the estimator.
Andrews (1991) presents Monte Carlo results that suggest that estimators of So
constructed in this manner perform well under most circumstances.

58
C. R. Harvey and C. Kirby
6. Closing remarks
Asset pricing models often imply that the expected return on an asset can be
written as a linear function of one or more beta coefficients that measure the
asset's sensitivity to sources of undiversifiable risk in the economy. This linear
tradeoff between risk and expected return makes such models both intuitively
appealing and analytically tractable. A number of different methods have been
proposed for estimating and testing beta pricing models, but the method of
instrumental variables is the approach of choice in most situations. The primary
advantage of the instrumental variables approach is that it provides a highly
tractable way of characterizing time-varying risk and expected returns.
This paper provides an introduction the econometric evaluation of both
conditional and unconditional beta pricing models. We present numerous
examples of how the instrumental variable methodology can be applied to
various models. We began with a discussion of the conditional version of the
Sharpe (1964) - Lintner (1965) CAPM and used it to illustrate how the
instrumental variables approach could be used to estimate and test single beta
models. Then we extended the analysis to models with multiple betas and
introduced the concept of latent variables. We also provided an overview of the
generalized method of moments approach (GMM) to estimation and testing. All
of the techniques developed in this paper have applications in other areas of asset
pricing as well.
References
Andrews, D. W. K. (1991). Heteroskedasticity and autocorrelation consistent covariance matrix
estimation. Econometrica 59, 817-858.
Bansal, R. and C. R. Harvey (1995). Performance evaluation in the presence of dynamic trading
strategies. Working Paper, Duke University, Durham, NC.
Beneish, M. D. and C. R. Harvey (1995). Measurement error and nonlinearity in the earnings-returns
relation. Working Paper, Duke University, Durham, NC.
Black, F. (1972). Capital market equilibrium with restricted borrowing. J. Business 45, 444-454.
Blake, I. F. and J. B. Thomas (1968). On a class of processes arising in linear estimation theory. IEEE
Transactions on Information Theory IT-14, 12-16.
Bollerslev, T., R. F. Engle and J. M. Wooldridge (1988). A capital asset pricing model with time
varying covariances. J. Politic. Econom. 96, 116-31.
Breeden, D. (1979). An intertemporal asset pricing model with stochastic consumption and investment
opportunities. J. Financ. Econom. 7, 265-296.
Campbell, J. Y. (1987). Stock returns and the term structure. J. Financ. Econom. 18, 373^00.
Carhart, M. and R. J. Krail (1994). Testing the conditional CAPM. Working Paper, University of
Chicago.
Chu, K. C. (1973). Estimation and decision for linear systems with elliptically random processes. IEEE
Transactions on Automatic Control AC-18, 499-505.
Cochrane, J. (1994). Discrete time empirical finance. Working Paper, University of Chicago.
Devlin, S. J. R. Gnanadesikan and J. R. Kettenring, Some multivariate applications of elliptical
distributions. In: S. Ideka et al., eds., Essays in probability and statistics, Shinko Tsusho, Tokyo,
365-393.

Instrumental variables estimation of conditional beta pricing models
59
Dybvig, P. H. and S. A. Ross (1985). Differential information and performance measurement using a
security market line. J. Finance 40, 383-400.
Dumas, B. and B. Solnik (1995). The world price of exchange rate risk. J. Finance 445^80.
Fama, E. F. and J. D. MacBeth (1973). Risk, return, and equilibrium: Empirical tests. J. Politic.
Econom. 81, 607-636.
Ferson, W. E. (1990). Are the latent variables in time-varying expected returns compensation for
consumption risk. J. Finance 45, 397-430.
Ferson, W. E. (1995). Theory and empirical testing of asset pricing models. In: Robert A. J. W. T.
Ziemba and V. Maksimovic, eds. North Holland 145-200
Ferson, W. E., S. R. Foerster and D. B. Keim (1993). General tests of latent variables models and
mean-variance spanning. J. Finance 48, 131-156.
Ferson, W. E. and C. R. Harvey (1991). The variation of economic risk premiums. J. Politic. Econom.
99, 285-315.
Ferson, W. E. and C. R. Harvey (1993). The risk and predictability of international equity returns.
Rev. Financ. Stud. 6, 527-566.
Ferson, W. E. and C. R. Harvey (1994a). An exploratory investigation of the fundamental
determinants of national equity market returns. In: Jeffrey Frankel, ed., The internationalization of
equity markets, Chicago: University of Chicago Press, 59-138.
Ferson, W. E. and R. A. Korajczyk (1995) Do arbitrage pricing models explain the predictability of
stock returns. J. Business, 309-350.
Ferson, W. E. and Stephen R. Foerster (1994). Finite sample properties of the Generalized Method of
Moments in tests of conditional asset pricing models. J. Financ. Econom. 36, 29-56.
Gallant, A. R. (1981). On the bias in flexible functional forms and an essentially unbiased form: The
Fourier flexible form. J. Econometrics 15, 211-224.
Gallant, A. R. (1987). Nonlinear statistical models. John Wiley and Sons, NY.
Gallant, A. R. and G. E. Tauchen (1989). Seminonparametric estimation of conditionally constrained
heterogeneous processes. Econometrica 57, 1091-1120.
Gallant, A. R. and H. White (1988). A unified theory of estimation and inference for nonlinear
dynamic models. Basil Blackwell, NY.
Gallant, A. R. and H. White (1990). On learning the derivatives of an unknown mapping with
multilayer feedforward networks. University of California at San Diego.
Gibbons, M. R. and W. E. Ferson (1985). Tests of asset pricing models with changing expectations
and an unobservable market portfolio. J. Financ. Econom. 14, 217-236.
Glodjo, A. and C. R. Harvey (1995). Forecasting foreign exchange market returns via entropy coding.
Working Paper, Duke University, Durham NC.
Hansen, L. P. (1982). Large sample properties of generalized method of moments estimators.
Econometrica 50, 1029-1054.
Hansen, L. P. and R. J. Hodrick (1983). Risk averse speculation in the forward foreign exchange
market: An econometric analysis of linear models. In: Jacob A. Frenkel, ed., Exchange rates and
international macroeconomics, University of Chicago Press, Chicago, IL.
Hansen, L. P. and R. Jagannathan (1991). Implications of security market data for models of dynamic
economies. J. Politic. Econom. 99, 225-262.
Hansen, L. P. and R. Jagannathan (1994). Assessing specification errors in stochastic discount factor
models. Unpublished working paper, University of Chicago, Chicago, IL.
Hansen, L. P. and S. F. Richard (1987). The role of conditioning information in deducing testable
restrictions implied by dynamic asset pricing models. Econometrica 55, 587-613.
Hansen, L. P. and K. J. Singleton (1982). Generalized instrumental variables estimation of nonlinear
rational expectations models. Econometrica, 50, 1269-1285.
Harvey, C. R. (1989). Time-varying conditional covariances in tests of asset pricing models. J. Financ.
Econom. 24, 289-317.
Harvey, C. R. (1991a). The world price of covariance risk. J. Finance 46, 111-157.
Harvey, C. R. (1991b). The specification of conditional expectations. Working Paper, Duke
University.

60
C. R. Harvey and C. Kirby
Harvey, C. R. (1995), Predictable Risk and returns in emerging markets, Rev. Financ. Stud. 773-816.
Harvey, C. R. and C. Kirby (1995). Analytic tests of factor pricing models. Working Paper, Duke
University, Durham, NC.
Harvey, C. R., B. H. Solnik and G. Zhou (1995). What determines expected international asset
returns? Working Paper, Duke University, Durham, NC.
Huang, R. D. (1989). Tests of the conditional asset pricing model with changing expectations.
Unpublished working Paper, Vanderbilt University, Nashville, TN.
Jagannathan, R. and Z. Wang (1996). The CAPM is alive and well. J. Finance 51, 3-53.
Kan, R. and C. Zhang (1995). A test of conditional asset pricing models. Working Paper, University of
Alberta, Edmonton, Canada.
Keim, D. B. and R. F. Stambaugh (1986). Predicting returns in the bond and stock market. J. Financ.
Econom. 17, 357-390.
Kelker, D. (1970). Distribution theory of spherical distributions and a location-scale parameter
generalization. Sankhya, series A, 419-430.
Kirby, C (1995). Measuring the predictable variation in stock and bond returns. Working Paper, Rice
University, Houston, Tx.
Lintner, J. (1965). The valuation of risk assets and the selection of risky investments in stock portfolios
and capital budgets. Rev. Econom. Statist. 47, 13-37.
Merton, R. C. (1973). An intertemporal capital asset pricing model. Econometrica 41, 867-887.
Newey, W. K. and K. D. West (1987). A simple, positive semi-definite, heteroskedasticity-consistent
covariance matrix. Econometrica 55, 703-708.
Potscher, B. M. and I. R. Prucha (1991a). Basic structure of the asymptotic theory in dynamic
nonlinear econometric models, part I: Consistency and approximation concepts. Econometric Rev.
10, 125-216.
Potscher, B. M. and I. R. Prucha (1991b). Basic structure of the asymptotic theory in dynamic
nonlinear econometric models, part II: Asymptotic normality. Econometric Rev. 10, 253-325.
Ross, S. A. (1976). The arbitrage theory of capital asset pricing. J. Econom. Theory 13, 341-360.
Shanken, J. (1990). Intertemporal asset pricing: An empirical investigation. J. Econometrics 45, 99-
120.
Sharpe, W. (1964). Capital asset prices: A theory of market equilibrium under conditions of risk. J.
Finance 19, 425^42.
Silverman, B. W. (1986). Density estimation for statistics and data analysis. London: Chapman and
Hall.
Solnik, B. (1991). The economic significance of the predictability of international asset returns.
Working Paper, HEC-School of Management.
Vershik, A. M. (1964). Some characteristics properties of Gaussian stochastic processes. Theory
Probab. Appl. 9, 353-356.
White, H. (1980). A heteroskedasticity consistent covariance matrix estimator and a direct test of
heteroskedasticity. Econometrica 48, 817-838.
Zhou, G. (1995). Small sample rank tests with applications to asset pricing. J. Empirical Finance 2, 71-
94.

G.S. Maddala and C.R. Rao, eds., Handbook of Statistics, Vol. 14
© 1996 Elsevier Science B.V. All rights reserved.
3
Semiparametric Methods for Asset Pricing Models
Bruce N. Lehmann
This paper discusses semiparametric estimation procedures for asset pricing
models within the generalized method of moments (GMM) framework. GMM is
widely applied in the asset pricing context in its unconditional form but the
conditional mean restrictions implied by asset pricing theory are seldom fully
exploited. The purpose of this paper is to take some modest steps toward
removing these impediments. The nature of efficient GMM estimation is cast in a
language familiar to financial economists: the language of maximum correlation
or optimal hedge portfolios. Similarly, a family of beta pricing models provides a
natural setting for identifying the sources of efficiency gains in asset pricing
applications. My hope is that this modest contribution will facilitate more routine
exploitation of attainable efficiency gains.
1. Introduction
Asset pricing relations in frictionless markets are inherently semiparametric. That
is, it is commonplace for valuation models to be cast in terms of conditional
moment restrictions without additional distributional assumptions. Accordingly,
a natural estimation strategy replaces population conditional moments with their
sample analogues. Put differently, the generalized method of moments (GMM)
framework of Hansen (1982) tightly links the economics and econometrics of
asset pricing relations.
While applications of GMM abound in the asset pricing literature, empirical
workers seldom make full use of the GMM apparatus. In particular, researchers
generally employ the unconditional forms of the procedures which do not exploit
all of the efficiency gains inherent in the moment conditions implied by asset
pricing models. There are two plausible reasons for this: (1) the information
requirements are often sufficiently daunting to make full exploitation seem
infeasible and (2) the literature on efficient semiparametric estimation is
somewhat dense.
The purpose of this paper is to take some modest steps toward removing these
impediments. The nature of efficient GMM estimation is cast in terms familiar to
financial economists: the language of maximum correlation or optimal hedge
61

62
B. N. Lehmann
portfolios. Similarly, a family of beta pricing models provides a natural setting for
identifying the sources of efficiency gains in asset pricing applications. My hope is
that this modest contribution will facilitate more routine exploitation of
attainable efficiency gains.
The layout of the paper is as follows. The next section provides an outline of
GMM basics with a view toward the subsequent application to asset pricing
models. The third section lays out the links between the economics of asset prices
when markets do not permit arbitrage opportunities and the econometrics of asset
pricing model estimation given the conditional moment restrictions implied by the
absence of arbitrage. The general efficiency gains discussed in these two sections
are worked out in detain in the fourth section, which documents the sources of
efficiency gains in beta pricing models. The final section provides some concluding
remarks.
2. Some relevant aspects of the generalized method of moments (GMM)
Before elucidating the links between GMM and asset pricing theory, it is
worthwhile to lay out some GMM basics with an eye toward the applications that
follow. The coverage is by no means complete. For example, the relevant large
sample theory is only sketched (and not laid out rigorously) and that which is
relevant is only a subset of the estimation and inference problems that can be
addressed with GMM. The interested reader is referred to the three surveys in
Volume 11 of this series Hall (1993), Newey (1993), and Ogaki (1993) for more
thorough coverage and references.
The starting point for GMM is a moment restriction of the form:
E[gt(60)\It-l]=E\gt(80)} = 0 (2.1)
where g (00) is the conditional mean zero random qxl vector in the model, 0o is
the associated pxl vector of parameters in the model, and It-\ is some
unspecified information set that at least includes lagged values of g (0O). The
restriction to zero conditional mean random variables means that g (0q) follows a
martingale difference sequence and, thus, is serially uncorrected.1
A variety of familiar econometric models take this form. Consider, for
example, the linear regression model:
Yt=xt% + et (2.2)
where yt is the tth observation on the dependent variable, xt is a pxl vector of
explanatory variables, and et is a random disturbance term. In this model,
suppose that the econometrician observes a vector zt for which it is known that
E[£f|if_i] = 0. Then this model is characterized by the conditional moment
condition:
1 The behavior of GMM estimators can be readily established when gjjf) is serially dependent so
long as a law of large numbers and central limit theorem apply to its time series average.

Semiparametric methods for asset pricing models
63
£»(&) = *&-*'> E[£^-ila-il = E[£^-i] = E[£fe-i = o ■ (2.3)
When z,_j = x, this is the linear regression model with possibly stochastic re-
gressors; otherwise, it is an instrumental variables estimator.
GMM involves setting sample analogues of these moment conditions as close
to zero as possible. Of course, they cannot all be set to zero if the number of
linearly independent moment conditions exceeds the number of unknown
parameters. Instead, GMM takes p linear combinations of these moment
conditions and seeks values of 6 for which these linear combinations are zero.
First, consider the unconditional version of the moment condition - that is,
E[g (Gq)] = 0. In order for the model to be identified, assume that a (Oq) possesses
a nonsingular population covariance matrix and that E[dgt(60)'/d6] has full row
rank. The GMM estimator can be derived in two ways. Following Hansen (1982),
the GMM estimator 6T minimizes the sample quadratic form based on a sample
of T observations on g (0O):
mragr(0)'0VC0o)£r(0) ; lr(0) =^^(0) (2.4)
- t=\
given a positive definite weighting matrix ^V(^o) converging in probability to a
positive definite limit W(60). In this variant, the econometrician chooses WT(60)
to give the GMM estimator desirable asymptotic properties.
Alternatively, we can simply define the estimator dT as the solution to the
equation system:
AT^ri^T) = ^MOo^iir) = 0 (2.5)
t=\
where^7-(0o) is a sequence of p x ^0^(1) matrices converging to a limit A(6) with
row rank p. In this formulation, AT(60) is chosen to give the resulting estimator
desirable asymptotic properties. The estimating equations for the two variants
are, of course, identical in form since:
At(0o)It(Ot) = GT8TWT(80)gT(§r) = 0 ;
_%(0)' 1^%(0O)' (2-6)
G
T(2)
T 2-~i
96 Tj^ 86
For my purpose, equation (2.5) is a more suggestive formulation.
The large sample behavior of 6T is straightforward, particularly in this case
where g^Q^) a martingale difference sequence.2 An appropriate weak law of large
numbers insures that gT(60)—>0, which, coupled with the identification conditions,
implies that 6T^>6a. So long as the necessary time series averages converge:
2 The standard reference on estimation and inference in this framework is Hansen (1982).

64
B. N. Lehmarm
M0o) = ^i>[£,(0o)0r(0o)lAs(0o)
TtJ "^ "~ (2.7)
\S(00)\>0 Gr(0o)-G(0o)
the standard first order Taylor expansion coupled with Slutsky's theorem yields:
and an appropriate central limit theorem for martingales ensures that
Vf(6T - 0o) ^N[0, D(eo)S(0o)D(eo)'] . (2.9)
Consistent standard error estimates that are robust to conditional hetero-
skedasticity can be calculated from this expression by replacing 0$ with 6T?
What choice of AT(Oo) or, equivalently, of WriOg) is optimal? All fixed weight
estimators - that is, those that apply the same matrix At{Qsj) to each g (60) for
fixed T - are consistent under the weak regularity conditions sketched above.
Accordingly, it is natural to compare the asymptotic variances of estimators, a
criterion that can, of course, be justified more formally by confining attention to
the class of regular estimators that rules out superefficient estimators. The
asymptotically optimal A^(60) is obtained by equating WT(0O) with SV(0o)~\
yielding an asymptotic covariance matrix of [G(0o)S(0o)_ G(0O)']_1. Once again,
St (So) can be estimated consistently by replacing 0 with 6T.4
The optimal unconditional GMM estimator has a clear connection with the
maximum likelihood estimator (MLE), even though we do not know the
probability law generating the data. Let ifr(0o,jj) denote the logarithm of the
population conditional distribution of the data underlying g (0q) where r\ is a
possibly infinite dimensional set of nuisance parameters. Similarly, let i?J(0o,»?)
denote the true score function, the vector of derivatives of i?f(0o>^)> with respect
to 6. Consider the unconditional population projection of Z£\{Q$,r\) on the
moment conditions g (0$):
3 Autocorrelation is not present under the hypothesis that (l(Q) has conditional mean zero and is
sampled only once per period (that is, the data are not overlapping). If the data are overlapping, the
moment conditions will have a moving average error structure. See Hansen and Hodrick (1980) for a
discussion of covariance matrix estimation in this case and Hansen and Singleton (1982) and Newey
and West (1987) for methods appropriate for more general autocorrelation.
4 The possible singularity of ST(6) is discussed indirectly in Section 4.3 as part of the justification
for factor structure assumptions. While my focus is not on hypothesis testing, the quadratic form in the
fitted value of the moment conditions and the optimal weighting matrix yields the test statistic
T 3j.(67-)'St(02-)~137.(02-)—>Z2(? ~ p) since p degrees of freedom are used in estimating Q. This test of
overidentifying conditions is known as Hansen's J test.

Semiparametric methods for asset pricing models
65
JS?;(0o)2) = Cov[if;(0Ol2),^(0o)']Var[0r(0o)]-10r(0o) + v^ut ;
= -4>xP-1gf(e0) + vj?ut ;
<?
Mo)''
(2.10)
do
v = E\Sx(e0)gt(e0)']
since E[i?J(0Q, jj)'0 (0o)'] = ~^ 1S zero given sufficient regularity to allow
differentiation of the moment condition E[g (0$)] = 0 under the integral sign. In this
notation, the asymptotic variance of the unconditional GMM estimator is
Hence, the optimal fixed linear combination of moment conditions A^,(6o) has
the largest unconditional correlation with the true, but unknown, conditional
score in finite samples. This fact does not lead to finite sample efficiency
statements for at least two reasons. First, the MLE itself has no obvious efficiency
properties in finite samples outside the case where the score takes the linear form
I{Qa){S- — So) where /(0O) is the Fisher information matrix. Second, the feasible
optimal estimator replaces 8$ with 0T in ^(0O), yielding a consistent estimator
with no obvious finite sample efficiency properties. Nevertheless, the optimal fixed
weight GMM estimator retains this optimality property in large samples.
Now consider the conditional version of the moment condition; that is,
E[g (0o)|/r_i] = 0. The prior information available to the econometrician is that
g (0O) is a martingale difference sequence. Hence, the econometrician knows only
that linear combinations of the g (6$) with weights based on information available
at time t—\ have zero means - nonlinear functions of g (0$) have unknown
moments given only the martingale difference assumption. Since the
econometrician is free to use time varying weights, consider estimators of the form:5
1 T
-Y,At-xgt{0T) = Q- 4_ie/,_i (2.11)
1 r=i
where At-\ is a sequence of p x q op{\) matrices chosen by the econometrician. In
order to identify the model, assume gf((0o) has a nonsingular population
conditional covariance matrix E[gr (0o)^ (0o)/[7r_1] and that E[dg (Qo)'/d6\It-i] has full
row rank.
The basic principles of asymptotically optimal estimation and inference in the
conditional and unconditional cases are surprisingly similar ignoring the
difficulties associated with the calculation of conditional expectations E[«|/r].6
Once again, under suitable conditional versions of the regularity conditions
sketched above:
5 The estimators could, in principle, involve nonlinear functions of these time series averages but
their asymptotic linearity means that their effect is absorbed in At-\.
6 Hansen (1985), Tauchen (1986), Chamberlain (2987), Hansen, Heaton, and Ogaki (1988), Newey
(1990), Robinson (1991), Chamberlain (1992), and Newey (1993) discuss efficient GMM estimation in
related circumstances.

66
B. N. Lehmann
1 T
-i
*<-i = E
as
dgt(80)'
1 T
t=i
>Dc(e)0 ;
80
\h
(2.12)
*Sc(So)
the sample moment condition (2.11) is asymptotically linear so that:
V^(gr_&)4-Z>c(&)-^£>_lgr
(So)
and
Vf(6T - Oa) ZN%Dc(eo)Sc{0o)Dc(0o)']
(2.12)
(2.13)
The econometrician can choose the weighting matrices At_\ to minimize the
asymptotic variance of this estimator. The weighting matrices A°t_\ which are
optimal in this sense are given by:
1t-i
*,_i«7-!i ; «Vi = e^)^)'!/,-!]
and the resulting minimal asymptotic variance is:
Var[VT(0r-0o)]^
7fP'-^'-v
(2.14)
(2.15)
4[E(*,_197-i*<-
i-i
The evaluation of A\'_{ need not be straightforward and doing so in asset pricing
applications is the main preoccupation of Section 4.7
The relations between the optimal conditional GMM estimator and the MLE
are similar to the relations arising in the unconditional case. The conditional
population projection of .S?J(0o, r\) on the moment conditions g (Qq) reveals that:
7 The implementation of this efficient estimator is straightforward given the ability to calculate the
relevant conditional expectations. Under weak regularity conditions, the estimator can be
implemented in two steps by first obtaining an initial consistent estimate (perhaps using the unconditional
GMM estimator (2.5)), estimating the optimal weighting matrix At_\ using this preliminary estimate,
and then solving (2.14) for the efficient conditional GMM estimator. Of course, equations (2.11) and
(2.14) can be iterated until convergence, although the iterative and two step estimators are
asymptotically equivalent to first order.

Semiparametric methods for asset pricing models
67
^(So.i?) = Cov[if;(0o,»Z),^(0o)'|/,-i]Var[^(0o)|/,_1]-1 gj^) +v#ct
since E[i?',(i90, jj)g (Qo)'+ dg {0^)'IdQ\It-\] is zero given sufficient regularity to
interchange the order of differentiation and integration of the conditional
moment condition E[g (0o)|7,_i] = 0. Hence, the optimal linear combination of
moment conditions A°t_x has the largest conditional correlation with the true, but
unknown, conditional score in finite samples. While this observation does not
translate into clear finite sample efficiency statements, the GMM estimator based
on A°t_x is that which is most highly correlated with the MLE asymptotically.
It is easy to characterize the relative efficiency of the optimal conditional and
unconditional GMM estimators. As is usual, the variance of the difference
between the optimal unconditional and conditional GMM estimators is the
difference in their variances since the latter is efficient relative to the former. The
difference in the optimal weights given to the martingale increments g (<90) is:
4-i -4(0<>) = [**-i - gt(6)}v;_\ + gt{6)[v;\ - srm1]
4[*/_1-*]«^i1 + *[«pr_11-y-1].
Note that the law of iterated expectations applies to both <P,_i and *f,_i
separately but not to the composite A°t_x so that E[A°t_x —A^Oq)] does not generally
converge to zero. In any event, the relative efficiency of the conditional estimator
is higher when there is considerable time variation in both $,_i and f^i.
Finally, the conventional application of the GMM procedure lies somewhere
between the conditional and unconditional cases. It involves the observation that
zero conditional mean random variables are also uncorrelated with the elements
of the information set. Let Z,_i e I,^\ denote an r x q(r > p) matrix of
predetermined variables and consider the revised moment conditions
E[Zt.lg4{60)\It-l] = EiZ^Oo)] = 0 V 3_! e /,_! . (2.18)
In the unconditional GMM procedure discussed above, Zt_\ is lq, the q x q
identity matrix. In many applications, the same predetermined variables z,_!
multiply each element of g (6$) so that Zt_\ takes the form lq®zt-\- Finally,
different subsets of the information available to the econometrician zit_{ g lt_\
can be applied to each element of g (0q) so that Zt_\ is given by
Zt-
(Zxt-i Q ■•• 0 \
V Q o ••• z^_J
(2.19)
While optimal conditional GMM can be applied in this case, the main point of
this procedure is to modify unconditional GMM. As before, the unconditional
population projection of i^(#o) on the moment conditions Z,_\g (6$) yields

68
B. N. Lehmarm
i?;(0o,2) = Cov[if;(0o,?I),0r(0o)'z;_i]Var[Zr_1gr(0o)]-1Zr_1£r(0o) + v^uZt
(2.20)
VguZt
$7.
VZ = E{Zr_l£r(0o)^(0o)Xi}
since E{i?'r(0o, r^g^)' Z^} = -4>z given sufficient regularity to allow
differentiation under the integral sign. The weights ^y^t-i can also be viewed as
a linear approximation to the optimal conditional weights A^{ = (P^if^.
Put differently, ^4°_1 would generally be a nonlinear function of Zr_i if Zr_i
were the relevant conditioning information from the perspective of the econ-
ometrician.
3. Asset pricing relations and their econometric implications
Modern asset pricing theory follows from the restrictions on security prices that
arise when markets do not permit arbitrage opportunities. That the absence of
arbitrage implies substantive restrictions is somewhat surprising. Outside of
international economics, it is not commonplace for the notion that two eggs should
sell for the same price in the absence of transactions costs to yield meaningful
economic restrictions on egg prices - after all, two eggs of equal grade and
freshness are obviously perfect substitutes.8 By contrast, the no-arbitrage
assumption yields economically meaningful restrictions on asset prices because of
the nature of close substitutes in financial markets. Different assets or, more
generally, portfolios of assets may be perfect substitutes in terms of their random
payoffs but this might not be obvious by inspection since the assets may represent
claims on seemingly very different cash flows.
The asset pricing implications of the absence of arbitrage have been elucidated
in a number of papers including Rubinstein (1976), Ross (1978b), Harrison and
Kreps (1979), and Chamberlain and Rothschild (1983), Hansen and Richard
(1987). Consider trade in a securities market on two dates: date t — 1 (i.e., today)
and date t (i.e., tomorrow). There are N risky assets, indexed by i = 1, • • •, N,
which need not exhaust the asset menu available to investors. The nominal price
of asset i today is Pu~\. Its value tomorrow - that is, its price tomorrow plus any
cash flow distribution between today and tomorrow - is uncertain from the
perspective of today and takes on the random value Pit + Dit tomorrow. Hence,
its gross return (that is, one plus its percentage return) is given by
Rit = (Pu + Dit)/Pit_i. Finally, the one period riskless asset, if one exists, has
the sure gross return Rft = \)Pft-\ and i always denotes a suitably conformable
vector of ones.
This observation was translated into a lively diatribe by Summers (1985, 1986).

Semiparametric methods for asset pricing models
69
The market has two crucial elements: one environmental and one behavioral.
First, the market is frictionless: trade takes place with no taxes, transactions costs,
or other restrictions such as short sales constraints.9 Second, investors vigorously
exploit any arbitrage opportunities, behavior that is facilitated by the no frictions
assumption, that is, investors are delighted to make something for nothing and
they can costlessly attempt to do so.
In order to illustrate the asset pricing implications of the absence of arbitrage,
suppose that a finite number of possible states of nature s = 1, ...,5 can occur
tomorrow and that the possible security values in these states are Pht + Dut.10
Clearly, there can be at most min [N,S] portfolios with linearly independent
payoffs. Hence, the prices of pure contingent claims - securities that pay one unit
of account if state s occurs and zero otherwise - are uniquely determined if N > S
and if there are at least S assets with linearly independent payoffs. If N < S, the
prices of such claims are not uniquely determined by arbitrage considerations
alone, although they are restricted to lie in an N-dimensional subspace if the asset
payoffs are linearly independent.
Let ^ist_\ denote the price of a pure contingent claim that pays one unit of
account if state s occurs tomorrow and zero otherwise. These state prices are all
positive so long as each state occurs with positive probability according to the
beliefs of all investors. The price of any asset is the sum of the values of its payoffs
state by state.11 In particular:
s s
or, equivalently:
s s
^st-xRis, = i ; R/t-i $>*-! = i • (3.2)
s=\ s=\
Since they are non-negative, scaling state prices so that they sum to one gives
them all of the attributes of probabilities. Hence, these risk neutral probabilities:
9 Some frictions can be easily accommodated in the no-arbitrage framework but general frictions
present nontrivial complications. For recent work that accommodates proportional transactions costs
and short sales constraints, see Hansen, Heaton, and Luttmer (1993), He and Modest (1993), and
Luttmer (1993).
10 The restriction to two dates involves little loss of generality as the abstract states of nature could
just as easily index both different dates and states of nature. In addition, most of the results for finite S
carry over to the infinite dimensional case, although some technical issues arise in the limit of
continuous trading. See Harrison and Kreps (1979) for a discussion.
11 The frictionless market assumption is implicit in this statement. In markets with frictions, the
return of a portfolio of contingent claims would not be the weighted average of the returns on the
component securities across states but would also depend on the trading costs or taxes incurred in this
portfolio.

70
B. N. Lehmarm
"st-l
■■Rfrt.
st-l
Pft-i
(3.3)
comprise the risk neutral martingale measure, so called because the price of any
asset under these probability beliefs is given by:
P„-i = Pft-i ^2 Kt-i (P*t + Dist)
(3.4)
s=l
that is, its expected present value. Risk neutral probabilities are one summary of
the implications of the absence of arbitrage; they exist if and only if there is no
arbitrage.
This formulation of the state pricing problem is extremely convenient for
pricing derivative claims. Under the risk neutral martingale measure, the riskless
rate is the expected return of any asset or portfolio that does not change the span of
the market and for which there is a deterministic mapping between its cash flows
and states of nature. However, it is not a convenient formulation for empirical
purposes. Actual return data is provided according to the true (objective)
probability measure. That is, actual returns are generated under rational expectations.
Accordingly, let nst-\ be the objective probability that state s occurs at time t
given some arbitrary set of information available at time t—\ denoted by lt-\. The
reformulation of the pricing relations (3.1) and (3.2) in terms of state prices per
unit probability qst-\ = \l/st_l/itst-i reveals:
*Vi=E
V»-i
^ qst-\ {Pist + Djst) \It-i
s=l
S
= E[Qt(Pit+Dit)\It
f-i
X^-'l7'-
.5=1
(3.5)
E[&|/«-
or, equivalently, in their expected return form:
s
UstUt-l
y~]qst-\RiSt\I,
s=l
S
/ ,qst-\Rft-\\it
s=l
E[&W-i] = 1
= RftE[Qt\It_l] = l
(3.6)
At this level of generality, these conditional moment restrictions are the only
implications of the hypothesis that markets are frictionless and that market prices
are marked by the absence of arbitrage.
Asset pricing theory endows these conditional moment conditions with
expirical content through models for the pricing kernel Qt couched in terms of

Semiparametric methods for asset pricing models
71
potential observables.12 Such models equate the state price per unit probability
qst-\, the cost per unit probability of receiving one unit of account in state s, with
some corresponding measure of the marginal benefit of receiving one unit of
account in state s.13 Most equilibrium models equate Qt, adjusted for inflation,
with the intertemporal marginal rate of substitution of a hypothetical,
representative optimizing investor.14 The most common formulation is additively
separable, constant relative risk aversion preferences for which Qt = p(c,/c,_i)~a
where p is the rate of time preference, c,/c,_i is the rate of consumption growth,
and a is the coefficient of relative risk aversion, all for the representative
agent.15
Accordingly, let xt denote the relevant observables that characterize these
marginal benefits in some asset pricing model. Hence, pricing kernel models take
the general form:
Qt = Q(xl,iQ) ; a>0 ; *,€/, (3.7)
where Oq is a vector of unknown parameters. To be sure, the parametric
component can be further weakened in settings where it is possible to estimate the
function g(«) nonparametrically given only observations on R^ and xt. However,
the bulk of the literature involves models in the form (3.7).16
Equations (3.5) through (3.7) are what make asset pricing theory inherently
semiparametric.17 The parametric component of these asset pricing relations is a
12 It is also possible to identify the pricing kernel nonparametrically with the returns of particular
portfolios. For example, the return of growth optimal portfolio which solves max
E{lnvi'(_15(|//_i;H'g/_i 6/(_i} is equal to Q~x. Of course, it is hard to solve this maximum problem
without parametric distributional assumptions. See Bansal and Lehmann (1955) for an application to
the term structure of interest rates. The addition of observables can serve to identify payoff relevant
states, giving nonparametric estimation a somewhat semiparametric flavor. Put differently, the econo-
metrician typically observes a sequence of returns without information on which states have been
realized; the vector x, provides is an indicator of the payoff relevant state of nature realized at time t
that helps identify similar outcomes (i.e., states with similar state prices per unit probability). Bansal
and Viswanathan (1993) estimate a model along these lines.
13 The marginal benefit side of this equation rationalizes the peculiar dating convention for Q,
when it is equal to the time t-\ state price per unit probability.
14 Embedding inflation in Q, eliminates the need for separate notation for real and nominal pricing
kernels. That is, Q, is equal to Q^PalPa~\ where Pa is an appropriate index for translating real cash
flows and the real pricing kernel 2fai into nominal cash flows and kernels.
15 More general models allow for multiple goods and nonseparability of preferences in
consumption over time and states as would arise from durability in consumption goods and from
preferences marked by habit formation and non-expected utility maximization. Constantinides and
Ferson (1991) summarize much of the durability and habit formation literatures, both theoretically
and empirically. See Epstein and Zin (1991a) and Epstein and Zin (1991b) for similar models for Q,
which do not impose state separability. Cochrane (1991) exploits the corresponding marginal
conditions for producers.
16 Exceptions include Bansal and Viswanathan (1993) and the linear model Q, = a>J_1x, with («,_,
unobserved, a model discussed in the next section.
17 To be sure, the econometrician could specify a complete parametric probability model for asset
returns and such models figure prominently in asset pricing theory. Examples include the Capital Asset
Pricing Model (CAPM) when it is based on normally distributed returns and the family of continuous
time intertemporal asset pricing models when prices are assumed to follow ltd processes.

72
B. N. Lehmann
model for the pricing kernel Q(x„0q). The conditional moment conditions (3.6)
can then be used to identify any unknown parameters in the model for Q, and to
test its overidentifying restrictions without additional distributional assumptions.
Note also that the structure of asset pricing theory confers an obvious
econometric simplification. The constructed variables QtRit — 1 constitute a
martingale difference sequence and, hence, are serially uncorrelated. This fact
greatly simplifies the calculation of the second moments of sample analogues of
(3.6), which in turn simplifies estimation and inference.18
Moreover, the economics of these relations constrains how these conditional
moment restrictions can be used for estimation and interference. Ross (1978b)
observed that portfolios are the only derivative assets that can be priced solely as
a function of observables, time, and primary asset values given only the absence
of arbitrage opportunities in frictionless markets. The same is true for
econometricians - for a given asset menu, the econometrician knows only the
prices and payoffs of portfolios with weights w/_1 Glt~i.
Hence, only linear combinations of the conditional moment conditions based
on information available at time t—\ can be used to estimate the model.
Accordingly, in the absence of distributional restrictions, the econometrician must
base estimation and inference on estimators of the form:
1 T
-Y,A,-x%Q{Xi,lQ) -i] = 0 ; A^ e /,_! (3.8)
1 (=i
where At-\ is a sequence of p x N op{\) matrices chosen by the econometrician
and p is the number of elements in Oq. The matrices At^\ can be interpreted as the
weights of p portfolios with random payoffs At-iE* that cost At-\i units of
account.
How would a financial econometrician choose A°t_{> An econometrician who
favors likelihood methods for their desirable asymptotic properties might prefer
the p portfolios with maximal conditional correlation with the true, but
unknown, conditional score. In this application, the conditional projection of
^",(Sjo,U) on %Q(x,,0Q) -i] is given by:
Coy[^(90,r,),S4Q(xt,eQ)'\It^]ySiT[RtQ(xt,eQ)\It^r1
x &efe,ee)-i] + t^ce(,
-*,_i ¥7-1 [&tQ(xt,eQ) - j] + v#cQt ; (3.9)
aE[8fe,ge)g,|/(.i]/
do
E{fee(xr, ee) - mtQ(xT,eQ) - i]'|/(~i}
18 This observation fails if returns and Q, are sampled more than once per period. For example,
consider the two period total return (i.e., with full reinvestment of intermediate cash flows)
Riv+\ =RitRit+\ which satisfies the two period moment condition E[Q,Ql+\Riltt+\ |/<_i] = 1. In this
case, the constructed random variable g,g!+iiJ„,(+i-l follows a first order moving average process.
See Hansen and Hodrick (1980) and Hansen, Heaton, and Ogaki (1988) for more complete
discussions.

Semiparametric methods for asset pricing models
73
since E{i^(0o,rf)[R,Q(xt,6Q) - i\'\It-i} = -$t-\ given sufficient regularity to
permit differentiation under the integral sign. The p portfolios with payoffs
^t-i^-i&t that cost 4>t-\f^il units of account have no obvious optimality
properties from the perspective of prospective investors. However, they are
definitely optimal from the perspective of financial econometricians - they are the
optimal hedge portfolios for the conditional score of the true, but unknown, log
likelihood function.
Put differently, the economics and the econometrics coincide here. The
econometrician can only observe conditional linear combinations of the
conditional moment conditions and seeks portfolios whose payoffs provide
information about the parameters of the pricing kernel Q(Xj,@q)- The optimal
portfolio weights are $/_iy,".1, and the payoffs $(_i f^1^ maximize the
information content of each observation, resulting in an incremental contribution
of ^-ilPjl1! <£,_!< to the information about 6Q. In other words, the Fisher
information matrix of the true score is Qt-xW'}^',^ — C and the positive
semidefinite matrix C is the smallest such matrix produced by linear combinations
of the conditional moment conditions.
This development conceals a host of implementation problems associated with
the evaluation of conditional expectations.19 To be sure, $r_i and fr-i can be
estimated with nonparametric methods when they are time invariant functions
#(Zr-i) and ^{zf-i) for zt_{ &It-\. The extension of the methods of Robinson
(1987), Newey (1990), Robinson (1991), and Newey (1993) to the present setting,
in which RTQ(JC,, 0q) — i is serially uncorrelated but not independently
distributed over time or homoskedastic, appears to be straightforward. However,
the circumstances in which A^ is a time invariant function of zt_x would appear
to be the exception rather than the rule. Accordingly, the econometrician
generally must place further restrictions on the no-arbitrage pricing model in
order to proceed with efficient estimation based on conditional moment
restrictions, a subject that occupies the next section.
Alternatively, the econometrician can work with weaker moment conditions
like the unconditional moment restrictions. The analysis of this case parallels that
of optimal conditional GMM. Once again, the fixed weight matrices At{@o) from
(2.10) are the weights of p portfolios with random payoffs AT(0o)Rt that cost
At(§jo)i units of account. As noted in the previous section, the price of these
random payoffs is 4>W~li which generally differs from E(A^_l)i. These portfolios
produce the fixed weight moment condition that has maximum unconditional
correlation with the derivatives of the true, but unknown, log likelihood function.
19 The nature of the information set itself is less of an issue. While investors might possess more
information than econometricians, this is not a problem because the law of iterated expectations
implies that E[iJ„g,|/^,j = lV/j^c/l-i. Of course, the conditional probabilities 7tj£_, implicit in this
moment condition generally differ from those implicit in E[Rt,Q,\It^\] = 1 as will the associated values
of the pricing kernel Qf (i.e., qft_x = "Psr-i/itf!,). The dependence of Qf on nft_x is broken in models
for Q, that equate the state price per unit probability qsl_\ with the marginal benefit of receiving one
unit of account in state s.

74
B. N. Lehmarm
Of course, conventional GMM implementations use conditioning information
within the optimal unconditional GMM procedure as discussed in the previous
section. Let Z,_i e It_\ denote anrxJV matrix of predetermined variables and
consider the revised moment conditions:
Eft_i(&e&,0e)--)|Ji-i]
= E[Z,_{ (RtQfa, Oq)-,)] =0 V Z,_! e /,_!.
In the preceding paragraph, Z,_i is /#, the N x N identity matrix; otherwise, it
could reflect identical or different elements of the information set available to
investors (i.e., z,_! in IN <£>?,_! and za_1 in (2.19), respectively) being applied to
each element of RtQ(xt,0o)—i as given in the previous section.
The introduction of z!t_l and zt_{ into the unconditional moment condition
(3.10) is often described as invoking trading strategies in estimation and inference
following Hansen and Jagannathan (1991) and Hansen and Jagannathan (1994).
This characterization arises because security returns are given different weights
temporally and, when zrY_j ^ zt_l; cross-sectionally after the fashion of an active
investor. In unconditional GMM, the returns weighted in this fashion are then
aggregated into p portfolios with weights that are refined as information is added
to (3.10) in the form of additional components of Zt-\.
Once again, there is an optimal fixed weight portfolio strategy for the revised
moment conditions based on Z,_i (RjQfa, 0q)— l)- From (2.20), the active portfolio
strategy with portfolio weights ^z^^A-i has random payoffs ^z^'Z,-]^ and
costs ^z^z^t-il units of account. The resulting moment conditions have the
largest unconditional correlation with the true, but unknown, unconditional score
in finite samples within the class of time varying portfolios with weights that are
fixed linear combinations of predetermined variables Zt-\. Of course, optimal
conditional weights can be obtained from the appropriate reformulation of (3.9)
above but the whole point of this approach is that the implementation of this
linear approximation to the optimal procedure is straightforward.
4. Efficiency gains within alternative beta pricing formulations
The moment condition E[Q(x„ 6Q)Rit\It-\] = 1 is often translated into the form of
a beta pricing model, so named for its resemblance to the expected return relation
that arises in the Capital Asset Pricing Model (CAPM). Beta pricing models serve
another purpose in the present setting; they highlight specific dimensions in which
fruitful constraints on the pricing kernel model can be added to facilitate more
efficient estimation and inference. Put differently, beta pricing models point to
assumptions that permit consistent estimation of the components of A°t_v
Accordingly, consider the population projection of the vector of risky asset
returns R^ on <3(x,,0g):

Semiparametric methods for asset pricing models
75
£ = «, + £fi&, 0g) + & ; E|c,|/,_i] = 0
^Covfe,efe,ge)|/(-i] (4.1)
& Var[(gfe)0G)|//_i]
and Var[«] and Cov[«] denote the variance and covariance of their arguments,
respectively. Asset pricing theory restricts the intercept vector a, in this projection
which are determined by substituting (4.1) into the moment condition (3.6):
I = E&e&.flg)!/,-!] = atE[Q(xt, flg) |/,_!] +PtE\Q(xt,eQf\It-l] (4.2)
which, after rearranging terms and insertion into (4.1), yields:
Rt = a0t + j^lQix,,§q) -^]+e,; Efel/^i] = 0 ;
ht = Efegfe, 0g)|/*-i]-1 ; AG/ = Ao,E02(xr»fiG)2|A-i] -
The riskless asset, if one exists, earns Ao<; otherwise, Ao* is the expected return of all
assets with returns uncorrelated with Qt. As noted earlier, the lack of serial
correlation in the residual vector et is econometrically convenient.
The bilinear form of (4.3) is a distinguishing characteristic of these beta pricing
models. Put differently, the moment conditions (3.6) constrain expected returns to
be linear in the covariances of returns with the pricing kernel. This linear structure
is a central feature of all models based on the absence of arbitrage in frictionless
markets; that is, the portfolio with returns that are maximally correlated with Qt
is conditionally mean-variance efficient.20 Hence, these asset pricing relations
differ from semiparametric multivariate regression models in their restrictions on
risk premiums like Xq( and lo<-21
The multivariate representation of these no-arbitrage models produces a
somewhat different, though arithmetically equivalent, description of efficient
GMM estimation. The estimator is based on the moment conditions:
1 T
- Y,Ai»-& = 0; k = Rt- dot - Pt[Q(xt, flg) - *&] (4.4)
t=i
and, after solving in terms of the expressions for Xot and Xq( (in particular, that
E[2(x„ Qq) - XQt\It-\\ = -XotVa.r[Q(xt, 6Q)\It-i]) and given sufficient regularity to
allow differentiation under the integral sign, the optimal choice of A\t_l is:
20 A portfolio is (conditionally) mean-variance efficient if it minimizes (conditional) variance for
given level of (conditional) mean return. A portfolio is (conditionally) mean-variance efficient for a
given set of assets if only if the (conditional) expected returns of all assets in the set are linear in their
(conditional) convariances with the portfolio. See Merton (1972), Roll (1977), and Hansen and
Richard (1987).
21 They differ in at least one other respect - most regression specifications with serially
uncorrelated errors have E[£,|g,] = 0, which need not satisfied by (4.3).

76
B. N. Lehmarm
*V-i -^/Var|fi(S()0G)|/,_1] = E[£,e/l^-i]
^(var[g(„ge)|^]| + 9Var^^)M,;) (4.5)
-^(l-Variefe.fle)!/,-!]^)'
°' 00 gg-d-cov[euc,,0e),&l^-i]) •
The last line in the expression for ##_! illustrates the relations with (3.9) in the
previous section. Note that the observation of the riskless rate eliminates the term
involving dAot/dOg.22
There is no generic advantage to casting no-arbitrage models in this beta
pricing form unless the econometrician is willing to make additional assumptions
about the stochastic processes followed by returns.23 As is readily apparent, there
are only three places where useful restrictions can be placed on beta pricing
models: (1) constraints on the behavior of the conditional betas, (2) additional
restrictions on the model QQc^Qq), and (3) on the regression residuals. We discuss
each of these in turn in the Sections 4.1-4.3 and these ingredients are combined in
Section 4.4.
4.1. Conditional beta models
The benefits of a model for conditional betas are obvious. Conditional beta
models facilitate the estimation of the pricing kernel model Q(xt,6o) by
sharpening the general moment restrictions (3.6) with a model for the covariances
embedded in them (i.e.,E|g(x„0G)*tf|/,_i] = CowlQ^^Ru^+^BiRu^-i]).
They also mitigate some of the problems associated with efficient of asset pricing
relations. Put differently, the econometrician is explicitly modeling some of the
components of $pt-\ in this case.
21 In the case of risk neutral pricing, $^_i collapses to -(dXo,/d9)i since Var[g(x„ 6g)|/,„jj is zero
and to zero if, in addition, the econometrician measures the riskless rate.
23 The law of iterated expectations does not apply to the second moments in these multivariate
regression models so that this representation alone does nothing to sharpen unconditional GMM
estimation. Additional covariances are introduced in the passage from conditional to unconditional
moments because of the bilinear form of beta pricing models. The unconditional moment condition for
security i is E[EaZjf-i|i<-i] = is[£j,z,(_[] = 0 Vz,(_j E /r_i and the sum of the two offending covariances
Cov{ft((E[e(x,,e)-Ae,)|/,_i], z,,_i} + Cov{ft„ (E[Q(x„6) - Aq^EIz,,^) cannot be separated without
further restrictions.
*Vi
Qpt-i =

Semiparametric methods for asset pricing models
11
Accordingly, suppose the econometrician observes a set of variables zt_x e It-\,
perhaps also contained in x, (i.e., zt_l e x,), and specifies a model of the form:
£ = /»&_, ,0p) ; &_, e/,_i (4.6)
where 0» is the vector of unknown parameters in the model for /?. In these
circumstances, the beta pricing model becomes:
R, = iht + £(&_!, ip) [Q(xt, Oq) - XQt] + spt . (4.7)
In the most common form of this model, the conditional betas are constant, the
Zf_! is simply the scalar 1, and 6p is the corresponding vector of constant
conditional betas p. All serial correlation in returns is mediated through the risk
premiums given constant conditional betas.24
Models for conditional betas make efficient GMM estimation more feasible by
refining the optimal weighting matrices since:
*^_, = e(^|/,_,} = ^(Var[g(x(,fle)|/t-i]^(g^'^)
~d~w y-~ Varts(&.%)i^-i]£fe-i.^)J (4-8)
where, as before, an observed riskless rate eliminates the last line of (4.8). Since
the parameter vector 6 is (OqO/)', $pzt-\ and $pt-\ in (4.5) differ in two respects:
=h,{ de m—£&-i.^)j
4^ ^ } = ^Varlefe,^)!/,-.] d-^M . (4.9)
24 Linear models of the form pit = 0;/S,-0z,-i are also common where Stp is a selection matrix that
picks the elements of z,_j relevant for fiu. Linear models for conditional betas naturally arise when the
APT holds both conditionally and unconditionally (cf., Lehmann (1992)). Some commercial risk
management models allow Oy to vary both across securities and over time; see Rosenberg (1974) and
Rosenberg and Marathe (1979) for early examples. Error terms can be added to these conditional beta
models when their residuals are orthogonal to the instruments z,_, e I»_j. Nonlinear models can be
thought of as specifications of the relevant components of $p,-\ by the econometrician.

78
B. N. Lehmann
A tedious calculation using partitioned matrix inversion verifies that the variance
of the efficient GMM estimator of 6Q falls after the imposition of the conditional
beta model, both because of the reduction in dimensionality in the transition from
the derivatives of Cov[g(x,,0g),i?,|/,_i] to the derivatives of Va.rlQfa,^)]!^] in
the first line of (4.9) and because of the additional moment conditions arising
from the conditional beta model in the second line of (4.9).
Hence, the problem of constructing estimates of the covariances between
returns and the derivatives of the pricing kernel in (3.9) is replaced by the
somewhat simpler problem of estimating the conditional variance of the pricing
kernel along with its derivatives in these models. Both formulations require
estimation of the conditional mean of Q(xj, 9q) and its derivatives through k0t, a
requirement eliminated by observation of a riskless asset. While stochastic
process assumptions are required to compute E[£>(x„ 0g)|/,_i], Var[£>(x„ 0g)|/,_i], and
their derivatives, a conditional beta model and, when possible, measurement of
the riskless rate simplifies efficient GMM estimation considerably.25
Note also that the optimal conditional weighting matrix ^^\W^_X has a
portfolio interpretation similar to that in the last section. The portfolio
interpretation in this case has a long standing tradition in financial econometrics.
Ignoring scale factors, the portfolio weights associated with the estimation of the
premium kgt are proportional to j6(S-ii^)- Similarly, the portfolio weights
associated with the estimation of the Xq( are proportional to i — fi{zt_x,d_^) after
scaling Var[g(xn0g)|/?_i] to equal one, as is appropriate when the econometrician
observes the return of portfolio perfectly correlated with Qt but not a model for
Qt itself (a case discussed briefly below). Such procedures have been used
assuming returns are independently and identically distributed with constant
betas beginning with Douglas (1968) and Lintner (1965) and maturing into a
widespread tool in Black, Jensen, and Scholes (1972), Miller and Scholes (1972),
and Fama and MacBeth (1973). Shanken (1992) provides a comprehensive and
rigorous description of the current state of the art for the independently and
identically distributed case.
Models for the determinants of conditional betas have another use-they make
it possible to identify aspects of the no-arbitrage model without an explicit model
for the pricing kernel Qt. Given only /^(z^j,^), expected returns are given by:
U]RMt-i] = Iht + Nzt-uOpWp, - h,} ■ (4.10)
The potentially estimable conditional risk premiums lot and Xpt are the expected
returns of conditionally mean-variance efficient portfolios since the expected
returns on the assets in this menu are linear in their conditional betas.26 However,
25 The presence of Var[g(x,,gg)|/,_i] and its derivatives in (4.8) arises because (4.6) is a model for
conditional betas, not for conditional covariances. In most applications, conditional beta models are
more appropriate.
26 The CAPM is the best known model which takes this form, in which portfolio p is the market
portfolio of all risky assets. The market portfolio return is maximally correlated with aggregate wealth
(which is proportional to Qt in this model) in the CAPM in general; it is perfectly correlated if markets
are complete.

Semiparametric methods for asset pricing models
79
these parameters are also the expected returns of any assets of portfolios that cost
one unit of account and have conditional betas of one and zero, respectively.
Portfolios constructed to have given betas are often called mimicking or basis
portfolios in the literature.27
Mimicking portfolios arise in the portfolio interpretation of efficient
conditional GMM estimation in this case and delimit what can be learned from
conditional beta models alone. Given only the beta model (4.6):
R, =iXo, + /?(£_!, dp)[Xp! - Xq,] + eppt ;
V/lpt-l = V[tpptgppt'\It-x]
(4.11)
$Ppt-i = (Xp! - ht) t^—^ + -00-[L-Hz-uOfi)]
+ -
dX
pt
M)t
dB
08
Note that if we treat the risk premiums as unknown parameters in each period,
the limiting parameter space is infinite dimensional. Ignoring this obvious
problem, the optimal conditional moment restrictions are given by:
E
(Xpt ~ Xq,)
d8R
Tppt-\
x Et- iht - jKzt-i, dp) {Xpt - X0t)
and the solution for each Xq, and Xp, — X0t is:
Xot
Xpt — Xq,
X (igfe-l,^))'^-!*,
(4.12)
(4.13)
27 See Grinblatt and Titman (1987), Huberman, Kandel, and Stambaugh (1987), Lehmann (1987),
Lehmann and Modest (1988), Lehmann (1990), and Shanken (1992) for related discussions. In
econometric terms, the portfolio weights that implicitly arise in cross-sectional regression models with
arbitrary matrices r solve the programming problems:
minw'rpt-l^rpt-l subject to w!rpt_{i = \ and w^,/?^!,^) = 1
min yJm_xrwm_, subject to }^r0t_il = 1 and w^or-l^fe-l^fl) = 0
Ordinary least squares corresponds to r = I,T = Diag{Var[jR,|/,-i]} to weighted least squares, and
r = Var^l/,-!] to generalized least squares.

80
B. N. Lehmann
which are, in fact, the actual, not the expected, returns of portfolios that cost one
and zero units of account and that have conditional betas of zero and one,
respectively.
Hence, there are three related limitations on what can be measured from risky
asset returns given only a conditional beta model. First, the conditional beta model
is identified only up to scale: P{zt_l,9p)(kpt — Ao») is observationally equivalent to
<p/?(z(_j, 0^) (Ap, - h)t)/(p for any <p ^ 0. Second, the portfolio returns Xo< and
hpt — hot have expected returns /lo< and kpt — Ao(, respectively, but the expected
returns can only be recovered with an explicit time series model for E[Rt\It^\].2&
Third, the pricing kernel Qt cannot be recovered from this model - only Rpt, the
return of the portfolio of these N risk assets that is maximally correlated with Qt,
can be identified from Xpt in the limit (i.e., as j8(z(_1( dp)—>j6(z(_1;^)).
4.2. Multifactor models
Another parametric assumption that facilitates estimation and inference is a
linear model for Qt. The typical linear models found in the literature
simultaneously strengthen and weaken the assumptions concerning the pricing kernel.
Clearly, linearity is more restrictive than possible nonlinear functional forms.
However, linear models generally involve weakening the assumption that Qt is
known up to an unknown parameter vector since the weights are usually treated
as unobservable variables.
Some equilibrium models restrict Qt to be a linear combination (that is, a
portfolio) of the returns of portfolios. In intertemporal asset pricing theory, these
portfolios let investors hedge against fluctuations in investment opportunities (cf.,
Merton (1973) and Breeden (1979)). Related results are available from portfolio
separation theory, in which such portfolios are optimal for particular preferences
(cf., Cass (1970)) or for particular distributions of returns (cf., Ross (1978a)).
Similarly, the Arbitrage Pricing Theory (APT) of Ross (1976) and Ross (1977)
combines the no-arbitrage assumption with distributional assumptions describing
diversification prospects to produce an approximate linear model for Qt.29
In these circumstances, the pricing kernel Qt (typically without any adjustment
for inflation) follows the linear model:
Qt = dxt-\h + e4(_i£m( ; Qt > 0 ; mxt_x, mmt^ e 7(_i (4.14)
where xt is a vector of variables that are not asset returns while R^, is a vector of
portfolio returns. These models typically place no restrictions on the (unobserved)
weights coxt_l and comt_l save for the requirement that they are based on
information available at time t— 1 and that they result in strictly positive values of
28 Moments of X0t and Xpt — A0( can be estimated. For example, the projection of Xq, and Xp, — Xq,
on z,_[ £ It-\ recovers the unconditional projection of Xq, and Xpt — X& on zJ_l £ 7,_i in large samples.
29 The APT as developed by Ross (1976) and Ross (1977) places insufficient restrictions on asset
prices to identify Q,. In order to obtain the formulation (4.14), sufficient restrictions must be placed on
preferences and investment opportunities so that diversifiable risk commands no risk premium.

Semiparametric methods for asset pricing models
81
£>,.30 Put differently, a model takes the more general form Q{xt,&) when (oxt—\ and
(Bint-i are parameterized as (ox(z4_l, 0) and co^fe-i, 0).
Accordingly, consider the linear conditional multifactor model:
R, = at+Bx (zt_,, 6Bx)xt + Bm (z,_,, 0Bm )£„, + &, ■ (4-15)
The imposition of the moment conditions (2.6) yields the associated restriction on
the intercept vector:
a, = [l -#m(?(_[, dBm)i]Xot -Bx(zt_{,Os^ka
r '1 (4-16)
4( = /W [Efjyc, |/(-i]«)*(_i + E[x(£m(J/,_i]fl)m(_1J
so that, in principle, coxt_{ and fl^^! can be inverted from the expression for Xxt.
Finally, insertion of this expected return relation into the multifactor model
yields:
R, = d0( +Bx(zt^,iBx)[xt - lxt]+Bm{zt_u6Bm)[Rmt ~ ik>t\
+ sBl;E[eBt\It-i}=0 . (4.17)
Once again, the residual vector has conditional mean zero because expected
returns are spanned by the factor loading matrix B(zi_l,6B) and a vector of ones.31
As is readily apparent, this model requires estimates of the conditional mean
vector and covariance matrix of (x/2?^,)'.
Note that, no restrictions are placed on E[/?m(|4_i] in (4.17). If the
econometrician observes the returns ^ and the variables x, with no additional
information on Qt, the absence of a model linking R^ with Qt eliminates the
restrictions on Efif^l/,-!] that arise from the moment condition E^^d^i-i] = I-
The same observation would hold if the returns of portfolio p were observed in
(4.10)—(4.13). Put differently, a linear combination of the returns R,„t or of the
return Rpt provides a scale-free proxy for Qt. In the absence of data on or of a
model for Qt, asset pricing relations explain relative asset prices and expected
returns, not the levels of asset prices and risk premiums.
As with the imposition of conditional beta models, linear factor models
simplify estimation and inference by weakening the information requirements.
Linearity of the pricing kernel confers three modest advantages compared with
the conditional beta models of the previous section: (1) the derivatives of the
conditional mean and variance of Q{xt,9_g) are no longer required; (2) the
conditional covariance matrices involving xt and R,„t contains no unknown model
parameters (in contrast to Var[£>(x(,0g)|/(_i]); and (3) the linear model permits
flirt^! and gymt_x to remain unobservable. The third point comes at a cost - the
30 Imposing the positivity constraint in linear models is sometimes quite difficult.
31 Since the multifactor models described above are cast in terms of Q,, [i — 5m(z,_1,6Sm)ji] will not
be identically zero. In multifactor models with no explicit link between Q, and the underlying common
factors, this remains a possibility. See Huberman, Kandel, and Stambaugh (1987), Huberman and
Kandel (1987), and Lehmann and Modest (1988) for a discussion of this issue.

82
B. N. Lehmann
model places no restrictions on the levels of asset prices and risk premiums. Once
again, additional simplifications arise if there is an observed riskless rate.
Multifactor models also take the form of prespecified beta models. The analysis
of these models parallels that of the single beta case in (4.10)—(4.13). A conditional
factor loading model B(zt_l, 6B) can only be identified up to scale and, at best, the
econometrician can estimate the returns of the minimum variance basis portfolios,
each with a loading of one on one factor and loadings of zero on the others. In
terms of the single beta representation, a portfolio of these optimal basis
portfolios with time-varying weights has returns that are maximally correlated
with Qt or, equivalently, a linear combination of Bf&^jdg) is proportional to the
conditional betas B in this multifactor prespecified beta model.
4.3. Diversifiable residual models and estimation in large cross-sections
One other simplifying assumption is often made in these models: that the residual
vectors are only weakly correlated cross-sectionally. This restriction is the
principal assumption of the APT and it implies that residual risk can be eliminated in
large, well-diversified portfolios. It is convenient econometrically for the same
reason; the impact of residuals on estimation can be eliminated through
diversification in large cross-sections.
In terms of efficient estimation of beta pricing models, this assumption
facilitates estimation of "P^_i, the remaining component of the efficient GMM
weighting matrix. To be sure, efficient estimation could proceed by postulating a
model for Wpt-i in (4.7) of the form ^z^). However, it is unlikely that an
econometrician, particularly one using semiparametric methods, would possess
reliable prior information of this form save for the factor models of Section 4.2.
Accordingly, consider the addition of a linear factor model to the conditional
beta models. Once again, consider the projection:32
Et = «, +^i,0/i)fife,0e) +Bx(zt-i,SBx)xt
+ Bm(zt_ueBm)Rml + £liBl (4.19)
and the application of the pricing relation to the intercept vector:
«u = \l-Bm(zt_u6Bm)L]ht - /%,_i,9p)XQt -Bxiz,^,0^)4, (4.20)
which, after rearranging terms and insertion into (4.19), yields:
Rt = do, + £&_i, 0p) [fife, fig) - ht\ + **&-i»&*) fe - 4J
+ Bm(z,_i, 6_Bm) [R^i - iXot] + £pBt
l& = h,E\Q{xt,6Q)2\It-i]; (4.21)
VpBt-i = E[e^£^/|/f_i]
X^ = k>tB[xiQ{xt,BjQ)\It-.\\ .
32 Of course, one element of (x/R^/) must be dropped if (x/Rm,') and g(x,,0o) are linearly
dependent.

Semiparametric methods for asset pricing models
83
When all of these components are present in the model, assume that a vector of
ones does not lie in the column span of either Bx(z4_l,6Bx) or 5m(zr_1, 0gm).
This formulation nests all of the models in the preceding subsections. When
Bx(zt-i,@.Bx) and Bm(zt_u6Bm) are identically zero, equations (4.21) yield the
conditional beta model (4.7) or, in the absence of the pricing kernel model
2(*f)0g)> tne prespecified beta model (4.11). Similarly, when ^(zt_l,6p) is
identically zero, equations (4.21) yield the observable linear factor model (4.17)
or, without observations on xt and B^,, the multifactor analogue of the
prespecified beta model. When all components are included simultaneously, the
conditional factor model places structure on the conditional covariance matrix of
the residuals f^_tin the conditional beta model (4.7).
This factor model represents more than mere elegant variation - it makes it
plausible to place a a priori restrictions on the conditional variance matrix "Ppst-i ■
In terms of the conditional beta model (4.7), the residual covariance matrix "Ppt-i
has an observable factor structure in this model given by:33
V/fc-i = (5,fe-i AJ^mfe-i Aj)var (!' )l4-i
x /**&-!. &*)' '
\Bm(zt_i,6Bm)'
= BpBt-\ VpBt-\BpBt-\ + *FpBt-l
and its inverse is given by:
V„
(4.22)
f;
-1 - r m-\ ~ vpBi-iBm-i{vPBt-i +BpBt-\xppBt-\Bt-\)
H/-1
TfiBl-
xBR
(4.23)
Hence, the factor model provides the final input necessary for the efficient
estimation of beta pricing models.
Chamberlain and Rothschild (1983) provide a convenient characterization of
diversifiability restrictions for residuals like £pBl. They assume that the largest
eigenvalue of the conditional residual covariance matrix *PpBt-i remains bounded
as the number of assets grows without bound. This condition is sufficient for a
weak law of large numbers to apply because the residual variance of a
portfolio with weights of order 1/iV (i.e., one for which vt/_,w,_| —> 0 as
N —> 00 V wt_
^maxC?'fiBt-i)
argument.
i^-i^
-1.2^-1
e It-\) converges to zero since <t^(.
-> 0 as N —> 00 where £max(») is the largest eigenvalue of its
-iwt_i < w,
t-i^t-i
33 Unobservable factor models can be imposed as well as long as the associated conditional betas
are constant. The methods developed for the iid case in Chamberlain and Rothschild (1983), Connor
and Korajczyk (1988) and Lehmann and Modest (1988) apply since the residuals in this application are
serially uncorrelated. Lehmann (1992) discusses the serially correlated case.

84
B. N. Lehmann
Unfortunately, there is no obvious way to estimate YpBt_i subject to this
boundedness condition.34 Hence, the imposition of diversification constraints in
practice generally involves the stronger assumption of a strict factor structure:
that is, that YpBt~i is diagonal. Of course, there is no guarantee that a diagonal
specification leads to an estimator of higher efficiency than an identity matrix
(that is, ordinary least squares) when generalized least squares is appropriate, as
would be the case if fpst-i is unrestricted save for the diversifiability condition
lim ^maxCPfiBt-i) < oo. While weighted least squares may in fact be superior in
most applications, conservative inference can be conducted assuming that this
specification is false. In any event, the econometrician can allow for a generous
amount of dependence in the idiosyncratic variances in the diagonal specification.
What is the large cross-section behavior of GMM estimators assuming that a
weak law applies to the residuals? To facilitate large N analysis, append the
subscript N to the residuals e^BNt and to the associated parameter vectors and
matrices PN{zt_l,dpN),BxN(zt_ueBxN),BmN(zt_ue£mN), and *PpBN,-\ and take all
limits as N grows without bound by adding elements to vectors and rows to
matrices as securities are added to the asset menu. An arbitrary conditional
GMM estimator can be calculated from:
1 T
-5«iv(a-i,lft»iv)[&tf-I^] • (4-24)
where Apsm-i is a sequence of p x Nop{l) matrices chosen by the econometrician
having full row rank for which {minG^M-i^BM-i) -> oo as N -> oo where ^min(»)
is the smallest eigenvalue of its argument. This latter condition ensures that the
weights are diversified across securities and not concentrated on only a few assets.
Examination of the estimating equations (4.24) reveals the benefits of large
cross-sections when residuals are diversifiable. The sample and population
residuals are related by:
IfiBNt =ZpBNt+l{^Ot ~ kit) + {P{Zt-\,ip)[Q{x„6e) ~ ^Qt]
+ [BxN(Zt-\, Q.Bxn) - 5xivfe-l > fiflxivfe
+ [BxN(Zt-\) Q.BxN)kct ~ BxN{Zt-\ i Q-Bxn)^]
+ [BmN (?(-1, iBmN) ~ BmN(z,_u 0BmN)]Rmt
+ {BmN{^\iS_BmN)ikjt -BmN(zt^u9BmN)i^it} (4-25)
the first component of which is the population residual vector e^BNt and the
remaining components of which represent the difference between the population and
34 Recently, Ledoit (1994) has proposed estimating covariance matrices using shrinkage estimators
of the eigenvalues, an approach that might work here.

Semiparametric methods for asset pricing models
85
fitted part of the model. Clearly, e^BN1 can be eliminated through diversification
and, hence, the application of ApBm-\ to e^BNt will do so since it places implicit
weights of order l/Non each asset as the number of assets grows without bound.
However, the benefits of diversification have a limit because of the difference
between the population and fitted part of the model. For example, the sampling
errors in Q(x„ 6Q), XQt, Xxl,BxN(zt_u 6BxN) and B^k,^,IW) generally cannot be
diversified away in a single cross-section. To be sure, some components of EpBNl
are amenable to diversification in some models. For example, if P{zt_l, Op) is
identically zero (i.e., if the pricing kernel Qt is given by coxt_{xt + comt_lRmt) and, if
the models for both BxN{zt_x,&BxN) and iJ^z,..!, 0gmJV)^ are linear, the
sampling errorsB^^^d^) ~ BxN(zt_uiBxN) and^fe.!,^,^) —JffmAr(^_i,0smAf)
can, in principle, be eliminated through diversification. In this case, the only risk
premium that can be consistently estimated from a single cross-section is X0t since
the difference Xxt — Xxt can only be eliminated in large time series samples.35
4.4. Feasible (nearly efficient) conditional GMM estimation
of beta pricing models
With these preliminaries in mind, we now consider efficient conditional GMM
estimation of the composite conditional beta model (4.21). In this model, the
optimal choice of A\Bt_x is $~jBt_\ where these matrices are given by:
*/i»-i = -~{i--^r[Q(xt,eQ)\it-l]^l,eli)~-Bx(zi_ueBx)
xCovfe, £>(*,, 0e)|/(_!]
-fl,fe-i,fl»,)i}'+mml - .w-ii'^k^'
+ Xot{Var[Q(xJAQ)\It-i}mitdleip)
dN3j[Q{xt,dQ)\It-X\ 0Covk,e(xtI0fi)|/,_1]'
+ m K^uk) + m
**fe-i, 0*)' + Covfe, gk,fle)!/,-i]'d*'(^,gfa)'} (4.26)
35 This point has resulted in much confusion in the beta pricing literature. The literature abounds
with inferences drawn from cross-sectional regressions of returns on the betas of individual assets
computed with respect to particular portfolios. If the betas in these prespecified beta models are
computed with respect to an efficient portfolio, the best one can do in a single cross-section (with a
priori knowledge of the population betas and return covariance matrix) is to recover the returns of the
efficient portfolio. Information on the risk premium of portfolios like p in Section 4.1 can only be
recovered over time while the return of portfolio 0 converges to the riskless rate in a single cross-
section if the residuals of the prespecified beta model are diversifiable given the population value of
<bppt-\. Shanken (1992) shows that this is the case using the sample analogue of <f^(_i in a model with
constant conditional betas and independently and identically distributed idiosyncratic disturbances
given appropriate corrections for biases induced by sampling error. See also Lehmann (1988) and
Lehmann (1990).

86
B. N. Lehmann
xBpBt-i'yjjBt-i ■
In the original formulation (3.6)-(3.9), efficient estimation required $t-\, the
derivatives of the conditional expectation of Q(xt, BqJR,, and "P,.;, the conditional
covariance matrix of RtQ(xt,(L^) — i. Equations (4.21) reflect the kinds of
assumptions that the econometrician can make to facilitate efficient estimation. The
conditional beta model eases the evaluation of the beta pricing version of $t^\
and the factor model assumption places structure on the associated analogue of
Vt-i.
Consistent estimation of ^am_\ requires the evaluation of a number of
conditional moments - Aot,E[Rmt\lt-\],Va.r[Q(xt,9o)\It-\], and Co v[x,, £>(*,, 0g)|
It-\] and their derivatives, when necessary, along with BpBt-i, Vpst-i, and Vpst-i-
The most common strategy by far is simply to assume that the relevant
conditional moments are time invariant functions of available informations. This
strategy was taken throughout this section in the models for conditional betas and
conditional factor loadings. For the evaluation of j$uBt_\, this approach requires
the econometrician to posit relations of the form:
^o(lt-i,i)
Amfe-1,£)
VfiB&-uS)
which permit the consistent estimation of A°.Bt_{ using initial consistent estimates
of 0.
It is far from obvious that a financial econometrician can be expected to have
reliable to prior information in this form. In most asset pricing applications, the
possession of such information about the conditional second moments Ogfe-i; S)
and ffgxfe-ijfi) is somewhat more plausible than the existence of the
corresponding conditional first moment specifications ^o(z(_i,0) and Am(^t-uS)
in its conditional mean from. However, observation of the riskless rate eliminates
the need to model Ao(z,_i,£) and models for Cov^,, £>(*,, 0g)|z,_i] seem no more
demanding than those for other conditional second moments. The conditional
covariance matrix Vpsfe-i, 9) is somewhat less problematic as well, although the
specification of multivariate conditional covariance models is in its infancy. The
discussion in Section 4.3 left some ambiguity concerning the availability of
plausible models of this sort for WpBt-\ due to the inability to impose the general
bounded eigenvalue condition. As noted there, the specification of idiosyncratic
= E[Q(xt,eQ)\zi_l]~l
= E&^-i] = -4>fe-i>£)(l- Cov&a.efe,^.,])
= Var[efe,0fi)|2/_1]
= Cov|xr)e(x,)0G) !*,_,] (4.27)
Var
Emt
lif-1
Var[£„B(|5
^-u

Semiparametric methods for asset pricing models
87
variances is comparatively straightforward if VpBt-i is diagonal. Finally,
conservative inference is always available through the use of the asymptotic
covariance matrix in (2.13).
Equations (4.27) can either represent parametric models for these conditional
moments or functions that are estimable by semiparametric or nonparametric
methods. Robinson (1987), Newwy (1990), Robinson (1991), and Newey (1993)
discuss bootstrap, nearest neighbor, and series estimation of functions such as
those appearing in (4.27). All of these methods suffer from the curse of
dimensionality so their invocation must be justified on a case by case basis.
Neural network approximations promising somewhat less impairment from this
source might be employed as well.36
5. Concluding remarks
This paper shows that efficient semiparametric estimation of asset pricing
relations is straightforward in principle if not in practice. Efficiency follows from the
maximum correlation property of the optimal GMM estimators described in the
second section, a property that has analogues in the optimal hedge portfolios that
arise in asset pricing theory. The semiparametric nature of asset pricing relations
naturally leads to a search for efficiency gains in the context of beta pricing
models.
The structure of these models suggests that efficient estimation is made feasible
by the imposition of conditional beta models and/or multifactor models with
residuals that satisfy a law of large numbers in the cross-section, models that exist
in various incarnations in the beta pricing literature. Hence, strategies that have
proved useful in the iid environment have natural, albeit nonlinear and perhaps
nonparametric, analogues in this more general setting, the details of which are
worked out in the paper. While it has offered no evidence on the magnitude of
possible efficiency gains, the paper has surely pointed to more straightforward
interpretation and implementation than has been heretofore attainable.
What remains is to extend there results in two dimensions. The analysis
sidestepped the development of the most general approximations of the
conditional moments that comprise the optimal conditional weighting matrices,
the subtleties of which arise from the martingale difference nature of the residuals
in no-arbitrage asset pricing models as opposed to the independence assumption
frequently made in other applications. The second dimension involves
examination of less parametric semiparametric estimators. In the asset pricing arena, this
amounts to semiparametric estimation of pricing kernels and state price densities,
a more ambitious and perhaps more interesting task.
36 Barron (1993) and Hornik et al. (1993) discuss the superior approximation properties of neural
networks in the multidimensional case.

88
B. N. Lehmann
References
Bansal, R. and B. N. Lehmann (1995). Bond returns and the prices of state contingent claims.
Graduate School of International Relations and Pacific Studies, University of California at San
Diego.
Bansal, R. and S. Viswanathan (1993). No arbitrage and arbitrage pricing: A new approach. J. Finance
48, pp. 1231-1262.
Barron, A. R. (1993). Universal approximation bounds for superpositions of a sigmoidal function.
IEEE Transactions on Information Theory 39, pp. 930-945.
Black, F., M. C. Jensen and M. Scholes (1972). The capital assest pricing model: Some empirical tests.
In: M. C. Jensen, ed., Studies in the Theory of Capital Markets, New York: Praeger.
Breeden, D. T. (1979). An intertemporal asset pricing model with stochastic consumption and
investment opportunities. J. Financ. Econom. 7, pp. 265-299.
Cass, D. and J. E. Stiglitz (1970). The structure of investor preferences and asset returns and
separability in portfolio allocation: A contribution to the pure theory of mutual funds. J. Econom.
Theory 2, pp. 122-160.
Chamberlain, G. (1987). Asymptotic efficiency in estimation with conditional moment conditions. J.
Econometrics 34, pp. 305-334.
Chamberlain, G. (1992). Efficiency bounds for semiparametric regression. Econometrica 60, pp. 567-
596.
Chamberlain, G. and M. Rothschild (1983). Arbitrage and mean-variance analysis on large asset
markets. Econometrica 51, pp. 1281-1304.
Cochrane, J. (1991). Production-based asset pricing and the link between stock returns and economic
fluctuations. J. Finance 146, pp. 207-234.
Connor, G. and R. A. Korajczyk (1988). Risk and return in an equilibrium APT: Application of a new
test methodology. J. Financ. Econom. 21, pp. 255-289.
Constantinides, G. and W. Ferson (1991). Habit persistence and durability in aggregate consumption:
Empirical tests. J. Financ. Econom. 29, pp. 199-240.
Douglas, G. W. (1968). Risk in the Equity Markets: An Empirical Appraisal of Market Efficiency. Ann
Arbor, Michigan: University Microfilms, Inc.
Epstein, L. G. and S. E. Zin (1991a). Substitution, risk aversion, and the temporal behavior of
consumption and asset returns: A theoretical framework. Econometrica 57, pp. 937-969.
Epstein, L. G. and S. E. Zin (1991b). Substitution, risk averison, and the temporal behavior of
consumption and asset returns: An empirical analysis. J. Politic. Econom. 96, pp. 263-286.
Fama, E. F. and J. D. MacBeth (1973). Risk, return, and equilibrium: Empirical tests. J. Politic.
Econom. 81, pp. 607-636.
Grinblatt, M. and S. Titman (1987). The relation between mean-variance efficiency and arbitrage
pricing. J. Business 60, pp. 97-112.
Hall, A. (1993). Some aspects of generalized method of moments estimation. In: G. S. Maddala, C. R.
Rao and H. D. Vinod, ed., Handbook of Statistics: Econometrics. Amsterdam, The Netherlands:
Elsevier Science Publishers, pp. 393-418.
Hansen, L. P. (1982). Large sample properties of generalized method of moments estimators.
Econometrica 50, pp. 1029-1054.
Hansen, L. P. (1985). A method for calculating bounds on the asymptotic covariance matrices of
generalized method of moments estimators. J. Econometrics 30, pp. 203-238.
Hansen, L. P., J. Heaton and E. Luttmer (1995). Econometric evaluation of assest pricing models. Rev.
Financ. Stud. 8 pp. 237-274.
Hansen, L. P., J. Heaton and M. Ogaki (1988). Efficiency bounds implied by multi-period conditional
moment conditions. J. Amer. Stat. Assoc. 83, pp. 863-871.
Hansen, L. P. and R. J. Hodrick (1980). Forward exchange rates as optimal predictors of future spot
rates: An econometric analysis. J. Politic. Econom. 88, pp. 829-853.
Hansen, L. P. and R. Jagannathan (1991). Implications of security market data for models of dynamic
Economies. J. Politic. Econom. 99, pp. 225-262.

Semiparametric methods for asset pricing models
89
Hansen, L. P. and R. Jagannathan (1994). Assessing specification errors in stochastic discount factor
models. Research Department, Federal Reserve Bank of Minneapolis, Staff" Report 167.
Hansen, L. P. and S. F. Richard (1987). The role of conditioning information in deducing testable
restrictions implied by dynamic asset pricing models. Econometrica 55, pp. 587-613.
Hansen, L. P. and K. J. Singleton (1982). Generalized instrumental variables estimation of nonlinear
rational expectations models. Econometrica 50, pp. 1269-1286.
Harrison, M. J. and D. Kreps (1979). Martingales and arbitrage in multiperiod securities markets. J.
Econom. Theory 20, pp. 381-^08.
He, H. and D. Modest (1995). Market frictions and consumption-based asset pricing. J. Politic.
Econom. 103, pp. 94-117.
Hornik, K., M. Stinchcombe, H. White and P. Auer (1993). Degree of approximation results for
feedforward networks approximating unknown mappings and their derivatives. Neural
Computation 6, pp. 1262-1275.
Huberman, G. and S. Kandel (1987). Mean-variance spanning. J. Finance 42, pp. 873-888.
Huberman, G., S. Kandel and R. F. Stambaugh (1987). Mimicking portfolios and exact asset pricing.
J. Finance 42, pp. 1-9.
Ledoit, O. (1994). Portfolio selection: Improved covariance matrix estimation. Sloan School of
Management, Massachusetts Institute of Technology,
Lehmann, B. N. (1987). Orthogonal portfolios and alternative mean-variance efficiency tests.
J. Finance 42, pp. 601-619.
Lehmann, B. N. (1988). Mean-variance efficiency tests in large cross-sections. Graduate School of
International Relations and Pacific Studies, University of California at San Diego.
Lehmann, B. N. (1990). Residual risk revisited. J. Econometrics 45, pp. 71-97.
Lehmann, B. N. (1992) Notes of dynamic factor pricing models. Rev. Quant. Finance Account. 2, pp.
69-87.
Lehmann, B. N. and David M. Modest (1988), The empirical foundations of the arbitrage pricing
theory. J. Financ. Econom. 21, pp. 213-254.
Lintner, J. (1965). Security prices and risk: The theory and a comparative analysis of A.T &T. and
leading industrials. Graduate School of Business, Harvard University.
Luttmer, E. (1993). Asset pricing in economies with frictions. Department of Finance, Northwestern
University.
Merton, R. C. (1972). An analytical derivation of the efficient portfolio frontier. J. Financ. Quant.
Anal. 7, pp. 1851-1872.
Merton, R. C. (1973). An intertemporal capital asset pricing model. Econometrica 41, pp. 867-887.
Miller, M. H. and M. Scholes (1972). Rates of return in relation to risk: A reexamination of some
recent findings. In: M.C. Jensen, ed., Studies in the Theory of Capital Markets, New York: Praeger,
pp. 79-121.
Newey, W. K. (1990). Efficient instrumental variables estimation of nonlinear models. Econometrica
58, pp. 809-837.
Newey, W. K. (1993). Efficient estimation of models with conditional moment restrictions. In: G. S.
Maddala, C. R. Rao and H. D. Vinod, eds., Handbook of Statistics: Econometrics. Amsterdam,
The Netherlands: Elsevier Science Publishers.
Newey, W. K. and K. D. West (1987). A simple, positive semi-definite, heteroskedasticity and
autocorrelation consistent covariance matrix. Econometrica 55, pp. 703-708.
Ogaki, M. (1993). Generalized method of moments: Econometric applications. In: G. S. Maddala, C.
R. Rao and H. D. Vinod, eds., Handbook of Statistics: Econometrics, Amsterdam, The Netherlands:
Elsevier Science Publishers, pp. 455-488.
Robinson, P. M. (1987). Asymptotically efficient estimation in the presence of heteroskedasticity of
unknown form. Econometrica 59, pp. 875-891.
Robinson, P. M. (1991). Best nonlinear three-stage least squares estimation of certain econometric
models. Econometrica 59, pp. 755-786.
Roll, R. W. (1977). A critique of the asset Pricing Theory's Tests - Part I: On past and potential
testability of the theory. J. Financ. Econom. 4, pp. 129-176.

90
B. N. Lehmann
Rosenberg, B. (1974). Extra-market components of covariance in security returns. /. Financ. Quant.
Anal. 9, pp. 262-274.
Rosenberg, B. and V. Marathe (1979). Tests of capital asset pricing hypotheses. Research in Finance:
A Research Annual 1, pp. 115-223.
Ross, S. A. (1976). The arbitrage theory of capital assest pricing. /. Economic Theory 13, pp. 341-360.
Ross, S. A. (1977). Risk, return, and arbitrage. In: I. Friend and J.L. Bicksler, eds., Risk and Return in
Finance. Cambridge, Mass.: Ballinger.
Ross, S. A. (1978a). Mutual fund separation and financial theory - the separating distributions. /.
Econom. Theory 17, pp. 254-286.
Ross, S. A. (1978b). A simple approach to the valuation of risky streams. /. Business 51, pp. 1^0
Rubinstein, M. (1976). The valuation of uncertain income streams and the pricing of options. Bell J.
Econom. Mgmt. Sci. 7, pp. 407-^125.
Shanken, J. (1992). On the estimation of beta pricing models. Rev. Financ. Stud. 5, pp. 1-33.
Summers, L. H. (1985). On economics and finance. /. Finance 40, pp. 633-636.
Summers, L. H. (1986). Does the stock market rationally reflect fundamental values? J. Finance 41, pp.
591-600.
Tauchen, G. (1986). Statistical properties of generalized method of moments estimators of structural
parameters obtained from financial market data. /. Business Econom. Statist. 4, pp. 397-^4-25.

G.S. Maddala and C.R. Rao, eds., Handbook of Statistics, Vol. 14
© 1996 Elsevier Science B.V. All rights reserved.
4
Modeling the term structure*
A. R. Pagan, A. D. Hall and V. Martin
1. Introduction
Models of the term structure of interest rates have assumed increasing importance
in recent years in line with the need to value interest rate derivative assets.
Economists and econometricians have long held an interest in the subject, as an
understanding of the determinants of the the term structure has always been
viewed as crucial to an understanding of the impact of monetary policy and its
transmission mechanism. Most of the approaches taken to the question in the
finance literature have revolved around the search for common factors that are
thought to underlie the term structure and little has been borrowed from the
economic or econometrics literature on the subject. The converse can also be said
about the small amount of attention paid in econometric research to the finance
literature models. The aim of the present chapter is to look at the connections
between the two literatures with the aim of showing that a synthesis of the two
may well provide some useful information for both camps.
The paper begins with a description of a standard set of data on the term
structure. This results in a set of stylized facts pertaining to the nature of the
stochastic processes generating yields as well as their spreads. Such a set of facts is
useful in forming an opinion of the likelihood of various approaches to term
structure modelling being capable of replicating the data. Section 3 outlines the
various models used in both the economics and finance literature, and assesses
how well these models perform in matching the stylized facts. Section 4 presents a
conclusion.
* We are grateful for comments on previous versions of this paper by John Robertson, Peter
Phillips and Ken Singleton. All computations were performed with a beta version of MICROFIT 4
and GAUSS 3.2
91

92
A. R. Pagan, A. D. Hall and V. Martin
2. Characteristics of term structure data
2.1. Univariate properties
The data set examined involves monthly observations on 1, 3, 6 and 9 month and
10 year zero coupon bond yields over the period December 1946 to February
1991, constructed by McCulloch and Kwon (1993); this is an updated version of
McCulloch (1989).
Table 1 records the autocorrelation characteristics of the series, with pj being
the /h autocorrelation coefficient, DF the Dickey-Fuller test, ADF(12) the
Augmented Dickey-Fuller test with 12 lags, rt(x) the yield on zero-coupon bonds
with maturity oft months and sp,(t) is the spread t,(t) — r,(l). It shows that there
is strong evidence of a unit root in all interest rate series, Because this would
imply the possibility of negative interest rates, finance modellers have generally
maintained that either there is no unit root and the series feature mean reversion
or, in continuous time, that an appropriate model is given by the stochastic
differential equation
dr, = adt + artdr\t ,
where, throughout the paper, dv\t is a Wiener process. Because of the "levels
effect" of r, upon the volatility of interest rate changes, we can think of this as an
equation in dlogr, with constant volatility, and the logarithmic transformation
ensures that r, remains positive.1 In any case, the important point to be made here
is that interest rates seem to behave as integrated processes, certainly over the
samples of data we possess. It may be that the autoregressive root is close to
unity, rather than identical to it, but such "near integrated" processes are best
handled with the integrated process technology rather than that for stationary
processes.
Table 1
Autocorrelation features, yields, full sample
r(l)
r(3)
r(6)
r(9)
#•(120)
*>(3)
sp{S)
sp{9)
sp{\20)
DF
-2.41
-2.15
-2.12
-2.12
-1.41
-15.32
-11.67
-10.38
-5.60
ADF(12)
-2.02
-1.89
-1.91
-1.89
-1.53
-3.37
-^.21
-^.38
-^.15
h
.98
.98
.99
.99
.99
.38
.59
.66
.89
h
.33
.51
.55
.80
h
.21
.26
.27
.55
Pl2
.38
.30
.26
.32
Pi(Ar)
.02
.11
.15
.15
.07
The 5% critical value for the DF and ADF tests is -2.87.
1 It is known that, if r, is replaced by r], the restriction y > .5 ensures a positive interest rate while,
if y = .5, a < 2a is needed.

Modeling the term structure
93
Instead of the yields one might examine the time series characteristics of the
forward rates. The forward rate F*(x) contracted at time t for a x period bond to
be bought at t + k is F*(x) = [\] [(x + k)r,(x + k) - krt{k)\. For a forward contract
one period ahead this becomes F}(x) = Ft(x) = ±[(t + l)n(x + 1) -rt{\)}. For
reasons that become apparent later it is also of interest to examine the properties
of the forward "spreads" Fp,(x,x- 1) = Ft(x - 1) -Ft_i(x). These results are to
be found in Table 2. Generally, the conclusions would be the same as for yields,
except that the persistence in forward rate spreads is not as marked, particularly
as the maturity gets longer.
As Table 1 also shows, there is a lot of persistence in spreads between short-
dated maturities; after fitting an AR(2) to spt(3) the LM test for serial correlation
over 12 lags is 80.71. This persistence shows up in other transformations of the
yield series, e.g. the realized excess holding yield ht+](x) = xrt(x) — (x — l)r,+i
(t — 1) — r,(l), when x = 3, has serial correlation coefficients of .188 (lag 1), .144
(lag 8), and .111 (lag 10). Such processes are persistent, but not integrated, as the
ADF(12) for ht+i(3) clearly shows with its value of -5.27. Papers have appeared
concluding that the excess holding yield is a non-stationary process-Evans and
Lewis (1994) and Hejazi (1994). That conclusion was reached by the authors
performing a Phillips-Hansen (1990) regression of ht+\(x) on Ft-\(x). Applying
the same test to our data, with McCulloch's forward rate series, produces an
estimated coefficient on Ft-\(x) of .11 with a t ratio of 10, quite consistent with
both Evans and Lewis' and Hejazi's results. However, it does not seem reasonable
to interpret this as evidence of non-stationarity. Certainly the series is persistent,
and an 1(1) series like the forward rate exhibits extreme persistence, so that
regressing one upon the other can be expected to lead to some "correlation", but
to conclude, therefore, that the excess holding yield is non-stationary is quite
incorrect. A fractionally integrated process that is stationary would also show
such a relationship with an 1(1) process. Indeed, the autocorrelation functions of
the spreads and excess yields are reminiscent of those for the squares of yield
changes, which have been modelled by fractionally integrated processes - see
Table 2
Autocorrelation features, forward rates, full sample
F(l)
F(3)
F{6)
F{9)
Fp(0,1)
Fp(2,3)
Fp(5,6)
Fp(i,9)
DF
-2.28
-2.18
-2.39
-2.14
-17.08
-19.52
-20.61
-19.73
ADF(12)
-1.92
-1.97
-1.91
-1.77
-4.07
-5.17
-5.77
-^.69
Pi
.98
.98
.98
.98
.29
.16
.11
.15
h
.18
.06
-.12
-.00
P6
.11
.01
-.05
.08
Pl2
.18
-.02
-.03
.02
Pi(Ar)
.07
.04
.09
.07
The 5% critical value for the DF and ADF tests is -2.87.

94
A. R. Pagan, A. D. Hall and V. Martin
Baillie et al. (1993). Nevertheless, the strong persistence in spreads is a
characteristic which is a substantial challenge to term structure models.2
As is well known, there was a switch in monetary policy in the US in October
1979 away from targeting interest rates, and this fact generally means that any
analyses have to be re-done to ensure that the results do not simply reflect
outcomes from 1979 to 1982. Table 3 therefore presents the same statistics as in
Table 1 but using only pre October 1979 data. It is apparent that the conclusions
drawn above are quite robust.
It is also well known that there is a substantial dependence of the conditional
volatility of Ar((t)upon the past, but the exact nature of this dependence has been
subject to much less analysis. As will become clear, the most important issue is
whether the conditional variance, a\v exhibits a levels effect and, if so, exactly
what relationship is likely to hold. Here we examine the evidence for a "levels
effect" in volatility, i.e. a\t depends on rt(i), concentrating upon the five yields
mentioned earlier. Evidence of the effect can be marshalled in a number of ways.
By far the simplest approach is to plot (Ar((t) - (i) against r(_i(T), (and this is
done in Fig. 1 for r((l)).3 The evidence of a levels effect looks very strong. A more
structured approach is to estimate the parameters of a diffusion process for yields
of the form
dr, = (o<i - $xrt)dt + rytldrjt
;i)
and to examine the estimate of yl. To estimate this requires some approximation
scheme. Chan et al. (1992) consider a discretization based on the Euler scheme
with h = 1 (ht being the discretized steps) producing
Table 3
Autocorrelation features, pre October 1979
r(l)
r(3)
r(6)
r(9)
r(120)
*>(3)
4P(6)
*>&)
sp(120)
DF
-.76
-.64
-.52
-.55
-.14
-11.99
-8.80
-8.20
-4.05
ADF(12)
-.79
-1.03
-1.00
-.89
.33
-2.64
-2.89
-3.10
-4.30
Pi
.97
.97
.98
.98
.99
.46
.70
.71
.90
Pi
.34
.56
.60
.84
k
.22
.39
.38
.59
Pl2
.34
.38
.36
.23
P.(Ar)
-.14
-.07
.08
.11
-.04
2 Throughout the paper we will take the term structure data as corresponding to actual
observations. In practice this may not be so, as a complete term structure is interpolated from
observations on parts of the curve. This may introduce some biases of unknown magnitude into
relationships between yields. McCulloch and Kwon (1993) interpolate with spline functions. Others
e.g. Gourieroux and Scaillet (1994), actually utilize some of the factor models discussed later to suggest
forms for the yield curve that may be used for interpolation.
3 Marsh and Rosenfeld (1983) also did this and commented on the relation.

Modeling the term structure
95
24.1132
15.3448 -
6.5763
-2.1921
.24900 5.5693 10.8897
Fig. 1. Plot of squared changes in one month yield against lagged yield
16.2100
Art = oci - fan-i + ar]]_
(2)
where here and in the remainder of the paper e, is n.i.d.{Q, 1).
Equation (2) can be estimated by OLS simply by defining the dependent
variable as krtr~]_\, while the regressors become xt= [r~]_\ rtZ{1], as the error
term is then <ret, which is n.i.d(0, a2). Because the conditional mean for rt depends
only on oci, jSj, while the conditional variance of ut = rt — Er_i(rr) is <r2rtl\, which
does not involve these parameters, we could estimate the parameters in the
following way.
1. Regress Art on 1 and rr_
2. Since
to get ai and fix
Er_, [u]] = ctr]l\
then
2 2 2i>,
+ vr ,
(3)
(4)
where Er_i(vr) = E[u2 — Et-i(u2)] = 0. Hence we can estimate yx by using a
nonlinear regression program.
3. We can re-estimate ai,/?! by then doing a weighted regression of Artr~]l
against r~2\ and rt~Jl.
The above steps would produce a maximum likelihood estimator if et was
taken to be Jf{0,1) and the estimation of yx was done by a weighted non-linear

96
A. R. Pagan, A. D. Hall and V. Martin
regression on (3) using the conditional standard deviation of vt as weights.4 Chan
et al. (1992) use a GMM estimator, which jointly estimates ai, f$x, yl and a from
the set of moments
E(£r) = 0, E(r,_i%) = 0, E(vt) = 0, E(r,_,v,) = 0 .
Their estimator would coincide with the one described above if the last moment
condition was replaced by ~E(r]_xvt) = 0. A potential problem with all the
estimators is that, if jSj is likely to be close to zero, the regressors in (2) and (4) will be
close to 1(1), and so non-standard distribution theory almost certainly applies to
the GMM estimator.
Table 4 presents estimates of the parameters <x\, p1 and yl found by using three
estimation methods. The first one is based on estimating the diffusion with an
Euler approximation,
Arth = ai/!-/?1/!r(r_1)A + ff/!1/2r[;_1)A£r , (5)
with h = 1. It is the estimator described above as GMM. The others stem from
the modern approach of indirect estimation proposed by Gourieroux et al. (1993)
Table 4
Estimates of diffusion process parameters
r,(l) r,(3) r,(6) r,(9) r,(120)
GMM
«i
ft
Ti
MLE
«i
ft
Tl
EGARCH
«i
ft
7i
.106
(2.19)
.020
(1.52)
1.351
(6.73)
.071
(2.17)
.012
(.74)
.583
(2.39)
.107
(1.63)
-.004
(2.15)
.838
(2.67)
.090
(1.82)
.015
(1.24)
1.424
(5.61)
.047
(2.48)
.007
(7.89)
.648
(1.92)
.044
(1.89)
-.010
(1.57)
.974
(5.73)
Asymptotic t-ratios in parentheses
4 Frydman (1994) argues that the distribution of the MLE of fly is non-standard when yx =1/2
and there is no drift.
.089
(1.80)
.015
(1.25)
1.532
(4.99)
.041
(3.00)
.005
(2.08)
.694
(2.31)
.045
(4.34)
-.008
(2.24)
.947
(7.21)
.091
(1.87)
.015
(1.31)
1.516
(5.12)
.037
(3.82)
.004
(1.08)
.753
(3.34)
.043
(1-67)
-.008
(2.04)
.941
(3.09)
.046
(1.77)
.006
(.98)
1.178
(9.80)
.015
(3.74)
-.001
(2.35)
1.136
(19.30)
.009
(3.30)
-.009
(2.36)
1.104
(4.88)

Modeling the term structure
97
and Gallant and Tauchen (1992). In these methods one simulates K multiple sets
of observations J*h (k = 1, ...,K) from (5), with given values of h (we use 1/100)
and 6' = («i f$x yx a2), and then finds the estimates of 6 that set Y%=i {K~l
!C*=i d(/>(r%;4>)} to zero> where (j> is an estimator of the parameters of some
auxiliary model found by solving ^J=l d^,(rt; $) = 0.5 The logic of the estimator is
that, if the model (5) is true, then </> —> </>*, where E[d,p(rt; 4>*)] = 0, and the term in
curly brackets estimates this expectation by simulation. Consistency and
asymptotic normality of the indirect estimator follows from the properties of </>
under mis-specification. It is important to note that the auxiliary model need not
be correct, but it should be a good representation of the data, otherwise the
indirect estimator will be very inefficient. We use two auxiliary models and, in
each instance, d$ are the scores for <j> from those models. The first is (5) with h = 1
and e, being assumed n.i.d.(0, \){MLE), while the second has r, being an AR(l)
with EGARCH(1,1) errors. The visual evidence of Figure 1 is strongly supported
by the estimated parametric models, although there is considerable diversity in
the estimates obtained. Perhaps the most interesting aspect of the table is the fact
that y! tends to increase with maturity. Based on the evidence from the indirect
estimators, yl = 1/2 seems a reasonable choice for the shortest maturity, which
would correspond to the diffusion process used by Cox et.al. (1985).
A problem in simply fitting a model with a "levels" effect is that the observed
conditional heteroskedasticity in the data might be better accounted for by a
GARCH process, and so the appropriate questions should either be whether there
is evidence of a levels effect after removing a GARCH process, or, whether a
levels representation fits the data better than a GARCH model does. To shed
some light on these questions, our strategy was to fit augmented EGARCH(1,1)
models to At,(t) = [i + oxtEtt, £%t ~./T(0,1), of the form
log ffj, = a0t + flit log o2t_x
+ alxEtt-\ + a3i I |ert-i | - \J- ) + Srt-i (t) . (6)
This specification is used to generate a diagnostic test for the presence of a levels
effect, and is not intended to be a good representation of the actual volatility.
Hence the /-statistic for testing if S is zero can be regarded as a valid test for more
general specifications, e.g. 8g{rt^\{z)), where g(-) is some function, provided
?7-i(t) is correlated with g(r,^\(x)). Table 5 gives the estimates of S and the
Table 5
S and t Ratios for Levels Effect
-5
t
*(1)
.050
3.73
0(3)
.025
3.51
0(6)
.023
3.42
0(9)
.021
3.04
o(120)
.019
2.42
5 A Mihlstein (1974) rather than Euler approximation of (5) was also tried, but there were very
minor differences in the results.

98
A. R. Pagan, A. D. Hall and V. Martin
associated t ratios. Every yield displays a levels effect, although with the 10 year
maturity it seems weaker.6 The same conclusion applies to the spreads between
forward rates, Fpt(x,x— 1). Fitting EGARCH(1,1) models to these series for
t = 1,3,6 and 9 months maturity, and allowing the levels effect to be a function of
F(_i(t), the f-ratios that this coefficient was zero were 3.85, 3.72, 17.25 and 12.07
respectively.
A number of studies have appeared that look at this phenomenon. Apart from
our own work, Chan et al. (1992), Broze et al. (1993), Koedijk et al. (1994), and
Brenner et al. (1994) have all considered the question, while Vetzal (1992) and
Kearns (1993) have tried to allow for stochastic volatility, i.e. of is not only a
function of the past history of yields. To date no formal comparison of the
different models is available, unlike the situation for stock returns e.g. Gallant
et al. (1994). All studies find strong evidence for a levels effect on volatility.
Brenner et al. provide ML estimates of the parameters of a discretized joint
GARCH/levels model in which the volatility function, of, is the product of a
GARCH(1,1) process and a levels effect i.e. of — (ao -\- a\<j^_^_^ +Q20f_i)'"J_i-
The estimated value of y falls to around .5, but remains highly significant. Koedijk
et al. (1993) have a similar formulation except that of is driven by £t_x rather than
of_i^_i- Again y is reduced but remains highly significant.
One might question the use of conventional significance levels for the "raw" t
ratios, owing to the fact that one of the regressors is a near-integrated process. To
examine the effects of this we simulated data from an estimated model, equation
(6) for r((l), treating the estimates obtained by MLE estimation as the true
parameter values, and then found the distribution of the t ratio for the hypothesis
that d = 0 using the MLE, constructed by taking one step from the true values of
the coefficients (this would be a simulation of the asymptotic distribution). The
results indicate that the distribution of the ?-ratio has fatter tails than the normal
with critical values for two tailed tests of 2.90 (5%) and 2.41 (10%), but use of
these would not change the decisions.
2.2. Multivariate properties
2.2.1. The level of the yield curve
As was mentioned in the introduction a great deal of work on the term structure
views yields as being driven by a set of M factors
M
6 It is interesting to observe that the distribution of the Dickey-Fuller test is very sensitive to
whether there is a levels effect or not. To see this we simulated a model in which Ar; = .001 + ti\r]_xet,
where e, ~ nid(0,1) and y either took the value of zero or unity. A small drift was added, although its
influence upon the distribution is likely to be small. The simulated critical values for 1 %, 2.5% and 5%
significance levels when 7 = 0,1 are (-3.14, -6.41), (-2.71, -4.97) and (-2.39, -4.03) respectively.
Clearly, the presence of a levels effect in volatility means that the critical values are much larger (in
absolute terms), strengthening the claim that Table 1 suggests a unit root in yields.

Modeling the term structure
99
and it is important to investigate whether this is a reasonable characterization of
the data. It is useful here to recognise that the modern econometrics literature on
multivariate relations admits just such a parameterization. Suppose the yields are
collected into an (n x 1) vector yt and that it is assumed that yt can be represented
as a VAR. Then, if yt are 1(1) and, in the n yields there are k co-integrating
vectors, Stock and Watson (1988) showed this to mean that the yields can be
described in the format
y_t=Jl + ut (g)
it = £t-i + vt ,
where 1t are the n — k common trends to the system, and ~Et-\vt = 0. The format
(8) is commonly referred to as the Beveridge-Nelson-Stock-Watson (BNSW)
representation. If there are (n — 1) co-integrating vectors, there will be a single
common factor, £y, that determines the level of the yields. How the yields relate to
one another is governed by yt — J£,\t = ut i.e. the yield curve is a function of ut.
Johansen's (1988) tests for the number of co-integrating vectors may be applied
to the data described earlier. Table 6 provides the two most commonly used - the
maximum eigenvalue (Max) and trace (Tr) tests - for the five yields under
investigation, and assuming a VAR of order one.7 From this table there appears to
be four co-integrating vectors, i.e. a single common trend. Johnson (1994),
Engsted and Tanggaard (1994) and Hall et al. (1992) reach the same conclusion.
Zhang (1993) argues that there are three common trends but Johnson shows that
this is due to Zhang's use of a mixture of yields from zero and non-zero coupon
bonds.
What is the common trend? There is no unique answer to this. One solution is
to find a yield that is determined outside of the system, as that will be the driving
force. For a small country, that rate is likely to be the "world interest rate", which
in practice either means a Euro-Dollar rate or some combination of the US,
German and Japanese interest rates. Another candidate for the common trend is
Table 6
Tests for cointegration amongst yields
5 vs 4 trends
4 vs 3 trends
3 vs 2 trends
2 vs 1 trends
1 vs 0 trends
Max
273.4
184.7
95.6
30.9
2.4
Crit.
33.5
27.1
21.0
14.1
3.8
Value (.05)
Tr.
586.9
313.5
128.8
33.3
2.4
Crit. Value (.05)
68.5
47.2
29.6
15.4
3.8
1 Changing this order to four does not affect any conclusions, but restricting it to unity fits in
better with the theoretical discussion.

100
A. R. Pagan, A. D. Hall and V. Martin
the simple average of the rates.8 In any case we will take this to be the first factor
Zu in (7).
2.2.2. The shape of the yield curve
The existence of k co-integrating vectors a (a is an (n x k) matrix), such that
Ct = v!yt is 7(0), means that any VAR in yt has the ECM format
Aj, = yCt-i + £>(i)Aj,_! + et , (9)
where E,_i (et) = 0 and D(L) is a polynomial in the lag operator. It is also possible
to show that ut in (8) can be written as a function of the k EC (error correction)
terms (t and this suggests that we might take these to be the remaining factors £Jt
(J = 2,..., K) in (7). To make the following discussion more concrete assume that
the expectations theory of the term structure holds i.e. a t period yield is the
weighted average of the expected one period yields into the future. In the case of
discount bonds the weights are equal to \ so that the theory states
l.-1
r,(T)=-£E,(r„*(l)) .
Tt=o
Of course this is an hypothesis, albeit one that seems quite sensible. It implies that
r,(t) - r,(l) = J ^E^+t(l) - E,r,(l) I
= {;EEE«^(1)}-
Now, if the yields are 7(1) processes, the yield spread rt{x) — rt{\) should be 7(0),
i.e. rt{x) and r,(l) should be co-integrated with co-integrating vector [1 — 1], and
these spreads are the EC terms. Therefore, to test the expectations hypothesis for
the five yields we need to test if the matrix of co-integrating vectors has the form
(10)
Johansen's (1988) test for this gives a #2(4) of 36.8, leading to a very strong
rejection of the hypothesis. Such an outcome has also been observed by Hall et al.
(1992), Johnson (1994) and Engsted and Tanggaard (1994).
A number of possible explanations for the rejection were canvassed in those
papers, involving the size of the test statistic being incorrect etc. One's inclination
is to examine the estimated matrix of co-integrating vectors given by Johansen's
a' =
"-1 1
-1 0
-1 0
-1 0
0
1
0
0
0
0
0
1
See Gonzalo and Granger (1991) for other alternatives.

Modeling the term structure
101
procedure, a, and to see how closely these correspond to the hypothesized values
but, unfortunately, the vectors are not unique and the estimated quantities will
always be linear combinations of the true values. Some structural information is
needed to recover the latter, and to this end we write a' = Aa, where
" -03 10 0"
-ft 0 1 0
a -09 0 0 0 '
.-^120 0 0 1
and then proceed to solve the equations a = Aa, where A is some non-singular
matrix. This produces 03 = 1.038, 06 = 1.063, 09 = 1.075 and 012O = 1.076, which
indicates that the point estimates are quite close to those predicted by the
expectations theory. It is also possible to estimate the 0T by "limited information"
rather than "full-information" methods. To that end the Phillips-Hansen (1990)
estimator was adopted with a Parzen kernel and eight lags being used to form the
long-run covariance matrices, producing 03 = 1.021, 06 = 1.034, 09 = 1.034 and
0120 = -91 • With the exception of the 10 year rate, neither set of estimates seems to
be greatly divergent from that predicted.
Some insight into why the rejection occurs may be had from (9). Given that co-
integration has been established, and working with a VAR(l), i.e. D(L) = 0 in (9),
the change in each yield should be governed by
5
t*tw = J2y^r<-iw - 0/vi(i))+««. (ii)
7=2
where j = 2,..., 5 maps one to one into the elements x = 3,6,9,120. If the
expectations theory is valid 0y = 1 and the system becomes
5
tot{v) = X^OviO') ~ 'ViO)) + *« ,
j=a
and the hypothesis Ho : fy = 1 can be tested by computing the likelihood ratio
statistic. It is well known that such a test will be distributed as a x2(4) under the
null hypothesis.
If the yields were taken to be 1(0), the simplest way to test if 0, = 1 would be to
re-write (11) as
tot{T)=J2yjArt-xU) -n-i(i)) + (E^(1 -A/)Vi(i) +e«> (i2)
7=2 \j=2 )
and to test if the coefficient of r?_i(l) in each of the equations for Art(x) was zero.
For a number of reasons this does not reproduce the x2(4) test cited above - there
are five coefficients being tested and rt-i(l) will be 1(1), making the distribution
non-standard. Nevertheless, the separate single equation tests might still be
informative. In this case the ^-values that rt-i(l) has a zero coefficient in each

102
A. R. Pagan, A. D. Hall and V. Martin
equation were -4.05, -1.77, -.72, -.24 and .55 respectively, suggesting that the
rejection of (10) lies in the behaviour of the one month rate i.e. the spreads are not
capable of fully accounting for its movement. Engsted and Tanggaard (1994) also
reach this conclusion. It may be that r(_i(l) is proxying for some omitted
variable, and the literature has in fact canvassed the possibility of non-linear effects
upon the short-term rate. Anderson (1994) makes the influence of spreads upon
Ar((l) non-linear, while Pfann et al. (1994) take the process driving rt(\) to be a
non-linear autoregression - in particular, the latter allow for a number of regimes
according to the magnitude of rt{\), with some of these regimes featuring 1(1)
behaviour of the rate while others do not. Another possibility, used in Conley
et.al. (1994) is that the "drift term" in a continuous time model has the form
Y0j=-m a-iA and this would induce a non-linearity into the relation between Art
and r(_i.
Instead of a mis-specification in the mean, rejection of (10) may be due to levels
effects in ezt. As noted earlier, the Dickey-Fuller test critical values are very
sensitive to this effect, and the test that ru^\ has a zero coefficient in the Ar((l)
equation in (12) is actually an ADF test, if the augmenting variables are taken to
be the spreads. This led us to produce a small Monte Carlo simulation of Jo-
hansen's test for (10) under different assumptions about levels effects in the errors
of the VAR. The example is a simplified version of the system above featuring
only two variables y\t and yu with co-integrating vector [1 — 1], and being
generated from the vector ECM,
Ayn = -.8(j>i,_i - y2t-\) + ^y\t^u
kyit = -•lO'if-i - yit-\) + -\y\t-\Zit ■
The 95% critical value for Johansen's test that the co-integrating vector is the true
one varies according to the value of y : 3.95(y = 0), 4.86(y = .5), 5.87(y = .6),
11.20(7 = .8), and 23.63(y = 1). Clearly, there is a major impact of the levels
effect upon the sampling distribution of Johansen's test, and the phenomenon
needs much closer investigation, but it is conceivable that rejection of (10) may
just be due to the use of critical values that are too small.
Even if one rejects the co-integrating vectors predicted by the expectations
theory, the evidence is still that there are k = n — 1 error correction terms. It is
natural to equate the remaining M — 1 factors in (7) (after elimination of the
common trend) with these EC terms, but this is not very helpful, as it would mean
thatM = n, i.e. the number of factors would equal the number of yields. Hall et al.
(1992) provide an example of forecasting the term structure using the ECM
relation (9), imposing the expectations theory co-integrating vectors to form £„ and
then regressing Ayt on Ct~\ and anY lags needed in Ayt. Hence their model is
equivalent to using a single factor, the common trend, to forecast the level, and
(n — 1) factors to forecast the slope (the EC or spread terms). In practice however
they impose the feature that some of the coefficients in y were zero, i.e. the number
of factors determining the yield varies with the maturity being examined. It is
interesting to note that their representation for Ar((4) has no EC terms i.e. it is

Modeling the term structure
103
effectively determined outside the system and plays the role of the "world interest
rate" mentioned earlier.
In an attempt to reduce the number of non-trend factors below » — 1, it is
tempting to assume that (say) only m — M - 1 of the (n - 1) terms in (, appear as
determinants of Art(x) and that these constitute the requisite common factors, but
such a restriction would necessitate m of the columns of y being zero, thereby
violating the rank condition, p(y) = n—\. Consequently the factors will need to
be combinations of the EC terms. Now, pre-multiplying (7) by a' gives
M
7=1
where M = [/? -j... fSjn] is a 1 x « vector. If we designate the first factor as the
common trend then it must be that a!b\ = 0 as the LHS is 1(0) by construction,
meaning that
t;t = a'f2bjZjt = x'BSt , (14)
7=2
where H, is the (K — 1) x 1 vector containing £2( • • ■ £&, and B is an n x (K — 1)
matrix with p(B) =K—\, where p(-) designates rank.
Equation (14) enables us to draw a number of interesting conclusions. Firstly,
p[co\(gt)] = min[p(a), p(B)], provided cov(E,) has rank K—\. Since K <n
implies K — 1 < « — 1, it must be that p(B) < p(a.), and therefore p[cov((,)] =K—\
i.e. the number of factors in the term structure (other than the common trend)
may be found by examining the rank of the covariance matrix of the
co-integrating errors. Secondly, since C = a'B has p{C) =K-l, Ft = {C'C)~XC%„
and hence the factors will be linear combinations of the EC terms. Applying
principal components to the data set composed of spreads spt(3), spt(6), spt(9)
and 5p((120), the eigenvalues of the covariance matrix are 4.1, .37, .02 and .002,
pointing to the fact that these four spreads can be summarized very well by three
components (at most).9 The three components are:
9 The principal components approach, or variants of it, has been used in a number of papers -
Litterman and Scheinkman (1991), Dybvig (1989) and Egginton and Hall (1993). This technique finds
linear combinations of the yields such that the variance of each combination is as small as possible.
Thus the i'th principal component of y, will be b^y,, where bt is a set of weights. Because one could
always multiply through by a scale factor the bt are normalized, i.e. 6Ji, = 1. With this restriction b
becomes the eigenvectors of var(y,). Since bt is an eigenvector it is clear that b'v&r{y,)b = A, where A is
a diagonal matrix with the eigenvalues (h ■■■ K) on it, and that tr[b'vaT(y,)b] = J^JLi h- It is
conventional to order the components according to the magnitude of A,-; the first principal component
having the largest Xt. There is a connection between principal components and common trends. Both
seek linear combinations of y, and, in many cases, one of the components can be interpreted as the
common trend, e.g. in Egginton and Hall (1993) the first component is effectively the average of the
interest rates, which we have mentioned as a possible common trend earlier.

104
A. R. Pagan, A. D. Hall and V. Martin
4>u = 32sPt(3) - .86.sp,(6) - 37sPt(9) + .17spt{l20)
4>2t = ~.78sPt(3) + .00sPt(6) - .55spt(9) + .29spt{l20)
4>* = .54spt{3) + .52spt(6) - .Sispt(9) + 37spt(120).
3. Models of the term structure
In this section we describe some popular ways of modelling the term structure. In
order to assess whether these models are capable of replicating observed term
structures, it is necessary to decide on some way to compare them to the data.
There is a small literature wherein formal statistical tests have been performed on
how well the models replicate the data in some designated dimension. Generally,
however, the reasons for any rejection of the models remain unclear, as many
characteristics are being tested at the one time. In contrast, this chapter uses the
method of "stylized facts", i.e. it seeks to match up the predictions of the model
with the nature of the data as summarized in Section 2. Thus, we look at whether
the models predict that yields are near-integrated, have levels effects in volatility,
exhibit specific co-integrating vectors, produce persistence in spreads, and would
be compatible with two or (at most) three factors in the term structure.10
3.1. Solutions from the consumer's Euler equations
Consider a consumer maximising expected utility over a period subject to a
budget constraint, i.e.
maxE,
X>(C,)/F
where P is a discount factor, and Cs is consumption at time s. It is well known that
a first order condition for this is
U'(Ct)vt = Et{ps-'U'(Cs)vs} ,
where vt is the value of an asset (or portfolio) in terms of consumption goods.
This can be re-arranged to give
E,
Vs ]r'U'(Cs)/U'(Q)
= 1 ■ (15)
Assuming that the asset is a discount bond, and the general price level is fixed,
consider setting s = t + z giving vt = fti?)- The solution of this equation will then
10 There are many other characteristics of these yields that we ignore in this paper but which are
challenging to explain e.g. the extreme leptokurtosis in the density of the change in yields and in the
spreads.

Modeling the term structure
105
provide a complete set of discount bond prices for any maturity. It is useful to re-
express (15) as
/,(t) = E([/r[/'(Q+T)/C/'(G)] ,
(16)
imposing the restriction that f,(t + t) = 1, so as to find the price of a zero coupon
bond paying SI at maturity. Hence the term structure would then be determined.
If the price level is not fixed (16) needs to be modified to
/,(t) = Et[FP,U'(Ct+z)/(U'(Ct)Ps+<)} ,
(17)
where Pt is the price level at time t.
There have been a few attempts to price bonds from (16) or (17). Canova
and Marrinan (1993) and Boudoukh (1993) do this by assuming that ct
= log (Q+i/Q) - 1 and pt = log (Pt+\/Pt) — 1, follow a VAR process with some
volatility in the errors, and that the utility function has the CRAA form,
U(Q) = C]~y/(l - y), where y is the coefficient of risk aversion.11
10g/,(T) •
It is necessary to evaluate (17) for the yield rt(x)
r,(r) = - - log E,\p*(Ct+z/Ct)-y(Pt/Pt+z)}
T
= --log Vt\p\\ + cH)-\\ + Pn)-x
where
cn = Ct+Z/Ct - 1 ~ log Cr+T - log C,
Pn = Pt+z/Pt ~ 1 ^ log A+t " log Pt .
Expanding around Et(cn) and E,(/>„), and ignoring all cross terms and terms of
higher order than a quadratic,12
where
~ _ logjS - 1 log { [(1 + E,(crt)P(l + MPn))'
+ aiT,var((c,T)+a2TfVar,(^rt)} ,
aut = 1/2(1 + 7)7(1 + Et(crT))-''-2(l + E,(At))"
^^(l+E,^))"^^^^))^
(18)
11 Canova and Marrinan actually use the Cambridge equation for the price level, Pt = M,/Yt, and
so their VAR involves the growth in money, output and consumption.
12 The conditional covariance terms between c,t and ptx are ignored as one is a real and the other a
nominal quantity and most general equilibrium models would make this zero. Boudoukh (1993)
however argues that the conditional covariance is important for explaining the term structure.

106
A. R. Pagan, A. D. Hall and V. Martin
~ - log j3 +1 log (1 + Et(cn)) + - log (1 + Et(PtT))
\ T (19)
- - log {Z>irtvar,(cft) + b2rt\ah{Ptr)} ,
z
where
but = \(l+ 7)7(1 +E((ctt)r2
b2Tt = (l+Et(Ptt))~2 .
Equation (19) points to a four factor model of the term structure with the level
being driven by the first two conditional moments of the inflation rate and
consumption growth. However, the relation is not easily interpreted as a linear
one, since the weights attached to volatilities are functions of the conditional
means.
The problem remains to evaluate the conditional moments. To complete the
model it is necessary to assume something about the evolution of zlt = ct\ and
zit = Pt\ -These are generally taken to be AR processes of the form
zJt = $oj + ®\jZjt-\ + ejt .
Canova and Marrinan (1993) take o^+1 = \aTt(eJt+i) to be GARCH processes of
the form
4+i = a0j + cijtft + aiie)t -
whereby the formulae in Baillie and Bollerslev (1992) can be used to evaluate
Et{zjH) and var,(z,tt), while Boudoukh (1993) has ajt as a stochastic volatility
process. For GARCH models var,(z,tt) is a linear function of o)t+\-
How well does this model perform in replicating the stylized facts of the term
structure? To produce a near unit root in yields it is necessary that
log(l +Et(pH)) ~ Et{pn) be near integrated i.e. inflation must be a near
integrated process, as it is the only one of the two series that has such persistence in
either mean or variance - see Boudoukh (1993) for a description of the time series
properties of the two series. Then the inflation rate becomes the common trend in
the term structure, and the spreads will depend upon consumption growth and
the two volatilities. As there is rather weak evidence for much dependence in
either inflation or consumption volatility - see the test statistics in Boudoukh- it is
difficult to see the persistence in spreads being explained by these models.13
Whether a levels effect in Art(z) can be produced is unclear; the GARCH
structures used by Canova and Marrinan will not produce it, but Boudoukh's
stochastic volatility formulation does allow for a levels effect in var, (p,). Moreover,
even if volatilities were constant, the conditional means enter the weights attached
13 Although Boudoukh finds much more in his estimated stochastic volatility specification than
GARCH specifications.

Modeling the term structure
107
to them, and this dependence might be used to induce a levels effect into Arf(r).
Whilst Et(cn) is likely to be close to a constant due to the weak autocorrelation in
consumption growth, there is strong serial correlation in inflation rates, and, with
inflation as the common trend, it is conceivable that the requisite effect could be
found in that way, although the question was not addressed by the authors.14
Another attempt at working within this framework is Constantinides (1992)
who writes (17) as
/,(t) = Et[Kt+x/Kt] ,
where Kt = P'U'(Ct)/Pt is referred to as a "pricing kernel". He then makes
assumptions about the evolution of Kt, in particular that
Kt = exp< - [g +-y-y + zQt+ ^2{zit - at)2 \ .
He works in continuous time and makes z0t a Weiner process while the other zit
are Ornstein-Uhlenbeck diffusion processes with parameters Xt and variances of.
Each of the zit are taken to be independent. Under these assumptions it turns out
that
f,(x) = {n^-Wr^cxpH-ff + E^T
+ fx'w^ - a<e"iT)2 - x> - a<-)2)»
i=\ i=l J
where Ht(x) = of/A, + (1 — of /'Xi)e2i-'t'. Consequently, rt(x) has the format
N N
r,(x) = <5o, + E di"(z'' ~ ^"f + E T_1 (z« " a<)2 ■
i=\ i=l
Terms such as (zit - a,)2 reflect the fact that the "variance" of the change in zit of
an Orstein-Uhlenbeck process depends upon the level of the variable zit.
Constantinides' model will have trouble producing the right outcomes. After
converting to yields his model has no factor that would be 1(1). The difficulty
arises from his specification of the "pricing kernel". The pricing kernel used to
evaluate (17) has an 1(2) variable Pt as it is the inflation rate which is 1(1).
Consequently it is the assumption implictly made by Constantinides that the
kernel is only 1(1) through the presence of the term z§t which is the root of the
problem with his model.
14 Essentially, these are "calibrated" models that emphasise the use of a highly specified theory to
explain an observed phenomenon. Hence, one should really distinguish between the model prediction
of yields, t*(x), and the observed outcomes, rt{%). The gap between the two variables is due to factors
not captured within the model, or perhaps to specification errors. Examination of the characteristics of
the gap may be very informative.

108
A. R. Pagan, A. D. Hall and V. Martin
3.2. One factor models from finance
Finance theory has developed by working with factor models to determine the
term structure. Common to the material just discussed is the use of models of an
economy in which there is inter-temporal optimization, but a notable difference is
the introduction of a production sector and a concern with ensuring that the
pricing formulae prohibit the possibility of arbitrage i.e the solution tends to be
closer to a general rather than partial equilibrium solution. The basic work horse
of the literature is the model due to Cox, Ingersoll and Ross (1985) (CIR).
Essentially they propose an economy driven by a number of processes that affect the
rate of return to assets e.g. technological change and (possibly) an inflation factor.
Dealing with the simplest case where there is just a single state vector, /j,t, perhaps
total factor productivity (TFP), it is assumed that this variable follows a diffusion
process of the form
dfit = (b — Kfit)dt + cpfj./ dr\t .
General equilibrium in asset markets for such an economy results in an expression
for the instantaneous rate of interest of the form
drt = (a — firt)dt + art' dr\t . (20)
Once one has the expression for the instantaneous rate the whole term
structure ft(x) is priced according to a partial differential equation
1/2 a2r /„ + {*- M fr + ft- MT ~ rf = 0 , (21)
where f„ = cff/drdr, fr = df/dr, ft = df/dt and the term Xrfr , which
depends upon the covariance of the change in the price of the factor with the
percentage change in the optimal portfolio, is the "market price of risk"
associated with that factor. This partial differential equation comes from the fact that
a zero coupon riskless bond maturing at t + x must be valued at
/rW = E,
'(-/
exp - / r(\j/)d\j/
(22)
Since the expected rate of change of the price of the bond is given by
r + krfr/f, it also can be interpreted as a liquidity premium. It is clear that we
could group together the terms (a - fir)fr and -Xrfr and treat the problem as
one of pricing an asset using a "hypothetical " instantaneous rate that is
generated by
dr, = (a - fir, - krt)dt + ar),2dr\t,
1/2 *■ '
= (a - yrt)dt + art' dy\t .
The distinction is between the true probability measure in (20) and the
"equivalent martingale measure " in (23).

Modeling the term structure
109
The analytic solution for the term structure in the CIR model is then (see Cox
et al. (p. 393))
f,{z) = Al(x)exp(-Bl(x)rt) ,
where
r 2dcW((d + y)z/2) l2^
|_(<5 + 7)(exp(<5T)-l) + 2<5j '
= 2(exp(<k) - 1)
l{ ' [(<5+y)(exp(<5T)-l) + 2<5
Converting to a yield
r,(T) = {-\og{Ax{z))+Bx{z)rt}/x . (24)
This is a single factor model with the instantaneous rate or, more fundamentally,
the "returns" factor, driving the whole term structure, i.e. the level of the term
structure depends on the value of rt at any point in time. The slope of the yield
curve depends upon the parameters of the diffusion equation as well as the market
price of risk.
Perhaps the biggest problem with this methodology is that it will never exactly
reproduce an observed yield curve. This bothers practitioners a lot. One response
has been to allow a to change according to t and t. What this does is to add on
"fudge factors" to the model based yield curve so that the modified curve equals
the observed yield structure. Then, after forecasting rt+\ and finding the predicted
term structure, the "fudge factors" from the previous period are added on. The
need for "fudge factors" suggests that there is substantial mis-specification in the
CIR model as a description of the term structure, just as "intercept corrections"
in macro econometric models were given such an interpretation.
Brown and Dybvig (1986) estimated the parameters of the CIR model by
maximum likelihood and then computed the residuals denned by the gap between
the observed bond prices (/,) and the predictions of the model (/*). Examination
of the residuals pointed to specification errors in the model.15 Looking at the CIR
model in the light of stylized facts, the data should posess the characteristic that
interest rates are near-integrated processes and possibly co-integrated with co-
integrating vectors between any pair of rates of [1 -1] i.e. the spreads should be
1(0). The question that arises is whether the CIR model would deliver such a
prediction. One problem to be overcome is quantifying the market price of risk, X,
in the CIR bond formulae. As CIR point out, X — 0 if the factor had no effect on
the real economy e.g. if it was some nominal quantity such as the inflation rate.
Accordingly, we will adopt this interpretation, allowing us to set X — 0. To induce
a unit root we set ft = 0, and we also put the drift term a = 0. This makes
15 Since there are n yields but only one factor they needed to add on a vector of errors to the model
to produce a non-singular covariance matrix for /*, in order to be able to form a likelihood. It may be
that the mis-specification reflects the assumptions made in this step.
,and<5 = (y + 2ff2)1/2.

110
A. R. Pagan, A. D. Hall and V. Martin
' w ' w <5[exp(<5T)-l)]+2<5
Now the spread spt(z) will be
rt(z)-rt(l) = [z-lBl(z)-Bl(l)}rt ,
so that we will not get spreads to be 1(0) unless the term in square brackets is zero.
Generally it will not be. Realistic values for a, /? and a might be the GMM
estimates for rt(\) of .049, .02 and .106. These produce values of z~xB\(z)
= .990, .967, .930, .890 and .181 for the five maturities. In the limit (z -► oo)Bi(z)
= 2/3, and so the spreads between adjoining yields tend to zero as the maturity
lengthens.
The source of the failure of the spreads to be 1(0) is the fact that 3 ^ 0. If 3 = 0
then, using L'Hopital's rule, B\ (z) = z, and so the spreads should be identically
zero. By making a very small we can always produce results in which the spreads
will be very close to being 1(0) i.e. even if a is not exactly zero it can be regarded
as sufficiently close to zero that the spreads are nearly non-integrated, although
the longer the maturity which the spread is based on the less likely we are to see
such an outcome.
Another way of understanding the problem is to look at a discrete form of the
fundamental pricing equation (22), ft(z) = E,[exp(— J2%]~1 r])\- Suppose that rt
is 1(1) with martingale difference innovations that are normally distributed. Then
ft(z)=exp(-zrt){H<z\[l/2(z-j)\art(Yl%]-{Art+j)]}. If the conditional
variance is a constant the spreads will therefore be 1(0). However, if it depends upon
the level of the instantaneous rate, the spreads at any maturity would be equal to
a non-linear function of rt. For example, substituting the "square-root"
formulation of CIR gives var((Ar(+y) = a2rt, and spt(z) = cnst — (1 — 1) log rt.
Thus, it is important to determine the nature of the conditional variances in the
data. Most econometric models of the term structure make these conditional
variances GARCH processes, which effectively means that they are functions of
Art_j. But, as seen in the section examining the term structure data, there is prima
facie evidence of a levels effect after allowing for a GARCH specification of the
conditional variance.
Given the conflicting evidence in Section 2, one might look at other
co-integrating vectors when performing the comparison with CIR. In general, the CIR
model points towards co-integrating vectors that are of the form
rt(z) = d(z)rt(\) ,
where d(z) < 1 and decreasing with z. As seen in Section 2, with one exception
both the Johansen and Phillips-Hansen estimates of d(z) have d(z) > 1 and

Modeling the term structure
111
increasing in z. The predictions from CIR type models are therefore diametrically
opposed to the data.16
3.3. Two factor models from finance
Another response to the discrepancy between the model based prediction of a
yield curve and the observed one, is to seek to make the model more complex. It is
not uncommon in this literature to see people "bypassing" the step between the
instantaneous rate and the fundamental driving forces and simply postulating a
process for the instantaneous rate, after which this is used to price all the bonds.
An example of this is the paper by Chen and Scott (1992) who assume that the
instantaneous rate is the sum of two factors
r, = ft, + & , (25)
where
dlu = (ai - h l\t)dt + a\£\lt2dr\u
d&t = (<*2 - ht,it)dt + (r2£l2t2dti2t ,
where dr\jt are independent, thereby making each factor independent. Then the
solution for the bond price is
ft(z) =Al(z)A2(z)Sxp{-Bl(z)^t-B2(z)i2t} ,
where A2 and B2 are defined analogously to A\ and B\. Obviously this framework
could be extended to encompass any number of factors, provided they are
assumed to be independent.
Another method is that of Longstaff and Schwartz (1992) who also have two
factors but these are related to the underlying rate of return process fj.t rather than
directly to the instantaneous rate. In particular they wish to have the two factors
being linear combinations of the instantaneous rate and its conditional variance.
The model is interesting because the second factor they use, £2?, affects only the
conditional variance of the fj.t process, whereas both factors affect the conditional
mean. This is unlike Chen and Scott's model which has £lt and £2t affecting both
the mean and variance. Empirically, the two factors are regarded as the short
term rate and its conditional volatility, where the latter is estimated by a GARCH
16 Brown and Schaefer (1994) find that the CIR model closely fits the term structure of real yields,
where these are computed from British government index-linked bonds. Note in constructing the
Johansen and Phillips-Hansen estimators that an intercept was allowed into the relations in order to
correspond to A(z).

112
A. R. Pagan, A. D. Hall and V. Martin
process when assessing the quality of the model.17 Tests of the model are limited
to how well it replicates the unconditional standard deviations of yield changes.
There are a number of other two factor models. Brennan and Schwartz (1979)
and Edmister and Madan (1993) begin with the long and short rates following a
joint diffusion process. After imposing the "no arbitrage condition" and assuming
that the long rate is a traded instrument, Brennan and Schwatz find that the price
of the instantaneous risk associated with the long rate can be eliminated, and the
two factors then effectively become the instantaneous rate and the yield spread
between that rate and the long rate. Eliminating the price of risk for the long rate
makes the model non-linear and they need to linearize to find a solution. Even
then there is no analytical solution for the yield curve as with CIR. Another
possibility for a two factor model might be to allow for stochastic volatility as a
factor. Edmister and Madan find closed form solutions for the term structure in
their formulation.
Suppose that the first factor in Chen and Scott's model is a "near 7(1)" process
whereas the second factor is 7(0) .Then the instantaneous rate has the common
trend format (compare (25) and (8) recognising that J can be regarded as the unit
column vector). Using the same parameter values for the first factor as the polar
case discussed in the preceding sub-section i.e. fix = 0, k\ = 0, g\ = 0, the first
factor disappears from the spreads, which now equal
rt{z) - r,(l) = log(^2(l)/^2(T)) + [t-1^) -52(l)]fc, ■
Hence, they are now stochastic and inherit the properties of the second factor.
For them to be persistent, it is necessary that the second factor have that
characteristic. Notice also that rt(z) — rt(z - 1) will tend to zero as z —> oo, and this
may make it implausible to use this model with a large range of maturities.
Consequently, this two factor model can be made to reproduce the standard
results of the co-integration approach in the sense that the EC terms are
decomposed into a smaller number of factors. Of course the model would predict
that the coefficients on the factors would be negative as z~lB2(z) < 52(1). The
conclusion of negative weights extends to any number of factors, provided they
are independent, so it is interesting to look at the evidence upon the signs of the
coefficients of the factors in our data set, where the non-trend factors are equated
with the principal components. Although one cannot uniquely move from the
principal components/spreads relation to a spreads/principal components
relation, a simple way to get some information on the relationship between spreads
and factors is to regress each of the spreads against the principal components.
Doing so the R2 are .999, .999, .98 and .99 respectively, showing that the spreads
are well explained by the three components. The results from the regressions are
17 Volatility affects the term structure here by its impact upon r, in (25). Shen and Starr (1992)
raise the interesting question of why volatility should be priced; if one thinks of bonds as part of a
larger portfolio only their covariances with the market portfolio would be relevant. To justify the
observed importance of volatility they note that the bid/ask spread will be a function of volatility and
that has an immediate effect upon yields.

Modeling the term structure
113
spt(3) = .36iAu - .83^ + .48^,
spt(6) = -J6il/u-. 09il/2t + .42^,
spt(9) = -1.28^,, + .33^ + .44^
5^(120) = -1.44^ + 1.84^ + 2.12^3, ,
where \j/jt are the first three principal components. It is clear that independent
factor models would not generate the requisite signs. Formal testing of two factor
pricing models is in its infancy. Pearson and Sun (1994) and Chen and Scott
(1993) estimate the parameters of the model by maximum likelihood and provide
some evidence that at least two factors are needed to capture the term structure
adequately.
The two factor model is also useful for examining some of the literature on the
validity of the expectations hypothesis. Campbell and Shiller (1991) pointed out
that the hypothesis implies that
r»+i(T - 1) - rt(z) = a0 +—-\rt{z) - rt{\)\ , (26)
if the liquidity premium was a constant. They found that this restriction was
strongly rejected by the data. With McCulloch and Kwon's data and z = 3, the
regression of rt+\{2) — r,(3) against rt(3) — rt{\) yields an estimated coefficient of
-.09, well away from the predicted value of .5. Of course, the assumption of a
constant premium is incorrect. Bond prices are determined by (22) which, when
discretized, would be,
tt-T-l
j=t
"■+*-! w (27)
exp (- iz rj)
L \ j=t /
Vt
where ff (t) is the bond price predicted by the expectations theory. Thus rt(z)
differs from that of the expectations theory by the term - z~l log v,, and this in
turn will be a function of the conditional moments of Art. In the case where Art is
conditionally normal it depends upon the conditional variance, and the equation
corresponding to (26) will now feature a time varying ao that depends on this
moment. If the conditional variance relates to the spreads with a negative
coefficient, then that could cause there to be a negative bias in the coefficient of
rt(z) — rt{\) in the Campbell and Shiller regressions. One scenario in which this
happens is if the conditional variance depended upon Art, as happens with an
EGARCH model. Then, due to cointegration amongst yields, Art could also be
replaced by the lagged spreads, and these will have negative coefficients. More
generally, since we observed in Section 2 that the factors influencing the term
structure, such as volatility, could be written as linear combinations of the

114
A. R. Pagan, A. D. Hall and V. Martin
spreads, there is a possibility that term structure anomalies might be explained in
this way.
3.4. Multiple non-independent factor models in finance
Duffle and Kan (1993) present a multi-factor model of the term structure where
the factors may not be independent. As for the two factor models it is assumed
that the instantaneous rate is a linear function of M factors, collected in an M x 1
vector £„ which evolves according to the diffusion process
where dr\t is a vector of standard Brownian motions and n(£t),a{£t) are vectors
and matrices corresponding to drift and volatility functions. They then ask what
type of functions /z(-) and a{-) are capable of producing a solution for the n bond
prices ft{z), z = 1, •. •, n, of the exponential affine form
ft(z)=exp[(A(z)+B(z)i;t)}
= exp
u
i=i
It turns out that n(£t) and a{£t) should be linear (affine) functions of £,.
Thereupon the solution for B[z) can be found by solving an ordinary differential
equation of the form
B{z) = B(B(z)), 5(0) =0.
In most cases only numerical solutions for B(z) are available. Duffle and Kan
consider some special cases, differing according to the evolution of £,. When the
£it are joint diffusions driven by Brownian motion with covariance matrix Q that
is not diagonal, there is the possibility that the weights attached to the factors can
have different signs, and so the principal defect with the two factor models of the
preceding sub-section might be overcome. To date little empirical work seems to
be available on these models, with the exception of El Karoui and Lacoste (1992)
who make £, Gaussian with constant volatility.
3.5. Forward rate models
In recent years it has become popular to model the forward rate structure directly
rather than the yields, e.g. in Ho and Lee (1986) and Heath, Jarrow and Morton
(1992) (HJM). Since the forward rates are linear combinations of the yields,
specifications based on the nature of the forward rate structure imply some
restriction upon the nature of the yield curve, and conversely. In the light of what is
known about the behavior of yields, this sub-section considers the likelihood that
popular models of forward rates can replicate the term structure. In what follows,
one step ahead forward rates are used along with the HJM framework. In the

Modeling the term structure
115
interest of space only a simple Euler discretization of the HJM stochastic
differential equations describing the evolution of the forward rate curve is used.
Many variants of these equations have emerged, but they have the common
format,
F,{z - 1) -F,-\(x) = c/]t_i + oviT_i£,,.T_i ,
where e,jT_i is n.i.d.(0,1). Differences among the models reflect differences in the
assumptions made about volatilities. Examples would be a constant volatility
model in which c/]T_i = a0 + a2z and ff/]t_i = a, or a proportional volatility model
that has ct<t-\ = -aF,(z)X + oFt(z)(^2"k=xF,(k)) and ov]T_i = oF,(z). The nature
of c/jT_i reflects the no-arbitrage assumption. After some manipulation it can be
shown that
Ft{z - \) -F,.x{z) =
1
Z + 1
sPt(z) (r<(T + 0 ~ ^(T))
fI±iAr/(T + l)--Ar/(l) ,
Z Z
so that the equation used by HJM for the evolution of the forward rate
incorporates spreads and changes in yields. In turn, using co-integration ideas,
Art(z + 1) depends upon spreads, and this shows quite clearly that the
characteristics of F,(z — 1) — F/_i(t) will be those of the spreads - see Table 2.
Consequently, at least for small z, constant volatility models with martingale
difference errors could not adequately describe the data. It is possible that
proportional volatility models might do so due to the dependence of their c/>T_i upon
Ft(z), as the latter is near integrated. To check this out we regressed
F,(2) — F,_i(3) against c/j2 and spt-\(3) for n = 9 and a variety of values for
the market price of risk X. For X = 0 the t ratio of the coefficient of spt-\ (3) was
-4.37, while for very large X it was -4.70. Adopting other values for X resulted in t
ratios between these extremes. Hence, the conditional mean for the forward rates
is far more complex than that found in HJM models. Moreover, the rank of the
covariance matrix of the errors e,T_i must reflect the number of factors in the
term structure, which appears to be two or three, so that the common assumption
of a single error to drive all forward spreads seems inaccurate.
A number of formal investigations have been made into the compatibility of
the HJM model with the data - Abken(1993) and Thurston(1994) fitted HJM
models to forward rate data by GMM whilst Amin and Morton(1994) used
options prices to recover implied volatilities whose evolution was compared to
those of the most popular variants of the HJM model. Abken and Thurston reach
conflicting conclusions - the latter favours a constant volatility formulation and
the former a proportional one, although his general conclusion was that all
models were rejected by the data. Consequently, it seems interesting to look at the
stylized facts regarding volatility and to compare them with model specifications.
Equation (28) is useful for this task. As it has been shown that there is a levels
effect in Art(k), in order to have constant volatility it would be necessary that

116
A. R. Pagan, A. D. Hall and V. Martin
there be some "co-levels" effect, analogous to the co-persistence phenomenon of
the GARCH literature - Bollerslev and Engle (1993) - i.e. even though Ar,(k)
displays a levels effect the linear combination ^Art(x + 1) — \Art(l) does not.
This contention is easily rejected - a plot of that variable squared against r(_i(3)
looks almost identical to Figure 1, and such an observation points to the
proportional volatility model as being the appropriate one.
4. Conclusion
This chapter has described methods of modeling the term structure that are to be
found in the econometrics and finance literatures. By utilizing a factor
representation we have been able to show that there are many similarities in the two
approaches. However, there were also some differences. Within the econometrics
literature it is common to assume that yields are integrated processes and that
spreads constitute the co-integrating relations. Although the finance literature
takes the stance that yields are near integrated but stationary, it emerges that the
models used in that literature would not predict that the spreads are
co-integrating errors if we actually replaced the stationarity assumption by one of a
unit root. The reason for this outcome is found to lie in the assumption that the
conditional volatility of yields is a function of the level of the yields. Empirical
work tends to support such an hypothesis and we suggest that the consequences
of such a relationship can be profound for testing propositions about the term
structure. We also document a number of stylized facts about a set of data on
yields that prove useful in assessing the likely adequacy of many of the models
that are used in finance for capturing the term structure
References
Abken, P. A. (1993). Generalized method of moments tests of forward rate processes. Working Paper,
93-7. Federal Reserve Bank of Atlanta.
Amin, K. I. and A. J. Morton (1994). Implied volatility functions in arbitrage-free term structure
models. /. Financ. Econom. 35, 141-180.
Anderson, H. M. (1994). Transaction costs and nonlinear adjustment towards equilibrium in the US
treasury bill market. Mimeo, University of Texas at Austin.
Baillie, R. T. and T. Bollerslev (1992). Prediction in dynamic models with time-dependent conditional
variances, /. Econometrics 52, 91-113.
Baillie, R. T., T. Bollerslev and H. O. Mikkelson (1993). Fractionally integrated autoregressive
conditional heteroskedasticity. Mimeo, Michigan State University.
Bollerslev T. and R. F. Engle (1993). Common persistence in conditional variances: Definition and
representation. Econometrica 61, 167-186.
Boudoukh, J. (1993). An equilibrium model of nominal bond prices with inflation-output correlation
and stochastic volatility. /. Money, Credit and Banking 25, 636-665.
Brennan M. J. and E. S. Schwartz (1979). A continuous time approach to the pricing of bonds.
J. Banking Finance 3, 133-155.
Brenner R. J., R. H. Harjes and K. F. Kroner (1994). Another look at alternative models of the short-
term interest rate. Mimeo, University of Arizona.

Modeling the term structure
117
Brown, S. J. and P. H. Dybvig (1986). The empirical implications of the Cox-Ingersoll-Ross theory of
the term structure of intestest rates. J. Finance XLI, 617-632.
Brown, R. H. and S. M. Schaefer (1994). The term structure of real interest rates and the Cox,
Ingersoll and Ross model. J. Financ. Econom. 35, 3-42.
Broze, L. O. Scaillet and J. M. Zakoian (1993). Testing for continuous-time models of the short-term
interest rates. CORE Discussion Paper 9331.
Campbell, J. Y. and R. J. Shiller (1991). Yield spreads and interest rate movements: A bird's eye view.
Rev. Econom. Stud. 58, 495-514.
Canova F. and J. Marrinan (1993). Reconciling the term structure of interest rates with the
consumption based ICAP model. Mimeo, Brown University.
Chan K. C, G. A. Karolyi, F. A. Longstaff and A. B. Sanders (1992). An empirical comparison of
alternative models of the short-term interest rate. J. Finance XLVII. 1209-1227.
Chen R. R. and L. Scott (1992). Pricing interest rate options in a two factor Cox-Ingersoll-Ross model
of the term structure. Rev. Financ. Stud. 5, 613-636.
Chen R. R. and L. Scott (1993). Maximum likelihood estimation for a multifactor equilibrium model
of the term structure of interest rates. J. Fixed Income 3, 14-31.
Conley T., L. P. Hansen, E. Luttmer and J. Scheinkman (1994). Estimating subordinated diffusions
from discrete time data. Mimeo, University of Chicago.
Constantinides, G. (1992). A theory of the nominal structure of interest rates. Rev. Financ. Stud. 5,
531-552.
Cox, J. C, J. E. Ingersoll and S. A. Ross. (1985). A theory of the term structure of interest rates.
Econometrica 53, 385-408.
DufBe, D. and R. Kan (1993). A yield-factor model of interest rates. Mimeo, Graduate School of
Business, Stanford University.
Dybvig, P. H. (1989). Bonds and bond option pricing based on the current term structure. Working
Paper, Washington University in St. Louis.
Edmister, R. O. and D. B. Madan (1993). Informational content in interest rate term structures. Rev.
Econom. Statist. 75, 695-699.
Egginton, D. M. and S. G. Hall (1993). An investigation of the effect of funding on the slope of the
yield curve. Working Paper No. 6, Bank of England.
El Karoui, N. and V. Lacoste, (1992). Multifactor models of the term structure of interest rates.
Working Paper. University of Paris VI.
Engsted, T. and C. Tanggaard (1994). Cointegration and the US term structure. J. Banking Finance 18,
167-181.
Evans, M. D. D. and K. L. Lewis (1994). Do stationary risk premia explain it all? Evidence from the
term structure. J. Monetary Econom. 33, 285-318.
Frydman, H. (1994). Asymptotic inference for the parameters of a discrete-time square-root process.
Math. Finance 4, 169-181.
Gallant, A. R. and G. Tauchen (1992). Which moments to match? Mimeo, Duke University.
Gallant, A. R., D. Hsieh and G. Tauchen (1994). Estimation of stochastic volatility models with
diagnostics. Mimeo, Duke University.
Gonzalo, J. and C. W. J. Granger, (1991). Estimation of common long-memory components in
cointegrated systems. UCSD, Discussion Paper 91-33.
Gourieroux, C, A. Monfort and E. Renault (1993). Indirect inference. J. Appl. Econometrics 8, S85-
Sl 18.
Gourieroux, C. and O. Scaillet (1994). Estimation of the term structure from bond data. Working
Paper No. 9415 CEPREMAP.
Hall, A. D., H. M. Anderson and C. W. J. Granger. (1992). A cointegration analysis of treasury bill
yields. Rev. Econom. Statist. 74, 116-126.
Heath, D., R. Jarrow and A. Morton (1992). Bond pricing and the term structure of interest rates: A
new methodology for contingent claims valuation. Econometrica 60, 77-105.
Hejazi, W. 1994. Are term premia stationary? Mimeo, University of Toronto.

118
A. R. Pagan, A. D. Hall and V. Martin
Ho, T. S. and S-B Lee (1986). Term structure movements and pricing interest rate contingent claims. J.
Finance 41, 1011-1029.
Johansen, S. (1988). Statistical analysis of cointegrating vectors. J. Econom. Dynamic Control 12, 231—
254.
Johnson, P. A. (1994). On the number of common unit roots in the term structure of interest rates.
Appl. Econom. 26, 815-820.
Kearns, P. (1993). Volatility and the pricing of interest rate derivative claims. Unpublished doctoral
dissertation, University of Rochester.
Koedijk, K. G., F. G. J. A. Nissen, P. C. Schotman and C. C. P. Wolff (1993). The dynamics of short-
term interest rate volatility reconsidered. Mimeo, Limburg Institute of Financial Economics.
Litterman, R and J. Scheinkman (1991). Common factors affecting bond returns. J. Fixed Income 1,
54-61.
Longstaff, F. and E. S. Schwartz (1992). Interest rate volatility and the term structure: A two factor
general equilibrium model. J. Finance XLVII 1259-1282.
Marsh, T. A. and E. R. Rosenfeld (1983). Stochastic processes for interest rates and equilibrium bond
prices. J. Finance XXXVIII, 635-650.
Mihlstein, G. N. (1974). Approximate integration of stochastic differential equations. Theory Probab.
Appl. 19, 557-562.
McCuUoch, J. H. (1989). US term structure data. 1946-1987, Handbook of Monetary Economics 1,
672-715.
McCuUoch, J. H. and H. C. Kwon (1993). US term structure data. 1947-1991. Ohio State University
Working Paper 93-6.
Pearson, N. D. and T-S Sun (1994). Exploiting the conditional density in estimating the term structure:
An application to the Cox, Ingersoll and Ross model. J. Fixed Income XLIX, 1279-1304.
Pfann, G. A., P. C. Schotman and R. Tschernig (1994). Nonlinear interest rate dynamics and
implications for the term structure. Mimeo, University of Limburg.
Phillips, P. C. B. and B. E. Hansen (1990). Statistical inference in instrumental variables regression
with 1(1) processes. Rev. Econom. Stud. 57, 99-125.
Shen, P. and R. M. Starr (1992). Liquidity of the treasury bill market and the term structure of interest
rates. Discussion paper 92-32. University of California at San Diego.
Stock, J. H. and M. W. Watson (1988). Testing for common trends. J. Amer. Statist. Assoc. 83, 1097-
1107.
Thurston, D. C. (1994). A generalized method of moments comparison of discrete Heath-Jarrow-
Morton interest rate models. Asia Pac. J. Mgmt. 11, 1-19.
Vetzal, K. R. (1992). The impact of stochastic volatility on bond option prices. Working Paper 92-08.
University of Waterloo. Institute of Insurance and Pension Research, Waterloo, Ontario.
Zhang, Z. (1993). Treasury yield curves and cointegration. Appl. Econom. 25, 361-367.

G. S. Maddala, and C. R. Rao, eds., Handbook of Statistics, Vol. 14
© 1996 Elsevier Science B.V. All rights reserved.
5
Stochastic Volatility*
Eric Ghysels, Andrew C. Harvey and Eric Renault
1. Introduction
The class of stochastic volatility (SV) models has its roots both in mathematical
finance and financial econometrics. In fact, several variations of SV models
originated from research looking at very different issues. Clark (1973), for instance,
suggested to model asset returns as a function of a random process of information
arrival. This so-called time deformation approach yielded a time-varying
volatility model of asset returns. Later Tauchen and Pitts (1983) refined this work
proposing a mixture of distributions model of asset returns with temporal
dependence in information arrivals. Hull and White (1987) were not directly
concerned with linking asset returns to information arrival but rather were interested
in pricing European options assuming continuous time SV models for the
underlying asset. They suggested a diffusion for asset prices with volatility following
a positive diffusion process. Yet another approach emerged from the work of
Taylor (1986) who formulated a discrete time SV model as an alternative to
Autoregressive Conditional Heteroskedasticity (ARCH) models. Until recently
estimating Taylor's model, or any other SV model, remained almost infeasible.
Recent advances in econometric theory have made estimation of SV models much
easier. As a result, they have become an attractive class of models and an
alternative to other classes such as ARCH.
Contributions to the literature on SV models can be found both in
mathematical finance and econometrics. Hence, we face quite a diverse set of topics. We
say very little about ARCH models because several excellent surveys on the
subject have appeared recently, including those by Bera and Higgins (1995),
Bollerslev, Chou and Kroner (1992), Bollerslev, Engle and Nelson (1994) and
* We benefited from helpful comments from Torben Andersen, David Bates, Frank Diebold, Rene
Garcia, Eric Jacquier and Neil Shephard on preliminary drafts of the paper. The first author would
like to acknowledge the financial support of FCAR (Quebec), SSHRC (Canada) as well as the
hospitality and support of CORE (Louvain-la-Neuve, Belgium). The second author wishes to thank the
ESRC for financial support. The third author would like to thank the Institut Universitaire de France,
the Federation Francaise des Societes d'Assurance as well as CIRANO and C.R.D.E. for financial
support.
119

120
E. Ghysels, A. C. Harvey and E. Renault
Diebold and Lopez (1995). Furthermore, since this chapter is written for the
Handbook of Statistics, we keep the coverage of the mathematical finance
literature to a minimum. Nevertheless, the subject of option pricing figures
prominently out of necessity. Indeed, Section 2, which deals with definitions of
volatility has extensive coverage of Black-Scholes implied volatilities. It also
summarizes empirical stylized facts and concludes with statistical modeling of
volatility. The reader with a greater interest in statistical concepts may want to
skip the first three subsections of Section 2 which are more finance oriented and
start with Section 2.4. Section 3 discusses discrete time models, while Section 4
reviews continuous time models. Statistical inference of SV models is the subject
of Section 5. Section 6 concludes.
2. Volatility in financial markets
Volatility plays a central role in the pricing of derivative securities. The Black-
Scholes model for the pricing of an European option is by far the most widely used
formula even when the underlying assumptions are known to be violated. Section
2.1 will therefore take the Black-Scholes model as a reference point from which to
discuss several notions of volatility. A discussion of stylized facts regarding
volatility and option prices will appear next in Section 2.2. Both sections set the scene
for a formal framework defining stochastic volatility which is treated in Section
2.3. Finally, Section 2.4 introduces the statistical models of stochastic volatility.
2.1. The Black-Scholes model and implied volatilities
More than half a century after the seminal work of Louis Bachelier (1900),
continuous time stochastic processes have become a standard tool to describe the
behavior of asset prices. The work of Black and Scholes (1973) and Merton (1990)
has been extremely influential in that regard. In Section 2.1.1 we review some of
the assumptions that are made when modeling asset prices by diffusions, in
particular to present the concept of instantaneous volatility. In Section 2.1.2 we
turn to option pricing models and the various concepts of implied volatility.
2.1.1. An instantaneous volatility concept
We consider a financial asset, say a stock, with today's (time t) market price
denoted by St.2 Let the information available at time t be described by I, and
consider the conditional distribution of the return St+h/St of holding the asset
over the period [t, t + h] given lt? A maintained assumption throughout this
chapter will be that asset returns have finite conditional expectation given I, or:
2 Here and in the remainder of the paper we will focus on options written on stocks or exchange
rates. The large literature on the term structure of interest rates and related derivative securities will
not be covered.
3 Section 2.3 will provide a more rigorous discussion of information sets. It should also be noted
that we will indifferently be using conditional distributions of asset prices St+h and of returns St+h/St
since S, belongs to /,.

Stochastic volatility
121
Et(St+h/St) = S^EtSt+h < +00 (2.1.1)
and likewise finite conditional variance given It, namely
vt{st+h/st) = s;2vtst+h < +00 . (2.1.2)
The continuously compounded expected rate of return will be characterized by
h~x \ogEt(St+f,/St). Then a first assumption can be stated as follows:
Assumption 2.1.1.A. The continuously compounded expected rate of return
converges almost surely towards a finite value fis(It) when h > 0 goes to zero.
From this assumption one has EtSt+h - S, ~ hfis{It)St or in terms of its differential
representation:
sE<<«
fis(It)St almost surely (2.1.3)
z=t
where the derivatives are taken from the right. Equation (2.1.3) is sometimes
loosely defined as: E,(dSt) = fis(I,)Stdt. The next assumption pertains to the
conditional variance and can be stated as:
Assumption 2.1.l.B. The conditional variance of the return h~xVt(St+h/St)
converges almost surely towards a finite value o2s(It) when h > 0 goes to zero.
Again, in terms of its differential representation this amounts to:
—Var,(ST) = a](I,)Sf almost surely (2.1.4)
®x t.=t
and one loosely associates with the expression Vt(dSt) = a^(It)S2dt.
Both assumptions 2.1.1.A and B lead to a representation of the asset price
dynamics by an equation of the following form:
dSt = ns(I,)Stdt + as(I,)StdW, (2.1.5)
where Wt is a standard Brownian Motion. Hence, every time a diffusion equation
is written for an asset price process we have automatically defined the so-called
instantaneous volatility process as (It) which from the above representation can
also be written as:
°si!t)
\imh-lVt(St+h/St)
h[o
1/2
(2.1.6)
Before turning to the next section we would like to provide a brief discussion of
some of the foundations for the Assumptions 2.1.1.A and B. It was noted that
Bachelier (1900) proposed Brownian Motion process as a model of stock price
movements. In modern terminology this amounts to the random walk theory of
asset pricing which claims that asset returns ought not to be predictable because
of the informational efficiency of financial markets. Hence, it assumes that returns

122
E. Ghysels, A. C. Harvey and E. Renault
on consecutive regularly sampled periods [t + k, t + k + 1], k = 0,2,..., h - 1 are
independently (identically) distributed. With such a benchmark in mind, it is
natural to view the expectation and the variance of the continuously
compounded rate of return log (St+h/St) as proportional to the maturity h of the
investment.
Obviously we no longer use Brownian Motions as a process for asset prices
but it is nevertheless worth noting that Assumptions 2.1.1. A and B also imply that
the expected rate of return and the associated squared risk (in terms of variance of
the rate of return) of an investment over an infinitely-short interval [t, t + h] is
proportional to h. Sims (1984) provided some rationale for both assumptions
through the concept of "local unpredictability".
To conclude, let us briefly discuss a particular special case of (2.1.5)
predominantly used in theoretical developments and also highlight an implicit
restriction we made. When ps(It) = ns and os(It) = as are constants for all t the
asset price is a Geometric Brownian Motion. This process was used by Black and
Scholes (1973) to derive their well-known pricing formula for European options.
Obviously, since os(It) is a constant we no longer have an instantaneous volatility
process but rather a single parameter as - a situation which undoubtedly greatly
simplifies many things including the pricing of options. A second point which
needs to be stressed is that Assumptions 2.1.1.A and B allow for the possibility of
discrete jumps in the asset price process. Such jumps are typically represented by a
Poisson process and have been prominent in the option pricing literature since the
work of Merton (1976). Yet, while the assumptions allow in principle for jumps,
they do not appear in (2.1.5). Indeed, throughout this chapter we will maintain
the assumption of sample path continuity and exclude the possibility of jumps as
we focus exclusively on SV models.
2.1.2. Option prices and implied volatilities
It was noted in the introduction that SV models originated in part from the
literature on the pricing of options. We have witnessed over the past two
decades a spectacular growth in options and other derivative security markets.
Such markets are sometimes characterized as places where "volatilities are
traded". In this section we will provide the rationale for such statements and
study the relationship between so-called options implied volatilities and the
concepts of instantaneous and averaged volatilities of the underlying asset
return process.
The Black-Scholes option pricing model is based on a Log-Normal or
Geometric Brownian Motion model for the underlying asset price:
dSt = UsStdt + asStdWt (2.1.7)
where ns and as are fixed parameters. A European call option with strike price K
and maturity t + h has a payoff:

Stochastic volatility
123
Since the seminal Black and Scholes (1973) paper, there is now a well
established literature proposing various ways to derive the pricing formula of such a
contract. Obviously, it is beyond the scope of this paper to cover this literature in
detail.4 Instead, the bare minimum will be presented here allowing us to discuss
the concepts of interest regarding volatility.
With continuous costless trading assumed to be feasible, it is possible to form
in the Black-Scholes economy a portfolio using one call and a short-sale strategy
for the underlying stock to eliminate all risk. This is why the option price can be
characterized without ambiguity, using only arbitrage arguments, by equating the
market rate of return of the riskless portfolio containing the call option with the
risk-free rate. Moreover, such arbitrage-based option pricing does not depend on
individual preferences.5
This is the reason why the easiest way to derive the Black-Scholes option
pricing formula is via a "risk-neutral world", where asset price processes are
specified through a modified probability measure, referred to as the risk neutral
probability measure denoted Q (as discussed more explicitly in Section 4.2). This
fictitious world where probabilities in general do not coincide with the Data
Generating Process (DGP), is only used to derive the option price which remains
valid in the objective probability setup. In the risk neutral world we have:
dSt/St = rtdt + asdWt (2.1.9)
Ct = C(St,K,h,t)=B(t,t + h)E?(St+h-K)+ (2.1.10)
where Ep is the expectation under Q, B(t, t + h) is the price at time t of a pure
discount bond with payoff one unit at time t + h and
rt = -limTLogB(t,t + h) (2.1.11)
is the riskless instantaneous interest rate.6 We have implicitly assumed that in this
market interest rates are nonstochastic (Wt is the only source of risk) so that:
ft+h
B(t, t + h)— exp
/t+h
rTd%
(2.1.12)
By definition, there are no risk premia in a risk neutral context. Therefore rt
coincides with the instantaneous expected rate of return of the stock and hence
4 See however Jarrow and Rudd (1983), Cox and Rubinstein (1985), Duffle (1989), Duffle (1992),
Hull (1993) or Hull (1995) among others for more elaborate coverage of options and other derivative
securities.
5 This is sometimes refered to as preference free option pricing. This terminology may somewhat be
misleading since individual preferences are implicitly taken into account in the market price of the
stock and of the riskless bond. However, the option price only depends on individual preferences
through the stock and bond market prices.
6 For notational convenience we denote by the same symbol W, a Brownian Motion under P (in
2.1.7) and under Q (in 2.1.9). Indeed, Girsanov's theorem establishes the link between these two
processes (see e.g. Duffle (1992) and section 4.2.1).

124
E. Ghysels, A. C. Harvey and E. Renault
the call option price Ct is the discounted value of its terminal payoff (St+h - K)+
as stated in (2.1.10).
The log-normality of St+h given St allows one to compute the expectation in
(2.1.10) yielding the call price formula at time t:
C, = St<t>{dt) - KB{t, t + h)4>(dt - osVh) (2.1.13)
where <f> is the cumulative standard normal distribution function while dt will be
defined shortly. Formula (2.1.13) is the so-called Black-Scholes option pricing
formula. Thus, the option price Ct depends on the stock price St, the strike price K
and the discount factor B(t, t + h). Let us now define:
x, = Log St/KB(t,t + h) . (2.1.14)
Then we have:
Q/St = <t>{dt) - e-x><t>{dt - as\fh) (2.1.15)
with dt = (xt/os\fJi) + 0SVh/2. It is easy to see the critical role played by the
quantity xt, called the moneyness of the option.
- If xt — 0, the current stock price St coincides with the present value of the strike
price K. In other words, the contract may appear to be fair to somebody who
would not take into account the stochastic changes of the stock price between t
and t + h. We shall say that we have in this case an at the money option.
- If xt > 0 (respectively xt < 0) we shall say that the option is in the money
(respectively out the money).7
It was noted before that the Black-Scholes formula is widely used among
practitioners, even when its assumptions are known to be violated. In particular
the assumption of a constant volatility as is unrealistic (see Section 2.2 for
empirical evidence). This motivated Hull and White (1987) to introduce an option
pricing model with stochastic volatility assuming that the volatility itself is a state
variable independent of Wt:&
dS,/St = rtdt + aStdWt (2 116}
{°st)t<$>,T\i{wt)t<z$,T\ independent Markovian . \- ■ )
It should be noted that (2.1.16) is still written in a risk neutral context since rt
coincides with the instantaneous expected return of the stock. On the other hand
the exogenous volatility risk is not directly traded, which prevents us from de-
7 We use here a slightly modified terminology with respect to the usual one. Indeed, it is more
common to call at the money /in the money/ out of the money options, when St = KjSt > K/St < K
respectively. From an economic point of view, it is more appealing to compare St with the present
value of the strike price K.
8 Other stochastic volatility models similar to Hull and White (1987) appear in Johnson and
Shanno (1987), Scott (1987), Wiggins (1987), Chesney and Scott (1989), Stein and Stein (1991) and
Heston (1993) among others.

Stochastic volatility
125
fining unambiguously a risk neutral probability measure, as discussed in more
detail in Section 4.2. Nevertheless, the option pricing formula (2.1.10) remains
valid provided the expectation is computed with respect to the joint probability
distribution of the Markovian process (S, as), given (St, ast)-9 We can then rewrite
(2.1.10) as follows:
Ct = B(t,t + h)Et(St+h-K)+
= B(t,t + h)Et{E[(St+h -K) + \KU<,+J}
where the expectation inside the brackets is taken with respect to the conditional
probability distribution of St+h given It and a volatility path oSt, t <% <t + h.
However, since the volatility process osz is independent of Wt, we obtain using
(2.1.15)
B(t,t + h)Et[{St+h -K)+\{aST)t^t+h] i
= StEt[4>(dlt) - e-*>4>(d2t)}
Here d\t and d2t are defined as follows:
d\t = (xt/y(t, t + h)Vh) + y(t, t + h)Vh/2
dn = di -y(t,t + h)Vh
where y(t,t + h) > 0 and:
1 ft+h
y2(t,t + h)=-Jt a2Sxdx . (2.1.19)
This yields the so-called Hull and White option pricing formula:
Ct = StEt[4>(dlt)-e-x>4>(d2t)} , (2.1.20)
where the expectation is taken with respect to the conditional probability
distribution (for the risk neutral probability measure) of y(t, t + h) given oSt-w
In the remainder of this section we will assume that observed option prices
obey Hull and White's formula (2.1.20). Then option prices would yield two types
of implied volatility concepts: (1) an instantaneous implied volatility, and (2) an
averaged implied volatility. To make this more precise, let us assume that the risk
neutral probability distribution belongs to a parametric family, Pg, 6 e ©. Then,
the Hull and White option pricing formula yields an expression for the option
price as a function:
Ct = StF[aSt,xt,eo] (2.1.21)
9 We implicitly assume here that the available information /, contains the past values (S,z,<rz)t<r
This assumption will be discussed in Section 4.2.
10 The conditioning is with respect to a, since it summarizes the relevant information taken from /,
(the process <r is assumed to be Markovian and independent of W).

126
E. Ghysels, A. C. Harvey and E. Renault
where 0o is the true unknown value of the parameters. Formula (2.1.21) reveals
why it is often claimed that "option markets can be thought of as markets trading
volatility" (see e.g. Stein (1989)). As a matter of fact, if for any given (xh6),
F(-,xt, 6) is one-to-one, then equation (2.1.21) can be inverted to yield an implied
instantaneous volatility:11
oi™v(6) = G[St,Ct,xt,6} . (2.1.22)
Bajeux and Rochet (1992), by showing that this one-to-one relationship
between option prices and instantaneous volatility holds, in fact formalize the use of
option markets as an appropriate instrument to hedge volatility risk. Obviously
implied instantaneous volatilities (2.1.22) could only be useful in practice for
pricing or hedging derivative instruments when we know the true unknown value
0o or, at least, are able to compute a sufficiently accurate estimate of it.
However, the difficulties involved in estimating SV models has for long
prevented their widespread use in empirical applications. This is the reason why
practitioners often prefer another concept of implied volatility, namely the so-
called Black-Scholes implied volatility introduced by Latane and Rendleman
(1976). It is a process w[mp(t,t + h) defined by:
' Ct = St[4>(dlt) - e-x'4>(d2t)}
< dit=(xt/(Qimv{t,t + h)y/h)+a)imv(t,t + h)^/2 (2.1.23)
Jit = dit- aF*(t,t + h)Vh
where Ct is the observed option price.12
The Hull and White option pricing model can indeed be seen as a theoretical
foundation for this practice; the comparison between (2.1.23) and (2.1.20) allows
us to interpret the Black-Scholes implied volatility a>imp(f, t + h) as an implied
averaged volatility since a>imp(f, t + h) is something like a conditional expectation
of y(t, t + h) (assuming observed option prices coincide with the Hull and White
pricing formula). To be more precise, let us consider the simplest case of at the
money options (the general case will be studied in Section 4.2). Since xt — 0 it
follows that du = —d\t and therefore: <j){d\t) — e~x'(f>(d2t) = 2(j>(d\t) — 1. Hence,
a>™p(f, t + h) (the index o is added to make explict that we consider at the money
options) is defined by:
0{&l+3^\ = ^fe'+»)^ . (2,.24)
Since the cumulative standard normal distribution function is roughly linear in
the neighborhood of zero, if follows that (for small maturities h):
11 The fact that F(-,x,,d) is one-to-one is shown to be the case for any diffusion model on trst under
certain regularity conditions, see Bajeux and Rochet (1992).
12 We do not explicitly study here the dependence between colmp(f, (+ h) and the various related
processes: C,, St, xt. This is the reason why, for sake of simplicity, this dependence is not apparent in
the notation <uimp(f, t + h).

Stochastic volatility
127
(J™V(t,t + h)*Ety(t,t + h) .
This yields an interpretation of the Black-Scholes implied volatility
co™p(f, t + h) as an implied average volatility:
rt+h _ -] 1/2
rdx
1 /"+
a^r(t,t + h)^Et- a2Szdx . (2.1.25)
2.2. Some stylized facts
The search for model specification and selection is always guided by empirical
stylized facts. A model's ability to reproduce such stylized facts is a desirable
feature and failure to do so is most often a criterion to dismiss a specification,
although one typically does not try to fit or explain all possible empirical
regularities at once with a single model. Stylized facts about volatility have been well
documented in the ARCH literature, see for instance Bollerslev, Engle and
Nelson (1994). Empirical regularities regarding derivative securities and implied
volatilities are also well covered, for instance, by Bates (1995a). In this section we
will summarize empirical stylized facts, complementing and updating some of the
material covered in the aforementioned references.
(a) Thick tails
Since the early sixties it was observed, notably by Mandelbrot (1963), Fama
(1963, 1965), among others that asset returns have leptokurtic distributions. As a
result, numerous papers have proposed to model asset returns as i.i.d. draws from
fat-tailed distributions such as Paretian or Levy.
(b) Volatility clustering
Any casual observations of financial time series reveal bunching of high and low
volatility episodes. In fact, volatility clustering and thick tails of asset returns are
intimately related. Indeed, the latter is a static explanation whereas a key insight
provided by ARCH models is a formal link between dynamic (conditional)
volatility behavior and (unconditional) heavy tails. ARCH models, introduced by
Engle (1982) and the numerous extensions thereafter, as well as SV models are
essentially built to mimic volatility clustering. It is also widely documented that
ARCH effects disappear with temporal aggregation, see e.g. Diebold (1988) and
Drost and Nijman (1993).
(c) Leverage effects
A phenomenon coined by Black (1976) as the leverage effect suggests that stock
price movements are negatively correlated with volatility. Because falling stock
prices imply an increased leverage of firms it is believed that this entails more
uncertainty and hence volatility. Empirical evidence reported by Black (1976),
Christie (1982) and Schwert (1989) suggests, however, that leverage alone is too

128
E. Ghysels, A. C. Harvey and E. Renault
small to explain the empirical asymmetries one observes in stock prices. Others
reporting empirical evidence regarding leverage effects include Nelson (1991),
Gallant, Rossi and Tauchen (1992, 1993), Campbell and Kyle (1993) and Engle
and Ng (1993).
(d) Information arrivals
Asset returns are typically measured and modeled with observations sampled at
fixed frequencies such as daily, weekly or monthly observations. Several authors,
including Mandelbrot and Taylor (1967) and Clark (1973) suggested linking asset
returns explicitly to the flow of information arrival. In fact it was already noted
that Clark proposed one of the early examples of SV models. Information arrival
is non-uniform through time and quite often not directly observable.
Conceptually, one can think of asset price movements as the realization of a process
Yt = Yz, where Zt is a so-called directing process. This positive nondecreasing
stochastic process Zt can be thought of as being related to the arrival of
information. This idea of time deformation or subordinated stochastic processes
was used by Mandelbrot and Taylor (1967) to explain fat tailed returns, by Clark
(1973) to explain volatility and was recently refined and further explored by
Ghysels, Gourieroux and Jasiak (1995a). Moreover, Easley and O'Hara (1992)
provide a microstructure model involving time deformation. In practice, it
suggests a direct link between market volatility and (1) trading volume, (2) quote
arrivals, (3) forecastable events such as dividend announcements or macro-
economic data releases, (4) market closures, among many other phenomena
linked to information arrival.
Regarding trading volume and volatility there are several papers documenting
stylized facts notably linking high trading volume with market volatility, see for
example Karpoff (1987) or Gallant, Rossi and Tauchen (1992).13 The intraday
patterns of volatility and market activity measured for instance by quote arrivals
are also well-known and documented. Wood, Mclnish and Ord (1985) and Harris
(1986) studied this phenomenon for securities markets and found a U-shaped
pattern with volatility typically high at the open and close of the market. The
around the clock trading in foreign exchange markets also yields a distinct
volatility pattern which is tied with the intensity of market activity and produces
strong seasonal patterns. The intraday patterns for FX markets are analyzed for
instance by Muller et al. (1990), Baillie and Bollerslev (1991), Harvey and Huang
(1991), Dacorogna et al. (1993), Bollerslev and Ghysels (1994), Andersen and
Bollerslev (1995), Ghysels, Gourieroux and Jasiak (1995b) among others.
Another related empirical stylized fact is that of overnight and weekend market
closures and their effect on volatility. Fama (1965) and French and Roll (1986)
have found that information accumulates more slowly when the NYSE and
AMEX are closed resulting in higher volatility on those markets after weekends
13 There are numerous models, theoretical and empirical, linking trading volume and asset returns
which we cannot discuss in detail. A partial list includes Foster and Viswanathan (1993a,b), Ghysels
and Jasiak (1994a,b), Hausman and Lo (1991), Huffman (1987), Lamoureux and Lastrapes (1990,
1993), Wang (1993) and Andersen (1995).

Stochastic volatility
129
and holidays. Similar evidence for FX markets has been reported by Baillie and
Bollerslev (1989). Finally, numerous papers documented increased volatility of
financial markets around dividend announcements (Cornell (1978), Patell and
Wolfson (1979,1981)) and macroeconomic data releases (Harvey and Huang
(1991, 1992), Ederington and Lee (1993)).
(e) Long memory and persistence
Generally speaking volatility is highly persistent. Particularly for high frequency
data one finds evidence of near unit root behavior of the conditional variance
process. In the ARCH literature numerous estimates of GARCH models for
stock market, commodities, foreign exchange and other asset price series are
consistent with an IGARCH specification. Likewise, estimation of stochastic
volatility models show similar patterns of persistence (see for instance Jacquier,
Poison and Rossi (1994)). These findings have led to a debate regarding modeling
persistence in the conditional variance process either via a unit root or a long
memory process. The latter approach has been suggested both for ARCH and SV
models, see Baillie, Bollerslev and Mikkelsen (1993), Breidt et al. (1993), Harvey
(1993) and Comte and Renault (1995). Ding, Granger and Engle (1993) studied
the serial correlations of \r(t, t + l)|c for positive values of c where r(t, t + 1) is a
one-period return on a speculative asset. They found \r(t,t+ \)\c to have quite
high autocorrelations for long lags while the strongest temporal dependence was
for c close to one. This result initially found for daily S&P500 return series was
also shown to hold for other stock market indices, commodity markets and
foreign exchange series (see Granger and Ding (1994)).
(/) Volatility comovements
There is an extensive literature on international comovements of speculative
markets. Concerns on whether globalization of equity markets increases price
volatility and correlations of stock returns has been the subject of many recent
studies including, von Fustenberg and Jean (1989), Hamao, Masulis and Ng
(1990), King, Sentana and Wadhwani (1994), Harvey, Ruiz and Sentana (1992),
and Lin, Engle and Ito (1994). Typically one uses factor models to model the
commonality of international volatility, as in Diebold and Nerlove (1989),
Harvey, Ruiz and Sentana (1992), Harvey, Ruiz and Shephard (1994) or explores so-
called common features, see e.g. Engle and Kozicki (1993) and common trends as
studied by Bollerslev and Engle (1993).
(g) Implied volatility correlations
Stylized facts are typically reported as model-free empirical observations.14
Implied volatilities are obviously model-based as they are calculated from a pricing
14 This is in some part fictitious even for macroeconomic data for instance when they are de-
trended or seasonally adjusted. Both detrending and seasonal adjustment are model-based. For the
potentially severe impact of detrending on stylized facts see Canova (1992) and Harvey and Jaeger
(1993) and for the effect of seasonal adjustment on empirical regularities see Ghysels et al. (1993).

130
E. Ghysels, A. C. Harvey and E. Renault
equation of a specific model, namely the Black and Scholes model as noted in
Section 2.1.3. Since they are computed on a daily basis there is obviously an
internal inconsistency since the model presumes constant volatility. Yet, since
many option prices are in fact quoted through their implied volatilities it is
natural to study the time series behavior of the latter. Often one computes a
composite measure since synchronous option prices with different strike prices
and maturities for the same underlying asset yield different implied volatilities.
The composite measure is usually obtained from a weighting scheme putting more
weight on the near-the-money options which are the most heavily traded in
organized markets.15
The time series properties of implied volatilities obtained from stock, stock
index and currency options are quite similar. They appear stationary and are well
described by a first order autoregressive model (see Merville and Pieptea (1989)
and Sheikh (1993) for stock options, Poterba and Summers (1986), Stein (1989),
Harvey and Whaley (1992) and Diz and Finucane (1993) for the S&P100 contract
and Taylor and Xu (1994), Campa and Chang (1995) and Jorion (1995) for
currency options). It was noted from equation (2.1.25) that implied (average)
volatilities are expected to contain information regarding future volatility and
therefore should predict the latter. One typically tests such hypotheses by
regressing realized volatilities on past implied ones.
The empirical evidence regarding the predictable content of implied volatilities
is mixed. The time series study of Lamoureux and Lastrapes (1993) considered
options on non-dividend paying stocks and compared the forecasting
performance of GARCH, implied volatility and historical volatility estimates and found
that implied volatility forecasts, although biased as one would expect from
(2.1.25), outperform the others. In sharp contrast, Canina and Figlewski (1993)
studied S&P100 index call options for which there is an extremely active market.
They found that implied volatilities were virtually useless in forecasting future
realized volatilities of the S&P100 index. In a different setting using weekly
sampling intervals for S&P100 option contracts and a different sample Day and
Lewis (1992) not only found that implied volatilities had a predictive content but
also were unbiased. Studies examining options on foreign currencies, such as
Jorion (1995), also found that implied volatilities were predicting future
realizations and that GARCH as well as historical volatilities were not outperforming
the implied measures of volatility.
(h) The term structure of implied volatilities
The Black-Scholes model predicts a flat term structure of volatilities. In reality,
the term structure of at-the-money implied volatilities is typically upward sloping
when short term volatilities are low and the reverse when they are high (see
Stein(1989)). Taylor and Xu (1994) found that the term structure of implied
15 Different weighting schemes have been suggested, see for instance Latane and Rendleman
(1976), Chiras and Manaster (1978), Beckers (1981), Whaley (1982), Day and Lewis (1988), Engle and
Mustafa (1992) and Bates (1995b).

Stochastic volatility
131
volatilities from foreign currency options reverses slope every few months. Stein
(1989) also found that the actual sensitivity of medium to short term implied
volatilities was greater than the estimated sensitivity from the forecast term
structure and concluded that medium term implied volatilities overreacted to
information. Diz and Finucane (1993) used different estimation techniques and
rejected the overreaction hypothesis, and instead reported evidence suggesting
underreaction.
(i) Smiles
If option prices in the market were conformable with the Black-Scholes formula,
all the Black-Scholes implied volatilities corresponding to various options written
on the same asset would coincide with the volatility parameter a of the underlying
asset. In reality this is not the case, and the Black-Scholes implied volatility
w™p(f, t + h) denned by (2.1.23) heavily depends on the calendar time t, the time
to maturity h and the moneyness xt = Log St/KB(t, t + h) of the option. This may
produce various biases in option pricing or hedging when BS implied volatilities
are used to evaluate new options with different strike prices K and maturities h.
These price distortions, well-known to practitioners, are usually documented in
the empirical literature under the terminology of the smile effect, where the so-
called "smile" refers to the U-shaped pattern of implied volatilities across
different strike prices. More precisely, the following stylized facts are extensively
documented (see for instance Rubinstein (1985), Clewlow and Xu (1993), Taylor
and Xu (1993)):
- The U-shaped pattern of wimp (t,t + h) as a function of K (or \ogK) has its
minimum centered at near-the-money options (discounted K close to St, i.e. xt
close to zero).
- The volatility smile is often but not always symmetric as a function of log K (or
of x,). When the smile is asymmetric, the skewness effect can often be described
as the addition of a monotonic curve to the standard symmetric smile: if a
decreasing curve is added, implied volatilities tend to rise more for decreasing
than for increasing strike prices and the implied volatility curve has its
minimum out of the money. In the reverse case (addition of an increasing curve),
implied volatilities tend to rise more with increasing strike prices and their
minimum is in the money.
- The amplitude of the smile increases quickly when time to maturity decreases.
Indeed, for short maturities the smile effect is very pronounced (BS implied
volatilities for synchronous option prices may vary between 15% and 25%)
while it almost completely disappears for longer maturities.
It is widely believed that volatility smiles have to be explained by a model of
stochastic volatility. This is natural for several reasons: First, it is tempting to
propose a model of stochastically time varying volatility to account for
stochastically time varying BS implied volatilities. Moreover, the decreasing
amplitude of the smile being a function of time to maturity is conformable with a
formula like (2.1.25). Indeed, it shows that, when time to maturity is increased,

132
E. Ghysels, A. C. Harvey and E. Renault
temporal aggregation of volatilities erases conditional heteroskedasticity, which
decreases the smile phenomenon. Finally, the skewness itself may also be
attributed to the stochastic feature of the volatility process and overall to the
correlation of this process with the price process (the so-called leverage effect).
Indeed, this effect, while sensible for stock prices data, is small for interest rate
and exchange rate series which is why the skewness of the smile is more often
observed for options written on stocks.
Nevertheless, it is important to be cautious about tempting associations:
stochastic implied volatility and stochastic volatility; asymmetry in stocks and
skewness in the smile. As will be discussed in Section 4, such analogies are not
always rigorously proven. Moreover, other arguments to explain the smile and its
skewness (jumps, transaction costs, bid-ask spreads, non-synchronous trading,
liquidity problems, ...) have also to be taken into account both for theoretical
reasons and empirical ones. For instance, there exists empirical evidence
suggesting that the most expensive options (the upper parts of the smile curve) are
also the least liquid; skewness may therefore be attributed to specific
configurations of liquidity in option markets.
2.3. Information sets
So far we left the specification of information sets vague. This was done on
purpose to focus on one issue at the time. In this section we need to be more
formal regarding the definition of information since it will allow us to clarify
several missing links between the various SV models introduced in the literature
and also between SV and ARCH models. We know that SV models emerged from
research looking at a very diverse set of issues. In this section we will try to define
a common thread and a general unifying framework. We will accomplish this
through a careful analysis of information sets and associate with it notions of
non-causality in the Granger sense. These causality conditions will allow us to
characterize in Section 2.4 the distinct features of ARCH and SV models.16
2.3.1. State variables and information sets
The Hull and White (1987) model is a simple example of a derivative asset pricing
model where the stock price dynamics are governed by some unobservable state
variables, such as random volatility. More generally, it is convenient to assume
that a multivariate diffusion process Ut summarizes the relevant state variables in
the sense that:
' dSt/St = fitdt + atdWt
. dUt = ytdt + 5,dW}J (2.3.1)
Cov(dWt,dWY) =ptdt
16 The analysis in this section has some features in common with Andersen (1992) regarding the
use of information sets to clarify the difference between SV and ARCH type models.

Stochastic volatility
133
where the stochastic processes fit,at,yt,St and pt are if = [UT,z<t] adapted
(Assumption 2.3.1). This means that the process U summarizes the whole
dynamics of the stock price process S (which justifies the terminology "state"
variable) since, for a given sample path (Ut)0<T<T of state variables, consecutive
returns Stk+1/Stk,0 < t\ < t2 < ■■■ < h < T are stochastically independent and log-
normal (as in the benchmark BS model).
The arguments of Section 2.1.2 can be extended to the state variables
framework (see Garcia and Renault (1995)) discussed here. Indeed, such an extension
provides a theoretical justification for the common use of the Black and Scholes
model as a standard method of quoting option prices via their implied
volatilities.17 In fact, it is a way of introducing neglected heterogeneity in the BS
option pricing model (see Renault (1995) who draws attention to the similarities
with introducing heterogeneity in microeconometric models of labor markets,
etc.).
In continuous time models, available information at time t for traders (whose
information determines option prices) is characterized by continuous time
observations of both the state variable sample path and stock price process sample
path; namely:
I, = <t[Ut,Sx; z<t] . (2.3.2)
2.3.2. Discrete sampling and Granger noncausality
In the next section we will treat explicitly discrete time models. It will necessitate
formulating discrete time analogues of equation (2.3.1). The discrete sampling
and Granger noncausality conditions discussed here will bring us a step closer to
building a formal framework for statistical modeling using discrete time data.
Clearly, a discrete time analogue of equation (2.3.1) is:
log St+l/St = fi(Ut) + a{Ut)et+l (2.3.3)
provided we impose some restrictions on the process et. The restrictions we want
to impose must be flexible enough to accommodate phenomena such as leverage
effects for instance. A setup that does this is the following:
Assumption 2.3.2.A. The process et in (2.3.3) is i.i.d. and not Granger-caused by
the state variable process Ut.
Assumption 2.3.2.B. The process et in (2.3.3) does not Granger-cause Ut.
Assumption 2.3.2.B is useful for the practical use of BS implied volatilities as it
is the discrete time analogue of Assumption 2.3.1 where it is stated that the
coefficients of the process U are 1^ adapted (for further details see Garcia and
17 Garcia and Renault (1995) argued that Assumption 2.3.1 is essential to ensure the homogeneity
of option prices with respect to the pair (stock price, strike price) which in turn ensures that BS implied
volatilities do not depend on the stock price level but only on the moneyness S/K. This homogeneity
property was first emphasized by Merton (1973).

134
E. Ghysels, A. C. Harvey and E. Renault
Renault (1995)). Assumption 2.3.2.A is important for the statistical interpretation
of the functions n(Ut) and o(Ut) respectively as trend and volatility coefficients,
namely,
E[log St+l/St\{Sx/Sx^t<t)]
= E[E[log St+l/St\(UT,eT;x < t)]\{Sx/Si-i;x < t)] (2.3.4)
= E\M(Ut)\(St/St-i;x<t)]
since E[£,+i | (£/t,£t;t < t)] = E[£,+i | et;z < t] = 0 due to the Granger non-
causality from Ut to et of Assumption 2.3.2.A. Likewise, one can easily show that
Var[log St+l/St - n(Ut)\(S,/^l;x < t)] ^
= E[a?(Ul)\(S,/S,-l;x<t)] •
Implicitly we have introduced a new information set in (2.3.4) and (2.3.5)
which, besides It denned in (2.3.2), will be useful as well for further analysis.
Indeed, one often confines (statistical) analysis to information conveyed by a
discrete time sampling of stock return series which will be denoted by the
information set
If = afo/Si-i : t = 0,1,..., / - 1, /] (2.3.6)
where the superscript R stands for returns. By extending Andersen (1994), we
shall adopt as the most general framework for univariate volatility modelling, the
setup given by the Assumptions 2.3.2.A, 2.3.2.B and:
Assumption 2.3.2.C. n{Ut) is if- measurable.
Therefore in (2.3.4) and (2.3.5) we have essentially shown that:
E[log St+l/St\I*] = n(Ut) (2.3.7)
Var[(log St+l/St)\l*] = E[o\Ut)\lf] . (2.3.8)
2.4. Statistical modelling of stochastic volatility
Financial time series are observed at discrete time intervals while a majority of
theoretical models are formulated in continuous time. Generally speaking there
are two statistical methodologies to resolve this tension. Either one considers for
the purpose of estimation statistical discrete time models of the continuous time
processes, or alternatively, the statistical model may be specified in continuous
time and inference done via a discrete time approximation. In this section we will
discuss in detail the former approach while the latter will be introduced in
Section 4. The class of discrete time statistical models discussed here is general. In
Section 2.4.1 we introduce some notation and terminology. The next section
discusses the so-called stochastic autoregressive volatility model introduced by

Stochastic volatility
135
Andersen (1994) as a rather general and flexible semi-parametric framework to
encompass various representations of stochastic volatility already available in the
literature. Identification of parameters and the restrictions required for it are
discussed in Section 2.4.3.
2.4.1. Notation and terminology
In Section 2.3, we left unspecified the functional forms which the trend /*(■) and
volatility <x(-) take. Indeed, in some sense we built a nonparametric framework
recently proposed by Lezan, Renault and de Vitry (1995) which they introduced
to discuss a notion of stochastic volatility of unknown form.18 This nonparametric
framework encompasses standard parametric models (see Section 2.4.2 for more
formal discussion). For the purpose of illustration let us consider two extreme
cases, assuming for simplicity that fi(Ut) = 0 : (i) the discrete time analogue of the
Hull and White model (2.1.16) is obtained when a{Ut) = at is a stochastic process
independent from the stock return standardized innovation process e and (ii) at
may be a deterministic function h(et, x < t) of past innovations. The latter is the
complete opposite of (z) and leads to a large variety of choices of parameterized
functions for h yielding X-ARCH models (GARCH, EGARCH, QTARCH,
Periodic GARCH, etc.).
Besides these two polar cases where Assumption 2.3.2.A is fulfilled in a trivial
degenerate way, one can also accommodate leverage effects.19 In particular the
contemporaneous correlation structure between innovations in U and the return
process can be nonzero, since the Granger non-causality assumptions deal with
temporal causal links rather than contemporaneous ones. For instance, we may
have a(Ut) = at with:
log St+i/S, = atet+i (2.4.1)
Cov(<x,+i,e(+i|lf)^0 . (2.4.2)
A negative covariance in (2.4.2) is a standard case of leverage effect, without
violating the non-causality Assumptions 2.3.2.A and B.
A few concluding observations are worth making to deal with the burgeoning
variety of terminology in the literature. First, we have not considered the
distinction due to Taylor (1994) between "lagged autoregressive random variance
models" given by (2.4.1) and "contemporaneous autoregressive random variance
models" defined by:
log St+i/S, = a,+iet+i . (2.4.3)
18 Lezan, Renault and de Vitry (1995) discuss in detail how to recover phenomena such as
volatility clustering in this framework. As a nonparametric framework it also has certain advantages
regarding (robust) estimation. They develop for instance methods that can be useful as a first
estimation step for efficient algorithms assuming a specific parametric model (see Section 5).
19 Assumption 2.3.2.B is fulfilled in case (i) but may fail in the GARCH case (ii). When it fails to
hold in the latter case it makes the GARCH framework not very well-suited for option pricing.

136
E. Ghysels, A. C. Harvey and E. Renault
Indeed, since the volatility process at is unobservable, the settings (2.4.1) and
(2.4.3) are observationally equivalent as long as they are not completed by precise
(non)-causality assumptions. For instance: (i) (2.4.1) and assumption 2.3.2.A
together appear to be a correct and very general definition of a SV model possibly
completed by Assumption 2.3.2.B for option pricing and (2.4.2) to introduce
leverage effects, (ii) (2.4.3) associated with (2.4.2) would not be a correct definition
of a SV model since in this case in general: E[log St+\/St | lf\ ^ 0, and the model
would introduce via the process a a forecast which is related not only to volatility
but also to the expected return.
For notational simplicity, the framework (2.4.3) will be used in Section 3 with
the leverage effect captured by Cov(<t(+i , et) ^ 0 instead of Cov(<r(+i, et+\) ^ 0.
Another terminology was introduced by Amin and Ng (1993) for option pricing.
Their distinction between "predictable" and "unpredictable" volatility is very
close to the leverage effect concept and can also be analyzed through causality
concepts as discussed in Garcia and Renault (1995). Finally, it will not be
necessary to make a distinction between weak, semi-strong and strong definitions of
SV models in analogy with their ARCH counterparts (see Drost and Nijman
(1993)). Indeed, the class of SV models as defined here can accommodate para-
meterizations which are closed under temporal aggregation (see also Section 4.1
on the subject of temporal aggregation).
2.4.2. Stochastic autoregressive volatility
For simplicity, let us consider the following univariate volatility process:
yt+\ = rt + °t£t+\ (2-4.4)
where \it is a measurable function of observables yt elf, x < t. While our
discussion will revolve around (2.4.4), we will discuss several issues which are general
and not confined to that specific model; extensions will be covered more explicitly
in Section 3.5. Following the result in (2.3.8) we know that:
Var[J(+1|/f]=E[^|/f] (2.4.5)
suggesting (1) that volatility clustering can be captured via autoregressive
dynamics in the conditional expectation (2.4.5) and (2) that thick tails can be
obtained in either one of three ways, namely (a) via heavy tails of the white noise et
distribution, (b) via the stochastic features of E \o2t \lf] and (c) via specific
randomness of the volatility process at which makes it latent i.e. o0f?® The
volatility dynamics that follow from (1) and (2) are usually an AR(1) model for some
nonlinear function of ot. Hence, the volatility process is assumed to be stationary
and Markovian of order one but not necessarily linear AR(1) in at itself. This is
20 Kim and Shephard (1994), using data on weekly returns on the S&P500 Index , found that a t-
GARCH model has an almost identical likelihood as the normal based SV model. This example shows
that a specific randomness in at may produce the same level of marginal kurtosis as a heavy tailed
student distribution of the white noise e.

Stochastic volatility
137
precisely what motivated Andersen (1994) to introduce the Stochastic Auto-
regressive Variance or SARV class of models where at (or of) is a polynomial
function g(Kt) of a Markov process Kt with the following dynamic specification:
K, = w + pK,-i + [y + aK,.i]u, (2.4.6)
where ut = ut — 1 is zero-mean white noise with unit variance. Andersen (1994)
discusses sufficient regularity conditions which ensure stationarity and ergodicity
for Kt. Without entering into the details, let us note that the fundamental non-
causality Assumption 2.3.2A implies that the ut process in (2.4.6) does not
Granger-cause et in (2.4.4). In fact, the non-causality condition suggests a slight
modification of Andersen's (1994) definition. Namely, it suggests assuming et+\
independent of ut-j, j > 0 for the conditional probability distribution, given et-j,
j > 0 rather than for the unconditional distribution. This modification does not
invalidate Andersen's SARV class of models as the most general parametric
statistical model studied so far in the volatility literature. The GARCH(1,1)
model is straightforwardly obtained from (2.4.6) by letting Kt = aj,y = 0 and
ut = ej. Note that the deterministic relationship ut — ej between the stochastic
components of (2.4.4) and (2.4.6) emphasizes that, in GARCH models, there is no
randomness specific to the volatility process. The Autoregressive Random
Variance model popularized by Taylor (1986) also belongs to the SARV class. Here:
log ov+i = £, + (j) log a, + r\t+x (2-4.7)
where r\t+l is a white noise disturbance such that Cov(^+1,£(+i) ^ 0 to
accommodate leverage effects. This is a SARV model with Kt = log at, a = 0 and
tlt+i =yut+i.21
2.4.3. Identification of parameters
Introducing a general class of processes for volatility, like the SARV class
discussed in the previous section prompts questions regarding identification.
Suppose again that
yt+i = <Jt£t+\
o? = g(Kt), g£{\,2} (2.4.8)
Kt = w + 0K,-i + [y + <jKt-i}u, .
Andersen (1994), noted that the model is better interpreted by considering the
zero-mean white noise process ut = ut — 1:
K, = (w + y) + (« + 0)K,-i + {y + aKt_x)ut . (2.4.9)
It is clear from the latter that it may be difficult to distinguish empirically the
constant w from the "stochastic" constant yut. Similarly, the identification of the
a and fi parameters separately is also problematic as (a + /?) governs the persis-
21 Andersen (1994) also shows that the SARV framework encompasses another type of random
variance model that we have considered as ill-specified since it combines (2.4.2) and (2.4.3).

138
E. Ghysels, A. C. Harvey and E. Renault
tence of shocks to volatility. These identification problems are usually resolved by
imposing (arbitrary) restrictions on the pairs of parameters (w,y) and (a, /?).
The GARCH(1,1) and Autoregressive Random Variance specifications assume
that 7 = 0 and a = 0 respectively. Identification of all parameters without such
restrictions generally requires additional constraints, for instance via some
distributional assumptions on £t+\ and ut, which restrict the semi-parametric
framework of (2.4.6) into a parametric statistical model.
To address more rigorously the issue of identification, it is useful to consider,
according to Andersen (1994), the following reparameterization (assuming for
notational convenience that a ^ 0):
(2.4.10)
Hence equation (2.4.9) can be rewritten as:
Kt=K + p{Kt.x -K) + (d + Kt-i)Ut
where Ut — aut.
It is clear from (2.4.10) that only three functions of the original parameters
a, /?, y, w may be identified and that the three parameters K, p, 8 are identified from
the first three unconditional moments of the process Kt for instance.
To give to these identification results an empirical content, it is essential to
know: (1) how to go from the moments of the observable process Yt to the
moments of the volatility process at, and (2) how to go from the moments of the
volatility process a, to the moments of the latent process Kt. The first point is
easily solved by specifying the corresponding moments of the standardized
innovation process e. If we assume for instance a Gaussian probability distribution,
we obtain:
= 2/n E(otot-j) (2.4.11)
= a/^ E(of<T(_y) .
The solution of the second point requires in general the specification of the
mapping g and of the probability distribution of ut in (2.4.6). For the so-called
Log-normal SARV model, it is assumed that a = 0 and Kt — log at (Taylor's
autoregressive random variance model) and that ut is normally distributed (Log-
normality of the volatility process). In this case, it is easy to show that:
Ecr? = exp[«E^ + n2Var^/2]
E(afa"t_j) = Eo?Eo?_Jexp[mnCov(Kt,Kt-J)] (2.4.12)
Co\(Kt,K,-j) = pJVarKt .
Without the normality assumption (i.e. QML, mixture of normal, Student
distribution ...) this model will be studied in much more detail in sections 3 and 5

Stochastic volatility
139
from both probabilistic and statistical points of view. Moreover, this is a template
for studying other specifications of the SARV class of models. In addition,
various specifications will be considered in Section 4 as proxies of continuous time
models.
3. Discrete time models
The purpose of this section will be to discuss the statistical handling of discrete
time SV models, using simple univariate cases. We start by defining the most basic
SV model corresponding to the autoregressive random variance model discussed
earlier in (2.4.7). We study its statistical properties in Section 3.2 and provide a
comparison with ARCH models in Section 3.3. Section 3.4 is devoted to filtering,
prediction and smoothing. Various extensions, including multivariate models, are
covered in the last section. Estimation of the parameters governing the volatility
process is discussed later in section 5.
3.1. The discrete time SV model
The discrete time SV model may be written as
yt = atet , t=l,...,T , (3.1.1)
where yt denotes the demeaned return process yt = log (St/St-i) — n and log of
follows an AR(1) process. It will be assumed that e, is a series of independent,
identically distributed random disturbances. Usually et is specified to have a
standard distribution so its variance of is known. Thus for a normal distribution
of is unity while for a ^-distribution with v degrees of freedom it will be v/(v - 2).
Following a convention often adopted in the literature we write ht = log of:
yt = aete0-5h- (3.1.2)
where o is a scale parameter, which removes the need for a constant term in the
stationary first-order autoregressive process
ht+l = j,ht + r,nr,t~IID(0,tf) , \4>\<l. (3.1.3)
It was noted before that if et and r\t are allowed to be correlated with each
other, the model can pick up the kind of asymmetric behavior which is often
found in stock prices. Indeed a negative correlation between et and r\t induces a
leverage effect. As in Section 2.4.1, the timing of the disturbance in (3.1.3) ensures
that the observations are still a martingale difference, the equation being written
in this way so as to tie in with the state space literature.
It should be stressed that the above model is only an approximation to the
continuous time models of Section 2 observed at discrete intervals. The accuracy
of the approximation is examined in Dassios (1995) using Edgeworth expansions
(see also Sections 4.1 and 4.3 for further discussion).

140
E. Ghysels, A. C. Harvey and E. Renault
3.2. Statistical properties
The following properties of the SV model hold even if e, and r\t are
contemporaneously correlated. Firstly, as noted, y, is a martingale difference.
Secondly, stationarity of ht implies stationarity of yt. Thirdly, if rjt is normally
distributed, it follows from the properties of the lognormal distribution that
E[exp(aA/)] = exp(a2cr2/2), where a is a constant and a\ is the variance of ht.
Hence, if e, has a finite variance, the variance of y, is given by
Var(Jr) = cr2^exp(cr2/2) . (3.2.1)
Similarly if the fourth moment of e, exists, the kurtosis of y, is Kexp(cr2), where k
is the kurtosis of et, so y, exhibits more kurtosis than et. Finally all the odd
moments are zero.
For many purposes we need to consider the moments of powers of absolute
values. Again, r\t is assumed to be normally distributed. Then for e, having a
standard normal distribution, the following expressions are derived in Harvey
(1993):
E|ylr = ^y/2r(c^/2)exp(y«i) , c >-I , c^O (3.2.2)
and
r(i/2)
Varl^r^^^exp^cr2'
T(c/2 + 1/2)12>
r(i/2)
c > -0.5, c ^ 0
Note that T(l/2) =y/n and T(l) = 1. Corresponding expressions may be
computed for other distributions of e, including Student's t and the General Error
Distribution (see Nelson (1991)).
Finally, the square of the coefficient of variation of of is often used as a
measure of the relative strength of the SV process. This is
Var(<72)/[E(c-2)]2 = exp(cr^) - 1. Jacquier, Poison and Rossi (1994) argue that this
is more easily interpretable than cr2. In the empirical studies they quote it is rarely
less than 0.1 or greater than 2.
3.2.1. Autocorrelation functions
If we assume that the disturbances e, and r\t are mutually independent, and r\t is
normal, the ACF of the absolute values of the observations raised to the power c
is given by
(e) _E(|yinyl-tn-{E(|yir)}2„, «p($«fa, j ~ 1
E(\yt)-{m<\C)}2 Kcexp(£cr2)-1 ' (3-2.3)
t> 1 , c> -0.5 , c^O

Stochastic volatility
141
where kc is
Kc = v{\yt)l{H\yt\c)}2 , (3-2.4)
and ph , t = 0,1,2,... denotes the ACF of ht. Taylor (1986) gives this expression
for c equal to one and two and et normally distributed. When c = 2, kc is the
kurtosis and this is three for a normal distribution. More generally,
Kc = r(c + i/2)r(i/2)/{r(c/2 +1/2)}2 , c ± 0 .
For Student's /-distribution with v degrees of freedom:
^ r{c + i/2)r(-c + v/2)r(i/2)r(v/2)
Kc ~~ {r(c/2 + l/2)r(-C/2 + v/2)}2 ' (3.2.5)
\c\ < v/2 , c^O
Note that v must be at least five if c is two.
The ACF, p\, has the following features. First, if a\ is small and/or pA is
close to one,
(»ccexp(^-o^)-l)
compare Taylor (1986, p. 74-5). Thus the shape of the ACF of ht is approximately
carried over to p\c' except that it is multiplied by a factor of proportionality,
which must be less than one for c positive as kc is greater than one. Secondly, for
the /-distribution, kc declines as v goes to infinity. Thus p^ is a maximum for a
normal distribution. On the other hand, a distribution with less kurtosis than the
normal will give rise to higher values of pi°'.
Although (3.2.6) gives an explicit relationship between pic' and c, it does not
appear possible to make any general statements regarding p\c' being maximized
for certain values of c. Indeed different values of a\ lead to different values of c
maximizing pf . If a\ is chosen so as to give values of p[ of a similar size to those
reported in Ding, Granger and Engle (1993) then the maximum appears to be
attained for c slightly less than one. The shape of the curve relating p\c' to c is
similar to the empirical relationships reported in Ding, Granger and Engle, as
noted by Harvey (1993).
3.2.2. Logarithmic transformation
Squaring the observations in (3.1.2) and taking logarithms gives
log y] = log a2 + ht + log e2 . (3.2.7)
Alternatively
log yj = m + h, + Zt , (3.2.8)

142
E. Ghysels, A. C. Harvey and E. Renault
where a> = log a2 + Elog e^,so that the disturbance £t has zero mean by
construction.
The mean and variance of log e1 are known to be -1.27 and 7t2/2 = 4.93 when
et has a standard normal distribution; see Abramovitz and Stegun (1970).
However, the distribution of log e1 is far from being normal, being heavily
skewed with a long tail.
More generally, if et has a ^-distribution with v degrees of freedom, it can be
expressed as:
where £t is a standard normal variate and Kt is independently distributed such that
vk{ is chi-square with v degrees of freedom. Thus
log e2 = log £ - log k,
and again using results in Abramovitz and Stegun (1970), it follows that the mean
and variance of log E2t are -1.27 -ij/(v/2) - log (v/2) and 4.93 + t^'(v/2)
respectively, where »/^(-) is the digamma function. Note that the moments of £r exist even
if the model is formulated in such a way that the distribution of et is Cauchy, that
is v — 1. In fact in this case t,t is symmetric with excess kurtosis two, compared
with excess kurtosis four when et is Gaussian.
Since log e^ is serially independent, it is straightforward to work out the ACF
of log yj for ht following any stationary process:
P<0)=Pv/{l + ^} , t>1 . (3.2.9)
The notation pi' reflects the fact that the ACF of a power of an absolute value
of the observation is the same as that of the Box-Cox transform, that is
{|jr|c-l}/c, and hence the logarithmic transform of an absolute value, raised to
any (non-zero) power, corresponds to c = 0. (But note that one cannot simply set
c = 0 in (3.2.3)).
Note that even if r\t and et are not mutually independent, the r\t and t,t
disturbances are uncorrected if the joint distribution of et and rjt is symmetric, that
is f{£t,qt) — fi~£ti —fit)'' see Harvey, Ruiz and Shephard (1994). Hence the
expression for the ACF in (3.2.9) remains valid.
3.3. Comparison with ARCH models
The GARCH(1,1) model has been applied extensively to financial time series. The
variance in (3.1.1) is assumed to depend on the variance and squared observation
in the previous time period. Thus
(x? = y-l-ay*.!+/»*?_, , t=l,...,T. (3.3.1)
The GARCH model was proposed by Bollerslev (1986) and Taylor (1986), and
is a generalization of the ARCH model formulated by Engle (1982). The

Stochastic volatility
143
ARCH(l) model is a special case of GARCH(1,1) with /? = 0. The motivation
comes from forecasting; in an AR(1) model with independent disturbances, the
optimal prediction of the next observation is a fraction of the current observation,
and in ARCH(l) it is a fraction of the current squared observation (plus a
constant). The reason is that the optimal forecast is constructed conditional on the
current information and in an ARCH model the variance in the next period is
assumed to be known. This construction leads directly to a likelihood function for
the model once a distribution is assumed for et. Thus estimation of the parameters
upon which of depends is straightforward in principle. The GARCH formulation
introduces terms analogous to moving average terms in an ARMA model, thereby
making forecasts a function of a distributed lag of past squared observations.
It is straightforward to show that yt is a martingale difference with
(unconditional) variance y/(l - a - /?). Thus a + /? < 1 is the condition for
covariance stationarity. As shown in Bollerslev (1986), the condition under which the
fourth moment exists in a Gaussian model is 2a2 + (a + fi)2 < 1. The model then
exhibits excess kurtosis. However, the fourth moment condition may not always
be satisfied in practice. Somewhat paradoxically, the conditions for strict
stationarity are much weaker and, as shown by Nelson (1990), even include the case
a + /J=l.
The specification of GARCH(1,1) means that we can write
y2 = y + ay2_x + jScr2,! + vt = y + (a + $)y)_x + vt - /to,_!
where vt = y2 - of is a martingale difference. Thus y2 has the form of an
ARMA(1,1) process and so its ACF can be evaluated in the same way. The ACF
of the corresponding ARMA model seems to be indicative of the type of patterns
likely to be observed in practice in correlograms of yj.
The GARCH model extends by adding more lags of of and y2. However,
GARCH(1,1) seems to be the most widely used. It displays similar properties to
the SV model, particularly if cf> is close to one. This should be clear from (3.2.6)
which has the pattern of an ARM A( 1,1) process. Clearly (f> plays a role similar to
that of a + p. The main difference in the ACFs seems to show up most at lag one.
Jacquier et al. (1994, p. 373) present a graph of the correlogram of the squared
weekly returns of a portfolio on the New York Stock Exchange together with the
ACFs implied by fitting SV and GARCH(1,1) models. In this case the ACF
implied by the SV model is closer to the sample values.
The SV model displays excess kurtosis even if <j> is zero since yt is a mixture of
distributions. The a2 parameter governs the degree of mixing independently of the
degree of smoothness of the variance evolution. This is not the case with a
GARCH model where the degree of kurtosis is tied to the roots of the variance
equation, a and /? in the case of GARCH(1,1). Hence, it is very often necessary to
use a non-Gaussian GARCH model to capture the high kurtosis typically found
in a financial time series.
The basic GARCH model does not allow for the kind of asymmetry captured
by a SV model with contemporaneously correlated disturbances, although it can

144
E. Ghysels, A. C. Harvey and E. Renault
be modified as suggested in Engle and Ng (1993). The EGARCH model,
proposed by Nelson (1991), handles asymmetry by taking log of to be a function of
past squares and absolute values of the observations.
3.4. Filtering, smoothing and prediction
For the purposes of pricing options, we need to be able to estimate and predict the
variance, of, which of course, is proportional to the exponent of ht. An estimate
based on all the observations up to, and possibly including, the one at time t is
called a filtered estimate. On the other hand an estimate based on all the
observations in the sample, including those which came after time t is called a
smoothed estimate. Predictions are estimates of future values. As a matter of
historical interest we may wish to examine the evolution of the variance over time
by looking at the smoothed estimates. These might be compared with the
volatilities implied by the corresponding options prices as discussed in Section 2.1.2.
For pricing "at the money" options we may be able to simply use the filtered
estimate at the end of the sample and the predictions of future values of the
variance, as in the method suggested for ARCH models by Noh, Engle and Kane
(1994). More generally, it may be necessary to base prices on the full distribution
of future values of the variance, perhaps obtained by simulation techniques; for
further discussion see Section 4.2.
One can think of constructing filtered and smoothed estimates in a very simple,
but arbitrary way, by taking functions (involving estimated parameters) of
moving averages of transformed observations. Thus:
£2 = A E wtjf(yt-j) 1 , t= l,.., t , (3.4.1)
where r = 0 or 1 for a filtered estimate and r = t - T for a smoothed estimate.
Since we have formulated a stochastic volatility model, the natural course of
action is to use this as the basis for filtering, smoothing and prediction. For a
linear and Gaussian time series model, the state space form can be used as the
basis for optimal filtering and smoothing algorithms. Unfortunately, the SV
model is nonlinear. This leaves us with three possibilities:
a. compute inefficient estimates based on a linear state space model;
b. use computer intensive techniques to estimate the optimal filter to a desired
level of accuracy;
c. use an (unspecified) ARCH model to approximate the optimal filter.
We now turn to examine each of these in some detail.
3.4.1. Linear state space form
The transformed observations, the log yfs, can be used to construct a linear state
space model as suggested by Nelson (1988) and Harvey, Ruiz and Shephard
(1994). The measurement equation is (3.2.8) while (3.1.3) is the transition equa-

Stochastic volatility
145
tion. The initial conditions for the state, ht, are given by its unconditional mean
and variance, that is zero and <72/(l - 4>2) respectively.
While it may be reasonable to assume that r\t is normal, £r would only be
normal if the absolute value of et were lognormal. This is unlikely. Thus
application of the Kalman filter and the associated smoothers yields estimators of the
state, h,, which are only optimal within the class of estimators based on linear
combinations of the log yfs. Furthermore, it is not the h'ts which are required, but
rather their exponents. Suppose h,\T denotes the smoothed estimator obtained
from the linear state space form. Then exp^y) is of the form (3.4.1), multiplied
by an estimate of the scaling constant, a2. It can be written as a weighted
geometric mean. This makes the estimates vulnerable to very small observations and
is an indication of the limitations of this approach.
Working with the logarithmic transformation raises an important practical
issue, namely how to handle observations which are zero. This is a reflection of
the point raised in the previous paragraph, since obviously any weighted
geometric mean involving a zero observation will be zero. More generally we wish to
avoid very small observations. One possible solution is to remove the sample
mean. A somewhat more satisfactory alternative, suggested by Fuller, and studied
by Breidt and Carriquiry (1995), is to make the following transformation based
on a Taylor series expansion:
log y] ~ log (y2 + cs2y) - cs2y/\y2 + cs2y) , t = 1, ■ ■ ■, T , (3.4.2)
where s2 is the sample variance of the /ts and c is a small number, the suggested
value being 0.02. The effect of this transformation is to reduce the kurtosis in the
transformed observations by cutting down the long tail made up of the negative
values obtained by taking the logarithms of the "inliers". In other words it is a
form of trimming. It might be more satisfactory, to carry out this procedure after
correcting the observations for heteroskedasticity by dividing by preliminary
estimates, a2's. The log afs are then added to the transformed observations. The
dt^s could be constructed from a first round or by using a totally different
procedure, perhaps a nonparametric one.
The linear state space form can be modified so as to deal with asymmetric
models. It was noted earlier that even if r\t and e, are not mutually independent,
the disturbances in the state space form are uncorrelated if the joint distribution
of e, and Tj, is symmetric. Thus the above filtering and smoothing operations are
still valid, but there is a loss of information stemming from the squaring of the
observations. Harvey and Shephard (1993) show that this information may be
recovered by conditioning on the signs of the observations denoted by st, a
variable which takes the value + 1 (-1) when y, is positive (negative). These signs
are, of course, the same as the signs of the er's. Let E+(E_) denote the expectation
conditional on et being positive (negative), and assign a similar interpretation to
variance and covariance operators. The distribution of £, is not affected by
conditioning on the signs of the e/s, but, remembering that E(f/,|e,) is an odd
function of e,,

146
E. Ghysels, A. C. Harvey and E. Renault
V* = E+fe) = E+[Et]t\£t} = -E_(jj() ,
and
f =Cov+(i/„&) = E+(i,,&) - E+(i/,)E(&) = E+(i/r&)
= -Cov_(jj(,^) ,
because the expectation of £, is zero and
E+(i/,&) = E+tE(i/,|eO log e,] - /i*E(log e,) = -E_(i/r&) .
Finally
Var+J?, - E+(i£) - (E+(i/r)]2 = ^ - ^*2 .
The linear state space form is now
log y* = w + ht + £t
h,+\ = 4>h, + s,/i* + n* ,
($MG)-(;uv))-
The Kalman filter may still be initialized by taking ho to have mean zero and
variance o^/(l — <t>2)-
The parameterization in (3.4.3) does not directly involve a parameter
representing the correlation between zt and r\t. The relationship between ff and y*
and the original parameters in the model can only be obtained by making a
distributional assumption about et as well as r\t. When e, and r\t are bivariate
normal with Corr(e(, jj() = p, E(jj(|er) = ponet, and so
/i* = E+fe) = pff,E+(er) - PSv^ = 0.7979^, . (3.4.4)
Furthermore,
y* = p«T,E(|e(| log e?) - 0.7979pcr^E(log e?) = 1.1061p<r, . (3.4.5)
When e, has a ^-distribution, it can be written as Ct^7°'5, and £( and r\t can be
regarded as having a bivariate normal distribution with correlation p, while Kt is
independent of both. To evaluate /i* and y* one proceeds as before, except that
the initial conditioning is on £, rather than on et, and the required expressions are
found to be exactly as in the Gaussian case.
The filtered estimate of the log volatility ht, written as ht+\\t, takes the form:
ht+\\t = 4>ht\t-\ + -ir-*—TT (lo8 >7 ~ « ~ ht\t-\) + W >
Pt\t-\ + iy st-\- a^
where pt\t~\ is the corresponding mean square error of the ht\t-\. If p < 0, then
y* < 0, and the filtered estimator will behave in a similar way to the EGARCH

Stochastic volatility
147
model estimated by Nelson (1991), with negative observations causing bigger
increases in the estimated log volatility than corresponding positive values.
3.4.2. Nonlinear filters
In principle, an exact filter may be written down for the original (3.1.2) and
(3.1.3), with the former taken as the measurement equation. Evaluating such a
filter requires approximating a series of integrals by numerical methods. Kita-
gawa (1987) has proposed a general method for implementing such a filter and
Watanabe (1993) has applied it to the SV model. Unfortunately, it appears to be
so time consuming as to render it impractical with current computer technology.
As part of their Bayesian treatment of the model as a whole, Jacquier, Poison
and Rossi (1994) show how it is possible to obtain smoothed estimates of the
volatilities by simulation. What is required is the mean vector of the joint
distribution of the volatilities conditional on the observations. However, because
simulating this joint distribution is not a practical proposition, they decompose it
into a set of univariate distributions in which each volatility is conditional on all
the others. These distributions may be denoted p(et\o-t, y), where &-t denotes all
the volatilities apart from at. What one would like to do is to sample from each of
these distributions in turn, with the elements of <r_f set equal to their latest
estimates, and repeat several thousand times. As such this is a Gibbs sampler.
Unfortunately, there are difficulties. The Markov structure of the SV model may
be exploited to write
p(ot\<r-t,y) = p{°t\°t-\,°t+\,yt) « p{yt\ht)p{h,\ht-\)p{h,+i\h,)
but although the right hand side of the above expression can be written down
explicitly, the density is not of a standard form and there is no analytic expression
for the normalizing constant. The solution adopted by Jacquier, Poison and Rossi
is to employ a series of Metropolis accept/reject independence chains.
Kim and Shephard (1994) argue that the single mover algorithm employed by
Jacquier, Poison and Rossi will be slow if (f> is close to one and/or a1 is small. This
is because at changes slowly; in fact when it is constant, the algorithm will not
converge at all. Another approach based on the linear state space form, is to
capture the non-normal disturbance term in the measurement equation, t,t, by a
mixture of normals. Watanabe (1993) suggested an approximate method based on
a mixture of two moments. Kim and Shephard (1994) propose a multimove
sampler based on the linear state space form. Blocks of the h'ts are sampled, rather
than taking them one at a time. The technique they use is based on mixing an
appropriate number of normal distributions to get the required level of accuracy in
approximating the disturbance in (3.2.7). Mahieu and Schotman (1994a) extend
this approach by introducing more degrees of freedom in the mixture of normals
where the parameters are estimated rather than fixed a priori. Note that the
distribution of the o'ts can be obtained from the simulated distribution of the h'ts.
Jacquier, Poison and Rossi (1994, p.416) argue that no matter how many
mixture components are used in the Kim and Shephard method, the tail behavior
of log z] can never be satisfactorily approximated. Indeed, they note that given

148
E. Ghysels, A. C. Harvey and E. Renault
the discreteness of the Kim and Shephard state space, not all states can be visited
in the small number of draws mentioned, i.e. the so called inlier problem (see also
Section 3.4.1 and Nelson (1994)) is still present.
As a final point it should be noted that when the hyperparameters are
unknown, the simulated distribution of the state produced by the Bayesian
approach allows for their sampling variability.
3.4.3. ARCH models as approximate filters
The purpose here is to draw attention to a subject that will be discussed in greater
detail in Section 4.3. In an ARCH model the conditional variance is assumed to
be an exact function of past observations. As pointed out by Nelson and Foster
(1994, p.32) this assumption is ad hoc on both economic and statistical grounds.
However, because ARCH models are relatively easy to estimate, Nelson (1992)
and Nelson and Foster (1994) have argued that a useful strategy is to regard them
as niters which produce estimates of the conditional variance. Thus even if we
believe we have a continuous time or discrete time SV model, we may decide to
estimate a GARCH(1,1) model and treat the afs as an approximate filter, as in
(3.4.1). Thus the estimate is a weighted average of past squared observations. It
delivers an estimate of the mean of the distribution of <rj, conditional on the
observations at time t—\. As an alternative, the model suggested by Taylor (1986)
and Schwert (1989), in which the conditional standard deviation is set up as a
linear combination of the previous conditional standard deviation and the
previous absolute value, could be used. This may be more robust to outliers as it is a
linear combination of past absolute values.
Nelson and Foster derive an ARCH model which will give the closest
approximation to the continuous time SV formulation (see Section 4.3 for more
details). This does not correspond to one of the standard models, although it is
fairly close to EGARCH. For discrete time SV models the filtering theory is not as
extensively developed. Indeed, Nelson and Foster point out that a change from
stochastic differential equations to difference equations makes a considerable
difference in the limit theorems and optimality theory. They study the case of near
diffusions as an example to illustrate these differences.
3.5. Extensions of the model
3.5.1. Persistence and seasonality
The simplest nonstationary SV model has ht following a random walk. The
dynamic properties of this model are easily obtained if we work in terms of the
logarithmically transformed observations, log yj. All we have to do is first
difference to give a stationary process. The untransformed observations are non-
stationary but the dynamic structure of the model will appear in the ACF of
\y,/y,-i\c, provided that c < 0.5.
The model is an alternative to IGARCH, that is (3.3.1) with a. + ft = 1. The
IGARCH model is such that the squared observations have some of the features
of an integrated ARM A process and it is said to exhibit persistence; see Bollerslev

Stochastic volatility
149
and Engle (1993). However, its properties are not straightforward. For example it
must contain a constant, y, otherwise, as Nelson (1990) has shown, of converges
almost surely to zero and the model has the peculiar feature of being strictly
stationary but not weakly stationary. The nonstationary SV model, on the other
hand, can be analyzed on the basis that h, is a standard integrated process of
order one .
Filtering and smoothing can be carried out within the linear state space
framework, since log y2 is just a random walk plus noise. The initial conditions are
handled in the same way as is normally done with nonstationary structural time
series models, with a proper prior for the state being effectively formed from the
first observation; see Harvey (1989). The optimal filtered estimate of h, within the
class of estimates which are linear in past log yj's, that is ht\t_\, is a constant plus
an equally weighted moving average (EWMA) of past log yf's. In IGARCH a2 is
given exactly by a constant plus an EWMA of past squared observations.
The random walk volatility can be replaced by other nonstationary
specifications. One possibility is the doubly integrated random walk in which A2h, is white
noise. When formulated in continuous time, this model is equivalent to a cubic
spline and is known to give a relatively smooth trend when applied in levels
models. It is attractive in the SV context if the aim is to find a weighting function
which fits a smoothly evolving variance. However, it may be less stable for
prediction.
Other nonstationary components can easily be brought into ht. For example, a
seasonal or intra-daily component can be included; the specification is exactly as
in the corresponding levels models discussed in Harvey (1989) and Harvey and
Koopman (1993). Again the dynamic properties are given straightforwardly by
the usual transformation applied to log y2, and it is not difficult to transform the
absolute values suitably. Thus if the volatility consists of a random walk plus a
slowly changing, nonstationary seasonal as in Harvey (1989, p. 40-3), the
appropriate transformations are A, log y2 and | yt/yt-s \c where s is the number of
seasons. The state space formulation follows along the lines of the corresponding
structural time series models for levels. Handling such effects is not so easy within
the GARCH framework.
Different approaches to seasonality can also be incorporated in SV models
using ideas of time deformation as discussed in a later sub-section. Such
approaches may be particularly relevant when dealing with the kind of abrupt
changes in seasonality which seem to occur in high frequency, like five minute or
tick-by-tick, foreign exchange data.
3.5.2. Interventions and other deterministic effects
Intervention variables are easily incorporated into SV models. For example, a
sudden structural change in the volatility process can be captured by assuming
that
log a2 = log cr2 + h, + Aw,

150
E. Ghysels, A. C. Harvey and E. Renault
where w, is zero before the break and one after, and A is an unknown parameter.
The logarithmic transformation gives (3.2.8) but with kw, added to the right hand
side. Care needs to be taken when incorporating such effects into ARCH models.
For example, in the GARCH(1,1) a sudden break has to be modelled as
(P-t = y + Aw, - (a + j?)M-i + aj'Li + P^-i
with k constrained so that of is always positive.
More generally observable explanatory variables, as opposed to intervention
dummies, may enter into the model for the variance.
3.5.3. Multivariate models
The multivariate model corresponding to (3.1.2) assumes that each series is
generated by a model of the form
yu = aieife0-5h' , t=l,...,T, (3.5.1)
with the covariance (correlation) matrix of the vector et = (e\t,..., eNt)' being
denoted by 2e. The vector of volatilities, ht, follows a VAR(l) process, that is
ht+i — ®h, + t]t ,
where r\t ~ /ZD(0,E,,). This specification allows the movements in volatility to be
correlated across different series via 2,. Interactions can be picked up by the off-
diagonal elements of <P.
The logarithmic transformation of squared observations leads to a
multivariate linear state space model from which estimates of the volatilities can be
computed as in Section 3.4.1.
A simple nonstationary model is obtained by assuming that the volatilities
follow a multivariate random walk, that is <P = /. If E, is singular, of rank K < N,
there are only K components in volatility, that is each hit in (3.5.1) is a linear
combination of K < N common trends, that is
h, = 0h} + h (3.5.2)
where h\ is the ^xl vector of common random walk volatilities, h is a vector of
constants and 0 is an N x K matrix of factor loadings. Certain restrictions are
needed on 0 and It to ensure identifiability; see Harvey, Ruiz and Shephard
(1994). The logarithms of the squared observations are "co-integrated" in the
sense of Engle and Granger (1987) since there are N — K linear combinations of
them which are white noise and hence stationary. This implies, for example, that
if two series of returns exhibit stochastic volatility, but this volatility is the same
with ©' = (1,1), then the ratio of the series will have no stochastic volatility. The
application of the related concept of "co-persistence" can be found in Bollerslev
and Engle (1993). However, as in the univariate case there is some ambiguity
about what actually constitutes persistence.

Stochastic volatility
151
There is no reason why the idea of common components in volatility should
not extend to stationary models. The formulation of (3.5.2) would apply, without
the need for \ and with h] modelled, for example, by a VAR(l).
Bollerslev, Engle and Wooldridge (1988) show that a multivariate GARCH
model can, in principle, be estimated by maximum likelihood, but because of the
large number of parameters involved computational problems are often
encountered unless restrictions are made. The multivariate SV model is much
simpler than the general formulation of a multivariate GARCH. However, it is
limited in that it does not model changing covariances. In this sense it is
analogous to the restricted multivariate GARCH model of Bollerslev (1986) in which
the conditional correlations are assumed to be constant.
Harvey, Ruiz and Shephard (1994) apply the nonstationary model to four
exchange rates and find just two common factors driving volatility. Another
application is in Mahieu and Schotman (1994b). A completely different way of
modelling exchange rate volatility is to be found in the latent factor ARCH model
of Diebold and Nerlove (1989).
3.5.4. Observation intervals, aggregation and time deformation
Suppose that a SV model is observed every 8 time periods. In this case, hz, where x
denotes the new observation (sampling) interval, is still AR(1) but with parameter
(f>d. The variance of the disturbance, qt, increases, but a\ remains the same. This
property of the SV model makes it easy to make comparisons across different
sampling intervals; for example it makes it clear why if (f> is around 0.98 for daily
observations, a value of around 0.9 can be expected if an observation is made
every week (assuming a week has 5 days).
If averages of observations are observed over the longer period, the
comparison is more complicated, as h% will now follow an ARMA(1,1) process. However,
the AR parameter is still 4>d. Note that it is difficult to change the observation
interval of ARCH processes unless the structure is weakened as in Drost and
Nijman (1993); see also Section 4.4.1.
Since, as noted in Section 2.4, one typically uses a discrete time approximation
to the continuous time model, it is quite straightforward to handle irregularly
spaced observations by using the linear state space form as described, for
example, in Harvey (1989). Indeed the approach originally proposed by Clark
(1973) based on subordinated processes to describe asset prices and their volatility
fits quite well into this framework. The techniques for handling irregularly spaced
observations can be used as the basis for dealing with time deformed
observations, as noted by Stock (1988). Ghysels and Jasiak (1994a,b) suggest a SV model
in which the operational time for the continuous time volatility equation is
determined by the flow of information. Such time deformed processes may be
particularly suited to dealing with high frequency data. If x = g{t) is the mapping
between calendar time x and operational time t, then
dS, = fJ,S,dt + o{g{t))StdWXt
and

152
E. Ghysels, A. C. Harvey and E. Renault
d\og <t(t) = a((b — log a(x))dx + cdW2x
where W\t and W2x are standard, independent Wiener processes. The discrete time
approximation generalizing (3.1.3), but including a term which in (3.1.2) is
incorporated in the constant scale factor a, is then
ht+1 = [1 - e-"^}b + e-"^ht + r,t
where \g(t) is the change in operational time between two consecutive calendar
time observations and nt is normally distributed with mean zero and variance
c2(l - e-2aAgW)/2a. Clearly if Ag(t) = 1, 0 = e~a in (3.1.3). Since the flow of
information, and hence Ag(t), is not directly observable, a mapping to calendar
time must be specified to make the model operational. Ghysels and Jasiak (1994a)
discuss several specifications revolving around a scaled exponential function
relating g(t) to observables such as past volume of trade and past price changes with
asymmetric leverage effects. This approach was also used by Ghysels and Jasiak
(1994b) to model return-volume co-movements and by Ghysels, Gourieroux and
Jasiak (1995b) for modeling intra-daily high frequency data which exhibit strong
seasonal patterns (cf. Section 3.5.1).
3.5.5. Long memory
Baillie, Bollerslev and Mikkelsen (1993) propose a way of extending the GARCH
class to account for long memory. They call their models Fractionally Integrated
GARCH (FIGARCH), and the key feature is the inclusion of the fractional
difference operator, (1 - L) , where L is the lag operator, in the lag structure of
past squared observations in the conditional variance equation. However, this
model can only be stationary when d = 0 and it reduces to GARCH. In a later
paper, Bollerslev and Mikkelsen (1995) consider a generalization of the
EGARCH model of Nelson (1991) in which log of is modelled as a distributed lag
of past e, 's involving the fractional difference operator. This FIEGARCH model
is stationary and invertible if | d |< 0.5.
Breidt, Crato and de Lima (1993) and Harvey (1993) propose a SV model with
ht generated by fractional noise
h, = ti,/(l-L)d , nt~NID(0,<fy , 0<d<l. (3.5.1)
Like the AR(1) model in (3.1.3), this process reduces to white noise and a
random walk at the boundary of the parameter space, that is d = 0 and 1
respectively. However, it is only stationary if d < 0.5. Thus the transition from
stationarity to nonstationarity proceeds in a different way to the AR(1) model. As
in the AR(1) case it is reasonable to constrain the autocorrelations in (3.5.1) to be
positive. However, a negative value of d is quite legitimate and indeed differencing
ht when it is nonstationary gives a stationary "intermediate memory" process in
which -0.5 < d < 0.
The properties of the long memory SV model can be obtained from the
formulae in sub-Section 3.2. A comparison of the ACF for ht following a long

Stochastic volatility
153
memory process with d = 0.45 and a\ = 2 with the corresponding ACF when ht is
AR(1) with (j> = 0.99 can be found in Harvey (1993). Recall that a characteristic
property of long memory is a hyperbolic rate of decay for the autocorrelations
instead of an exponential rate, a feature observed in the data (see Section 2.2e).
The slower decline in the long memory model is very clear and, in fact, for
t = 1000, the long memory autocorrelation is still 0.14, whereas in the AR case it
is only 0.000013. The long memory shape closely matches that in Ding, Granger
and Engle (1993, p. 86-8).
The model may be extended by letting r\t be an ARMA process and/or by
adding more components to the volatility equation.
As regards smoothing and filtering, it has already been noted that the state
space approach is approximate because of the truncation involved and is
relatively cumbersome because of the length of the state vector. Exact smoothing and
filtering, which is optimal within the class of estimators linear in the log y\ 's , can
be carried out by a direct approach if one is prepared to construct and invert the
T x T covariance matrix of the log y\ 's .
4. Continuous time models
At the end of Section 2 we presented a framework for statistical modelling of SV
in discrete time and devoted the entire Section 3 to specific discrete time SV
models. To motivate the continuous time models we study first of all the exact
relationship (i.e. without approximation error) between differential equations and
SV models in discrete time. We examine this relationship in Section 4.1 via a class
of statistical models which are closed under temporal aggregation and proceed
(1) from high frequency discrete time to lower frequencies and (2) from
continuous time to discrete time. Next, in Section 4.2, we study option pricing and
hedging with continuous time models and elaborate on features such as the smile
effect. The practical implementation of option pricing formulae with SV often
requires discrete time SV and/or ARCH models as filters and forecasters of the
continuous time volatility processes. Such filters, covered in Section 4.3, are in
general discrete time approximations (and not exact discretizations as in Section
4.1) of continuous time SV models. Section 4.4 concludes with extensions of the
basic model.
4.1. From discrete to continuous time
The purpose of this section is to provide a rigorous discussion of the relationship
between discrete and continuous time SV models. The presentation will proceed
first with a discussion of temporal aggregation in the context of the SARV class of
models and focus on specific cases including GARCH models. This material is
covered in Section 4.1.1. Next we turn our attention to the aggregation of
continuous time SV models to yield discrete time representations. This is the subject
matter of Section 4.1.2.

154
E. Ghysels, A. C. Harvey and E. Renault
4.1.1. Temporal aggregation of discrete time models
Andersen's SARV class of models was presented in Section 2.4 as a general
discrete time parametric SV statistical model. Let us consider the zero-mean case,
namely:
y,+\ = a,£t+i (4.1.1)
and of for q = 1 or 2 is a polynomial function g(Kt) of the Markov process Kt
with stationary autoregressive representation:
Kt = co + pKt^ + vt (4.1.2)
where |/?| < 1 and
E[e(+i|eT,uTT<f]=0 (4.1.3a)
E[e^+l\eT,v,x<t} =1 (4.1.3b)
E[u(+i|eT,uTT<f] = 0 . (4.1.3c)
The restrictions (4.1.3a-c) imply that v is a martingale difference sequence with
respect to the filtration Jf= <r[eT,uT,T < t].22 Moreover, the conditional moment
conditions in (4.3.1a-c) also imply that e in (4.1.1) is a wliite noise process in a
semi-strong sense, i.e. E[e(+i|eT,T < t] = 0 and E[e2+1|eT,T < t\ = 1, and is not
Granger-caused by u.23 From the very beginning of Section 2 we choose the
continuously compounded rate of return over a particular time horizon as the
starting point for continuous time processes. Therefore, let yt+\ in (4.1.1) be the
continuously compounded rate of return for [t, t + 1] of the asset price process Sh
consequently:
j(+i=log St+l/St . (4.1.4)
Since the unit of time of the sampling interval is to a large extent arbitrary, we
would surely want the SV model defined by equations (4.1.1) through (4.1.3), (for
given q and function g) to be closed under temporal aggregation. As rates of
return are flow variables, closure under temporal aggregation means that for any
integer m:
-l
ytn = log StmlStm-m = ^ ytm~k
k=0
is again conformable to a model of the type (4.1.1) through (4.1.3) for the same
choice of q and g involving suitably adapted parameter values. The analysis in this
section follows Meddahi and Renault (1995) who study temporal aggregation of
SV models in detail, particularly the case of = Kt, i.e. q = 2 and g is the identity
22 Note that we do not use here the decomposition appearing in (2.4.9) namely, u, = [y + aK,-i]ut.
23 The Granger noncausality considered here for e, is weaker than Assumption 2.3.2.A as it
applies only to the first two conditional moments.

Stochastic volatility
155
function. It is related to the so called continuous time GARCH approach of Drost
and Werker (1994). Hence, we have (4.1.1) with:
o? = © + /&»?_!+o, (4.1.5)
With conditional moment restrictions (4.1.3a-c) this model is closed under
aggregation. For instance, for m = 2:
with:
where:
v(2) _ v , , v _ J2) .(2)
yt+\ - yt+i +yt — <rt-\£t+\
(^)W> + ^>(^2+,£>
= 2©(1+/?)
= (/?+l)D?0,_2 + V,-i] .
Moreover, it also worth noting that whenever a leverage effect is present at the
aggregate level, i.e.:
Cov
,(2) J2)
^o
with ef\ = (j^-i + yt-2)/of\, it necessarily appears at the disaggregate level, i.e.
Cov(o,, et) ± 0.
For the general case Meddahi and Renault (1995) show that model (4.1.5)
together with conditional moment restrictions (4.1.3a-c) is a class of processes
closed under aggregation. Given this result, it is of interest to draw a comparison
with the work of Drost and Nijman (1993) on temporal aggregation of GARCH.
While establishing this link between Meddahi and Renault (1995) and Drost and
Nijman (1993) we also uncover issues of leverage properties in GARCH models.
Indeed, contrary to what is often believed, we find leverage effect restrictions
in GARCH processes. Moreover, we also find from the results of Meddahi
and Renault that the class of weak GARCH processes includes certain SV
models.
To find a class of GARCH processes which is closed under aggregation Drost
and Nijman (1993) weakened the definition of GARCH, namely for a positive
stationary process a,:
a] = w + ay]_x + ba]_x (4.1.6)
where a + b < 1, they defined:
- strong GARCH if yt+\/at is i.i.d. with mean zero and variance 1

156
E. Ghysels, A. C. Harvey and E. Renault
- semi-strong GARCH if E {yt+i\y%,x < t] = 0 and E[y2+1\yz,x < t] = a2
- weak GARCH if EL[yt+l \y%, y2,x < t] = 0; EL [y2+l \y%, y2, x < t] = a2.24
Drost and Nijman show that weak GARCH processes temporally aggregate
and provide explicit formulae for their coefficients. In Section 2.4 it was noted
that the framework of SARV includes GARCH processes whenever there is no
randomness specific to the volatility process. This property will allow us to show
that the class of weak GARCH processes - as defined above - in fact includes
more general SV processes which are strictly speaking not GARCH. The
arguments, following Meddahi and Renault (1995), require a classification of the
models defined by (4.1.3) and (4.1.5) according to the value of the correlation
between ut and yj, namely:
(a) Models with perfect correlation: This first class, henceforth denoted C\, is
characterized by a linear correlation between ut and yj conditional on
(eT, \>%,x < t) which is either 1 or -1 for the model in (4.1.5).
(b) Models without perfect correlation: This second class, henceforth denoted C2,
has the above conditional correlation less than one in absolute value.
The class C\ contains all semi-strong GARCH processes, indeed whenever
Var[y^|£f,\>%,x < t] is proportional to Var[uf|£T, vz,x < t] in C\ we have a semi-
strong GARCH. Consequently, a semi-strong GARCH processes is a model
(4.1.5) with (1) restrictions (4.1.3), (2) a perfect conditional correlation as in C\,
and (3) restrictions on the conditional kurtosis dynamics.25
Let us consider now the following assumption:
Assumption 4.1.1. The following two conditional expectations are zero:
E[etvt\e%,vz,x<t] = 0 (4.1.7a)
E[e3t\e%,v%,x<t] = 0 . (4.1.7ft)
This assumption amounts to an absence of leverage effects, where the latter is
defined in a conditional covariance sense to capture the notion of instantaneous
causality discussed in Section 2.4.1 and applied here in the context of weak white
noise.26 It should also be noted that (4.1.7a) and (4.1.7b) are in general not
equivalent except for the processes of class C\.
The class C2 allows for randomness proper to the volatility process due to the
imperfect correlation. Yet, despite this volatility-specific randomness one can
24 For any Hilbert space H of L2, EL[x,|z, z e H] is the best linear predictor of x, in terms of 1 and
z e H. It should be noted that a strong GARCH process is a fortiori semi-strong which itself is also a
weak GARCH process.
25 In fact, Nelson and Foster (1994) observed that the most commonly used ARCH models
effectively assume that the variance of the variance rises linearly in of, which is the main drawback of
ARCH models in approximating SV models in continuous time (see also Section 4.3).
26 The conditional expectation (4.1.7b) can be viewed as a conditional covariance between e, and
e2. It is this conditional covariance which, if nonzero, produces leverage effects in GARCH.

Stochastic volatility
157
show that under Assumption 4.1.1 processes of C2 satisfy the weak GARCH
definition. A fortiori, any SV model conformable to (4.1.3a-c), (4.1.5), (4.7.1a-b)
and Assumption 4.1.1 is a weak GARCH process. It is indeed the symmetry
assumptions (4.1.7a-b), or restrictions on leverage in GARCH, that make
EL[j^+1|}'.c,}^,t < f] = of (together with the conditional moment restrictions
(4.1.3a-c)) and yield the internal consistency for temporal aggregation found by
Drost and Nijman (1993, example 2, p. 915) for the class of so called symmetric
weak GARCH(1,1). Hence, this class of weak GARCH( 1,1) processes can be
viewed as a subclass of processes satisfying (4.1.3) and (4.1.5).27
4.1.2. Temporal aggregation of continuous time models
To facilitate our discussion we will specialize the general continuous time model
(2.3.1) to processes with zero drift, i.e.:
d\ogSt = atdWt (4.1.8a)
da, = ytdt + btdWat (4.1.8b)
Cov (dWh dWt) = Ptdt (4.1.8c)
where the stochastic processes at,yt,8t and pt are If = [az;x < t] adapted. To
ensure that at is a nonnegative process one typically follows either one of two
strategies: (1) considering a diffusion for log of or (2) describing of as a CEV
process (or Constant Elasticity of Variance process following Cox (1975) and Cox
and Ross (1976)).28 The former is frequently encountered in the option pricing
literature (see e.g. Wiggins (1987)) and is also clearly related to Nelson (1991),
who introduced EGARCH, and to the log-Normal SV model of Taylor (1986).
The second class of CEV processes can be written as
da) = k(6 - a))dt + y(a2t)&dW? (4.1.9)
where 8 < 1/2 ensures that of is a stationary process with nonnegative values.
Equation (4.1.9) can be viewed as the continuous time analogue of the discrete
time SARV class of models presented in Section 2.4. This observation establishes
links with the discussion of the previous Section 4.1.1 and yields exact
discretization results of continuous time SV models. Here, as in the previous section,
it will be tempting to draw comparisons with the GARCH class of models, in
particular the diffusions proposed by Drost and Werker (1994) in line with the
temporal aggregation of weak GARCH processes.
27 As noted before, the class of processes satisfying (4.1.3) and (4.1.5) is closed under temporal
aggregation, including processes with leverage effects not satisfying Assumption 4.1.1.
28 Occasionally one encounters specifications which do not ensure nonnegativity of the ot process.
For the sake of computational simplicity some authors for instance have considered Ornstein-Uh-
lenbeck processes for a, or of (see e.g. Stein and Stein (1991)).

158
E. Ghysels, A. C. Harvey and E. Renault
Firstly, one should note that the CEV process in (4.1.9) implies an auto-
regressive model in discrete time for a\ , namely:
aj+At = 0(1 - e~kA<) + e-^a] + e~kAl J ek^y{<j2rfdW°u .
(4.1.10)
Meddahi and Renault (1995) show that whenever (4.1.9) and its discretization
(4.1.10) govern volatility, the discrete time process log St+(k+i)At/St+kAt,k G Z is a
SV process satisfying the model restrictions (4.1.3a-c) and (4.1.5). Hence, from the
diffusion (4.1.9) we obtain the class of discrete time SV models which is closed
under temporal aggregation, as discussed in the previous section. To be more
specific, consider for instance At = 1 , then from (4.1.10) it follows that:
yt+\ = log St+i/S, = <7,(1)£(+i
(^)2=-+K'<(1))2+* (4'U1)
where from (4.1.10):
p = e-k,w = e(l-e-k),
/ x * (4.1.12)
It is important to note from (4.1.12) that absence of leverage effect in
continuous time, i.e. p, = 0 in (4.1.8c), means no such effect at low frequencies and
the two symmetry conditions of Assumption 4.1.1 are fulfilled. This line of
reasoning also explains the temporal aggregation result of Drost and Werker (1994),
but one more generally can interpret discrete time SV models with leverage effects
as exact discretizations of continuous time SV models with leverage.
4.2. Option pricing and hedging
Section 4.2.1 is devoted to the basic option pricing model with SV, namely the
Hull and White model of Section 2. We are better equipped now to elaborate on
its theoretical foundations. The practical implications appear in Section 4.2.2
while 4.2.3 concludes with some extensions of the basic model.
4.2.1. The basic option pricing formula
Consider again formula (2.1.10) for a European option contract maturing at time
t + h = T. As noted in Section 2.1.2, we assume continuous and frictionless
trading. Moreover no arbitrage profits can be made from trading in the
underlying asset and riskless bonds ; interest rates are nonstochastic so that B(t, T)
defined by (2.1.12) denotes the time t price of a unit discount bond maturing at
time T. Consider now the probability space (Q ,J*,P), which is the fundamental
space of the underlying asset price process S:

Stochastic volatility
159
dS,jSt = n(t,S„ U,)dt + <jtdWf
a2 = f(Ut) (4-2-1)
dUt = a(t, U,)dt + b(t, Ut)dWta
where Wt = (Wf, Wf) is a standard two dimensional Brownian Motion (Wf and
Wf are independent, zero-mean and unit variance) defined on (Q ,#,.P). The
function /, called the volatility function, is assumed to be one-to-one. In this
framework (under suitable regularity conditions) the no free lunch assumption is
equivalent to the existence of a probability distribution Q on (Q,#"), equivalent to
P, under which discounted price processes are martingales (see Harrison and
Kreps (1979)). Such a probability is called an equivalent martingale measure and
is unique if and only if the markets are complete (see Harrison and Pliska
(1981)).29 From the integral form of martingale representations (see Karatzas and
Shreve (1988), p. 184), the (positive) density process of any probability measure Q
equivalent to P can be written as:
[^dWsu-\j\xsu)2du
[Kdw:-\j\x:fdu
ft 1 rt
Mt = exp
)o LJo
(4.2.2)
where the processes Xs and X" are adapted to the natural filtration
ot = a[Wz,x <t],t> 0, and satisfy the integrability conditions (almost surely):
I (Xsufdu < + oo and / (X°)2du < +oo .
Jo Jo
~ ~ ~ '
By Girsanov's theorem the process W = (Ws, W") defined by:
Wf =Wf + I Xsudu and Wta = Wt" + f Xaudu (4.2.3)
Jo Jo
is a two dimensional Brownian Motion under Q. The dynamic of the underlying
asset price under Q is obtained directly from (4.2.1) and (4.2.3). Moreover, the
discounted asset price process StB(0, t), 0 < t < T, is a g-martingale if and only if
for rt defined in (2.1.11):
xs^{t,St,Ut)-rt
Since S is the only traded asset, the process X" is not fixed. The process Xs
defined by (4.2.4) is called the asset risk premium. By analogy, any process X"
satisfying the required integrability condition can be viewed as a volatility risk
29 Here, the market is seen as incomplete (before taking into account the market pricing of the
option) so that we have to characterize a set of equivalent martingale measures.

160
E. Ghysels, A. C. Harvey and E. Renault
premium and for any choice of X" , the probability Q(X") defined by the density
process M in (4.2.2) is an equivalent martingale measure. Therefore, given the
volatility risk premium process Xa:
Cf = B(t, 7,)Ep(r) [Max[0, ST - K]] , 0<t<T (4.2.5)
is an admissible price process of the European call option.30
The Hull and White option pricing model relies on the following assumption,
which restricts the set of equivalent martingale measures:
Assumption 4.2.1. The volatility risk premium Xat only depends on the current
value of the volatility process: X"t = Xa(t, Ut),Vt e [0, T}.
This assumption is consistent with an intertemporal equilibrium model where
the agent preferences are described by time separable isoelastic utility functions
(see He (1993) and Pham and Touzi (1993)). It ensures that Ws and W" are
independent, so that the Q(X") distribution of log ST/St, conditionally on if and
the volatility path (ah0 <t<T) is normal with mean Jt rudu — \y2(t,T) and
variance y2(t, T) = Jt o\du. Under Assumption 4.2.1 one can compute the
expectation in (4.2.5) conditionally on the volatility path, and obtain finally:
Cf = StT$(n[<$>{du) - e-X'Md*)} (4.2.6)
with the same notation as in (2.1.20). To conclude it is worth noting that many
option pricing formulae available in the literature have a feature common with
(4.2.6) as they can be expressed as an expectation of the Black-Scholes price over a
heterogeneous distribution of the volatility parameter (see Renault (1995) for an
elaborate discussion on this subject).
4.2.2. Pricing and hedging with the Hull and White model
The Markov feature of the process (S, a) implies that the option price (4.2.6) only
depends on the contemporaneous values of the underlying asset prices and its
volatility. Moreover, under mild regularity conditions, this function is differ-
entiable. Therefore, a natural way to solve the hedging problem in this stochastic
volatility context is to hedge a given option of price C\ by A* units of the
underlying asset and £)* units of any other option of price Cj where the hedging
ratios solve:
r dc}/ast - a; - e; dcj/ast = 0
\dCl/dat-J2*tdCf/dat = 0 .
Such a procedure, known as the delta-sigma hedging strategy, has been studied
by Scott (1991). By showing that any European option completes the market, i.e.
dCf/dot + 0, 0 < t < T, Bajeux and Rochet (1992) justify the existence of an
30 Here elsewhere Ep(-) = Ee(-|#",) stands for the conditional expectation operator given J5",
when the price dynamics are governed by Q.

Stochastic volatility
161
unique solution to the delta-sigma hedging problem (4.2.7) and the implicit
assumption in the previous sections that the available information It contains the
past values (St, at), x < t. In practice, option traders often focus on the risk due to
the underlying asset price variations and consider the imperfect hedging strategy
Y,t = 0 and A, = dC}/dSt. Then, the Hull and White option pricing formula
(4.2.6) provides directly the theoretical value of A,:
At = dC?/dSt = E?{n4>(du) ■ (4.2.8)
This theoretical value is hard to use in practice since: (1) even if we knew the
Q(X") conditional probability distribution of d\t given It (summarized by at), the
derivation of the expectation (4.2.8) might be computationally demanding and (2)
the conditional probability is directly related to the conditional probability
distribution of y2(t, T) = J a\du given at, which in turn may involve nontrivially the
parameters of the latent process at. Moreover, these parameters are those of the
conditional probability distribution of y2(t, T) given at under the risk-neutral
probability Q{X") which is generally different from the Data Generating Process
P. The statistical inference issues are therefore quite complicated. We will argue in
Section 5 that only tools like simulation-based inference methods involving both
asset and option prices (via an option pricing model) may provide some
satisfactory solutions.
Nevertheless, a practical way to avoid these complications is to use the Black-
Scholes option pricing model, even though it is known to be misspecified. Indeed,
option traders know that they cannot generally obtain sufficiently accurate option
prices and hedge ratios by using the BS formula with historical estimates of the
volatility parameters based on time series of the underlying asset price. However,
the concept of Black-Scholes implied volatility (2.1.23) is known to improve the
pricing and hedging properties of the BS model. This raises two issues: (1) what is
the internal consistency of the simultaneous use of the BS model (which assumes
constant volatility) and of BS implied volatility which is clearly time-varying and
stochastic and (2) how to exploit the panel structure of option pricing errors?31
Concerning the first issue, we noted in Section 2 that the Hull and White
option pricing model can indeed be seen as a theoretical foundation for this
practice of pricing. Hedging issues and the panel structure of option pricing errors
are studied in detail in Renault and Touzi (1992) and Renault (1995).
4.2.3. Smile or smirk?
As noted in Section 2.2, the smile effect is now a well documented empirical
stylized fact. Moreover the smile becomes sometimes a smirk since it appears
more or less lopsided (the so called skewness effect). We cautioned in Section 2
that some explanations of the smile/smirk effect are often founded on tempting
analogies rather than rigorous proofs.
31 The value of a which equates the BS formula to the observed market price of the option heavily
depends on the actual date t, the strike price K, the time to maturity (T - i) and therefore creates a
panel data structure.

162
E. Ghysels, A. C. Harvey and E. Renault
To the best of our knowledge, the state of the art is the following: (i) the first
formal proof that a Hull and White option pricing formula implies a symmetric
smile was provided by Renault and Touzi (1992), (ii) the first complete proof that
the smile/smirk effects can alternatively be explained by liquidity problems (the
upper parts of the smile curve, i.e. the most expensive options are the least liquid)
was provided by Platten and Schweizer (1994) using a micro structure model,
(iii) there is no formal proof that asymmetries of the probability distribution of
the underlying asset price process (leverage effect, non-normality,...) are able to
capture the observed skewness of the smile. A different attempt to explain the
observed skewness is provided by Renault (1995). He showed that a slight
discrepancy between the underlying asset price St used to infer BS implied volatilities
and the stock price St considered by option traders may generate an empirically
plausible skewness in the smile. Such nonsynchronous St and St may be related to
various issues: bid-ask spreads, non-synchronous trading between the two
markets, forecasting strategies based on the leverage effect, etc.
Finally, to conclude it is also worth noting that a new approach initiated by
Gourieroux, Monfort, Tenreiro (1994) and followed also by Ait-Sahalia, Bickel,
Stoker (1994) is to explain the BS implied volatility using a nonparametric
function of some observed state variables. Gourieroux, Monfort, Tenreiro (1995)
obtain for example a good nonparametric fit of the following form:
at(St,K) = a(K)+b(K)(log St/St.xf .
A classical smile effect is directly observed on the intercept a(K) but an inverse
smile effect appears for the path-dependent effect parameter b(K). For American
options a different nonparametric approach is pursued by Broadie, Detemple,
Ghysels and Torres (1995) where, besides volatility, exercise boundaries for the
option contracts are also obtained.32
4.3. Filtering and discrete time approximations
In Section 3.4.3 it was noted that the ARCH class of models could be viewed as
filters to extract the (continuous time) conditional variance process from discrete
time data. Several papers were devoted to the subject, namely Nelson (1990,
1992, 1995a,b) and Nelson and Foster (1994, 1995). It was one of Nelson's
seminal contributions to bring together ARCH and continuous time SV.
Nelson's first contribution in his 1990 paper was to show that ARCH models, which
model volatility as functions of past (squared) returns, converge weakly to a
diffusion process, either a diffusion for log aj or a CEV process as described in
Section 4.1.2. In particular, it was shown that a GARCH(1,1) model observed at
finer and finer time intervals At = h with conditional variance parameters
a>h = hco, ah = a(h/2)1'2 and Ph = 1 - a.(h/2)x'2-Qh and conditional mean
32 See also Bossaerts and Hillion (1995) for the use of a nonparametric hedging procedure and the
smile effect.

Stochastic volatility
163
Hh = hcaj converges to a diffusion limit quite similar to equations (4.1.8a)
combined with (4.1.9) with <5 = 1, namely
d logS; = cajdt + OtdWt
d a2 = (co - 9a2) dt + a\dW°t .
Similarly, it was also shown that a sequence of AR(1)-EGARCH(1,1) models
converges weakly to an Ornstein-Uhlenbeck diffusion for In a2:
d In a) = u(P- In a2)dt + dW? .
Hence, these basic insights showed that the continuous time stochastic
difference equations emerging as diffusion limits of ARCH models were no longer
ARCH but instead SV models. Moreover, following Nelson (1992), even when
misspecified, ARCH models still kept desirable properties regarding extracting
the continuous time volatility. The argument was that for a wide variety of
misspecified ARCH models the difference between the ARCH filter volatility
estimates and the true underlying diffusion volatilities converges to zero in
probability as the length of the sampling time interval goes to zero at an
appropriate rate. For instance the GARCH(1,1) model with cot,, ctf, and fih described
before estimates a2 as follows:
oo
i=o
where yt = log St/St_h- This filter can be viewed as a particular case of equation
(3.4.1). The GARCH(1,1) and many other models, effectively achieve consistent
estimation of at via a lag polynomial function of past squared returns close to
time t.
The fact that a wide variety of misspecified ARCH models consistently extract
at from high frequency data raises questions regarding efficiency of niters. The
answers to such questions are provided in Nelson (1995a,b) and Nelson and
Foster (1994, 1995). In Section 3.4 it was noted that the linear state space Kalman
filter can also be viewed as a (suboptimal) extraction filter for at. Nelson and
Foster (1994) show that the asymptotically optimal linear Kalman filter has
asymptotic variance for the normalized estimation error /i_1/4[ln(6f) -ln<r2]
equal to lY(l/2)^2 where Y(x) = d[\nr(x)]/dx and X is a scaling factor. A
model, closely related to EGARCH of the following form:
H°2t+h) = H^) + pKSt+h - st)a;x
+i(i - p^'^ni/if^rwif^-s^ - 2-v*]
yields the asymptotically optimal ARCH filter with asymptotic variance for the
normalized estimation error equal to l[2(l — p2)] where the parameter p
measures the leverage effect. These results also show that the differences between

164
E. Ghysels, A. C. Harvey and E. Renault
the most efficient suboptimal Kalman filter and the optimal ARCH filter can be
quite substantial. Besides filtering one must also deal with smoothing and
forecasting. Both of these issues were discussed in Section 3.4 for discrete time SV
models. The prediction properties of (misspecified) ARCH models were studied
extensively by Nelson and Foster (1995). Nelson (1995) takes ARCH models a
step further by studying smoothing filters, i.e. ARCH models involving not only
lagged squared returns but also future realizations, i.e. r = t—T in equation
(3.4.1).
4.4. Long memory
We conclude this section with a brief discussion of long memory in continuous
time SV models. The purpose is to build continuous time long memory stochastic
volatility models which are relevant for high frequency financial data and for
(long term) option pricing. The reasons motivating the use of long memory
models were discussed in sections 2.2 and 3.5.5. The advantage of considering
continuous time long memory is their relative ability to provide a more structural
interpretation of the parameters governing short term and long term dynamics.
The first subsection defines fractional Brownian Motion. Next we will turn our
attention to the fractional SV model followed by a section on filtering and discrete
time approximations.
4.4.1. Stochastic integration with respect to fractional Brownian Motion
We recall in this subsection a few definitions and properties of fractional and long
memory processes in continuous time, extensively studied for instance in Comte
and Renault (1993). Consider the scalar process:
x,= f a(t-s)dWs . (4.4.1)
Jo
Such a process is asymptotically equivalent in quadratic mean to the stationary
process:
yt= [ a{t-s)dWs (4.4.2)
J — oo
whenever j^00 a2(x)dx < +oo. Such processes are called fractional processes if
a(x) =xaa(x)/r(l + a)for |a| < 1/2, a continuously differentiable on [0,7] and
where r(l + a) is a scaling factor useful for normalizing fractional derivative
operators on [0,T\. Such processes admit several representations, and in
particular they can also be written:
xt= [ c{t-s)dWas, Wat= [ rf~*\dWs (4.4.3)
Jo Jo / (1 + «)
where Wa is the so-called fractional Brownian Motion of order a (see Mandelbrot
and Van Ness (1968)).

Stochastic volatility
165
The relation between the functions a and c is one-to-one. One can show that
Wa is not a semi-martingale (see e.g. Rogers (1995)) but stochastic integration with
respect to Wa can be defined properly. The processes xt are long memory if:
lim xa(x) = a^, 0 < a < 1/2 and 0 < ax < +oo , (4.4.4)
x—>+oo v '
for instance,
dxt = -kxtdt + adWat xt = 0, k > 0 , 0 < a < 1/2 (4.4.5)
with its solution given by:
xt = [ (t - s)a(r(l + a))"1^ (4.4.6a)
Jo
x\a) = f e-W-tadW, . (4.4.6b)
Jo
Note that, xy the derivative of order a of xr, is a solution of the usual SDE:
dzt = —kztdt + adWt-
4.4.2. The fractional SV model
To facilitate comparison with both the FIEGARCH model and the fractional
extensions of the log-Normal SV model discussed in Section 3.5.5 let us consider
the following fractional SV model (henceforth FSV):
dSt/St = otdWt (4.4.7a)
d log at = -klog atdt + ydWat (4.4.7b)
where k > 0 and 0 < a. < 1/2. If nonzero, the fractional exponent a will provide
some degree of freedom in the order of regularity of the volatility process, namely
the greater a the smoother the path of the volatility process. If we denote the
autocovariance function of a by ra{-) then:
a > 0 =>• (ra(h) - ra(0))/h -> 0 as A -> 0 .
This would be incorrectly interpreted as near-integrated behavior, widely
found in high frequency data for instance, when:
re{h) - ra(0)/h = (ph - \)/h -f log p as h -► 0 ,
and at is a continuous time AR(1) with correlation p near 1.
The long memory continuous time approach allows us to model persistence
with the following features:(l) the volatility process itself (and not just its
logarithm) has hyperbolic decay of the correlogram ; (2) the persistence of volatility
shocks yields leptokurtic features for returns which vanishes with temporal

166
E. Ghysels, A. C. Harvey and E. Renault
aggregation at a slow hyperbolic rate of decay.33 Indeed for rate of return on
[0,h]:
Epog St+h/ S, - E(log St+h/St)}4
(Epog St+h/St - E(log St+h/St)}2)
as h —» oo at a rate /j2*-1 if a e [0,1/2] and a rate exp(-M/2) if a = 0.
4.4.J. Filtering and discrete time approximations
The volatility process dynamics are described by the solution to the SDE (4.4.5),
namely:
log <jt = f (t - s)x/r( 1 + a)d log <jW (4.4.6)
Jo
where log a^ follows the O-U process:
d log <j{ta) = -kloga^dt + ydW, . (4.4.7)
To compute a discrete time approximation one must evaluate numerically the
integral (4.4.6) using only values of the process log a^ on a discrete partition of
[o,t] at points j/nj = 0,1..., [nt].34 A natural way to proceed is to use step
functions, generating the following proxy process:
M
logo? = £(* - (/ - l)/»)7r(l + «)Alog<T$ (4.4.8)
where Alog^ = log^"),-logff(^_1)/n. Comte and Renault (1995) show that
log <Tnt converges to the log at process for n —> oo uniformly on compact sets.
Moreover, by rearranging (4.4.8) one obtains:
log o% (4.4.9)
los fyn = E([(« + iT-n/n«r(i + «))4
L >'=o
where Ln is the lag operator corresponding to the sampling scheme j/n, i.e.
L„Zj/„ =Zq_i)/„. With this sampling scheme loga^ is a discrete time AR(1)
deduced from the continuous time process with the following representation:
(l-pnLn)\oga^n = uJ/n (4.4.10)
where pn = exp(-£/n) and uj/„ is the associated innovations process. Since the
process
J < 0):
process is stationary we are allowed to write (assuming logOj) = iij/„ = 0 for
33 With usual GARCH or SV models, it vanishes at an exponential rate (see Drost and Nijman
(1993) and Drost and Werker (1994) for these issues in the short memory case).
34 [z] is the integer k such that k < z < k + 1.

Stochastic volatility
167
.(»)
kg*£
(1-/>„!„)«;/„ (4.4.11)
which gives a parameterization of the volatility dynamics in two parts: (1) a long
memory part which corresponds to the filter Y^oSi>L'„/na with
a, = [(i + \)a~ia]/r(\ + a) and (2) a short memory part which is characterized by
the AR(1) process: (1 - pnLn)~lUj/n. Indeed, one can show that the long memory
filter is "long-term equivalent" to the usual discrete time long memory filters
(1 - L)~a in the sense that there is a long term relationship (a cointegration
relation) between the two types of processes. However, this long-term equivalence
between the long-memory filter and the usual discrete time one (1 - L)~a does not
imply that the standard parametrization FARIMA(l,a,0) is well-suited in our
framework. Indeed, one can show that the usual discrete time filter (1 — L)"a
introduces some mixing between long and short term characteristics whereas the
parsimonious continuous time model doesn't.35 This feature clearly puts the
continuous time FSV at an advantage with regard to the discrete time SV and
GARCH long-memory models.
5. Statistical inference
Evaluating the likelihood function of ARCH models is a relatively
straightforward task. In sharp contrast for SV models it is impossible to obtain explicit
expressions for the likelihood function. This is a generic feature common to
almost all nonlinear latent variable models. The lack of estimation procedures for
SV models made them for a long time an unattractive class of models in
comparison to ARCH. In recent years, however, remarkable progress has been made
regarding the estimation of nonlinear latent variable models in general and SV
models in particular. A flurry of methods are now available and are up and
running on computers with ever increasing CPU performance. The early attempts
to estimate SV models used a GMM procedure. A prominent example is Melino
and Turnbull (1990). Section 5.1 is devoted to GMM estimation in the context of
SV models. Obviously, GMM is not designed to handle continuous time
diffusions as it requires discrete time processes satisfying certain regularity conditions.
A continuous time GMM approach, developed by Hansen and Scheinkman
(1994), involves moment conditions directly drawn from the continuous time
representation of the process. This approach is discussed in Section 5.3. In
between, namely in Section 5.2, we discuss the QML approach suggested by Harvey,
Ruiz and Shephard (1994) and Nelson (1988). It relies on the fact that the
nonlinear (Gaussian) SV model can be transformed into a linear non-Gaussian state
space model as in Section 3, and from this a Gaussian quasi-likelihood can be
computed. None of the methods covered in Sections 5.1 through 5.3 involve
simulation. However, increased computer power has made simulation-based es-
Namely, (1 -Z„)"logo^, is not an AR(1) process.

168
E. Ghysels, A. C. Harvey and E. Renault
timation techniques increasingly popular. The simulated method of moments, or
simulation-based GMM approach proposed by Duffie and Singleton (1993), is a
first example which is covered in Section 5.4. Next we discuss the indirect
inference approach of Gourieroux, Monfort and Renault (1993) and the moment
matching methods of Gallant and Tauchen (1994) in Section 5.5. Finally, Section
5.6 covers a very large class of estimators using computer intensive Markov Chain
Monte Carlo methods applied in the context of SV models by Jacquier, Poison
and Rossi (1994) and Kim and Shephard (1994), and simulation based ML
estimation proposed in Danielsson (1994) and Danielsson and Richard (1993).
In each section we will only try to limit our focus to the use of estimation
procedures in the context of SV models and avoid details regarding econometric
theory. Some useful references to complement the material which will be covered
are (1) Hansen (1992), Gallant and White (1988), Hall (1993) and Ogaki (1993)
for GMM estimation, (2) Gourieroux and Monfort (1993b) and Wooldridge
(1994) for QMLE, (3) Gourieroux and Monfort (1995) and Tauchen (1995) for
simulation based econometric methods including indirect inference and moment
matching, and finally (4) Geweke (1995) and Shephard (1995) for Markov Chain
Monte Carlo methods.
5.7. Generalized method of moments
Let us consider the simple version of the discrete time SV as presented in
equations (3.1.2) and (3.1.3) with the additional assumption of normality for the
probability distribution of the innovation process (et,nt). This log-normal SV
model has been the subject of at least two extensive Monte Carlo studies on
GMM estimation of SV models. They were conducted by Andersen and Serensen
(1993) and Jacquier, Poison and Rossi (1994). The main idea is to exploit the
stationary and ergodic properties of the SV model which yield the convergence of
sample moments to their unconditional expectations. For instance, the second
and fourth moments are simple expressions of a2 and a\, namely <72exp(<7|/2) and
3(74exp(2<7|) respectively. If these moments are computed in the sample, o\ can be
estimated directly from the sample kurtosis, k, which is the ratio of the fourth
moment to the second moment squared. The expression is just a\ = log(ic/3). The
parameter a2 can then be estimated from the second moment by substituting in
this estimate of o\. We might also compute the first-order autocovariance of y\,
or simply the sample mean of y\y\^\ which has expectation <r4exp({l + 4>}<r\) and
from which, given the estimate of a2 and o\ , it is straightforward to get an
estimate of 4>.
The above procedure is an example of the application of the method of
moments. In general terms, m moments are computed. For a sample of size T, let
gr(P) denote the m x 1 vector of differences between each sample moment and its
theoretical expression in terms of the model parameters fi. The generalized method
of moments (GMM) estimator is constructed by minimizing the criterion function
fiT = Arg min gT(PJWTgT(P)
P

Stochastic volatility
169
where WT is an m x m weighting matrix reflecting the importance given to
matching each of the moments. When et and t\t are mutually independent, Jac-
quier, Poison and Rossi (1994) suggest using 24 moments. The first four are given
by (3.2.2) for c = 1,2,3,4, while the analytic expression for the others is:
E[|y^T|] = |^2e[r0 + i)] /Aexp(j<Tl[l + <P]) ,
c= 1,2 , x-- ...
In the more general case when et and r\t are correlated, Melino and Turnbull
(1990) included estimates of: E[| yt \ yt^],x = 0, ±1, ±2,..., 10. They presented
an explicit expression in the case of x = 1 and showed that its sign is entirely
determined by p.
The GMM method may also be extended to handle a non-normal distribution
for et. The required analytic expressions can be obtained as in Section 3.2. On the
other hand, the analytic expression of unconditional moments presented in
Section 2.4 for the general SARV model may provide the basis of GMM estimation
in more general settings (see Andersen (1994)).
From the very start we expect the GMM estimator not to be efficient. The
question is how much inefficiency should be tolerated in exchange for its relative
simplicity. The generic setup of GMM leaves unspecified the number of moment
conditions, except for the minimal number required for identification, as well as
the explicit choice of moments. Moreover, the computation of the weighting
matrix is also an issue since many options exist in practice. The extensive Monte
Carlo studies of Andersen and Sorensen (1993) and Jacquier, Poison and Rossi
(1994) attempted to answer these outstanding questions. In general they find that
GMM is a fairly inefficient procedure primarily stemming from the stylized fact,
noted in Section 2.2, that <f> in equation (3.1.3) is quite close to unity in most
empirical findings because volatility is highly persistent. For parameter values of 0
close to unity convergence to unconditional moments is extremely slow suggesting
that only large samples can rescue the situation. The Monte Carlo study of
Andersen and S0rensen (1993) provides some guidance on how to control the extent
of the inefficiency, notably by keeping the number of moment conditions small.
They also provide specific recommendations for the choice of weighting matrix
estimators with data-dependent bandwidth using the Bartlett kernel.
5.2. Quasi maximum likelihood estimation
5.2.1. The basic model
Consider the linear state space model described in sub-Section 3.4.1, in which
(3.2.8) is the measurement equation and (3.1.3) is the transition equation. The
36 A simple way to derive these moment conditions is via a two-step approach similar in spirit to
(2.4.8) and (2.4.9) or (3.2.3).

170
E. Ghysels, A. C. Harvey and E. Renault
QML estimators of the parameters <f>, a2 and the variance of £„ a2, are obtained
by treating ^ and r\t as though they were normal and maximizing the prediction
error decomposition form of the likelihood obtained via the Kalman filter. As
noted in Harvey, Ruiz and Shephard (1994), the quasi maximum likelihood
(QML) estimators are asymptotically normal with covariance matrix given by
applying the theory in Dunsmuir (1979, p. 502). This assumes that t]t and & have
finite fourth moments and that the parameters are not on the boundary of the
parameter space.
The parameter co can be estimated at the same time as the other parameters.
Alternatively, it can be estimated as the mean of the log yf's, since this is
asymptotically equivalent when 0 is less than one in absolute value.
Application of the QML method does not require the assumption of a specific
distribution for et. We will refer to this as unrestricted QML. However, if a
distribution is assumed, it is no longer necessary to estimate a2, as it is known,
and an estimate of the scale factor, a2, can be obtained from the estimate of m.
Alternatively, it can be obtained as suggested in sub-Section 3.4.1.
If unrestricted QML estimation is carried out, a value of the parameter
determining a particular distribution within a class may be inferred from the
estimated variance of £,. For example in the case of the Student's t,v may be
determined from the knowledge that the theoretical value of the variance of £, is
4.93 + ij/'(v/2) (where *P(-) is the digamma function introduced in Section 3.2.2).
5.2.2. Asymmetric model
In an asymmetric model, QML may be based on the modified state space form in
(3.4.3). The parameters a2, o2v 0, ju*, and y* can be estimated via the Kalman filter
without any distributional assumptions, apart from the existence of fourth
moments of r\t and £t and the joint symmetry of ^ and r\t. However, if an estimate of
p is wanted it is necessary to make distributional assumptions about the
disturbances, leading to formulae like (3.4.4) and (3.4.5). These formulae can be used
to set up an optimization with respect to the original parameters a2,a2^> and p.
This has the advantage that the constraint \p\ < 1 can be imposed. Note that any
^-distribution gives the same relationship between the parameters, so within this
class it is not necessary to specify the degrees of freedom.
Using the QML method with both the original disturbances assumed to be
Gaussian, Harvey and Shephard (1993) estimate a model for the CRSP daily
returns on a value weighted US market index for 3rd July 1962 to 31st December
1987. These data were used in the paper by Nelson (1991) to illustrate his
EGARCH model. The empirical results indicate a very high negative correlation.
5.2.3. QML in the frequency domain
For a long memory SV model, QML estimation in the time domain becomes
relatively less attractive because the state space form (SSF) can only be used by
expressing h, as an autoregressive or moving average process and truncating at a
suitably high lag. Thus the approach is cumbersome, though the initial state
covariance matrix is easily constructed, and the truncation does not affect the

Stochastic volatility
171
asymptotic properties of the estimators. If the autoregressive approximation, and
therefore the SSF, is not used, time domain QML requires the repeated
construction and inversion of the T x T covariance matrix of the log yj's; see Sowell
(1992). On the other hand, QML estimation in the frequency domain is no more
difficult than it is in the AR(1) case. Cheung and Diebold (1994) present
simulation evidence which suggests that although time domain estimation is more
efficient in small samples, the difference is less marked when a mean has to be
estimated.
The frequency domain (quasi) log-likelihood function is, neglecting constants,
logL = -^loggj - nY,I{Xj)lgj (5.2.1)
where I(2.j) is the sample spectrum of the log yfs and gj is the spectral generating
function (SGF), which for (3.5.1) is
gj = (T>[2(l-cos).j)]-d + ff1( .
Note that the summation in (5.2.1) is from j = 1 rather than j = 0. This is because
go cannot be evaluated for positive d . However, the omission of the zero
frequency does remove the mean. The unknown parameters are <tjj, <7^ and d, but <P^
may be concentrated out of the likelihood function by a reparameterisation in
which a* is replaced by the signal-noise ratio q = a^/a^. On the other hand if a
distribution is assumed for st, then a^ is known. Breidt, Crato and de Lima (1993)
show the consistency of the QML estimator.
When d lies between 0.5 and one, ht is nonstationary, but differencing the
log yj 's yields a zero mean stationary process, the SGF of which is
gj = o*[2(l - cos^-)]1^ + 2(1 - cos kj) a\ .
One of the attractions of long memory models is that inference is not affected by
the kind of unit root issues which arise with autoregressions. Thus a likelihood
based test of the hypothesis that d = 1 against the alternative that it is less than
one can be constructed using standard theory; see Robinson (1993).
5.2.4. Comparison of GMM and QML
Simulation evidence on the finite sample performance of GMM and QML can be
found in Andersen and Sorensen (1993), Ruiz (1994), Jacquier, Poison and Rossi
(1994), Breidt and Carriquiry (1995), Andersen and Sorensen (1996) and Harvey
and Shephard (1996). The general conclusion seems to be that QML gives
estimates with a smaller MSE when the volatility is relatively strong as reflected in a
high coefficient of variation. This is because the normally distributed volatility
component in the measurement equation, (3.2.8), is large relative to the non-
normal error term. With a lower coefficient of variation, GMM dominates.
However, in this case Jacquier, Poison and Rossi (1994, p. 383) observe that "...
the performance of both the QML and GMM estimators deteriorates rapidly." In

172
E. Ghysels, A. C. Harvey and E. Renault
other words the case for one of the more computer intensive methods outlined in
Section 5.6 becomes stronger.
Other things being equal, an AR coefficient, 4>, close to one tends to favor
QML because the autocorrelations are slow to die out and are hence captured less
well by the moments used in GMM. For the same reason, GMM is likely to be
rather poor in estimating a long memory model.
The attraction of QML is that it is very easy to implement and it extends easily
to more general models, for example nonstationary and multivariate ones. At the
same time, it provides filtered and smoothed estimates of the state, and
predictions. The one-step ahead prediction errors can also be used to construct
diagnostics, such as the Box-Ljung statistic, though in evaluating such tests it must be
remembered that the observations are non-normal. Thus even if the hyperpara-
meters are eventually estimated by another method, QML may have a valuable
role to play in finding a suitable model specification.
5.3. Continuous time GMM
Hansen and Scheinkman (1995) propose to estimate continuous time diffusions
using a GMM procedure specifically tailored for such processes. In Section 5.1 we
discussed estimation of SV models which are either explicitly formulated as
discrete time processes or else are discretizations of the continuous time diffusions.
In both cases inference is based on minimizing the difference between
unconditional moments and their sample equivalent. For continuous time processes
Hansen and Scheinkman (1995) draw directly upon the diffusion rather than its
discretization to formulate moment conditions. To describe the generic setup of
the method they proposed let us consider the following (multivariate) system of n
diffusion equations:
dyt = n{yt]6)dt + o{yt;6)dWt . (5.3.1)
A comparison with the notation in Section 2 immediately draws attention to
certain limitations of the setup. First, the functions ne{-) = n(-;9) and
ag{-) = a{-\0) are parameterized by yt only which restricts the state variable
process Ut in Section 2 to contemporaneous values of yt. The diffusion in (5.3.1)
involves a general vector process yt, hence yt could include a volatility process to
accommodate SV models. Yet, the yt vector is assumed observable. For the
moment we will leave these issues aside, but return to them at the end of the
section. Hansen and Scheinkman (1995) consider the infinitesimal operator A
defined for a class of square integrable functions q>: W —> U as follows:
AeHy) = "pM ATi^^^W) ■ (53"2)
Because the operator is defined as a limit, namely:
Aecp{y) = \imCl[fc.{q>{yt)\y0 = y) - y] ,

Stochastic volatility
173
it does not necessarily exist for all square integrable functions (p but only for a
restricted domain D. A set of moment conditions can now be obtained for this
class of functions (p E D. Indeed, as shown for instance by Revuz and Yor (1991),
the following equalities hold:
EAe<p(yt) = 0 , (5.3.3)
E[Ag(p(yt+l)(p(yt) - (p(yt+i)A*g(p{yt)] = 0 , (5.3.4)
where A*e is the adjoint infinitesimal operator of Ag for the scalar product
associated with the invariant measure of the process y?1 By choosing an appropriate
set of functions, Hansen and Scheinkman exploit moment conditions (5.3.3) and
(5.3.4) to construct a GMM estimator of 6.
The choice of the function cp e D and q> € D* determines what moments of the
data are used to estimate the parameters. This obviously raises questions
regarding the choice of functions to enhance efficiency of the estimator but first and
foremost also the identification of 9 via the conditions (5.3.3) and (5.3.4). It was
noted in the beginning of the section that the multivariate process yt, in order to
cover SV models, must somehow include the latent conditional variance process.
Gourieroux and Monfort (1994, 1995) point out that since the moment conditions
based on cp and q> cannot include any latent process it will often (but not always)
be impossible to attain identification of all the parameters, particularly those
governing the latent volatility process. A possible remedy is to augment the model
with observations indirectly related to the latent volatility process, in a sense
making it observable. One possible candidate would be to include in yt both the
security price and the Black-Scholes implied volatilities obtained through option
market quotations for the underlying asset. This approach is in fact suggested by
Pastorello, Renault and Touzi (1993) although not in the context of continuous
time GMM but instead using indirect inference methods which will be discussed
in Section 5.5.38 Another possibility is to rely on the time deformation
representation of SV models as discussed in the context of continuous time GMM
by Conley et al. (1995).
5.4. Simulated method of moments
The estimation procedures discussed so far do not involve any simulation
techniques. From now on we cover methods combining simulation and estimation
beginning with the simulated method of moments (SMM) estimator, which is
covered by Duffie and Singleton (1993) for time series processes.39 In Section 5.1
37 Please note that A% is again associated with a domain D* so that q> 6 D and <j> 6 D* in (5.3.4).
38 It was noted in section 2.1.3 that implied volatilities are biased. The indirect inference
procedures used by Pastorello, Renault and Touzi (1993) can cope with such biases, as will be explained in
section 5.5. The use of option price data is further discussed in section 5.7.
39 SMM was originally proposed for cross-section applications, see Pakes and Pollard (1989) and
McFadden (1989). See also Gourieroux and Monfort (1993a).

174
E. Ghysels, A. C. Harvey and E. Renault
we noted that GMM estimation of SV models is based on minimizing the distance
between a set of chosen sample moments and unconditional population moments
expressed as analytical functions of the model parameters. Suppose now that such
analytical expressions are hard to obtain. This is particularly the case when such
expressions involve marginalizations with respect to a latent process such a
stochastic volatility process. Could we then simulate data from the model for a
particular value of the parameters and match moments from the simulated data
with sample moments as a substitute? This strategy is precisely what SMM is all
about. Indeed, quite often it is fairly straightforward to simulate processes and
therefore take advantage of the SMM procedure. Let us consider again as point
of reference and illustration the (multivariate) diffusion of the previous section
(equation (5.3.1)) and conduct H simulations i = l,...,H using a discretization:
A#(0) = ji(#(0); 0) + o(yt(6); B)et and i = 1,... ,H and t = 1,..., T
where yt(6) are simulated given a parameter 9 and et is i.i.d. Gaussian.40 Subject
to identification and other regularity conditions one then considers
9HT=ATgr^n\\f(yl,...yT)-j-J2f{y\(e),...,yiT(e))\\
with a suitable choice of norm, i.e. weighting matrix for the quadratic form as in
GMM, and function / of the data, i.e. moment conditions. The asymptotic
distribution theory is quite similar to that of GMM, except that simulation
introduces an extra source of random error affecting the efficiency of the SMM
estimator in comparison to its GMM counterpart. The efficiency loss can be
controlled by the choice of H.41
5.5. Indirect inference and moment matching
The key insight of the indirect inference approach of Gourieroux, Monfort and
Renault (1993) and the moment matching approach of Gallant and Tauchen
(1994) is the introduction of an auxiliary model parameterized by a vector, say /?,
in order to estimate the model of interest. In our case the latter is the SV model.42
In the first subsection we will describe the general principle while the second will
focus exclusively on estimating diffusions.
5.5.7. The principle
We noted at the beginning of Section 5 that ARCH type models are relatively easy
to estimate in comparison to SV models. For this reason an ARCH type model
40 We discuss in detail the simulation techniques in the next section. Indeed, to control for the
discretization bias, one has to simulate with a finer sampling interval.
41 The asymptotic variance of the SMM estimator depends on H through a factor(l +H~l), see
e.g. Gourieroux and Monfort (1995).
42 It is worth noting that the simulation based inference methods we will describe here are
applicable to many other types of models for cross-sectional, time series and panel data.

Stochastic volatility
175
may be a possible candidate as an auxiliary model. An alternative strategy would
be to try to summarize the features of the data via a SNP density as developed by
Gallant and Tauchen (1989). This empirical SNP density, or more specifically its
score, could also fulfill the role of auxiliary model. Other possibilities could be
considered as well. The idea is then to use the auxiliary model to estimate /?, so that:
T
PT = Arg max V log/*(* I y,-ij) (5-5.1)
where we restrict our attention here to a simple dynamic model with one lag for
the purpose of illustration. The objective function f* in (5.5.1) can be a pseudo-
likelihood function when the auxiliary model is deliberately misspecified to
facilitate estimation. As an alternative /* can be taken from the class of SNP
densities.43 Gourieroux, Monfort and Renault then propose to estimate the same
parameter vector fi not using the actual sample data but instead using samples
{y't(®)},-i smlulated i = 1, ...H times drawn from the model of interest given 6.
This yields a new estimator of /?, namely:
Pm{e) = Arg max(l//O25>gr$(0) I tf-iW./O • (5-5-2)
fi 1=1 «=i
The next step is to minimize a quadratic distance using a weighting matrix WT to
choose an indirect estimator of 6 based on H simulation replications and a sample
of T observations, namely:
0HT = Arg min(j8r - jM0)) Vr(j8r - jM0)) (5.5.3)
The approach of Gallant and Tauchen (1994) avoids the step of estimating
Pht(@) by computing the score function of f* and minimizing a quadratic
distance similar to (5.5.3) but involving the score function evaluated at fiT and
replacing the sample data by simulated series generated by the model of interest.
Under suitable regularity conditions the estimator GHT is root T consistent and
asymptotically normal. As with GMM and SMM there is again an optimal
weighting matrix. The resulting asymptotic covariance matrix depends on the
number of simulations in the same way the SMM estimator depends on H.
Gourieroux, Monfort and Renault (1993) illustrated the use of indirect
inference estimator with a simple example that we would like to briefly discuss here.
Typically AR models are easy to estimate while MA models require more
elaborate procedures. Suppose the model of interest is a moving average model of
order one with parameter 9. Instead of estimating the MA parameter directly
from the data they propose to estimate an AR(p) model involving the parameter
43 The discussion should not leave the impression that the auxiliary model can only be estimated
via ML-type estimators. Any root T consistent asymptotically normal estimation procedure may be
used.

176
E. Ghysels, A. C. Harvey and E. Renault
vector /?. The next step then consists of simulating data using the MA model and
proceeding further as described above.44 They found that the indirect inference
estimator for 9HT appeared to have better finite sample properties than the more
traditional maximum likelihood estimators for the MA parameter. In fact the
indirect inference estimator exhibited features similar to the median unbiased
estimator proposed by Andrews (1993). These properties were confirmed and
clarified by Gourieroux, Renault and Touzi (1994) who studied the second order
asymptotic expansion of indirect inference estimators and their ability to reduce
finite sample bias.
5.5.2. Estimating diffusions
Let us consider the same diffusion equation as in Section 5.3 which dealt with
continuous time GMM, namely:
dy, = n(yt; &)dt + o{yt; 6)dWt . (5.5.4)
In Section 5.3 we noted that the above equation holds under certain
restrictions such as the functions \i and a being restricted to yt as arguments. While
these restrictions were binding for the setup of Section 5.3 this will not be the case
for the estimation procedures discussed here. Indeed, equation (5.5.4) is only used
as an illustrative example. The diffusion is then simulated either via exact
discretizations or some type of approximate discretization (e.g. Euler or Mil'shtein,
see Pardoux and Talay (1985) or Kloeden and Platten (1992) for further details).
More precisely we define the process y) ' such that:
j&o, - $ + *{&<>)* + °{yfhey,24U ■ (5-5-5)
Under suitable regularity conditions (see for instance Strook and Varadhan
(1979)) we know that the diffusion admits a unique solution (in distribution) and
the process y) ' converges to yt as 5 goes to zero. Therefore one can expect to
simulate yt quite accurately for S sufficiently small. The auxiliary model may be a
discretization of (5.5.4) choosing 5 = 1. Hence, one formulates a ML estimator
based on the nonlinear AR model appearing in (5.5.5) setting d = 1. To control
for the discretization bias one can simulate the underlying diffusion with 8 = 1/10
or 1/20, for instance, and aggregate the simulated data to correspond with the
sampling frequency of the DGP. Broze, Scaillet and Zako'ian (1994) discuss the
effect of the simulation step size on the asymptotic distribution.
The use of simulation-based inference methods becomes particularly
appropriate and attractive when diffusions involve latent processes, such as is the case
44 Again one could use a score principle here, following Gallant and Tauchen (1994). In fact in a
linear Gaussian setting the SNP approach to fit data generated by a MA (1) model would be to
estimate an AR(p) model. Ghysels, Khalaf and Vodounou (1994) provide a more detailed discussion of
score-based and indirect inference estimators of MA models as well as their relation with more
standard estimators.

Stochastic volatility
177
with SV models. Gourieroux and Monfort (1994, 1995) discuss several examples
and study their performance via Monte Carlo simulation. It should be noted that
estimating the diffusion at a coarser discretization is not the only possible choice
of auxiliary model. Indeed, Pastorello, Renault and Touzi (1993), Engle and Lee
(1994) and Gallant and Tauchen (1994) suggest the use of ARCH-type models.
There have been several successful applications of these methods to financial
time series. They include Broze et al. (1995), Engle and Lee (1994), Gallant, Hsieh
and Tauchen (1994), Gallant and Tauchen (1994, 1995), Ghysels, Gourieroux and
Jasiak (1995b), Ghysels and Jasiak (1994a,b), Pastorello et al. (1993), among
others.
5.6. Likelihood-based and Bayesian methods
In a Gaussian linear state space model the likelihood function is constructed from
the one step ahead prediction errors. This prediction error decomposition form of
the likelihood is used as the criterion function in QML, but of course it is not the
exact likelihood in this case. The exact filter proposed by Watanabe (1993) will, in
principle, yield the exact likelihood. However, as was noted in Section 3.4.2,
because this filter uses numerical integration, it takes a long time to compute and
if numerical optimization is to be carried out with respect to the hyperparameters
it becomes impractical.
Kim and Shephard (1994) work with the linear state space form used in QML
but approximate the log(x2) distribution of the measurement error by a mixture
of normals. For each of these normals, a prediction error decomposition
likelihood function can be computed. A simulated EM algorithm is used to find the
best mixture and hence calculate approximate ML estimates of the hyperpar-
amaters.
The exact likelihood function can also be constructed as a mixture of
distributions for the observations conditional on the volatilities, that is
L{y-4>,c1vG1) = J p{y\h)p{h)dh
where y and h contain the T elements of yt and ht respectively. This expression
can be written in terms of the a2 's, rather than their logarithms, the ht 's, but it
makes little difference to what follows. Of course the problem is that the above
likelihood has no closed form, so it must be calculated by some kind of simulation
method. Excellent discussions can be found in Shephard (1995) and in Jacquier,
Poison and Rossi (1994), including the comments. Conceptually, the simplest
approach is to use Monte Carlo integration by drawing from the unconditional
distribution of h for given values of the parameters,(0, a2, a2), and estimating the
likelihood as the average of the p(y\h) 's. This is then repeated, searching over
0, a2 until the maximum of the simulated likelihood is found. As it stands this
procedure is not very satisfactory, but it may be improved by using ideas of
importance sampling. This has been implemented for ML estimation of SV

178
E. Ghysels, A. C. Harvey and E. Renault
models by Danielsson and Richard (1993) and Danielsson (1994). However, the
method becomes more difficult as the sample size increases.
A more promising way of attacking likelihood estimation by simulation
techniques is to use Markov Chain Monte Carlo (MCMC) to draw from the
distribution of volatilities conditional on the observations. Ways in which this can
be done were outlined in sub-Section 3.4.2 on nonlinear filters and smoothers.
Kim and Shephard (1994) suggest a method of computing ML estimators by
putting their multimove algorithm within a simulated EM algorithm. Jacquier,
Poison and Rossi (1994) adopt a Bayesian approach in which the specification of
the model has a hierarchical structure in which a prior distribution for the hy-
perparameters, q> = (<x,,,<^,<x)', joins the conditional distributions, y\h and h\q>.
(Actually the at 's are used rather than the ht 's). The joint posterior of h and q> is
proportional to the product of these three distributions, that is
p(h, q>\y) oc p(y\h)p{h\(p)p{q>). The introduction of h makes the statistical
treatment tractable and is an example of what is called data augmentation; see
Tanner and Wong (1987). From the joint posterior, p(h,q>\y), the marginal
p{h\y) solves the smoothing problem for the unobserved volatilities, taking
account of the sampling variability in the hyperparameters. Conditional on h, the
posterior of q>, p(q>\h, y) is simple to compute from standard Bayesian treatment
of linear models. If it were also possible to sample directly from p(h\q>, y) at low
cost, it would be straightforward to construct a Markov chain by alternating back
and forth drawing from p(q>\h,y) and p(h\q>,y). This would produce a cyclic
chain, a special case of which is the Gibbs sampler. However, as was noted in sub-
Section 3.4.2, Jacquier, Poison and Rossi (1994) show that it is much better to
decompose p{h\<p,y) into a set of univariate distributions in which each ht, or
rather ct, is conditioned on all the others.
The prior distribution for co, the parameters of the volatility process in JPR
(1994), is the standard conjugate prior for the linear model, a (truncated) Normal-
Gamma. The priors can be made extremely diffuse while remaining proper. JPR
conduct an extensive sampling experiment to document the performance of this
and more traditional approaches. Simulating stochastic volatility series, they
compare the sampling performances of the posterior mean with that of the QML
and GMM point estimates. The MCMC posterior mean exhibit root mean
squared errors anywhere between half and a quarter of the size of the GMM and
QML point estimates. Even more striking are the volatility smoothing
performance results. The root mean squared error of the posterior mean of h, produced
by the Bayesian filter is 10% smaller than the point estimate produced by an
approximate Kalman filter supplied with.the true parameters.
Shephard and Kim in their comment of JPR (1994) point out that for very high
(j> and small an, the rate of convergence of the JPR algorithm will slow down.
More draws will then be required to obtain the same amount of information.
They propose to approximate the volatility disturbance with a discrete mixture of
normals. The benefit of the method is that a draw of the vector h is then possible,
faster than T draws from each h,. However this is at the cost that the draws
navigate in a much higher dimensional space due to the discretisation effected.

Stochastic volatility
179
Also, the convergence of chains based upon discrete mixtures is sensitive to the
number of components and their assigned probability weights. Mahieu and
Schotman (1994) add some generality to the Shephard and Kim idea by letting the
data produce estimates of the characteristics of the discretized state space
(probabilities, mean and variance).
The original implementation of the JPR algorithm was limited to a very basic
model of stochastic volatility, AR(1) with uncorrected mean and volatility
disturbances. In a univariate setup, correlated disturbances are likely to be
important for stock returns, i.e., the so called leverage effect. The evidence in
Gallant, Rossi, and Tauchen (1994) also points at non normal conditional errors
with both skewness and kurtosis. Jacquier, Poison, and Rossi (1995a) show how
the hierarchical framework allows the convenient extension of the MCMC
algorithm to more general models. Namely, they estimate univariate stochastic
volatility models with correlated disturbances, and skewed and fat-tailed variance
disturbance, as well as multivariate models. Alternatively, the MCMC algorithm
can be extended to a factor structure. The factors exhibit stochastic volatility and
can be observable or non-observable.
5.7. Inference and option price data
Some of the continuous time SV models currently found in the literature were
developed to answer questions regarding derivative security pricing. Given this
rather explicit link between derivates and SV diffusions it is perhaps somewhat
surprising that relatively little attention has been paid to the use of option price
data to estimate continuous time diffusions. Melino (1994) in his survey in fact
notes: "Clearly, information about the stochastic properties of an asset's price is
contained both in the history of the asset's price and the price of any options written
on it. Current strategies for combining these two sources of information, including
implicit estimation, are uncomfortably ad hoc. Statistically speaking, we need to
model the source of the prediction errors in option pricing and to relate the
distribution of these errors to the stock price process". For example implicit
estimation, like computation of BS implied volatilities, is certainly uncomfortably ad
hoc from a statistical point of view. In general, each observed option price
introduces one source of prediction error when compared to a pricing model. The
challenge is to model the joint nondegenerate probability distribution of options
and asset prices via a number of unobserved state variables. This approach has
been pursued in a number of recent papers, including Christensen (1992), Renault
and Touzi (1992), Pastorello et al. (1993), Duan (1994) and Renault (1995).
Christensen (1992) considers a pricing model for n assets as a function of a
state vector xt which is (/ + n) dimensional and divided into a /-dimensional
observed (zt) and «-dimensional unobserved (cot) components. Let pt be the price
vector of the n assets, then:
Pt = m(z„ co„ 6) .
(5.7.1)

180
E. Ghysels, A. C. Harvey and E. Renault
Equation (5.7.1) provides a one-to-one relationship between the n latent state
variables cot and the n observed prices pt, for given zt and 0. From a financial
viewpoint, it implies that the n assets are appropriate instruments to complete the
markets if we assume that the observed state variables zt are already mimicked by
the price dynamics of other (primitive) assets. Moreover, from a statistical
viewpoint it allows full structural maximum likelihood estimation provided the
log-likelihood function for observed prices can be deduced easily from a statistical
model for xt. For instance, in a Markovian setting where, conditionally on xq , the
joint distribution of x\ = [xt)x<t<T is given by the density:
T
/x(*f|*O,0) =n/(z,,o>,|zr_,,«»,_,, 0) (5.7.2)
t=\
the conditional distribution of data D\ = (pt,zt)i<t<T given D0 = (po,zo) is
obtained by the usual Jacobian formula:
T
fD{DTx\D0,e) =Y[f[zt,mgi{zt, pt)\zt-i,mg\zt-u pt-\),0]x
t=\
x \Vam(zt,nqx{zt,pt),e)\~X (5.7.3)
where nigl(z,.) is the co-inverse of m(z,.,6) denned formally by
mg\z,m(z,co,6)) = co while Vram (■) represents the columns corresponding to co
of the Jacobian matrix. This MLE using price data of derivatives was proposed
independently by Christensen (1992) and Duan (1994). Renault and Touzi (1992)
were instead more specifically interested in the Hull and White option pricing
formula with: zt = St observed underlying asset price, and cot — at unobserved
stochastic volatility process. Then with the joint process xt = (St, at) being
Markovian we have a call price of the form:
Ct = m{x„e,K)
where 0 = (a',y') involves two types of parameters: (1) the vector a of parameters
describing the dynamics of the joint process xt — (St, at) which under the
equivalent martingale measure allows to compute the expectation with respect to
the (risk-neutral) conditional probability distribution of y2(t,t + h) given o>; and
(2) the vector y of parameters which characterize the risk premia determining the
relation between the risk neutral probability distribution of the x process and the
Data Generating Process.
Structural MLE is often difficult to implement. This motivated Renault and
Touzi (1992) and Pastorello, Renault and Touzi (1993) to consider less efficient
but simpler and more robust procedures involving some proxies of the structural
likelihood (5.7.3).
To illustrate these procedures let us consider the standard log-normal SV
model in continuous time:

Stochastic volatility
181
d logff, = k(a - logat)dt + cdWat . (5.7.4)
Standard option pricing arguments allow us to ignore misspecifications of the
drift of the underlying asset price process. Hence, a first step towards simplicity
and robustness is to isolate from the likelihood function the volatility dynamics,
namely:
IK2
7rc2)-1/2exp
-(2c2)-1 (log ex, - e-^logcx,,, - 0(1 - e~m))
(5.7.5)
associated with a sample atj,i = 1,..., n and u — f,-_, = At. To approximate this
expression one can consider a direct method, as in Renault and Touzi (1992) or an
indirect method, as in Pastorello et al. (1993). The former involves calculating
implied volatilities from the Hull and White model to create pseudo samples ctl
parameterized by k, a and c and computing the maximum of (5.7.5) with respect
to those three parameters.45 Pastorello et al. (1993) proposed several indirect
inference methods, described in Section 5.5, in the context of (5.7.5). For instance,
they propose to use an indirect inference strategy involving GARCH(1,1)
volatility estimates obtained from the underlying asset (also independently suggested
by Engle and Lee (1994)). This produces asymptotically unbiased but rather
inefficient estimates. Pastorello et al. indeed find that an indirect inference
simplification of the Renault and Touzi direct procedure involving option prices is far
more efficient. It is a clear illustration of the intuition that the use of option price
data paired with suitable statistical methods should largely improve the accuracy
of estimating volatility diffusion parameters.
5.8. Regression models with stochastic volatility
A single equation regression model with stochastic volatility in the disturbance
term may be written
y, = x'tp + ut , t=l,...,T , (5.8.1)
where yt denotes the tth observation, xt is a k x 1 vector of explanatory variables,
P is a k x 1 vector of coefficients and ut = aet exp(0.5/z,) as discussed in Section 3.
As a special case, the observations may simply have a non-zero mean so that
x\p =]i yt.
Since ut is stationary, an OLS regression of yt on xt yields a consistent
estimator of p. However it is not efficient.
45 The direct maximization of (5.7.5) using BS implied volatilities has also been proposed, see e.g.
Heynen, Kemna and Vorst (1994). Obviously the use of BS implied volatility induces a misspecification
bias due to the BS model assumptions.

182
E. Ghysels, A. C. Harvey and E. Renault
For given values of the SV parameters, $ and a1, a smoothed estimator of ht,
ht\T, can be computed using one of the methods outlined in Section 3.4.
Multiplying (5.8.1) through by exp(-.5Afir) gives
yt
1 + u,, t = l,
(5.8.2)
where the u/s can be thought of as heteroskedasticity corrected disturbances.
Harvey and Shephard (1993) show that these disturbances have zero mean,
constant variance and are serially uncorrelated and hence suggest the
construction of a feasible GLS estimator
(=i
t\T
XtXt
(=1
~A||7
xtyt
(5.8.3)
In the classical heteroskedastic regression model ht is deterministic and
depends on a fixed number of unknown parameters. Because these parameters can
be estimated consistently, the feasible GLS estimator has the same asymptotic
distribution as the GLS estimator. Here ht is stochastic and the MSE of its
estimator is of 0(1). The situation is therefore somewhat different. Harvey and
Shephard (1993) show that, under standard regularity conditions on the sequence
of xt, P is asymptotically normal with mean P and a covariance matrix which can
be consistently estimated by
avar(jS) =
-i -i
,-h,
■l\T
xtxt
J2(yt-x'tp)2e-
2h*Txtx't
t=\
^ e^Tx,x't
(=1
(5.8.4)
When ht\T is the smoothed estimate given by the linear state space form, the
analysis in Harvey and Shephard (1993) suggests that, asymptotically, the feasible
GLS estimator is almost as efficient as the GLS estimator and considerably more
efficient than the OLS estimator. It would be possible to replace exp(A!|r) by a
better estimate computed from one of the methods described in Section 3.4 but
this may not have much effect on the efficiency of the resulting feasible GLS
estimator of p.
When ht is nonstationary, or nearly nonstationary, Hansen (1995) shows that it
is possible to construct a feasible adaptive least squares estimator which is
asymptotically equivalent to GLS.
Conclusions
No survey is ever complete. There are two particular areas we expect will flourish
in the years to come but which we were not able to cover. The first is the area of
market microstructures which is well surveyed in a recent review paper by
Goodhart and O'Hara (1995). With the ever increasing availability of high fre-

Stochastic volatility
183
quency data series, we anticipate more work involving game theoretic models.
These can now be estimated because of recent advances in econometric methods,
similar to those enabling us to estimate diffusions. Another area where we expect
interesting research to emerge is that involving nonparametric procedures to
estimate SV continuous time and derivative securities models. Recent papers
include Ait-Sahalia (1994), Ait-Sahalia et al. (1994), Bossaerts, Hafner and Hardle
(1995), Broadie et al. (1995), Conley et al. (1995), Elsheimer et al. (1995), Gourie-
roux, Monfort and Tenreiro (1994), Gourieroux and Scaillet (1995), Hutchinson,
Lo and Poggio (1994), Lezan et al. (1995), Lo (1995), Pagan and Schwert (1992).
Research into the econometrics of Stochastic Volatility models is relatively
new. As our survey has shown, there has been a burst of activity in recent years
drawing on the latest statistical technology. As regards the relationship with
ARCH, our view is that SV and ARCH are not necessarily direct competitors, but
rather complement each other in certain respects. Recent advances such as the use
of ARCH models as filters, the weakening of GARCH and temporal aggregation
and the introduction of nonparametric methods to fit conditional variances,
illustrate that a unified strategy for modelling volatility needs to draw on both
ARCH and SV.
References
Abramowitz, M. and N. C. Stegun (1970). Handbook of Mathematical Functions. Dover Publications
Inc., New York.
Ait-Sahalia, Y. (1994). Nonparametric pricing of interest rate derivative securities. Discussion Paper,
Graduate School of Business, University of Chicago.
Ait-Sahalia, Y. S. J. Bickel and T. M. Stoker (1994). Goodness-of-Fit tests for regression using kernel
methods. Discussion Paper, University of Chicago.
Amin, K. L. and V. Ng (1993). Equilibrium option valuation with systematic stochastic volatility. J.
Financed, 881-910.
Andersen, T. G. (1992). Volatility. Discussion paper, Northwestern University.
Andersen, T. G. (1994). Stochastic autoregressive volatility: A framework for volatility modeling.
Math. Finance 4, 75-102.
Andersen, T. G. (1996). Return volatility and trading volume: An information flow interpretation of
stochastic volatility. J. Finance, to appear.
Andersen, T. G. and T. Bollerslev (1995). Intraday seasonality and volatility persistence in financial
Markets. J. Emp. Finance, to appear.
Andersen, T. G. and B. Sarensen (1993). GMM estimation of a stochastic volatility model: A Monte
Carlo study. J. Business Econom. Statist, to appear.
Andersen, T. G. and B. Sarensen (1996). GMM and QML asymptotic standard deviations in
stochastic volatility models: A response to Ruiz (1994). J. Econometrics, to appear.
Andrews, D. W. K. (1993). Exactly median-unbiased estimation of first order autoregressive unit root
models. Econometrica 61, 139-165.
Bachelier, L. (1900). Theorie de la speculation. Ann. Sci. Ecole Norm. Sup. 17, 21-86, [On the Random
Character of Stock Market Prices (Paul H. Cootner, ed.) The MIT Press, Cambridge, Mass. 1964].
Baillie, R. T. and T. Bollerslev (1989). The message in daily exchange rates: A conditional variance
tale. J. Business Econom. Statist. 7, 297-305.
Baillie, R. T. and T. Bollerslev (1991). Intraday and Interday volatility in foreign exchange rates. Rev.
Econom. Stud. 58, 565-585.

184
E. Ghysels, A. C. Harvey and E. Renault
Baillie, R. T., T. Bollerslev and H. O. Mikkelsen (1993). Fractionally integrated generalized auto-
regressive conditional heteroskedasticity. J. Econometrics, to appear.
Bajeux, I. and J. C. Rochet (1992). Dynamic spanning: Are options an appropriate instrument? Math.
Finance, to appear.
Bates, D. S. (1995a). Testing option pricing models. In: G. S. Maddala ed., Handbook of Statistics,
Vol. 14, Statistical Methods in Finance. North Holland, Amsterdam, in this volume.
Bates, D. S. (1995b). Jumps and stochastic volatility: Exchange rate processes implicit in PHLX
Deutschemark options. Rev. Financ. Stud., to appear.
Beckers, S. (1981). Standard deviations implied in option prices as predictors of future stock price
variability. J. Banking Finance 5, 363-381.
Bera, A. K. and M. L. Higgins (1995). On ARCH models: Properties, estimation and testing. In: L.
Exley, D. A. R. George, C. J. Roberts and S. Sawyer eds., Surveys in Econometrics. Basil Blackwell:
Oxford, Reprinted from J. Econom. Surveys.
Black, F. (1976). Studies in stock price volatility changes. Proceedings of the 1976 Business Meeting of
the Business and Economic Statistics Section, Amer. Statist. Assoc. 177-181.
Black, F. and M. Scholes (1973). The pricing of options and corporate liabilities. J. Politic. Econom.
81, 637-654.
Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. J. Econometrics 31,
307-327.
Bollerslev, T., Y. C. Chou and K. Kroner (1992). ARCH modelling in finance: A selective review of the
theory and empirical evidence. J. Econometrics 52, 201-224.
Bollerslev, T. and R. Engle (1993). Common persistence in conditional variances. Econometrica
61, 166-187.
Bollerslev, T., R. Engle and D. Nelson (1994). ARCH models. In: R. F. Engle and D. McFadden eds.,
Handbook of Econometrics, Volume IV. North-Holland, Amsterdam.
Bollerslev, T., R. Engle and J. Wooldridge (1988). A capital asset pricing model with time varying
covariances. J. Politic. Econom. 96, 116-131.
Bollerslev, T. and E. Ghysels (1994). On periodic autoregression conditional heteroskedasticity. J.
Business Econom. Statist., to appear.
Bollerslev, T. and H. O. Mikkelsen (1995). Modeling and pricing long-memory in stock market
volatility. J. Econometrics, to appear.
Bossaerts, P., C. Hafner and W. Hardle (1995). Foreign exchange rates have surprising volatility.
Discussion Paper, CentER, University of Tilburg.
Bossaerts, P. and P. Hillion (1995). Local parametric analysis of hedging in discrete time. J.
Econometrics, to appear.
Breidt, F. J., N. Crato and P. de Lima (1993). Modeling long-memory stochastic volatility. Discussion
paper, Iowa State University.
Breidt, F. J. and A. L. Carriquiry (1995). Improved quasi-maximum likelihood estimation for
stochastic volatility models. Mimeo, Department of Statistics, University of Iowa.
Broadie, M., J. Detemple, E. Ghysels and O. Torres (1995). American options with stochastic
volatility: A nonparametric approach. Discussion Paper, CIRANO.
Broze, L., O. Scaillet and J. M. Zakoian (1994). Quasi indirect inference for diffusion processes.
Discussion Paper CORE.
Broze, L., O. Scaillet and J. M. Zakoian (1995). Testing for continuous time models of the short term
interest rate. J. Emp. Finance, 199-223.
Campa, J. M. and P. H. K. Chang (1995). Testing the expectations hypothesis on the term structure of
implied volatilities in foreign exchange options. J. Finance 50, to appear.
Campbell, J. Y. and A. S. Kyle (1993). Smart money, noise trading and stock price behaviour. Rev.
Econom. Stud. 60, 1-34.
Canina, L. and S. Figlewski (1993). The informational content of implied volatility. Rev. Financ. Stud.
6, 659-682.
Canova, F. (1992). Detrending and Business Cycle Facts. Discussion Paper, European University
Institute, Florence.

Stochastic volatility
185
Chesney, M. and L. Scott (1989). Pricing European currency options: A comparison of the modified
Black-Scholes model and a random variance model. J. Financ. Quant. Anal. 24, 267-284.
Cheung, Y.-W. and F. X. Diebold (1994). On maximum likelihood estimation of the differencing
parameter of fractionally-integrated noise with unknown mean. J. Econometrics 62, 301-316.
Chiras, D. P. and S. Manaster (1978). The information content of option prices and a test of market
efficiency. J. Financ. Econom. 6, 213-234.
Christensen, B. J. (1992). Asset prices and the empirical martingale model. Discussion Paper, New
York University.
Christie, A. A. (1982). The stochastic behavior of common stock variances: Value, leverage, and
interest rate effects. J. Financ. Econom. 10, 407-432.
Clark, P. K. (1973). A subordinated stochastic process model with finite variance for speculative
prices. Econometrica 41, 135-156.
Clewlow, L and X. Xu (1993). The dynamics of stochastic volatility. Discussion Paper, University of
Warwick.
Comte, F. and E. Renault (1993). Long memory continuous time models. J. Econometrics, to appear.
Comte, F. and E. Renault (1995). Long memory continuous time stochastic volatility models. Paper
presented at the HFDF-I Conference, Zurich.
Conley, T., L. P. Hansen, E. Luttmer and J. Scheinkman (1995). Estimating subordinated diffusions
from discrete time data. Discussion paper, University of Chicago.
Cornell, B. (1978). Using the options pricing model to measure the uncertainty producing effect of
major announcements. Financ. Mgmt. 7, 54-59.
Cox, J. C. (1975). Notes on option pricing I: Constant elasticity of variance diffusions. Discussion
Paper, Stanford University.
Cox, J. C. and S. Ross (1976). The valuation of options for alternative stochastic processes. J. Financ.
Econom. 3, 145-166.
Cox, J. C. and M. Rubinstein (1985). Options Markets. Englewood Cliffs, Prentice-Hall, New Jersey.
Dacorogna, M. M, U. A. Miiller, R. J. Nagler, R. B. Olsen and O. V. Pictet (1993). A geographical
model for the daily and weekly seasonal volatility in the foreign exchange market. J. Internal.
Money Finance 12, 413-438.
Danielsson, J. (1994). Stochastic volatility in asset prices: Estimation with simulated maximum
likelihood. J. Econometrics 61, 375-400.
Danielsson, J. and J. F. Richard (1993). Accelerated Gaussian importance sampler with application to
dynamic latent variable models. J. Appl. Econometrics 3, S153-S174.
Dassios, A. (1995). Asymptotic expressions for approximations to stochastic variance models. Mimeo,
London School of Economics.
Day, T. E. and C. M. Lewis (1988). The behavior of the volatility implicit in the prices of stock index
options. J. Financ. Econom. 22, 103-122.
Day, T. E. and C. M. Lewis (1992). Stock market volatility and the information content of stock index
options. J. Econometrics 52, 267-287.
Diebold, F. X. (1988). Empirical Modeling of Exchange Rate Dynamics. Springer Verlag, New York.
Diebold, F. X. and J. A. Lopez (1995). Modeling Volatility Dynamics. In: K. Hoover ed.,
Macroeconomics: Developments, Tensions and Prospects.
Diebold, F. X. and M. Nerlove (1989). The dynamics of exchange rate volatility: A multivariate latent
factor ARCH Model. J. Appl. Econometrics 4, 1-22.
Ding, Z., C. W. J. Granger and R. F. Engle (1993). A long memory property of stock market returns
and a new model. J. Emp. Finance 1, 83-108.
Diz, F. and T. J. Finucane (1993). Do the options markets really overreact? J. Futures Markets 13,
298-312.
Drost, F. C. and T. E. Nijman (1993). Temporal aggregation of GARCH processes. Econometrica
61, 909-927.
Drost, F. C. and B. J. M. Werker (1994). Closing the GARCH gap: Continuous time GARCH
modelling. Discussion Paper CentER, University of Tilburg.
Duan, J. C. (1994). Maximum likelihood estimation using price data of the derivative contract. Math.
Financed 155-167.

186
E. Ghysels, A. C. Harvey and E. Renault
Duan, J. C. (1995). The GARCH option pricing model. Math. Finance 5, 13-32.
Duffle, D. (1989). Futures Markets. Prentice-Hall International Editions.
Duffle, D. (1992). Dynamic Asset Pricing Theory. Princeton University Press.
Duffie, D. and K. J. Singleton (1993). Simulated moments estimation of Markov models of asset
prices. Econometrica 61, 929-952.
Dunsmuir, W. (1979). A central limit theorem for parameter estimation in stationary vector time series
and its applications to models for a signal observed with noise. Ann. Statist. 7, 490-506.
Easley, D. and M. O'Hara (1992). Time and the process of security price adjustment. J. Finance, 47,
577-605.
Ederington, L. H. and J. H. Lee (1993). How markets process information: News releases and
volatility. J. Finance 48, 1161-1192.
Elsheimer, B., M. Fisher, D. Nychka and D. Zirvos (1995). Smoothing splines estimates of the discount
function based on US bond Prices. Discussion Paper Federal Reserve, Washington, D.C.
Engle, R. F. (1982). Autoregressive conditional heteroskedasticity with estimates of the variance of
United Kingdom inflation. Econometrica 50, 987-1007.
Engle, R. F. and C. W. J. Granger (1987). Co-integration and error correction: Representation,
estimation and testing. Econometrica 55, 251-576.
Engle, R. F. and S. Kozicki (1993). Testing for common features. J. Business Econom. Statist. 11, 369-
379.
Engle, R. F. and G. G. J. Lee (1994). Estimating diffusion models of stochastic volatility. Discussion
Paper, Univeristy of California at San Diego.
Engle, R. F. and C. Mustafa (1992). Implied ARCH models from option prices. J. Econometrics 52,
289-311.
Engle, R. F. and V. K. Ng (1993). Measuring and testing the impact of news on volatility. J. Finance
48, 1749-1801.
Fama, E. F. (1963). Mandelbrot and the stable Paretian distribution. J. Business 36, 420-429.
Fama, E. F. (1965). The behavior of stock market prices. J. Business 38, 34-105.
Foster, D. and S. Viswanathan (1993a). The effect of public information and competition on trading
volume and price volatility. Rev. Financ. Stud. 6, 23-56.
Foster, D. and S. Viswanathan (1993b). Can speculative trading explain the volume volatility relation.
Discussion Paper, Fuqua School of Business, Duke University.
French, K. and R. Roll (1986). Stock return variances: The arrival of information and the reaction of
traders. J. Financ. Econom. 17, 5-26.
Gallant, A. R., D. A. Hsieh and G. Tauchen (1994). Estimation of stochastic volatility models with
suggestive diagnostics. Discussion Paper, Duke University.
Gallant, A. R., P. E. Rossi and G. Tauchen (1992). Stock prices and volume. Rev. Financ. Stud. 5,199-
242.
Gallant, A. R., P. E. Rossi and G. Tauchen (1993). Nonlinear dynamic structures. Econometrica
61, 871-907.
Gallant, A. R. and G. Tauchen (1989). Semiparametric estimation of conditionally constrained
heterogeneous processes: Asset pricing applications. Econometrica 57, 1091-1120.
Gallant, A. R. and G. Tauchen (1992). A nonparametric approach to nonlinear time series analysis:
Estimation and simulation. In: E. Parzen, D. Brillinger, M. Rosenblatt, M. Taqqu, J. Geweke and
P. Caines eds., New Dimensions in Time Series Analysis. Springer-Verlag, New York.
Gallant, A. R. and G. Tauchen (1994). Which moments to match. Econometric Theory, to appear.
Gallant, A. R. and G. Tauchen (1995). Estimation of continuous time models for stock returns and
interest rates. Discussion Paper, Duke University.
Gallant, A. R. and H. White (1988). A Unified Theory of Estimation and Inference for Nonlinear
Dynamic Models. Basil Blackwell, Oxford.
Garcia, R. and E. Renault (1995). Risk aversion, intertemporal substitution and option pricing.
Discussion Paper CIRANO.
Geweke, J. (1994). Comment on Jacquier, Poison and Rossi. J. Business Econom. Statist. 12, 397-399.

Stochastic volatility
187
Geweke, J. (1995). Monte Carlo simulation and numerical integration. In: H. Amman, D. Kendrick
and J. Rust eds., Handbook of Computational Economics. North Holland.
Ghysels, E., C. Gourieroux and J. Jasiak (1995a). Market time and asset price movements: Theory and
estimation. Discussion paper CIRANO and C.R.D.E., Univeriste de Montreal.
Ghysels, E., C. Gourieroux and J. Jasiak (1995b). Trading patterns, time deformation and stochastic
volatility in foreign exchange markets. Paper presented at the HFDF Conference, Zurich.
Ghysels, E. and J. Jasiak (1994a). Comments on Bayesian analysis of stochastic volatility models. J.
Business Econom. Statist. 12, 399-401.
Ghysels, E. and J. Jasiak (1994b). Stochastic volatility and time deformation an application of trading
volume and leverage effects. Paper presented at the Western Finance Association Meetings, Santa
Fe.
Ghysels, E., L. Khalaf and C. Vodounou (1994). Simulation based inference in moving average
models. Discussion Paper, CIRANO and C.R.D.E.
Ghysels, E., H. S. Lee and P. Siklos (1993). On the (mis)specification of seasonality and its
consequences: An empirical investigation with U.S. Data. Empirical Econom. 18, 747-760.
Goodhart, C. A. E. and M. O'Hara (1995). High frequency data in financial markets: Issues and
applications. Paper presented at HFDF Conference, Zurich.
Gourieroux, C. and A. Monfort (1993a). Simulation based Inference: A survey with special reference
to panel data models. J. Econometrics 59, 5-33.
Gourieroux, C. and A. Monfort (1993b). Pseudo-likelihood methods in Maddalaet al. ed., Handbook
of Statistics Vol. 11, North Holland, Amsterdam.
Gourieroux, C. and A. Monfort (1994). Indirect inference for stochastic differential equations.
Discussion Paper CREST, Paris.
Gourieroux, C. and A. Monfort (1995). Simulation-Based Econometric Methods. CORE Lecture
Series, Louvain-la-Neuve.
Gourieroux, C, A. Monfort and E. Renault (1993). Indirect inference. J. Appl. Econometrics 8, S85-
S118.
Gourieroux, C, A. Monfort and C. Tenreiro (1994). Kernel M-estimators: Nonparametric diagnostics
for structural models. Discussion Paper, CEPREMAP.
Gourieroux, C, A. Monfort and C. Tenreiro (1995). Kernel M-estimators and functional residual
plots. Discussion Paper CREST - ENSAE, Paris.
Gourieroux, C, E. Renault and N. Touzi (1994). Calibration by simulation for small sample bias
correction. Discussion Paper CREST.
Gourieroux, C. and O. Scaillet (1994). Estimation of the term structure from bond data. J. Emp.
Finance, to appear.
Granger, C. W. J. and Z. Ding (1994). Stylized facts on the temporal and distributional properties of
daily data for speculative markets. Discussion Paper, University of California, San Diego.
Hall, A. R. (1993). Some aspects of generalized method of moments estimation in Maddala et al. ed.,
Handbook of Statistics Vol. 11, North Holland, Amsterdam.
Hamao, Y., R. W. Masulis and V. K. Ng (1990). Correlations in price changes and volatility across
international stock markets. Rev. Financ. Stud. 3, 281-307.
Hansen, B. E. (1995). Regression with nonstationary volatility. Econometrica 63, 1113-1132.
Hansen, L. P. (1982). Large sample properties of generalized method of moments estimators.
Econometrica 50, 1029-1054.
Hansen, L. P. and J. A. Scheinkman (1995). Back to the future: Generating moment implications for
continuous-time Markov processes. Econometrica 63, 767-804.
Harris, L. (1986). A transaction data study of weekly and intradaily patterns in stock returns. J.
Financ. Econom. 16, 99-117.
Harrison, M. and D. Kreps (1979). Martingale and arbitrage in multiperiod securities markets. J.
Econom. Theory 20, 381-408.
Harrison, J. M. and S. Pliska (1981). Martingales and stochastic integrals in the theory of continuous
trading. Stochastic Processes and Their Applications 11, 215-260.

188
E. Ghysels, A. C. Harvey and E. Renault
Harrison, P. J. and C. F. Stevens (1976). Bayesian forecasting (with discussion). J. Roy. Statis. Soc,
Ser. B, 38, 205-247.
Harvey, A. C. (1989). Forecasting, Structural Time Series Models and the Kalman Filter. Cambridge
University Press.
Harvey, A. C. and A. Jaeger (1993). Detrending, stylized facts and the business cycle. J. Appl.
Econometrics 8, 231-247.
Harvey, A. C. (1993). Long memory in stochastic volatility. Discussion Paper, London School of
Economics.
Harvey, A. C. and S. J. Koopman (1993). Forecasting hourly electricity demand using time-varying
splines. J. Amer. Statist. Assoc. 88, 1228-1236.
Harvey, A. C, E. Ruiz and E. Sentana (1992). Unobserved component time series models with ARCH
Disturbances, J. Econometrics 52, 129-158.
Harvey, A. C, E. Ruiz and N. Shephard (1994). Multivariate stochastic variance models. Rev.
Econom. Stud. 61, 247-264.
Harvey, A. C. and N. Shephard (1993). Estimation and testing of stochastic variance models, STI-
CERD Econometrics. Discussion paper, EM93/268, London School of Economics.
Harvey, A. C. and N. Shephard (1996). Estimation of an asymmetric stochastic volatility model for
asset returns. J. Business Econom. Statist, to appear.
Harvey, C. R. and R. D. Huang (1991). Volatility in the foreign currency futures market. Rev. Financ.
Stud. 4, 543-569.
Harvey, C. R. and R. D. Huang (1992). Information trading and fixed income volatility. Discussion
Paper, Duke University.
Harvey, C. R. and R. E. Whaley (1992). Market volatility prediction and the efficiency of the S&P 100
index option market. J. Financ. Econom. 31, 43-74.
Hausman, J. A. and A. W. Lo (1991). An ordered probit analysis of transaction stock prices.
Discussion paper, Wharton School, University of Pennsylvania.
He, H. (1993). Option prices with stochastic volatilities: An equilibrium analysis. Discussion Paper,
University of California, Berkeley.
Heston, S. L. (1993). A closed-form solution for options with stochastic volatility with applications to
bond and currency options. Rev. Financ. Stud. 6, 327-343.
Heynen, R., A. Kemna and T. Vorst (1994). Analysis of the term structure of implied volatility. J.
Financ. Quant. Anal.
Hull, J. (1993). Options, futures and other derivative securities. 2nd ed. Prentice-Hall International
Editions, New Jersey.
Hull, J. (1995). Introduction to Futures and Options Markets. 2nd ed. Prentice-Hall, Englewood Cliffs,
New Jersey.
Hull, J. and A. White (1987). The pricing of options on assets with stochastic volatilities. J. Finance 42,
281-300.
Huffman, G. W. (1987). A dynamic equilibrium model of asset prices and transactions volume. J.
Politic. Econom. 95, 138-159.
Hutchinson, J. M., A. W. Lo and T. Poggio (1994). A nonparametric approach to pricing and hedging
derivative securities via learning networks. J. Finance 49, 851-890.
Jacquier, E., N. G. Poison and P. E. Rossi (1994). Bayesian analysis of stochastic volatility models
(with discussion). J. Business Econom. Statist. 12, 371-417.
Jacquier, E., N. G. Poison and P. E. Rossi (1995a). Multivariate and prior distributions for stochastic
volatility models. Discussion paper CIRANO.
Jacquier, E., N. G. Poison and P. E. Rossi (1995b). Stochastic volatility: Univariate and multivariate
extensions. Rodney White center for financial research. Working Paper 19-95, The Wharton
School, University of Pennsylvania.
Jacquier, E., N. G. Poison and P. E. Rossi (1995c). Efficient option pricing under stochastic volatility.
Manuscript, The Wharton School, University of Pennsylvania.
Jarrow, R. and Rudd (1983). Option Pricing. Irwin, Homewood III.
Johnson, H. and D. Shanno (1987). Option pricing when the variance is changing. J. Financ. Quant.
Anal. 22, 143-152.

Stochastic volatility
189
Jorion, P. (1995). Predicting volatility in the foreign exchange market. J. Finance 50, to appear.
Karatzas, I. and S. E. Shreve (1988). Brownian Motion and Stochastic Calculus. Springer-Verlag: New
York, NY.
Karpoff, J. (1987). The relation between price changes and trading volume: A survey. J. Financ. Quant.
Anal. 22, 109-126.
Kim, S. and N. Shephard (1994). Stochastic volatility: Optimal likelihood inference and comparison
with ARCH Model. Discussion Paper, Nuffield College, Oxford.
King, M., E. Sentana and S. Wadhwani (1994). Volatility and links between national stock markets.
Econometrica 62, 901-934.
Kitagawa, G. (1987). Non-Gaussian state space modeling of nonstationary time series (with
discussion). J. Amer. Statist. Assoc. 79, 378-389.
Kloeden, P. E. and E. Platten (1992). Numerical Solutions of Stochastic Differential Equations.
Springer-Verlag, Heidelberg.
Lamoureux, C. and W. Lastrapes (1990). Heteroskedasticity in stock return data: Volume versus
GARCH effect. J. Finance 45, 221-229.
Lamoureux, C. and W. Lastrapes (1993). Forecasting stock-return variance: Towards an
understanding of stochastic implied volatilities. Rev. Financ. Stud. 6, 293-326.
Latane, H. and R. Jr. Rendleman (1976). Standard deviations of stock price ratios implied in option
prices. J. Finance 31, 369-381.
Lezan, G., E. Renault and T. deVitry (1995) Forecasting foreign exchange risk. Paper presented at 7th
World Congres of the Econometric Society, Tokyo.
Lin, W. L., R. F. Engle and T. Ito (1994). Do bulls and bears move across borders? International
transmission of stock returns and volatility as the world turns. Rev. Financ. Stud., to appear.
Lo, A. W. (1995). Statistical inference for technical analysis via nonparametric estimation. Discussion
Paper, MIT.
Mahieu, R. and P. Schotman (1994a). Stochastic volatility and the distribution of exchange rate news.
Discussion Paper, University of Limburg.
Mahieu, R. and P. Schotman (1994b). Neglected common factors in exchange rate volatility. J. Emp.
Finance 1,279-311.
Mandelbrot, B. B. (1963). The variation of certain speculative prices. J. Business 36, 394--416.
Mandelbrot, B. and H. Taylor (1967). On the distribution of stock prices differences. Oper. Res. 15,
1057-1062.
Mandelbrot, B. B. and J.W. Van Ness (1968). Fractal Brownian motions, fractional noises and
applications. SI AM Rev. 10, 422-437.
McFadden, D. (1989). A method of simulated moments for estimation of discrete response models
without numerical integration. Econometrica 57, 1027-1057.
Meddahi, N. and E. Renault (1995). Aggregations and marginalisations of GARCH and stochastic
volatility models. Discussion Paper, GREMAQ.
Melino, A. and M. Turnbull (1990). Pricing foreign currency options with stochastic volatility. J.
Econometrics 45, 239-265.
Melino, A. (1994). Estimation of continuous time models in finance. In: C.A. Sims ed., Advances in
Econometrics (Cambridge University Press).
Merton, R. C. (1973). Rational theory of option pricing. Bell J. Econom. Mgmt. Sci. 4, 141-183.
Merton, R. C. (1976). Option pricing when underlying stock returns are discontinuous. J. Financ.
Econom. 3, 125-144.
Merton, R. C. (1990). Continuous Time Finance. Basil Blackwell, Oxford.
Merville, L. J. and D. R. Pieptea (1989). Stock-price volatility, mean-reverting diffusion, and noise. J.
Financ. Econom. 242, 193-214.
Metropolis, N., A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller and E. Teller (1954). Equation of
state calculations by fast computing machines. J. Chem. Physics 21, 1087-1092.
Miiller, U. A., M. M. Dacorogna, R. B. Olsen, W. V. Pictet, M. Schwarz and C. Morgenegg (1990).
Statistical study of foreign exchange rates. Empirical evidence of a price change scaling law and
intraday analysis. J. Banking Finance 14, 1189-1208.

190
E. Ghysels, A. C. Harvey and E. Renault
Nelson, D. B. (1988). Time series behavior of stock market volatility and returns. Ph.D. dissertation,
MIT.
Nelson, D. B. (1990). ARCH models as diffusion approximations. J. Econometrics 45, 7-39.
Nelson, D. B. (1991). Conditional heteroskedasticity in asset returns: A new approach. Econometrica
59, 347-370.
Nelson, D. B. (1992). Filtering and forecasting with misspecified ARCH Models I: Getting the right
variance with the wrong model. J. Econometrics 25, 61-90.
Nelson, D. B. (1994). Comment on Jacquier, Poison and Rossi. J. Business Econom. Statist. 12, 403-
406.
Nelson, D. B. (1995a). Asymptotic smoothing theory for ARCH Models. Econometrica, to appear.
Nelson, D. B. (1995b). Asymptotic filtering theory for multivariate ARCH models. J. Econometrics, to
appear.
Nelson, D. B. and D. P. Foster (1994). Asymptotic filtering theory for univariate ARCH models.
Econometrica 62, 1-41.
Nelson, D. B. and D. P. Foster (1995). Filtering and forecasting with misspecified ARCH models II:
Making the right forecast with the wrong model. J. Econometrics, to appear.
Noh, J., R. F. Engle and A. Kane (1994). Forecasting volatility and option pricing of the S&P 500
index. J. Derivatives, 17-30.
Ogaki, M. (1993). Generalized method of moments: Econometric applications. In: Maddalaet al. ed.,
Handbook of Statistics Vol. 11, North Holland, Amsterdam.
Pagan, A. R. and G. W. Schwert (1990). Alternative models for conditional stock volatility. J.
Econometrics 45, 267-290.
Pakes, A. and D. Pollard (1989). Simulation and the asymptotics of optimization estimators.
Econometrica 57, 995-1026.
Pardoux, E. and D. Talay (1985). Discretization and simulation of stochastic differential equations.
Acta Appl. Math. 3, 23-47.
Pastorello, S., E. Renault and N. Touzi (1993). Statistical inference for random variance option
pricing. Discussion Paper, CREST.
Patell, J. M. and M. A. Wolfson (1981). The ex-ante and ex-post price effects of quarterly earnings
announcement reflected in option and stock price. J. Account. Res. 19, 434-458.
Patell, J. M. and M. A. Wolfson (1979). Anticipated information releases reflected in call option prices.
J. Account. Econom. 1, 117-140.
Pham, H. and N. Touzi (1993). Intertemporal equilibrium risk premia in a stochastic volatility model.
Math. Finance, to appear.
Platten, E. and Schweizer (1995). On smile and skewness. Discussion Paper, Australian National
University, Canberra.
Poterba, J. and L. Summers (1986). The persistence of volatility and stock market fluctuations. Amer.
Econom. Rev. 76, 1142-1151.
Renault, E. (1995). Econometric models of option pricing errors. Invited Lecture presented at 7th
W.C.E.S., Tokyo, August.
Renault, E. and N. Touzi (1992). Option hedging and implicit volatility. Math. Finance, to appear.
Revuz, A. and M. Yor (1991). Continuous Martingales andBrownian Motion. Springer-Verlag, Berlin.
Robinson, P. (1993). Efficient tests of nonstationary hypotheses. Mimeo, London School of
Economics.
Rogers, L. C. G. (1995). Arbitrage with fractional Brownian motion. University of Bath, Discussion
paper.
Rubinstein, M. (1985). Nonparametric tests of alternative option pricing models using all reported
trades and quotes on the 30 most active CBOE option classes from August 23, 1976 through
August 31, 1978. J. Finance 40, 455-480.
Ruiz, E. (1994). Quasi-maximum likelihood estimation of stochastic volatility models. J. Econometrics
63, 289-306.
Schwert, G. W. (1989). Business cycles, financial crises, and stock volatility. Camegie-Rochester
Conference Series on Public Policy 39, 83-126.

Stochastic volatility
191
Scott, L. O. (1987). Option pricing when the variance changes randomly: Theory, estimation and an
application. J. Financ. Quant. Anal. 22, 419-438.
Scott, L. (1991). Random variance option pricing. Advances in Futures and Options Research, Vol. 5,
113-135.
Sheikh, A. M. (1993). The behavior of volatility expectations and their effects on expected returns. J.
Business 66, 93-116.
Shephard, N. (1995). Statistical aspect of ARCH and stochastic volatility. Discussion Paper 1994,
Nuffield College, Oxford University.
Sims, A. (1984). Martingale-like behavior of prices. University of Minnesota.
Sowell, F. (1992). Maximum likelihood estimation of stationary univariate fractionally integrated time
series models. J. Econometrics 53, 165-188.
Stein, J. (1989): Overreactions in the options market. J. Finance 44, 1011-1023.
Stein, E. M. and J. Stein (1991). Stock price distributions with stochastic volatility: An analytic
approach. Rev. Financ. Stud. 4, 727-752.
Stock, J. H. (1988). Estimating continuous time processes subject to time deformation. J. Amer.
Statist. Assoc. 83, 77-84.
Strook, D. W. and S. R. S. Varadhan (1979). Multi-dimensional Diffusion Processes. Springer-Verlag,
Heidelberg.
Tanner, T. and W. Wong (1987). The calculation of posterior distributions by data augmentation. J.
Amer. Statist. Assoc. 82, 528-549.
Tauchen, G. (1995). New minimum chi-square methods in empirical finance. Invited Paper presented
at the 7th World Congress of the Econometric Society, Tokyo.
Tauchen, G and M. Pitts (1983). The price variability-volume relationship on speculative markets.
Econometrica 51, 485-505.
Taylor, S. J. (1986). Modeling Financial Time Series. John Wiley: Chichester.
Taylor, S. J. (1994). Modeling stochastic volatility: A review and comparative study. Math. Finance 4,
183-204.
Taylor, S. J. and X. Xu (1994). The term structure of volatility implied by foreign exchange options. J.
Financ. Quant Anal. 29, 57-74.
Taylor, S. J. and X. Xu (1993). The magnitude of implied volatility smiles: Theory and empirical
evidence for exchange rates. Discussion Paper, University of Warwick.
Von Furstenberg, G. M. and B. Nam Jeon (1989). International stock price movements: Links and
messages. Brookings Papers on Economic Activity 1,125-180.
Wang, J. (1993). A model of competitive stock trading volume. Discussion Paper, MIT.
Watanabe, T. (1993). The time series properties of returns, volatility and trading volume in financial
markets. Ph.D. Thesis, Department of Economics, Yale University.
West, M. and J. Harrison (1990). Bayesian Forecasting and Dynamic Models. Springer-Verlag, Berlin.
Whaley, R. E. (1982). Valuation of American call options on dividend-paying stocks. J. Financ.
Econom. 10, 29-58.
Wiggins, J. B. (1987). Option values under stochastic volatility: Theory and empirical estimates. J.
Financ. Econom. 19, 351-372.
Wood, R. T. Mclnish and J. K. Ord (1985). An investigation of transaction data for NYSE Stocks. J.
Finance 40, 723-739.
Wooldridge, J. M. (1994). Estimation and inference for dependent processes. In: R.F. Engle and D.
McFadden eds., Handbook of Econometrics Vol. 4. North Holland, Amsterdam.

G. S. Maddala and C. R. Rao, eds., Handbook of Statistics, Vol. 14
© 1996 Elsevier Science B.V. All rights reserved
6
Stock Price Volatility
Stephen F. LeRoy
1. Introduction
In the early days of the efficient capital markets literature, discourse between
finance academics and practitioners was characterized by mutual
incomprehension. Academics held that security prices were governed exclusively by their
prospective payoffs - in fact, the former equaled the discounted expected value of
the latter. Practitioners, on the other hand, made no secret of their opinion that
only naive academics could take the present value relation seriously as a theory of
asset pricing: everyone knows that traders routinely ignore cash flows, and that
large price changes often occur in the complete absence of news about future cash
flows. Academics, at least since Samuelson's (1965) paper, responded that
rejection of the present value relation implies the existence of profitable trading rules.
Given that no one appeared to be identifying a trading rule that significantly
outperforms buy-and-hold, academics saw no grounds for rejecting the present-
value relation.
Prior to the 1980's, empirical tests of market efficiency were conducted on the
home court of the academics: one searched for evidence of return predictability;
failing to find it, one concluded in favor of market efficiency. The variance-
bounds tests introduced by Shiller (1981) and LeRoy and Porter (1981), however,
can be interpreted as shifting the locus of the debate from the home court of the
academics to that of the practitioners - instead of looking for patterns in returns
that are ruled out by market efficiency, one looked for the price patterns that are
implied by market efficiency. Specifically, one asked whether security price
changes are of about the magnitude one would expect if they were generated
exclusively by fundamentals.
The implications of this shift from returns tests to price-level tests were at first
difficult to sort out since finding a predictable pattern has opposite interpretations
in the two cases: finding that fundamentals predict future security returns argues
against market efficiency, whereas finding that fundamentals predict current
prices supports market efficiency. In both cases the early evidence suggested that
the correlation being sought was not in the data; hence the returns tests accepted
market efficiency, whereas the variance-bounds tests rejected efficiency.
193

194
S. F. LeRoy
To understand the relation between returns and variance-bounds tests of
market efficiency, note that the simplest specification of the efficient markets
model (applied to stock prices) says that
E,(r,+i) = p, (1.1)
where rt is the (gross) rate of return on stock, p is a constant greater than one, and
E, denotes mathematical expectation conditional on some information set /,.
Equation 1.1 says that no matter what agents' information is, the conditional
expected rate of return on stock is p; past information, such as past realized stock
returns, should not be correlated with future returns. Conventional efficiency tests
directly investigated this implication.
Variance-bounds tests, on the other hand, used the definition of the rate of
return,
_ dt+\ +pt+\ . ..
r,+i= , (1.2)
Pt
to derive from 1.1 the relation
p, = PE,(d,+i + p,+i) , (1.3)
where /? = 1/(1 + p). After successive substitution and application of the law of
iterated expectations, (1.3) may be written as
pt = E,(M+i + fdt+2 + ... + Pn+ldl+n+l + Pn+lpt+n+i) ■ (1.4)
Assuming the convergence condition
lim/f+1E,(^+„+1)=0 (1.5)
n—»oo
is satisfied, sending n to infinity in (1.4) results in
Pt = W) , (1-6)
where p* is the ex-post rational stock price; i.e., the value stock would have if
future dividends were perfectly forecastable:
oo
p*(=Y,Pndt+n ■ (1-7)
n=\
Because the conditional expectation of any random variable is less volatile
than that random variable itself, (1.6) implies the variance bounds inequality
V(pt) < V(p*) . (1.8)
Both Shiller and LeRoy-Porter reported reversal of the empirical counterpart to
inequality (1.8): prices appear to be more volatile than the upper bound implied
by the volatility of dividends under market efficiency.

Stock price volatility
195
2. Statistical issues
Several statistical issues must be considered in interpreting the fact that the
empirical counterpart of inequality 1.8 is apparently reversed. These are (1) bias in
parameter estimation, (2) nuisance parameter problems, and (3) sample variation
of parameter estimates. Of these, discussion in the variance-bounds literature has
concentrated almost exclusively on bias. However, bias is not a serious problem in
the absence of nuisance parameter or sample variability problems since the
rejection region can always be modified to allow for bias. In contrast, nuisance
parameter problems - which occur whenever the sample distribution of the test
statistic is materially affected by a parameter which is unrestricted under the null
hypothesis - make it difficult or impossible to set rejection regions so that
rejection will occur with prespecified probability if the null hypothesis is true.
Therefore they are much more serious. High sample variability in the test statistic
is also a serious problem since it diminishes the ability of the test to distinguish
between the null and the alternative, therefore reducing the power of the test for
given size.
In testing (1.8) one immediately encounters the fact that/7* cannot be directly
constructed from any finite sample since dividends after the end of the sample
are unobservable. The problems of bias, nuisance parameters and sample
variability in testing (1.8) take different forms depending on how this problem is
addressed. Two methods for estimating V(p*) are available, the model-free
estimator used by Shiller and the model-based estimator used by LeRoy-Porter.
The model-free estimator simply replaces the unobservable p* with the expected
value of p* conditional on the sample, which is observable. This is given by
setting the terminal value p*j,T of the observable proxy series p*,T equal to actual
pT:
P*t\t=Pt (2.1)
and computing earlier values p*, T from the backward recursion
P:\T = P(P*,+x\T + dt+i) , (2-2)
which has the required property:
E(p*\T\pi,du...,pT,dT)=p* (2.3)
(under the assumption that the population value of ft is used in the discounting).
The estimated series is model-free in the sense that its construction requires no
assumptions about how dividends are generated, an attractive property.
Using the model-free /jjjr series to construct V(p*) has several less attractive
consequences. Most important, if the model-builder is unwilling to commit to a
model for dividends, there is no prospect of evaluating the sample variability of
V(p*t), rendering construction of confidence intervals impossible. Thus it was no
accident that Shiller reported point estimates of V{pt) and V{p*t), but no ?-sta-
tistics.

196
S. F. LeRoy
One can, however, investigate the statistical properties of V(p*t) under
particular models of dividends, and this has been done. As Flavin (1983) and Kleidon
(1986) showed, because of the very high serial correlation of p*,T, V(p*t) is severely
biased downward as an estimator of V(p*t); see Gilles and LeRoy (1991) for
intuitive interpretation. As noted above, this by itself is not a problem since the
rejection region can always be modified to offset the effect of bias. However, such
modification cannot be implemented without committing to a dividends model, so
if one takes this route the advantage of the model-free estimator is foregone. Also,
it is known that the model-free estimator V(p*t) has higher sample variability than
its model-based counterpart, discussed below.
A model-based estimator of V(p*t) can be constructed if one is willing to specify
a statistical model assumed to generate dividends. For example, suppose
dividends are generated by a first-order autoregression:
dt+i = kdt + er+i . (2.4)
Then an expression for the population value of V(p*) is readily computed as a
function of 1, c\ and /?, and a model-based estimator V(p*) can be constructed by
substituting parameter estimates for their population counterparts. Assuming the
dividends model is correctly specified, the model-based estimator has little bias (at
least in some settings) and, more important, very low sample variability (LeRoy-
Parke (1992)). In the setting of LeRoy-Parke the model-based point estimate of
V(p*) is about three times greater than the estimate of V(pt), suggesting
acceptance of (1.8). However, due to the nuisance-parameter problem to be discussed
now, this result is not of much importance.
Besides the ambiguities resulting from the various methods of constructing
V(p*t), an even more basic problem arises from the fact that 1.8 is an inequality
rather than an equality. Assuming that the null hypothesis is true, the population
value of V(pt) depends on the magnitude of the error in investors' estimates of
future dividends. Therefore the same is true of the volatility parameter
V(p*t) - V{pt), the sample counterpart of which constitutes the test statistic. This
error variance is not restricted by the assumption of market efficiency, leading to
its characterization as a nuisance parameter. In LeRoy-Parke it is argued that this
problem is very serious quantitatively: there is no way to set a rejection region for
the volatility statistic V(p*t) — V(pt)- It is argued there that because of this
nuisance parameter problem, directly testing Eq. (1.8) is essentially impossible. Since
(1.8) is the best-known of the variance-bounds relations, this is not a minor
conclusion.
There exist other variance-bounds tests that are better-behaved econometri-
cally than inequality (1-8). To develop these, define e,+i as the innovation in stock
payoffs:
e,+i =dt+i +p,+i -Et(dt+i +p,+\) , (2.5)
so that the present-value relation (1.3) can be written as
pt = j?Er(dr+i +Pt+\) = P{d,+i +pt+\ - £r+i) • (2-6)

Stock price volatility
197
Substituting recursively, using the definition 1.7 of p* and assuming convergence,
(2.6) becomes
oo
tf =/>« + £ A+/ , (2-7)
(=1
so that the difference between p* and pt is expressible as a weighted sum of payoff
innovations. Equation (2.7) implies
V(p*t) = V(j>t)+J^j1V{Et) . (2.8)
Put this result aside for the moment.
The upper bound for price volatility is derived by considering the volatility of a
hypothetical price series that would obtain if investors had perfect information
about future dividends. LeRoy-Porter also showed that a lower bound on price
volatility could be derived if one was willing to specify that investors have at least
some minimal information about future dividends. Suppose that one assumes that
investors know at least current and past dividends; they may or may not have
access to other variables that predict future dividends. Let p, denotes the stock
price that would prevail under this minimal information specification:
pt = E(p*t\dt,dt-Udt-2,...) . (2.9)
Then because It is a refinement of the information partition induced by
dt,dt-\,dt-i,..., we have
pt=E({E(p*t\It)]\dt,dt-Udt-2,...) , (2.10)
by the law of iterated expectations, or
pt = E(pt\d„dt-Udt-2,---) , (2.11)
using (1.6). Therefore, by exactly the same reasoning used to derive (1.8), we
obtain
V(p,) < V(p,) , (2.12)
so the variance of pt is a lower bound for the variance of pt.
This lower bound is without direct empirical interest since no one has seriously
suggested that stock prices are less volatile than is implied by the present-value
model under the assumption that investors know current and past dividends.
However, the lower bound may be put to a more interesting use. By defining lt+\
as the payoff innovation under the information set generated by du dt~\, dt-i,...,
e(+i =dt+i +pt+i -E(dt+i +p,+l\dt,dt-udt-2,---) , (2.13)
we derive
oo
pl=~Pt + J2F~£<+i ■ (2-14)

198
S. F. LeRoy
by following exactly the derivation of (2.7). Equation (2.14) implies
V(p*t) = V(pt)+j£-pV(lt) . (2.15)
Equations (2.8) and (2.15) plus the lower bound inequality (2.12) imply
V(l,) > V(et) . (2.16)
Thus the present-value relation implies not just that prices are less volatile than
they would be if investors had perfect information, but also that net one-period
payoffs are less volatile than they would be if investors had less information than
they (by assumption) do.
To test (2.16), one simply fits a univariate time-series model to dividends and
uses it to compute V(st), while V(et) is just the estimated residual variance in the
regression
d,+p, = prlPt-i+et ■ (2.17)
This adaptation of LeRoy-Porter's lower bound on price volatility to the formally
equivalent - but much more interesting econometrically - upper bound on payoff
volatility is due to West (1988). The West test, like Shiller and LeRoy-Porter's
upper bound tests on price volatility, resulted in rejection. West reported
statistically significant rejection (as noted, Shiller did not compute confidence intervals,
while LeRoy-Porter's rejections were only of borderline statistical significance).
Generally, the West test is free of the most serious econometric problems that
beset the price bounds tests. Most important, under the null hypothesis payoff
innovations are serially uncorrelated, so sample means yield good estimates of
population means (recall that model-free tests of price volatility are subject to the
problem that pt and p* are highly serially correlated). Further, the associated
^-statistics can be used to compute rejection regions. Finally, there is no need to
specify investors' information since a model-free estimate of V(et) is used,
implying that the nuisance parameter problem that occurs under model-based price
bounds tests does not appear here.
3. Dividend-smoothing and nonstationarity
One objection sometimes raised against the variance-bounds tests is that
corporate managers smooth dividends. That being the case, and because the ex-post
rational stock price is in turn a highly smoothed average of dividends, it is argued
that we should not be surprised that actual stock prices are choppier than ex-post
rational prices. This point was raised most forcefully by Marsh and Merton
(1983), (1986).l Marsh-Merton asserted that the variance-bounds theorems re-
1 This discussion is drawn from the 1988 version of Gilles-LeRoy (1991), available from the
author. Discussion of Marsh-Merton was deleted from the published version of that paper in response
to an editor's request.

Stock price volatility
199
quire for their derivation the assumption that dividends are exogenous, and also
that the resulting series is stationary.
If these assumptions are not satisfied the variance-bounds theorems are
reversed. To prove this, Marsh-Merton (1986) assumed that managers set dividends
as a distributed lag on past stock prices:
iV
dt = Y^hPt-t . (3.1)
/=i
Further, from (1.7) the ex-post rational stock price can be written as
p^^dM + f-%. (3.2)
i=i
Finally, Marsh-Merton took the terminal ex-post rational stock price to be given
by the sample average stock price:
p*T = l2=±2L . (3.3)
Substituting (3.1) and (3.3) into (3.2), it is seen that p* is expressible as a weighted
average of the in-sample p/s. Using this result, Marsh-Merton proved that in
every sample p* has lower variance than pt, just the opposite of the variance-
bounds theorem.
Questions emerge about Marsh-Merton's assertion that the variance-bounds
inequality is reversed if managers smooth dividends. The most important
question arises from the fact that none of the rigorous derivations of the variance-
bounds theorems available in the literature make use, explicitly or implicitly, of
any assumption of exogeneity or stationarity: instead, the theorems depend only
on the fact that the conditional expectation of a random variable is less volatile
than the random variable itself. How, then, does dividend smoothing reverse the
variance-bounds theorem? It turns out that Marsh-Merton are not in fact
asserting that the variance-bounds theorems are incorrect, but only that in the
setting they specify the sample counterparts of the variance of;?* and pt reverse
the population inequality; Marsh-Merton's failure to use notation that
distinguishes population from sample moments renders careful reading of their paper
needlessly difficult.
Marsh-Merton's dividend specification implies that dividends and prices are
necessarily nonstationary (this is proved explicitly in Shiller's (1986) comment on
Marsh-Merton). Sample moments cannot be expected to satisfy the same
inequalities as population moments if the latter are infinite (or time-varying,
depending on the interpretation). In nonstationary populations, in fact, there is
essentially no relation between population moments and the corresponding
sample moments2 - indeed, the very idea that there is a correspondence between
2 Gilles-LeRoy (1991) set out an example, adapted from Kleidon (1986), in which the martingale
convergence theorem implies that the sample counterpart of the variance-bounds inequality is reversed
with arbitrarily high probability in arbitrarily long samples despite being troe at each date in the
population. As with Marsh-Merton, nonstationarity is the culprit.

200
S. F. LeRoy
sample and population moments in time-series analysis derives its meaning from
the analysis of stationary series. Thus there is no inconsistency whatever between
the assertion that the population variance-bounds inequality is satisfied at every
date, as it is in Marsh-Merton's model, and Marsh-Merton's demonstration that
under their specification its sample counterpart is reversed for every possible
sample.
What Marsh-Merton's example demonstrates is that if one uses analytical
methods appropriate under stationarity when the data under investigation are
nonstationary, one can be misled. Thus formulated, Marsh-Merton's conclusion
is surely correct. The logical implication is that one wishes to make progress with
the analysis of stock price volatility, one should go on to formulate statistical
procedures that are appropriate in the nonstationary setting they assume. Marsh-
Merton did not do so, and no easy extension of their model would have allowed
them to take this next step. The reason is that Marsh-Merton's model does not
contain any specification of what exogenous variables drive their model; the only
behavior they model is managers' response to stock prices, treated as exogenous,
in setting dividends.
Marsh-Merton made two criticisms of the variance-bounds tests: (1) that they
depend on the assumption that dividends are stationary, and (2) that they depend
on the assumption that dividends are exogenous, as opposed to being smoothed
by managers (this second criticism is especially prominent in Marsh-Merton's
unpublished paper (1983) dealing with LeRoy-Porter (1981)). Marsh-Merton
treated the two points as interchangeable, so that exogeneity was taken to imply
stationarity, and dividend-smoothing nonstationarity. In fact dividend exogeneity
neither implies nor is implied by stationarity, and the variance-bounds theorems
require neither one, as we saw above.
It is true that the specific empirical implementation adopted by Shiller has
attractive econometric properties only when dividends are stationary in levels.3
However, whether or not the analyst chooses to model the dividend-payout
decision, as Marsh-Merton did, or directly assigns dividends a probabilistic model,
as LeRoy-Porter did, is immaterial: if the assumed dividends model under the
latter coincides with the behavior implied for dividends in the former case, the
two are equivalent. It follows that any implementation of the variance-bounds
tests that accurately characterizes dividend behavior is acceptable, regardless of
whether corporate managers are smoothing dividends and regardless of whether
such behavior, if occurring, is modeled.
Whether or not Shiller's assumption of trend-stationarity is acceptable has
been controversial: many analysts believe that major macroeconomic time series,
such as GNP, have a unit root. The debate about trend-stationarity vs. unit roots
in macroeconomic time-series is not reviewed here, except to note that (1) of all
3 LeRoy-Porter used a trend correction based on reversing the effect of earnings retention that
should have resulted in stationary data, but in fact produced series with a downward trend (which
explained why their rejections of the variance-bounds theorems were of only marginal statistical
significance). The reasons for the failure of LeRoy-Porter's trend correction are unclear.

Stock price volatility
201
the major macroeconomic time series, aggregate dividends appears closest to
trend-stationarity, and (2) many econometricians believe that it is difficult to
distinguish empirically between the trend-stationary and unit-root cases.
Kleidon (1986) showed that if dividends have a unit root, so that dividend
shocks have a permanent component, then stock prices should be more volatile
than they would be if dividends were stationary. Kleidon expressed the opinion
that the evidence of excess volatility reflects nothing more than the
nonstationarity of dividends. However, this opinion cannot be sustained. First, the
West test is valid if dividends are generated by a linear time-series process with a
unit root, so that, if the expected present-value model is correct, dividends and
stock prices are cointegrated. West, it is recalled, found significant excess
volatility. Other tests, of which Campbell and Shiller (1988) was the first to be
published, dealt with dividend nonstationarity by working with the price-dividend
ratio instead of price levels. Again the conclusion was that stock prices are
excessively volatile. LeRoy-Parke (1992) showed that the variance equality that
LeRoy-Porter had used,
VW) = V(P,)+J^ , (3-4)
could be adapted to apply to the intensive price-dividend variables, yielding
V(p*t/dt)=V(p,/dt) + dV{rt) , (3.5)
where 5 is a function of various parameters, under the assumption that all
variances of the intensive variables pt/d,, p*/d, and r, remain constant over time
(this is the counterpart of the assumption, required to derive (3.4), that variances
of extensive variables like p,, p* and e, remain constant over time). LeRoy-Parke
also found excess volatility (see also LeRoy and Steigerwald, 1993).
Thus the debate about whether dividends are trend-stationary or have a unit
root is, from the point of view of the variance-bounds tests, irrelevant: either way,
volatility exceeds that predicted by the present-value model.
4. Bubbles
These results show that excess volatility occurs under at least some forms of
dividend nonstationarity. However, they do not necessarily completely dispose of
Marsh-Merton's criticisms; any model-based variance-bounds test requires some
specification of the probability law, stationary or nonstationary, assumed to
generate dividends, and critics can always question this specification. For
example, LeRoy-Parke assumed that dividends follow a geometric random walk, a
characterization that appears not to do great violence to the data. However, it
may be that the dividend-smoothing behavior of managers results in a less
parsimonious model for dividends, in which case LeRoy-Parke's results may reflect
nothing more than misspecification of the dividends model.

202
S. F. LeRoy
Two sets of circumstances might invalidate variance-bounds tests based on
particular dividend specification such as the geometric random walk. First, it may
be that even data sets as long as a century (the length of Shiller's 1981 data set,
which was also used in several of the subsequent variance-bounds papers) are too
short to allow accurate estimation of dividend volatility. Regime shift models, for
example, require very long data sets for accurate estimation. Alternatively, the
stock market may be subject to a "peso problem" - investors might attach time-
varying probabilities to an event which did not occur in the finite sample.
The second circumstance that might invalidate variance-bounds tests is
rational speculative bubbles. Thus consider an extreme case of Marsh-Merton's
dividend-smoothing behavior: suppose that firms pay some positive (but low)
level of dividends that is deterministic.4 Thus all fluctuations in earnings show up
as additions to (or subtractions from) capital. In this setting the market value of
the firm will reflect the value of its capital, which by assumption does not depend
on past dividends. Price volatility will obviously exceed the volatility implied by
dividends, since the latter is zero, so the variance-bounds theorem is violated.
Theoretically, what is happening in this case is that the limiting condition (1.5) is
not satisfied, so that stock prices do not equal the limit of the present value of
dividends. Models in which (1.5) fails are defined as rational speculative bubbles:
prices are higher than the present value of future dividends but, because they are
expected to rise still higher, (1.3) is satisfied. Thus insofar as they are suggesting
that dividend smoothing invalidates empirical tests of the variance-bounds
relations even in infinite samples, Marsh-Merton are asserting the existence of
rational speculative bubbles.
Bubbles have received much study in the recent economics literature, partly
because of their potential role in resolving the excess volatility puzzle (for
theoretical studies of rational bubbles, see Gilles and LeRoy (1992) and the sources
cited there; for a summary of the empirical results as they apply to variance-
bounds, see Flood and Hodrick (1990)). This is not the place for a complete
discussion of bubbles; we remark only that the widely-held impression that
bubbles cannot occur in models incorporating rationality is incorrect. This
impression is fostered by the practice of referring incorrectly to (1.5) as a trans-
versality condition (a transversality condition is associated with an optimization
problem; no such problem has been specified here), suggesting that its satisfaction
is somehow virtually automatic. In fact, (1) there exist well-posed optimization
problems that do not have necessary transversality conditions, and (2)
transversality conditions, even when necessary for optimization, do not always imply
(1.5.) Examples are found in Gilles-LeRoy (1992). These examples, it is true,
appear recondite. However, recall that the goal here is to explain behavior -
4 This specification conflicts with limited liability, which in conjunction with random earnings
implies that firm managers may not be able to commit to paying positive dividends with certainty into
the infinite future. This objection, while valid, is extraneous to the present concern, and hence is set
aside.

Stock price volatility
203
excess volatility - that is itself counterintuitive; given this, we should not readily
dismiss out of hand counterintuitive specifications of preferences.
If (1.3) is satisfied but (1.5) fails, then the price of stock differs from the
expected present value of dividends by a bubble term that satisfies
bt+i =^+p)bt + n,+x , (4-1)
so that a bubble is a martingale with drift p. Since the bubble increases in value at
average rate p, which exceeds the growth rate of dividends (otherwise stock prices
would be infinite), stock prices rise more rapidly than dividends. Therefore the
dividend-price ratio will decrease over time. Informal examination of a plot of the
dividend-price ratio shows no clear downward trend, and the majority of the
empirical studies surveyed by Flood-Hodrick (1990) do not find evidence of
bubbles. This literature is under rapid development, however, from both the
theoretical and empirical sides, and this conclusion may shortly be reversed. For
now, however, it is difficult to find support for the contention that firms are
smoothing dividends in such a way as to invalidate the stationarity presumed in
the variance-bounds tests.
5. Time-varying discount rates
One possible explanation for the apparent excess volatility of securities prices is
that conditionally expected rates of return depend on the values taken on by the
conditioning variables, contradicting (1.1). There is no reason, other than a desire
for simplification, to adopt the restriction that the conditional expected return on
stock is constant over time, as implied by (1.1). If agents are risk averse, one
would expect the conditions of equilibrium in asset markets to reflect a risk-return
tradeoff, so that (1.1) would be replaced by a term involving the higher moments
of return distributions as well as the conditional mean (consider CAPM, for
example). Thus equilibrium conditions like (1.1) are best interpreted as obtaining
in efficient markets under the additional assumption of risk-neutrality (LeRoy
(1973), Lucas (1978)).
Further, in simple models in which agents are risk averse, price volatility is
likely to exceed that predicted by risk-neutrality. The intuition is simple: under
risk aversion agents try to transfer consumption from dates when income and
consumption are high to dates when they are low. Decreasing returns in
production mean that this transfer is increasingly costly, so security prices must
behave in such a way as to penalize agents who make this transfer. If stock prices
are high (low) when income is high (low), then agents are motivated to adapt their
saving or dissaving to the production technology, as they must in equilibrium.
Thus the more risk averse agents are, the more choppy equilibrium stock prices
will be (LaCivita and LeRoy (1981), Grossman and Shiller (1981)). This raises the
possibility that the apparent volatility is nothing more than an artifact of the
misspecification of risk neutrality implicit in (1.1).

204
S. F. LeRoy
A very simple modification of the efficient markets model is seen to be, in
principle, sufficient to explain existing price volatility. Providing other
explanations subsequently became a minor cottage industry, perhaps because it is so easy
to modify the characterization of market efficiency so as to alter its volatility
prediction (1.8) (see Eden and Jovanovic 1994, Romer 1993 or Allen and Gale
1994, for example, for recent contributions).
For example, consider an overlapping generations model in which the
aggregate endowment is deterministic, but some stochastic factor like a random
wealth transfer or monetary shock affects individual agents. In general this
random shock will affect equilibrium stock prices. This juxtaposition of deterministic
aggregate dividends and stochastic prices contradicts the simplest formulation of
market efficiency, since deterministic dividends means that the right-hand side of
(1.8) is zero, while the left-hand side is strictly positive. Evidently, however, such
models are efficient in any reasonable sense of the word: transactions costs are
excluded and agents are assumed to be rational and to have rational expectations.
Models with asymmetric information can be shown to predict price volatility that
exceeds that associated with the conventional market efficiency definition.
These efforts have been instructive, but should not be viewed as disposing of
the volatility puzzle. The variance-bounds literature was never properly
interpreted as pointing to a puzzle for which potential theoretical explanations were in
short supply. Rather, it consisted in showing that a simple model which had
served well in some contexts did not appear to serve so well in another context.
Resolving the puzzle would consist not in pointing out that other more general
models do not generate the volatility implication that the data contradict - this
was never in doubt - but in showing that these models actually explain the
observed variations in security prices. Such explanations have not been
forthcoming. For example, attempts to incorporate the effects of risk aversion in
security pricing have not succeeded (Hansen and Singleton (1983), Mehra and
Prescott (1985)), nor have any of the other proposed explanations of excess
volatility been successfully implemented empirically.
The enduring interest of the variance-bounds controversy lies in the fact that it
was here that it was first pointed out that we do not have good explanations, even
ex post, for why security prices behave as they do. It is hard to imagine a more
important conclusion, and nothing in the recent development of empirical finance
has altered it.
6. Interpretation
Variance-bounds tests as currently formulated appear to be essentially free of
major econometric problems - for example, LeRoy-Parke (1992) relied on Monte
Carlo simulations to assess the behavior of test statistics, thus ensuring that any
econometric biases in the real-world statistics appears equally in the simulated
statistics. Therefore econometric problems are automatically accommodated in

Stock price volatility
205
setting the rejection region. These reformulated variance-bounds tests have
continued to find excess price volatility.
The debate about statistical problems with the variance-bounds tests has died
out in recent years: it is no longer seriously argued that there does not exist excess
price volatility relative to that implied by the simplest expected present-value
relation. As important as the above-mentioned refinements of the variance-
bounds tests were in leading to this outcome, another development was still more
important: conventional market efficiency tests were themselves evolving at the
same time as the variance-bounds tests were being developed. The most important
modification of the conventional return market efficiency tests was that they
investigated return autocorrelations over much longer time horizons than had the
earlier tests. Fama and French (1988) found significant predictability in returns.
These return autocorrelations are most significant when returns are averaged over
five to ten years; earlier studies, such as those reported in Fama (1970), had
investigated return autocorrelations over weeks or months rather than years.
There are several general methodological lessons to be learned from
comparison of conventional market efficiency tests and variance bounds tests about
econometric testing of economic theories. Since the same null hypothesis is tested,
one would presume that there exist no grounds for a different interpretation of
rejection in one case relative to the other. Yet it is extraordinarily difficult to keep
this in mind: the existence of excess volatility suggests the conclusion that "we
cannot explain security prices", whereas the return autocorrelation results suggest
the more workaday conclusion that "average security returns are subject to
gradual shifts over time".
To bring home the point that this difference in interpretation is unjustified,
assume that security prices equal those predicted by the present-value model plus
a random term independent of dividends which has low innovation variance, but
is highly autocorrelated. One can interpret that random term either as
representing an irrational fad or as capturing smooth shifts in security returns due to
changes in investment opportunities, shifts in social conditions, or whatever. This
modification will generate excess volatility, and will also generate return
autocorrelations of the type observed. With the same alternative hypothesis generating
both the excess volatility and the return autocorrelations by assumption, there
can be no justification for attaching different verbal interpretations to the two
rejections. The lesson to be learned is that rejection of a model is just that:
rejection of a model. One must be careful about basing interpretations of the
rejection on the particular test leading to the rejection, rather than on the model
being rejected.
Despite being generally aware of the possibility that excess price volatility is
the same thing statistically as long-horizon return autocorrelation, many financial
economists nonetheless dismiss the possibility that excess price volatility has
anything to do with capital market efficiency. Fama (1991) is a good example.
Fama began his 1991 update of his survey (1970) by reemphasizing the point
(made also in his 1970 survey) that any test of market efficiency is necessarily a
joint test with a particular returns model. He then surveyed the evidence (to which

206
S. F. LeRoy
he has been a major contributor) that there exists high negative autocorrelation in
returns at long horizons, remarking that this is statistically equivalent to "long
swings away from fundamental value" (p. 1581). However, in discussing the
variance-bounds tests, Fama expressed the opinion that, despite the fact that they
are "another useful way to show that expected returns vary through time",
variance-bounds tests "are not informative about market efficiency". Contrary to
this, it would seem that the joint-hypothesis problem applies no less or more to
variance-bounds tests than to return autocorrelation tests: if one type of evidence
is relevant to market efficiency, so is the other.
Another lesson is that one must be careful about applying implicit
psychological metrics that seem appropriate, but in fact are not. For example, it is easy to
regard the apparently spectacular rejections of the variance bounds tests as
justifying a strong verbal characterization, whereas the extraneous random term that
accounts for return autocorrelations appears too small to justify a similar
interpretation. This too is incorrect: a random term that adds and subtracts two or
three percentage points, on average, to real stock returns (which average some six
or eight per cent) will, if it is highly autocorrelated, routinely translate into a large
increase in price variance. The small change in real stock returns is the same thing
arithmetically as the large increase in price volatility, so the two should be
accorded a similar verbal characterization.
7. Conclusion
In the introduction it was noted that the early interchanges between academics
and finance practitioners about capital market efficiency generated more heat
than light. Models derived from market efficiency, such as CAPM-based portfolio
management models, made some inroads among practitioners, but for the most
part the debate between proponents and opponents of rationality in financial
markets died down. Parties on both sides agreed to disagree. The evidence of
excess price volatility reopened the debate, since it seemed at first to give
unambiguous testimony to the existence of irrational elements in security price
determination. Now it is clear that there exist other more conservative ways to
interpret the evidence of excess volatility: for example, that we simply do not
know what causes changes in the rates at which future expected dividends are
discounted.
The variance-bounds controversy, together with parallel developments in
financial economics, permit a considerable narrowing of the gap separating
proponents and opponents of market efficiency. The existence of excess volatility
implies that there are profitable trading rules, but it is known that these generate
only small utility gains to those employing them. In fact, this juxtaposition
between large departures from present-value pricing and small gains to those who
try to exploit these departures provides the key to finding some middle ground in
the efficiency debate. Proponents of market efficiency are vindicated because no
one has identified trading rules that are more than marginally profitable. De-

Stock price volatility
207
tractors of market efficiency are vindicated because a large proportion of the
variation in security prices remains unexplained by market fundamentals. Both
are correct; both are discussing the same sets of stylized facts.
Some proponents of market efficiency go to great lengths to argue that it is
unscientific to interpret excess volatility as evidence in favor of the importance of
psychological elements in security price determination; see, for example, Co-
chrane's otherwise excellent review (1991) of Shiller's (1989) book. On this view,
evidence is scientific only when it is incontrovertible and, presumably, not
susceptible to interpretations other than that proposed. At best this is an
unconventional use of the term "scientific". Indeed, if the term "unscientific" is to be
applied at all, should it not be to those who feel no embarrassment about the
continuing presence in their models of an uninterpreted residual that accounts for
most of the variation in the data? Given the continuing failure of financial models
based exclusively on received neoclassical economics to provide ex-post
explanations of security price behavior, why does being scientific rule out
broadening the field of inquiry to include psychological considerations?
References
Allen, F. and D. Gale (1994). Limited market participation and volatility of asset prices. Amer.
Econom. Rev. 84, 933-955.
Campbell, J. Y. and R. J. Shiller (1988). The dividend-price ratio and expectations of future dividends
and discount factors. Rev. Financ. Stud. 1, 195-228.
Cochrane, J. (1991). Volatility tests and efficient markets: A review essay. /. Monetary Econom. 27,
463-485.
Eden, B. and B. Jovanovic (1994). Asymmetric information and the excess volatility of stock prices.
Economic Inquiry 32, 228-235.
Fama, E. F. (1970). Efficient capital markets: A review of theory and empirical work. /. Finance 25,
283-417.
Fama, E. F. (1991). Efficient capital markets: II. /. Finance 46, 1575-1617.
Fama, E. F. and K. R. French (1988). Permanent and transitory components of stock prices. /. Politic.
Econom. 96, 246-273.
Flavin, M. (1983). Excess volatility in the financial markets: A reassessment of the empirical evidence.
/. Politic. Econom. 91, 929-956.
Flood, R. P. and R. J. Hodrick (1990). On testing for speculative bubbles. /. Econom. Perspectives 4,
85-101.
Gilles, C. and S. F. LeRoy (1992). Bubbles and charges. Internat. Econom. Rev. 33, 323-339.
Gilles, C. and S. F. LeRoy (1991). Economic aspects of the variance-bounds tests: A survey. Rev.
Financ. Stud. 4, 753-791.
Grossman, S. J. and R. J. Shiller (1981). The determinants of the variability of stock prices. Amer.
Econom. Rev. Papers Proc. 71, 222-227.
Hansen, L. and K. J. Singleton (1983). Stochastic consumption, risk aversion, and the temporal
behavior of asset returns. Econometrica 91, 249-265.
Kleidon, A. W. (1986). Variance bounds tests and stock price valuation models. /. Politic. Econom. 94,
953-1001.
LaCivita, C. J. and S. F. LeRoy (1981). Risk aversion and the dispersion of asset prices. /. Business 54,
535-547.

208
S. F. LeRoy
LeRoy, S. F. (1973). Risk aversion and the martingale model of stock prices. Internal. Econom. Rev.
14, 436-446.
LeRoy, S. F. and W. R. Parke (1992). Stock price volatility: Tests based on the geometric random
walk. Amer. Econom. Rev. 82, 981-992.
LeRoy, S. F. and A. D. Porter (1981). Stock price volatility: Tests based on implied variance bounds.
Econometrica 49, 555-574.
LeRoy, S. F. and D. G. Steigerwald (1993). Volatility. University of Minnesota.
Lucas, R. E. (1978). Asset prices in an exchange economy. Econometrica 46, 1429-1445.
Marsh, T. A. and R. C. Merton (1986). Dividend variability and variance bounds tests for the
rationality of stock market prices. Amer. Econom. Rev. 76, 483^498.
Marsh, T. A. and R. E. Merton (1983). Earnings variability and variance bounds tests for stockmarket
prices: A comment. Reproduced, MIT
Mehra, R. and E. C. Prescott (1985). The equity premium: A puzzle. /. Monetary Econom. 15, 145—
161.
Romer, D. (1993). Rational asset price movements without news. Amer. Econom. Rev. 83, 1112-1130.
Samuelson, P. A. (1965). Proof that properly anticipated prices flutuate randomly. Indust. Mgmt. Rev.
6, 41^19.
Shiller, R. J. (1981). Do stock prices move too much to be justified by subsequent changes in
dividends? Amer. Econom. Rev. 71, 421^436.
Shiller, R. J. (1989). Market Volatility. MIT Press, Cambridge, MA.
Shiller, R. J. (1986). The Marsh-Merton model of managers' smoothing of dividends. Amer. Econom.
Rev. 76, 499-503.
West, K. (1988), Bubbles, fads and stock price volatility: A partial evaluation. /. Finance 43, 636-656.

G. S. Maddala and C. R. Rao, eds., Handbook of Statistics, Vol. 14
© 1996 Elsevier Science B. V. All rights reserved.
7
GARCH Models of Volatility*
F. C. Palm
1. Introduction
Until some fifteen years ago, the focus of statistical analysis of time series centered
on the conditional first moment. The increased role played by risk and
uncertainty in models of economic decision making and the finding that common
measures of risk and volatility exhibit strong variation over time lead to the
development of new time series techniques for modeling time-variation in second
moments.
In line with Box-Jenkins type models for conditional first moments, Engle
(1982) put forward the Autoregressive Conditional Heteroskedastic (ARCH)
class of models for conditional variances which proved to be extremely useful for
analyzing economic time series. Since then an extensive literature has been
developed for modeling higher order conditional moments. Many applications can
be found in the field of financial time series. This vast literature on the theory and
empirical evidence from ARCH modeling has been surveyed in Bollerslev et al.
(1992), Nijman and Palm (1993), Bollerslev et al. (1994), Diebold and Lopez
(1994), Pagan (1995) and Bera and Higgings (1995). A detailed treatment of
ARCH models at a textbook level is also given by Gourieroux (1992).
The purpose of this chapter is to provide a selective account of certain aspects
of conditional volatility modeling in finance using ARCH and GARCH
(generalized ARCH) models and to compare the ARCH approach to alternatives lines
of research. The emphasis will be on recent developments for instance in
multivariate modeling using factor-ARCH models. Finally, an evaluation of the state
of the art will be given.
In Section 2, we introduce the univariate and multivariate GARCH models
(including ARCH models), discuss their properties and the choice of the
functional form and compare them with alternative volatility models. Section 3 will be
devoted to problems of inference in these models. In Section 4, the statistical
properties of GARCH models, their relationships with continuous time diffusion
* The author acknowledges many helpful comments by G. S. Maddala on an earlier version of the
paper.
209

210
F. C. Palm
models and the forecasting volatility will be discussed. Finally in Section 5 we
conclude and comment on potentially fruitful directions of future research.
2. GARCH models
2.1. Motivation
GARCH models have been developed to account for empirical regularities in
financial data. As emphasized by Pagan (1995) and Bollerslev et al. (1994), many
financial time series have a number of characteristics in common. First, asset
prices are generally nonstationary, often have a unit root whereas returns are
usually stationary. There is increasing evidence that some financial series are
fractionally integrated. Second, return series usually show no or little
autocorrelation. Serial independence between the squared values of the series however
is often rejected pointing towards the existence of nonlinear relationships between
subsequent observations. Volatility of the return series appears to be clustered.
Heavy fluctuations occur for longer periods. Small values for returns tend to be
followed by small values. These phenomena point towards time-varying
conditional variances. Third, normality has to be rejected frequently in favor of some
thick-tailed distribution. The presence of unconditional excess kurtosis in the
series could be related to the time-variation in the conditional variance. Fourth,
some series exhibit so-called leverage effects [see Black (1976)], that is changes in
stock prices tend to be negatively correlated with changes in volatility. Some
series have skewed unconditional empirical distributions pointing towards the
inappropriateness of the normal distribution. Fifth, volatilities of different
securities very often move together, indicating that there are linkages between markets
and that some common factors may explain the temporal variation in conditional
second moments. In the next subsection, we shall present several models which
account for temporal dependence in conditional variances, for skewness and
excess kurtosis.
2.2. Univariate GARCH models
Consider stochastic models of the form
yt = Bth]/2 , (2.1)
p i
/*r = a0 + ]T PA-i + J2 arf-i (2-2)
e=i i=i
with Eer = 0, Var(et) = 1, a0 > 0, ft > 0, «, > 0, and £f=1 ft + £?=i «* < L This
is the (p,q)th order GARCH model introduced by Bollerslev (1986). When
ft = 0, i = 1,2, ...p, it specializes to the ARCH(^) model put forward in a seminal
paper by Engle (1982). The nonnegativity conditions imply a nonnegative
variance, while the condition on the sum of the a,'s and ft's is required for wide sense

GARCH models of volatility
211
stationarity. These sufficient conditions for a nonnegative conditional variance
can be substantially weakened as shown by Nelson and Cao (1992). The
conditional variance of yt can become larger than the unconditional variance given by
a2 = ao/(l— J2jLi $i ~ 121=1 ad if past realizations of y2 have been larger than a2.
As shown by Anderson (1992), the GARCH model belongs to the class of
deterministic conditional heteroskedasticity models in which the conditional
variance is a function of variables that are in the information set available at time
t. Adding the assumption of normality, the model can be written as
j,,|*,_, ~JV(0,A,) , (2.3)
with ht being given by (2.2) and <P,^\ being the set of information available a