Application Of Random Matrix Theory To Study Cross-Correlations Of Stock Prices

We address the question of how to precisely identify correlated behavior between different firms in the economy by applying methods of random matrix theory (RMT). Specifically, we use methods of random matrix theory to analyze the cross-correlation matrix$\bf {\mathsf C}$of price changes of the largest 1000 US stocks for the 2-year period 1994–1995. We find that the statistics of most of the eigenvalues in the spectrum of$\bf {\mathsf C}$agree with the predictions of random matrix theory, but there are deviations for a few of the largest eigenvalues. To prove that the rest of the eigenvalues are genuinely random, we test for universal properties such as eigenvalue spacings and eigenvalue correlations. We demonstrate that$\bf {\mathsf C}$shares universal properties with the Gaussian orthogonal ensemble of random matrices. In addition, we quantify the number of significant participants, that is companies, of the eigenvectors using the inverse participation ratio, and find eigenvectors with large inverse participation ratios at both edges of the eigenvalue spectrum — a situation reminiscent of results in localization theory.