Random matrix ensembles of time-lagged correlation matrices: derivation of eigenvalue spectra and analysis of financial time-series

We derive the exact form of the eigenvalue spectra of correlation matrices derived from a set of time-shifted, finite Brownian random walks (time-series). These matrices can be seen as real, asymmetric random matrices where the time-shift superimposes some structure. We demonstrate that, for large matrices, the associated eigenvalue spectrum is circular symmetric in the complex plane. This fact allows us to exactly compute the eigenvalue density via an inverse Abel-transform of the density of the symmetrized problem. We demonstrate the validity of this approach numerically. Theoretical findings are then compared with eigenvalue densities obtained from actual high-frequency (5 min) data of the S&P 500 and the observed deviations are discussed. We identify various non-trivial, non-random patterns and find asymmetric dependencies associated with eigenvalues departing strongly from the Gaussian prediction in the imaginary part. For the same time-series, with the market contribution removed, we observe strong clustering of stocks into causal sectors. We finally comment on the stability of the observed patterns.