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

Abstract

We study testing high-dimensional covariance matrices under a generalized elliptical model. The model accommodates several stylized facts of real data including heteroskedasticity, heavy-tailedness, asymmetry, etc. We consider the high-dimensional setting where the dimension p and the sample size n grow to infinity proportionally, and establish a central limit theorem for the linear spectral statistic of the sample covariance matrix based on self-normalized observations. The central limit theorem is different from the existing ones for the linear spectral statistic of the usual sample covariance matrix. Our tests based on the new central limit theorem neither assume a specific parametric distribution nor involve the kurtosis of data. Simulation studies show that our tests work well even when the fourth moment does not exist. Empirically, we analyze the idiosyncratic returns under the Fama-French three-factor model for S&P 500 Financials sector stocks, and our tests reject the hypothesis that the idiosyncratic returns are uncorrelated.

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

JEL Classification: C12, C55