A bayesian approach to testing the arbitrage pricing theory

Abstract

We consider new methods of testing the restrictions imposed on asset returns implied by an equilibrium version of the Arbitrage Pricing Theory. In the competitive equilibrium version of the APT [see, for example, Connor and Korajczyk (1988)], excess returns on a collection of portfolios or individual assets are related to measured factors in a multivariate regression model without intercepts. The traditional significance testing approach uses estimates of these intercept terms and a likelihood ratio procedure for statistical testing. In our approach, we quantify differences between the restricted and unrestricted models by computing posterior model probabilities. These posterior model probabilities are combined to form posterior odds ratios which represent the weight of prior and sample evidence for or against the pricing restrictions. A dramatic simplification of the odds ratio calculation is achieved by using the Savage density ratio method. The special structure of the APT model with factors extracted by principal components is exploited to produce a simple prior. The sensitivity of the odds ratio to different prior specifications is explored. We apply these new methodologies to tests of the APT with a collection of ten size-based portfolios. The factors are extracted by a principal components technique developed by Connor and Korajczyk using all listed AMEX/NYSE firms over the period 1964–1983. The odds ratios show that the sample evidence is weak and does not favor the APT pricing restrictions. We are able to gauge the strength of the evidence in the data by examining the sensitivity of the odds ratio to the prior.