Bayes Factors for Testing the Equality of Covariance Matrix Eigenvalues

Abstract

We use the methods of McCulloch and Rossi (1992) to construct a Bayes Factor for the hypothesis of the equality of eigenvalues of a covariance matrix. If the smallest s eigenvalues of a covariance matrix are equal, then we can think of the p dimensional sample data as arising from a reduced rank model. Tests of this sort are often used to identify the number of components used in a Principal Components analysis. The Bayes Factor approach requires specification of prior distributions for both the restricted and unrestricted covariance matrices. The problem of specifying a prior distribution on the restricted covariance matrix is solved via a projection method. We exploit the duality between sufficient statistics and parameters in exponential families to use the restricted MLE as a projection device. We illustrate this method with simulated and actual data. Our real data example addresses the question of the number of factors underlying stock return data.