A random matrix theory approach to financial cross-correlations

It is common knowledge that any two firms in the economy are correlated. Ev en firms belonging to different sectors of an industry may be correlated be cause of "indirect" correlations. How can we analyze and understand these c orrelations? This article reviews recent results regarding cross-correlatio ns between stocks. Specifically, we use methods of random matrix theory (RM T), which originated from the need to understand the interactions between t he constituent elements of complex interacting systems, to analyze the cros s-correlation matrix C of returns. We analyze 30-min returns of the largest 1000 US stocks for the 2-year period 1994-1995. We find that the statistic s of approximately 20 of the largest eigenvalues (2%) show deviations from the predictions of RMT. To test that the rest of the eigenvalues are genuin ely random, we test for universal properties such as eigenvalue spacings an d eigenvalue correlations, and demonstrate that C shares universal properti es with the Gaussian orthogonal ensemble of random matrices. The statistics of the eigenvectors of C confirm the deviations of the largest few eigenva lues from the RMT prediction. We also find that these deviating eigenvector s are stable in time. In addition, we quantify the number of firms that par ticipate significantly to an eigenvector using the concept of inverse parti cipation ratio, borrowed from localization theory. (C) 2000 Published by El sevier Science B.V. All rights reserved.