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Bayesian inference in asset pricing tests

__Journal of Financial Economics__, 26 (1990): 221-254

**
Campbell R. Harvey**

*
Duke University, Durham, NC 27708, USA*

**
Guofu Zhou**

*
Washington University, St. Louis, MO 63130, USA*

Abstract

We test the mean-variance efficiency of a given portfolio using a Bayesian
framework. Our test is more direct than Shanken's (1987b), because we impose a
prior on all the parameters of the multivariate regression model. The approach
is also easily adapted to other problems. We use Monte Carlo numerical
integration to accurately evaluate 90-dimensional integrals. Posterior-odds
ratios are calculated for 12 industry portfolios from 1926-1987. The
sensitivity of the inferences to the prior is investigated by using three
different distributions. The Probability that the given portfolio is
mean-variance efficient is small for a range of plausible priors.