On Comparing Asset Pricing Models

We revisit the framework of Barillas and Shanken (2018) (BS henceforth) to point out that the Bayesian marginal likelihood based model comparison method in that paper is unsound. We show that in this comparison of asset pricing models, 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 Jeffreys priors used in the BS method. Hence, the BS "marginal likelihoods" are non-comparable across models and cannot be used to locate the (traded) risk factors. We conduct extensive simulation exercises in two designs: one with 8 potential pricing factors and a second with 12 factors, in each case matching the factors to real world factors that arise in this setting. As expected, the BS method performs unsatisfactorily, even when epic (and practically unattainable) sample sizes of .12 and 1.2 million are used to conduct the model comparisons. In a notable advance, we derive a new class of improper priors on the nuisance parameters, starting from a single improper prior, which leads to valid marginal likelihoods, and valid model comparisons. The empirical performance of our marginal likelihoods is substantially better, opening doors to reliable Bayesian work on which factors are risk factors in asset pricing models