On Comparing Asset Pricing Models

• Author: LINGXIAO ZHAO,XIAMING ZENG,SIDDHARTHA CHIB
• DOI: http://doi.org/10.1111/jofi.12854
• Date: 01 February 2020
• Published date: 01 February 2020

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

On Comparing Asset Pricing Models

SIDDHARTHA CHIB, XIAMING ZENG, and LINGXIAO ZHAO∗

ABSTRACT

Revisiting the framework of (Barillas, Francisco, and Jay Shanken, 2018, Comparing

asset pricing models, The Journal of Finance 73, 715–754). BS henceforth, we show

that the Bayesian marginal likelihood-based model comparison method in that pa-

per is unsound : the priors on the nuisance parameters across models must satisfy a

change of variable property for densities that is violated by the Jeffreys priors used in

the BS method. Extensive simulation exercises confirm that the BS method performs

unsatisfactorily. We derive a new class of improper priors on the nuisance parame-

ters, starting from a single improper prior, which leads to valid marginal likelihoods

and model comparisons. The performance of our marginal likelihoods is significantly

better, allowing for reliable Bayesian work on which factors are risk factors in asset

pricing models.

IN THIS PAPER,WE REVISIT the framework of Barillas and Shanken (2018), BS

henceforth, and show that the Bayesian marginal likelihood-based model com-

parison method in that paper is unsound. In particular, we show that in this

comparison of asset pricing models, in which the nuisance parameters {ηj}

across models are connected by invertible mappings, the priors on the nuisance

parameters across models must satisfy a certain change of variable property

for densities that is violated by the off-the-shelf Jeffrey’ priors used in the BS

method. Hence, the BS “marginal likelihoods” each depend on an arbitrary con-

stant, which voids the ranking of models by the size of the marginal likelihoods

and invalidates any conclusions drawn from such a method about the underly-

ing data-generating process (DGP). In the online appendix of their paper, BS

discuss an alternative method for calculating marginal likelihoods with their

improper priors, which they call the permutation method. This more involved

method is not used in the paper but, as we show below, it is also unsound and

as a result leads to invalid marginal likelihoods.

We conduct extensive simulation exercises using two experiments. In the

first, we match eight potential risk factors to the excess market return (Mkt),

size (SMB), value (HML), profitability (RMW), and investment (CMA) factors

proposed by Fama and French (1993,2015), the profitability (ROE) and in-

vestment (IA) factors in the q-factor model proposed by Hou, Xue, and Zhang

∗Siddhartha Chib is at the Olin Business School, Washington University in St. Louis. Xiaming

Zeng is an Investment Professional. Lingxiao Zhao is at the Department of Economics, Washington

University in St. Louis. We are grateful to the Editor (Stefan Nagel) and two anonymous reviewers

for their constructive and helpful comments. Wehave read The Journal of Finance disclosure policy

and have no conflicts of interest to disclose.

DOI: 10.1111/jofi.12854

C2019 the American Finance Association

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552 The Journal of Finance R

(2015), and the Carhart (1997) momentum (MOM) factor. In the second, we

match 12 potential risk factors to the eight factors above as well as the Asness,

Frazzini, and Pedersen (2014) quality minus junk (QMJ) factor, the P´

astor

and Stambaugh (2003) liquidity (LIQ) factor, the Frazzini and Pedersen (2014)

betting against beta (BAB) factor, and another version of value factor (HMLD)

proposed by Asness and Frazzini (2013). Given the prejudged status of the

Mkt factor as a risk factor, we have 27=128 candidate models in the first

experiment and 211 =2,048 candidate models in the second. We repeat our

comparison exercises over 100 simulated replications of the data for sample

sizes of 600, 1,200 and 12,000, 120,000 and 1.2 million for each of 30 (55) true

DGPs in the first (second) experiment. In the first experiment, the BS method

has some success when the sample size is 1.2 million, but in the second ex-

periment the BS method fails to locate any of the true DGPs even once in 100

replications for any sample size, including the epic sample size of 1.2 million.

In a significant advance, we derive a new class of improper priors on the

nuisance parameters, starting from a single improper prior, with the property

that the improper priors in this class necessarily share the same arbitrary

constant c. This class of priors leads to valid marginal likelihoods and, in

turn, valid model comparisons. The construction of this class of improper

priors is summarized in Proposition 2. As we detail, the ability of the resulting

marginal likelihoods to pick the true DGPs is significantly better.

We also discuss an extension of our method to the more general class of model

comparisons in which the status of the Mkt factor as a risk factor is also in

doubt. Chib and Zeng (2019) have recently developed a method for conducting

such comparisons that is based on proper priors, each derived from a single

proper prior, and student-tdistributions of the factors. The approach in this

paper, though closely related to that of Chib and Zeng (2019), requires fewer

prior inputs, and together pave the way for reliable Bayesian work on which

factors are risk factors in asset pricing models.

The rest of the paper is organized as follows. In Section I, we outline the

BS method for calculating marginal likelihoods. In Section II, we discuss the

issues that arise in calculating marginal likelihoods with improper priors, and

in Proposition 2we provide a class of improper priors on nuisance parameters

that lead to valid marginal likelihoods. In Section III, we derive the priors and

marginal likelihoods that satisfy Proposition 2(which we refer to as the Chib,

Zeng, and Zhao priors and marginal likelihoods) for the problem of compar-

ing asset pricing models. Section IV contains further critical discussion of the

BS method, and Section Vand VI present results from extensive simulation

experiments on the performance of the BS and Chib, Zeng, and Zhao meth-

ods, respectively. Section VII concludes. Appendices contain additional details

relevant for the discussion in the main text.

I. BS Method

In the method of BS, one starts with a collection of K(traded) potential risk

factors. The market factor (Mkt) is one of these Kfactors and is prejudged to be