Collective Origin of the Coexistence of Apparent RMT Noise and Factors in Large Sample Correlation Matrices

TL;DR

This paper demonstrates that the coexistence of RMT noise and large eigenvalues in large correlation matrices arises from a collective effect of the underlying data, not from external factors, revealing intrinsic information in the spectrum.

Contribution

It introduces a simple analytical framework showing the collective origin of spectral features in large correlation matrices, challenging the traditional factor-based interpretation.

Findings

Eigenvalue spectrum coexistence results from collective effects.

Large eigenvalues are not solely due to external factors.

Bulk spectrum contains relevant information.

Abstract

Through simple analytical calculations and numerical simulations, we demonstrate the generic existence of a self-organized macroscopic state in any large multivariate system possessing non-vanishing average correlations between a finite fraction of all pairs of elements. The coexistence of an eigenvalue spectrum predicted by random matrix theory (RMT) and a few very large eigenvalues in large empirical correlation matrices is shown to result from a bottom-up collective effect of the underlying time series rather than a top-down impact of factors. Our results, in excellent agreement with previous results obtained on large financial correlation matrices, show that there is relevant information also in the bulk of the eigenvalue spectrum and rationalize the presence of market factors previously introduced in an ad hoc manner.