The bulk of the stock market correlation matrix is not pure noise

Abstract

We analyse the structure of the distribution of eigenvalues of the stock market correlation matrix with increasing length of the time series representing the price changes. We use 100 highly-capitalized stocks from the American market and relate result to the corresponding ensemble of Wishart random matrices. It turns out that systematically more eigenvalues stay beyond the borders prescribed by this variant of the Random Matrix Theory (RMT). This may indicate that even the bulk of the spectrum of the stock market correlation matrix carries some sort of correlations that are masked by a measurement noise when the time series used to construct the matrix are short. We also study some other characteristics of the "noisy" eigensignals, like their return distributions, temporal correlations or their multifractal spectra and the results support the above conclusions.Comment: updated version to appear in Physica

Cross-Correlations in Warsaw Stock Exchange

“…Clearly, for Groups 1, 2, 3 the corresponding shift is stronger than for Group 4. As regards the rest of the eigenvalues, they are close to the random matrix region; the observed discrepancies between their position and the RMT bounds can be attributed to the "squeezing" effect of large λ 1 exerted on the smaller eigenvalues [2,7] which are shifted towards zero. The larger magnitude of λ 1 , the stronger is this effect.…”

“…It occurs that a vast majority of these eigenvalues are concordant with the eigenvalue distribution of the relevant random matrix ensemble (the Wishart ensemble) [2,6], which according to a common belief suggests that (561) these RMT modes do not carry any market-specific information beyond being a pure noise. Validity of this belief, however, has recently been challenged in some works [7]. As regards the remaining minority of the eigenvalues which deviate from the RMT predictions, there is a general agreement that they express the actual non-random linear dependences between the price fluctuations of different assets.…”

Power law deformation of Wishart–Laguerre ensembles of random matrices

“…In the finance and risk management domain, the empirical covariance of a set of N assets over a temporal window of size M has been under scrutiny for some time [9], and its eigenvalues were shown to be distributed in reasonably good agreement with the MP law, as if they were originated by a completely uncorrelated data series. However, the same analysis repeated by several groups [21,22,23] on different data sets have shown that either the part of the spectrum corresponding to extremely low eigenvalues -the most interesting for portfolio selections -or the fat tails are not reproduced by this crude approach. This has led to the appearance of more sophisticated models [23,24,25], e.g.…”

Cross-Correlation Dynamics in Financial Time Series

“…In this paper, the authors demonstrate that the existence of factors such as an overall market effect, firm size and industry type is due to collective influence of the assets. More evidence that the RMT fit is not perfect was provided, [14], where it was shown that the dispersion of "noise" eigenvalues is inflated, indicating that the bulk of the eigenvalue spectrum contains correlations masked by measurement noise.…”