Testing the Diagonality of a Large Covariance Matrix in a Regression Setting

In multivariate analysis, the covariance matrix associated with a set of vari- ables of interest (namely response variables) commonly contains valuable infor- mation about the dataset. When the dimension of response variables is con- siderably larger than the sample size, it is a non-trivial task to assess whether they are linear relationships between the variables. It is even more challenging to determine whether a set of explanatory variables can explain those relation- ships. To this end, we develop a bias-corrected test to examine the significance of the off-diagonal elements of the residual covariance matrix after adjusting for the contribution from explanatory variables. We show that the resulting test is asymptotically normal. Monte Carlo studies and a numerical example are pre- sented to illustrate the performance of the proposed test.