Unconditional Tests of Linear Asset Pricing Models with Time‐Varying Betas

Published date01 August 2017
Date01 August 2017
The Financial Review 52 (2017) 373–404
Unconditional Tests of Linear Asset Pricing
Models with Time-Varying Betas
Ji Zhou
University of Manitoba
Alex Paseka
University of Manitoba
In conditional affine factor models, estimated risk prices should satisfy certain uncondi-
tional constraints. Specifically,a cross-sectional estimate of the unconditional slope associated
with a risk factor should equal the average price of risk of the factor. The estimated slope
associated with the product of a risk factor and an instrument should be equal to the covariance
of the factor risk premium with the instrument. We show that the constraints only apply to
the conditional models with time-varying betas. We identify an unconditional constraint on
unconditional betas for time-varying beta models and incorporate it into model tests. Weshow
that imposing this unconditional constraint changes estimates of unconditional betas and risk
prices significantly.
Keywords: asset pricing tests, conditional linear factor model, time-varying betas
JEL Classification:G12
Corresponding author: Department of Accounting and Finance, University of Manitoba, Winnipeg,
Manitoba, Canada. E-mail: jizhou86@outlook.com.
We would like to thank Aerambamoorthy Thavaneswaran,Steven Zheng, and seminar participants at the
University of Manitoba for invaluable comments and insightful discussions. We also benefited greatly
from the comments of Srinivasan Krishnamurthy (the Editor) and the anonymous referees who helped us
improve the quality of our paper.In addition, we thank the participants of the Midwest Finance Association
2015 annual meeting and Eastern Finance Association 2015 annual meeting for their helpful comments.
We are solely responsible for all remaining errors.
C2017The Eastern Finance Association 373
374 J. Zhou and A. Paseka/The Financial Review52 (2017) 373–404
1. Introduction
Previous studies show that although unconditional factor models have gener-
ally failed in explaining the cross section of average stock returns, dynamic asset
pricing models may succeed in the task. By allowing factor loadings in a stochastic
discount factor (SDF) or factor betas to vary over time, conditional models (such as
Jagannathan and Wang,1996’s [hereafter JW] conditional capital-asset pricing model
(CAPM); Lettau and Ludvigson, 2001b’s[hereafter LL01] scaled consumption-based
CAPM (CCAPM); and Santos and Veronesi,2006’s [hereafter SV] labor income ratio
model) are able to explain a large proportion of variation in average stock returns.
These models suggest that identifying proper instruments to describe time variation in
an investment opportunity set in conditional linear factor models is a fruitful avenue
for future research in asset pricing.
However, Lewellen and Nagel (2006) (hereafter LN) warn that the success of
the conditional models may be illusory and should be treated with caution. LN argue
that conditional models mentioned above may not be able to produce statistically
small unconditional alphas. Their analysis implies that good performance of those
conditional models in cross-sectional tests may be due to the fact that some important
unconditional constraints are omitted in the estimation procedure.
LN start with beta representation E(Rt+1|Ft)=βtλt. If we assume that con-
ditional factor beta βtis an affine function of a (zero-mean) instrument zt(i.e.,
βt=β0+δzzt, the “affine beta” constraint), we can derive unconditional moments
by taking unconditional expectation on both sides of the conditional model:
Based on the unconditional moment above, LN argue that the slope on β0should
equal E(λt) and slopes on δzshould equal cov(ztλt) in cross-sectional regressions.
LN claim that neglecting these constraints on unconditional risk prices in model
estimation may be responsible for the good reported performance of conditional
affine models. Ludvigson (2011) takes a closer look at LN’s critique. She shows that
LN’s critique cannot be applied to CCAPM.1LN’s critique is a direct consequence
of writing time-varying betas as functions of instruments and unconditional betas.
Such relation does not exist in the conditional CCAPM. More specifically, in the
conditional CCAPM, pricing kernel’s loadings on factors, rather than factor betas,
are time-varying functions of instruments. Therefore, LN’s critique does not apply to
the model. We return to the technical details in the later part of the paper.
While Ludvigson (2011) shows that CCAPM is immune to LN’s critique, it
is still not clear if models with time-varying beta assumptions are subject to LN’s
critique. In this paper, we follow LN’s work and examine the unconditional moment
implications of conditional factor models in which factor betas are assumed to be
1We conduct a simple empirical test and confirmher results in the Appendix.
J. Zhou and A. Paseka/The Financial Review52 (2017) 373–404 375
time-varying or, more specifically, affine functions of instruments. The last assump-
tion is not implied in any way by linear SDF models. Instead, it must be considered
as an additional constraint on the model. Theoretically, the assumption that betas are
affine functions of instruments is likely invalid. The “affine beta” constraint does
not necessarily correctly map back into the SDF affine in factors, which is a cen-
tral assumption of linear asset pricing models in general and the models we test,
in particular. Even if the affine beta assumption is not valid theoretically, it may
still be a valid empirical approximation. We examine the empirical validity of this
Once a conditional model is conditioned down to its unconditional version, the
affine restriction on conditional factor betas leads to unconditional constraints on
unconditional betas and risk prices (LN’s critique). This feature allows us to as-
sess the empirical validity of the affine beta assumption by comparing estimation
results with or without imposing these unconditional constraints. To do so, we de-
velop a test that allows us to incorporate the constraints on factor betas and risk
To impose the “affine beta” constraint in empirical tests, we propose a three-
stage regression procedure. Wefirst estimate time-varying betas using rolling-window
regressions. In the second stage, we run a regression of the estimated time-varying
betas on instruments to retrieve unconditional betas. The last step is to run a cross-
sectional regression to estimate factor risk premiums. Our method produces the
estimates of unconditional betas from estimated time series of time-varying betas
based on the affine relation between them. Our empirical results of the time-varying
beta model (SV’s labor income ratio model is used as an example in our tests) show
that imposing the constraint changes estimates of unconditional betas and those of
factor risk premiums significantly and has a strong negative impact on the model
performance (measured by the root mean squared error [RMSE]).
Our extension of the work of LN is to identify and test unconditional moment
implications of conditional models with time-varying betas. Our work is related to
Nagel and Singleton (2011), who expand a set of conditional restrictions and de-
vise an optimal Generalized Method of Moments (GMM) procedure to test these
restrictions. Our work is also related to Maio and Santa-Clara (2012), who examine
intertemporal CAPM (ICAPM)-based theoretical restrictions on multifactor models.
While Nagel and Singleton (2011) and Maio and Santa-Clara (2012) focus on con-
ditional constraints on factor models, we test unconditional constraints arising at the
conditioning-down stage of conditional model tests.
This paper is organized as follows. In Section 2, we present LN’s critique of
cross-sectional tests of conditional models and our discussion of the critique. Section
3 describes our testing approach and presents main empirical results. We conclude
in Section 4. For most technical details and robustness checks, we refer the reader to

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