Testing High-Dimensional Covariance Matrices Under the Elliptical Distribution and Beyond

Abstract

We develop tests for high-dimensional covariance matrices under a generalized elliptical model. Our tests are based on a central limit theorem 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.

Keywords: covariance matrix, high-dimension, elliptical model, linear spectral statistics, central limit theorem, self-normalization

JEL Classification: C12, C55