  The salient properties of large empirical covariance and correlation matrices
are studied for three datasets 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.
