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Empirical properties of large covariance matrices

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  • Gilles Zumbach

Abstract

The salient properties of large empirical covariance and correlation matrices are studied for three datasets of size 54, 55 and 330. The covariance is defined as a simple cross product of the returns, with weights that decay logarithmically slowly. The key general properties of the covariance matrices are the following. The spectrum of the covariance is very static, except for the top three to 10 eigenvalues, and decay exponentially fast toward zero. The mean spectrum and spectral density show no particular feature that would separate 'meaningful' from 'noisy' eigenvalues. The spectrum of the correlation is more static, with three to five eigenvalues that have distinct dynamics. The mean projector of rank k on the leading subspace shows that a large part of the dynamics occurs in the eigenvectors. Together, this implies that the reduction of the covariance to a few leading static eigenmodes misses most of the dynamics. Finally, all the analysed properties of the dynamics of the covariance and correlation are similar. This indicates that a covariance estimator correctly evaluates both volatilities and correlations, and separate estimators are not required.

Suggested Citation

  • Gilles Zumbach, 2011. "Empirical properties of large covariance matrices," Quantitative Finance, Taylor & Francis Journals, vol. 11(7), pages 1091-1102.
  • Handle: RePEc:taf:quantf:v:11:y:2011:i:7:p:1091-1102
    DOI: 10.1080/14697688.2010.508047
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    1. Szilard Pafka & Marc Potters & Imre Kondor, 2004. "Exponential Weighting and Random-Matrix-Theory-Based Filtering of Financial Covariance Matrices for Portfolio Optimization," Papers cond-mat/0402573, arXiv.org.
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    3. Paolella, Marc S. & Polak, Paweł & Walker, Patrick S., 2021. "A non-elliptical orthogonal GARCH model for portfolio selection under transaction costs," Journal of Banking & Finance, Elsevier, vol. 125(C).

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