Testing Equality of Covariance Matrices via Pythagorean Means

Abstract

We provide a new test for equality of covariance matrices that leads to a convenient mechanism for testing specification using the information matrix equality. The test relies on a new characterization of equality between two k dimensional positive-definite matrices A and B: the traces of AB^{–1} and BA^{–1} are equal to k if and only if A = B. Using this criterion, we introduce a class of omnibus test statistics for equality of covariance matrices and examine their null, local, and global approximations under some mild regularity conditions. Monte Carlo experiments are conducted to explore the performance characteristics of the test criteria and provide comparisons with existing tests under the null hypothesis and local and global alternatives. The tests are applied to the classic empirical models for voting turnout investigated by Wolfinger and Rosenstone (1980) and Nagler (1991, 1994). Our tests show that all classic models for the 1984 presidential voting turnout are misspecified in the sense that the information matrix equality fails.

Keywords: Matrix equality, Trace, Determinant, Arithmetic mean, Geometric mean, Harmonic mean, Information matrix, Eigenvalues, Parametric bootstrap

JEL Classification: C01, C12, C52