The Analysis of the Cross Section of Security Returns
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The Analysis of the Cross Section
of Security Returns
Ravi Jagannathany
Georgios Skoulakisz
Zhenyu Wangx
This paper will appear as a chapter in the forthcoming Handbook of Financial Econometrics edited by Yacine
At-Sahalia and Lars P. Hansen. The authors wish to thank Bob Korajczyk, Ernst Schaumburg and Jay Shanken for
comments and Aiyesha Dey for editorial assistance.
yDepartment of Finance, Kellogg School of Management, Northwestern University, Evanston, IL 60201, USA.
E-mail: rjaganna@kellogg.northwestern.edu
zDepartment of Finance, Kellogg School of Management,Northwestern University, Evanston, IL 60201, USA.
E-mail: g-skoulakis@kellogg.northwestern.edu
xDepartment of Finance and Economics, Graduate School of Business, Columbia University, New York, NY 10027,
USA. E-mail: zwang@columbia.edu
Contents
1 Introduction 1
2 Linear Beta Pricing Models, Factors and Characteristics 3
2.1 Linear beta pricing models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Factor selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.3 Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3 Cross-Sectional Regression Methods 8
3.1 Description of the CSR method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.2 Consistency and asymptotic normality of the CSR estimator . . . . . . . . . . . . . . 10
3.3 Fama-MacBeth variance estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.4 Conditionally homoscedastic residuals given the factors . . . . . . . . . . . . . . . . 14
3.5 Using security characteristics to test factor pricing models . . . . . . . . . . . . . . . 17
3.5.1 Consistency and asymptotic normality of the CSR estimator . . . . . . . . . 19
3.5.2 Misspeci cation bias and protection against spurious factors . . . . . . . . . . 20
3.6 Time-varying security characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.6.1 No pricing restrictions imposed on traded factors . . . . . . . . . . . . . . . . 21
3.6.2 Traded factors with imposed pricing restrictions . . . . . . . . . . . . . . . . 25
3.6.3 Using time-average characteristics to avoid the bias . . . . . . . . . . . . . . . 29
3.7 N-consistency of the CSR estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4 Maximum Likelihood Methods 36
4.1 Nontraded factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.2 Some factors are traded . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.3 Single risk-free lending and borrowing rates with portfolio returns as factors . . . . . 39
5 The Generalized Method of Moments 40
5.1 An overview of the GMM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.2 Evaluating beta pricing models using the beta representation . . . . . . . . . . . . . 42
5.3 Evaluating beta pricing models using the stochastic discount factor representation . 46
5.4 Models with time-varying betas and risk premia . . . . . . . . . . . . . . . . . . . . . 50
6 Conclusions 56
1 Introduction
Financial assets exhibit wide variation in their historical average returns. For example, during the
period from 1926 to 1999 large stocks earned an annualized average return of 13.0%, whereas longterm
bonds earned only 5.6%. Small stocks earned 18.9% - substantially higher than large stocks.
These di erences are statistically and economically signi cant (Jagannathan and McGrattan (1996),
Ferson and Jagannathan (1996)). Furthermore, such signi cant di erences in average returns are
also observed among other classes of stocks. If investors were rational, they would have anticipated
such di erences. Nevertheless, they still preferred to hold nancial assets with such widely di erent
expected returns. A natural question that arises is why this is the case. A variety of asset pricing
models have been proposed in the literature for understanding why di erent assets earn di erent
expected rates of return. According to these models di erence assets earn di erent expected returns
only because they di er in their systematic risk. The models di er based on the stand they take
regarding what constitutes systematic risk. Among them, the linear beta pricing models form an
important class.
According to linear beta pricing models, a few economy-wide pervasive factors are sucient to
represent systematic risk, and the expected return on an asset is a linear function of its factor betas
(Ross (1976), Connor (1984)). Some beta pricing models specify what the risk factor should be
based on theoretical arguments. According to the standard Capital Asset Pricing Model (CAPM)
of Sharpe (1964) and Lintner (1965), the return on the market portfolio of all assets that are in
positive net supply in the economy is the relevant risk factor. Other models specify factors based
on economic intuition and introspection. For example, Chen, Roll and Ross (1986) specify unanticipated
changes in the term premium, default premium, the growth rate of industrial production
and in
ation as the factors, whereas Fama and French (1993) construct factors that capture the
size and book-to-market e ects documented in the literature and examine if these are sucient to
capture all economy-wide pervasive sources of risk. Campbell (1996) and Jagannathan and Wang
(1996) use innovations to labor income as an aggregate risk factor. Another approach is to identify
the pervasive risk factors based on systematic statistical analysis of historical return data as in
Connor and Korajczyk (1988) and Lehmann and Modest (1988).
In this chapter we discuss econometric methods that have been used to evaluate linear beta
pricing models using historical return data on a large cross section of stocks. Three approaches
have been suggested in the literature for examining linear beta pricing models: (a) Cross sectional
regressions method; (b) Maximum Likelihood (ML) methods; and (c) Generalized method of moments
(GMM). Shanken (1992) and MacKinlay and Richardson (1991) show that the cross-sectional
method is asymptotically equivalent to ML and GMM when returns are conditionally homoscedastic.
In view of this, we focus our attention primarily on the cross-sectional regression method since
1
it is more robust and easier to implement in large cross sections, and provide only a brief overview
of the use of ML and GMM.
Fama and MacBeth (1973) developed the two pass cross sectional regression method to examine
whether the relation between expected return and factor betas are linear. Betas are estimated using
time series regression in the rst pass and the relation between returns and betas are estimated
using a second pass cross sectional regression. The use of estimated betas in the second pass
introduces the classical errors-in-variables problem. The standard method for handling errors in
variables problem is to group stocks into portfolios following Black, Jensen and Scholes (1972).
Since each portfolio has a large number of individual stocks, portfolio betas are estimated with
sucient precision and this fact allows one to ignore the errors-in-variables problem as being of
second order in importance. One, however, has to be careful to ensure that the portfolio formation
method does not highlight or mask characteristics in the data that have valuable information about
the validity of the asset pricing model under examination. Put in other words, one has to avoid
data snooping biases discussed in Lo and MacKinlay (1990).
Shanken (1992) provided the rst comprehensive analysis of the statistical properties of the
classical two-pass estimator under the assumption that returns and factors exhibit conditional homoscedasticity.
He demonstrated ed how to take into account the sampling errors in the betas
estimated in the rst pass and generalized-least-squares in the second stage cross-sectional regressions.
Given these adjustments, Shanken (1992) conjectured that it may not be necessary to group
securities into portfolios in order to address the errors in variables problem. Brennan, Chordia and
Subrahmanyam (1998) make the interesting observation that the errors in variables problem can
be avoided without grouping securities into portfolios by using risk-adjusted returns as dependent
variables in tests of linear beta pricing models, provided all the factors are excess returns on traded
assets. However, the relative merits of this approach as compared to portfolio grouping procedures
has not been examined in the literature.
Jagannathan andWang (1998) extended Shanken's analysis to allow for conditional heteroscedasticity
and consider the case where the model is misspeci ed. This may happen even when the model
holds in the population, if the econometrician uses the wrong factors or misses factors in computing
factor betas. When the linear factor pricing model is correctly speci ed, rm characteristics such
as rm size should not be able to explain expected return variations in the cross section of stocks.
In the case of misspeci ed factor models, Jagannathan and Wang (1998) showed that the t-values
associated with rm characteristics will typically be large. Hence, model misspeci cation can be
detected using rm characteristics in cross-sectional regression. Such a test does not require that
the number of assets be small relative to the length of the time series of observations on asset
returns, as is the case with standard multivariate tests of linearity.
2
Gibbons (1982) showed that the classical maximum likelihood method can be used to estimate
and test linear beta pricing models when stock returns are i.i.d and jointly normal. Kandel (1984)
developed a straight forward computational procedure for implementing the maximum likelihood
method. Shanken (1992) extended it further and showed that the cross sectional regression approach
can be made asymptotically as ecient as the maximum likelihood method. Kim (1985) developed
a maximum likelihood procedure that allows for the use of betas estimated using past data. Jobson
and Korkie (1982) and MacKinlay (1987) developed exact multivariate tests for the CAPM and
Gibbons, Ross and Shanken (1989) exact multivariate tests for linear beta pricing models when
there is a risk free asset.
MacKinlay and Richardson (1991) show how to estimate the parameters of the CAPM by
applying the GMM to its beta representation. They illustrate the bias in the tests based on
standard maximum likelihood methods when stock returns exhibit contemporaneous conditional
heteroscedasticity and show that the GMM estimator and the maximum likelihood method are
equivalent under conditional homoscedasticity. An advantage of using the GMM is that it allows
estimation of model parameters in a single pass thereby avoiding the error-in-variables problem.
Linear factor pricing models can also be estimated by applying the GMM to their stochastic discount
factor (SDF) representation. Jagannathan and Wang (2001) show that parameters estimated by
applying the GMM to the SDF representation and the beta representation of linear beta pricing
models are asymptotically equivalent.
The rest of the chapter is organized as follows. In Section 2 we set up the necessary notation and
describe the general linear beta pricing model. We discuss in detail the two pass cross sectional
regression method in Section 3 and provide an overview of the maximum likelihood methods in
Section 4 and the GMM in Section 5. We summarize in Section 6.
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