There are two competing approaches to statistical inference: classical and Bayesian. The fundamental difference between them is the notion of probability. In the classical framework, the probability of an event is defined by the limit of its relative frequency. Estimators and test procedures are evaluated in repeated samples. In the Bayesian framework, probability is defined by a degree of belief.
The Bayesian approach makes it possible to incorporate a belief about the hypothesis being tested and its alternative in the form of a prior-odds ratio. When we look at the data, we get a posterior-odds ratio, which summarizes all the evidence (prior and sample) in favor of the hypothesis or its alterna-tive. The posterior-odds ratio can be interpreted as the ratio of the probability that the hypothesis is valid to the probability that its alternative is valid.
Many applications in finance involve prior beliefs about the behavior of the data. However, almost all empirical analysis has been carried out in the classical framework. The slow adoption of Bayesian econometrics is a result of two practical difficulties in implementing the approach. One is how to choose a prior. The critical difficulty arises, however, in evaluating the posterior distribution. This may involve high-dimensional integration, which is analytically intractable. For example, some of our empirical work involves 90-dimensional integration. Fortunately, with the recent development of Monte Carlo numerical integration, high-order integration problems are routinely solved with a high degree of accuracy
This paper examines multivariate tests of the mean-variance efficiency of a given portfolio. Usually, these tests are done in a classical framework. Bayesian inference about mean-variance efficiency has received relatively little attention. An exception is an important paper by Shanken (1987b) who uses a result in Gibbons, Ross, and Shanken (1989) to develop a computationally convenient way to calculate the posterior-odds ratio.
Shanken (1987b) uses the posterior-odds ratio to test the restriction imposed by the Sharpe (1964) - Lintner (1965) capital asset pricing model (CAPM) that the intercepts in the multivariate regression of excess returns on the market excess return are equal to zero. Shanken's test is indirect, however. He replaces the intercepts with a function of the intercepts and tests whether this function is zero. With this method, he can impose a prior only on the function, not on the intercepts. More importantly, Shanken's test cannot easily be applied to other problems because it critically relies on the sampling distribution of the classical F statistic proposed in Gibbons, Ross, and Shanken (1989). For example, we cannot apply Shanken's results to test the restrictions implied in the Black (1972) CAPM. Indeed, Shanken realizes that:
- A more ambitious and much more complicated approach to this problem would start with a joint prior distribution for all parameters in the multivariate linear regression of returns.
Our paper addresses this challenge and proposes a full Bayesian specification of the asset pricing model tests. We use the algorithm suggested by Geweke (1988, 1989) to evaluate the posterior distributions. Posterior-odds ratios are calculated using both a diffuse prior and an informative prior to test the restrictions implied by the Sharpe-Lintner CAPM. Further, we check the sensitivity of the inference to the choice of prior distributions. Finally, we calculate Bayesian confidence intervals for the parameters of interest. Tests are carried out on monthly returns from 1926 to 1987 on 12 industry portfolios. The evidence suggests that the value-weighted New York Stock Exchange (NYSE) market portfolio is not mean-variance efficient.
The paper is organized as follows. In the second section, we present the Bayesian framework for testing the asset pricing restrictions. The empirical results are included in the third section. Some concluding remarks are offered in section 4. A brief introduction to Monte Carlo integration appears in appendix A.
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