Robust Inference for Consumption‐Based Asset Pricing

• Published date: 01 February 2020
• Author: ZHAOGUO ZHAN,FRANK KLEIBERGEN
• Date: 01 February 2020
• DOI: http://doi.org/10.1111/jofi.12855

THE JOURNAL OF FINANCE •VOL. LXXV, NO. 1 •FEBRUARY 2020

Robust Inference for Consumption-Based

Asset Pricing

FRANK KLEIBERGEN and ZHAOGUO ZHAN∗

ABSTRACT

The reliability of traditional asset pricing tests depends on: (i) the correlations be-

tween asset returns and factors; (ii) the time series sample size Tcompared to the

number of assets N. For macro-risk factors, like consumption growth, (i) and (ii)

are often such that traditional tests cannot be trusted. We extend the Gibbons-Ross-

Shanken statistic to test identification of risk premia and construct their 95% confi-

dence sets. These sets are wide or unbounded when Tand Nare close, but show that

average returns are not fully spanned by betas when Texceeds Nconsiderably. Our

findings indicate when meaningful empirical inference is feasible.

CLASSICAL CONSUMPTION-BASED ASSET PRICING THEORY relates asset returns to

consumption risk. Yet a worrisome phenomenon is that different measures of

consumption lead to different empirical findings. It has been well documented

that the canonical consumption measure from the National Income and Prod-

uct Accounts (NIPA) leads to small correlations between consumption growth

and asset returns that could be improved by adopting alternative consump-

tion measures, including the three-year consumption measure in Parker and

Julliard (2005), the fourth-quarter to fourth-quarter consumption measure in

Jagannathan and Wang (2007), the garbage measure in Savov (2011), and the

unfiltered NIPA consumption measure in Kroencke (2017).

The credibility of these consumption measures for asset pricing is commonly

tested using two methodologies: (i) the two-pass regression for linear asset

pricing models, where expected asset returns are expressed by the beta rep-

resentation (Fama and MacBeth (FM, 1973)), and (ii) the generalized method

of moments (GMM) of Hansen (1982) for nonlinear asset pricing models in a

stochastic discount factor (SDF) representation. Empirical findings based on

∗Frank Kleibergen is at the University of Amsterdam. Zhaoguo Zhan is at Kennesaw State

University. We thank the Editor, Stefan Nagel, the Associate Editor,and two anonymous referees

for their comments that substantially improved the paper. Our thanks for helpful discussions also

go to Lucy F. Ackert, Tomislav Ladika, and participants at the seminars as well as conferences

where the paper has been presented. We thank Alexi Savov, Tim A. Kroencke, Martin Lettau, and

Sydney Ludvigson for sharing their data. We have read The Journal of Finance disclosure policy

and have no conflicts of interest to disclose.

This is an open access article under the terms of the Creative Commons Attribution License, which

permits use, distribution and reproduction in any medium, provided the original work is properly

cited.

DOI: 10.1111/jofi.12855

C2019 The Authors. The Journal of Finance published by the American Finance Association

507

508 The Journal of Finance R

these two standard methodologies appear to favor some consumption measures

for asset pricing over others (Kroencke (2017)).

It is well known, however, that the reliability of asset pricing tests in both

the FM and the GMM settings depends on the statistical quality of consump-

tion measures, or more generally, risk factors. For instance, Kan and Zhang

(1999a) and Kleibergen (2009) warn that the t-test in the FM two-pass pro-

cedure can spuriously favor risk factors that are independent of or weakly

correlated with asset returns, respectively. Similar spurious outcomes arise

in the GMM setting (Stock and Wright (2000), Kleibergen (2005), Gospodinov,

Kan, and Robotti (2017)). Moreover, Kan and Zhang (1999b), Lewellen, Nagel,

and Shanken (2010), and Kleibergen and Zhan (2015,2018) provide intuitive

explanations for why poor quality risk factors can induce seemingly promising

empirical outcomes. In light of this literature, it is important not to misinter-

pret potentially spurious outcomes as evidence in support of factor pricing.

In addition to the statistical quality of various consumption measures, lim-

ited sample sizes impose challenges for empirical asset pricing tests. For esti-

mation results of asset pricing models to be representative of the span of the

market, existing studies typically use a considerable number of test assets. At

the same time, macro-risk factors such as consumption growth are commonly

measured at annual or quarterly frequencies, in which case the number of ob-

servations is often not much larger than the number of test assets. The limited

number of time series observations Tcompared to the number of cross-section

observations Nleads to large estimation error of the covariance matrix of the

test assets. This “limited Tversus large N” problem is not accounted for in

standard asymptotic approximations of tand Wald statistics (Bekker (1994),

Newey and Windmeijer (2009)) and thus provides a second argument for why

researchers cannot rely on standard t/Wald-tests to conduct dependable statis-

tical inference.

The credibility of traditional asset pricing tests on the parameters of interest,

such as risk premia, in FM and GMM thus depends on (i) the strength of

identification, which is reflected by the correlations between risk factors and

returns on test assets, and (ii) the number of time series observations of the

risk factors relative to the number of test assets, that is, the “limited Tversus

large N” problem. For asset pricing models that involve macroeconomic factors,

both of these issues can threaten the reliability of standard asset pricing tests.

In this paper, we propose two straightforward asset pricing tests that, unlike

traditional tests, are valid for all possible strengths of identification of the

risk premia and for scenarios in which the time series sample size exceeds the

number of test assets. Our tests are simple extensions of the Gibbons, Ross,

and Shanken (GRS, 1989) test, which tests the joint significance of the so-

called alphas, or constant terms, in a set of regressions of test asset returns

on a constant and risk factors. The first of our proposed statistics tests the

identification of the risk premia, while the second tests for specific values of

the risk premia to allow for the construction of confidence sets.

In the linear beta representation of expected asset returns, risk premia are

identified through the variation of the betas over the cross-section of test assets.

Robust Inference for Consumption-Based Asset Pricing 509

Our first proposed identification statistic therefore examines whether there is

sufficient variation in the betas to identify risk premia. In the case of a single

risk factor, the risk premium is not identified in the beta representation that

includes a zero-beta return if the betas do not vary over the cross-section, so

our identification test statistic simply tests whether all betas are identical. The

GRS statistic that we use for this purpose allows the number of cross-section

and time series observations to be close and has an F-distribution in finite

samples, as in GRS.

We apply our identification test to data from Kroencke (2017) to investi-

gate the identification of the risk premia on the five consumption measures

mentioned above (i.e., NIPA, Parker and Julliard (2005), Jagannathan and

Wan g (2007), Savov (2011), and Kroencke (2017)). At the 5% significance level,

our identification test cannot reject the possibility that the betas of these five

consumption measures are constant. Consequently, despite the significance

of the betas for some consumption measures, we cannot reject that the risk

premia are unidentified under these consumption measures. Simulation stud-

ies calibrated to these data further show that the FM t-test on risk premia

is indeed unreliable for each of the five consumption growth series. The rea-

son is that when the betas are almost constant, they basically become pro-

portional to the constant term in the second pass of the FM two-pass pro-

cedure. The resulting near multicollinearity violates the assumptions needed

for the asymptotic validity of the FM t-test and explains why its rejection

frequencies do not accord with conventional asymptotics in the simulation

experiment.

Unlike the FM t-test, whose validity is subject to the questionable asymp-

totic normal approximation of the risk premia estimator, our second proposed

test statistic does not involve a risk premia estimator but rather considers all

restrictions imposed by factor pricing at the hypothesized value of the risk

premia. It does so by noting that factor pricing implies zero alphas in an appro-

priately specified set of regressions that use the prespecified value of the risk

premia. We test the joint significance of all of these alphas using the GRS test,

which for this purpose, we refer to as the GRS-Factor Anderson-Rubin (GRS-

FAR) statistic to indicate that it also provides an extension of the well-known

Anderson-Rubin statistic in instrumental variables regression (Anderson and

Rubin (1949), Kleibergen (2009)). Since it is just a GRS test, the GRS-FAR

statistic has an exact F-distribution so its reliability does not depend on either

the identification of the risk premia or the number of time series observations

compared to the number of test assets.

We report confidence sets that consist of the hypothesized values of the risk

premia for which the GRS-FAR statistic is insignificant at the appropriate

significance level. By varying the value of the risk premia used in the GRS-

FAR statistic, we can map out the confidence set, but we can also compute it

with a closed-form expression (Dufour and Taamouti (2005)). Using the data

from Kroencke (2017), we find that the 95% confidence sets for the risk premia

do not exclude any pre-set value for all five consumption measures. Our 95%

confidence sets are thus unbounded. This might seem odd but it should come