Testing super-diagonal structure in high dimensional covariance matrices

The covariance matrices are essential quantities in econometric and statistical applications including portfolio allocation, asset pricing and factor analysis. Testing the entire covariance under high dimensionality endures large variability and causes a dilution of the signal-to-noise ratio and hence a reduction in the power. We consider a more powerful test procedure that focuses on testing along the super-diagonals of the high dimensional covariance matrix, which can infer more accurately on the structure of the covariance. We show that the test is powerful in detecting sparse signals and parametric structures in the covariance. The properties of the test are demonstrated by theoretical analyses, simulation and empirical studies.