A Test for Kronecker Product Structure Covariance Matrix

We propose a test for a covariance matrix to have Kronecker Product Structure (KPS). KPS implies a reduced rank restriction on an invertible transformation of the covariance matrix and the new procedure is an adaptation of the Kleibergen-Paap (2006) reduced rank test. The main extension concerns the singularity of the covariance matrix estimator involved in the rank test which complicates the derivation of its limiting distribution. We show this limiting distribution to be χ² with degrees of freedom corresponding to the number of restrictions tested. Re-examining sixteen highly cited papers conducting IV regressions, we find that KPS is not rejected in 24 of 30 specifications for moderate sample sizes at the 5% nominal size