Testing High-dimensional Covariance Matrices under the Elliptical Distribution and Beyond

TL;DR

This paper introduces new statistical tests for high-dimensional covariance matrices under elliptical distributions, using a CLT for spectral statistics that do not rely on parametric assumptions, applicable even to non-invertible matrices.

Contribution

The authors develop distribution-free tests for high-dimensional covariance matrices based on a CLT for spectral statistics, extending applicability beyond traditional parametric models.

Findings

Tests perform well in empirical studies

Can test for uncorrelatedness among returns

Applicable to non-invertible matrices

Abstract

We develop tests for high-dimensional covariance matrices under a generalized elliptical model. Our tests are based on a central limit theorem (CLT) for linear spectral statistics of the sample covariance matrix based on self-normalized observations. For testing sphericity, our tests neither assume specific parametric distributions nor involve the kurtosis of data. More generally, we can test against any non-negative definite matrix that can even be not invertible. As an interesting application, we illustrate in empirical studies that our tests can be used to test uncorrelatedness among idiosyncratic returns.