The Empirical Properties of Large Covariance Matrices

The knowledge acquired on single financial time series can now be applied to study the multivariate case. The first objective is to understand the generic properties of the covariance and correlation matrices. The definition of the covariance matrix uses the univariate long-memory kernel, hence providing the best short-term forecast suitable to build processes. The eigenvalue decomposition of the covariance matrix is presented, in order to study the properties of the spectrum and eigenvectors. For financial data, the dynamics of the eigenvalues is studied and compared to analytical results obtained from random matrix theory, while the eigenvectors dynamics point to the absence of clear invariant subspaces that would correspond to the established market modes.