The Empirical Properties of Large Covariance Matrices

Abstract

The salient properties of large empirical covariance and correlation matrices are studied for three data sets of size 54, 55 and 330. The covariance is defined as a simple cross product of the returns, with weights that decay logarithmically slowly. The key general properties of the covariance matrices are the following. The spectrum of the covariance is very static, except for the top three to ten eigenvalues, and decay exponentially fast toward zero.

The mean spectrum and spectral density show no particular feature that would separate "meaningful'' from "noisy'' eigenvalues. The spectrum of the correlation is more static, with three to five eigenvalues that have distinct dynamics. The mean projector of rank k on the leading subspace shows instead that most of the dynamics occur in the eigenvectors, including deep in the spectrum. Together, this implies that the reduction of the covariance to a few leading eigenmodes misses most of the dynamics, and that a covariance estimator correctly evaluates both volatilities and correlations.

Keywords: Covariance matrix, spectrum, spectral density