A Test for Kronecker Product Structure Covariance Matrix

TL;DR

This paper introduces a new statistical test to determine if a covariance matrix has Kronecker Product Structure, adapting existing reduced rank testing methods and demonstrating good finite-sample properties through simulations.

Contribution

It develops a novel Wald-type test for KPS in covariance matrices, addressing challenges in deriving its distribution and applying it to real data.

Findings

Test has chi-square null distribution with correct degrees of freedom.

Monte Carlo simulations show good size and power.

KPS not rejected in most real data applications.

Abstract

We propose a test for a covariance matrix to have Kronecker Product Structure (KPS). KPS implies a reduced rank restriction on a certain transformation of the covariance matrix and the new procedure is an adaptation of the Kleibergen and Paap (2006) reduced rank test. To derive the limiting distribution of the Wald type test statistic proves challenging partly because of the singularity of the covariance matrix estimator that appears in the weighting matrix. We show that the test statistic has a chi square limiting null distribution with degrees of freedom equal to the number of restrictions tested. Local asymptotic power results are derived. Monte Carlo simulations reveal good size and power properties of the test. Re-examining fifteen highly cited papers conducting instrumental variable regressions, we find that KPS is not rejected in 56 out of 118 specifications at the 5% nominal size.