Title: 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.