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The empirical properties of large covariance matrices

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  • Gilles Zumbach
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    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 ten 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 instead that most of the dynamics occur in the eigenvectors, including deep in the spectrum. Together, this implies that the reduction of the covariance to a few leading eigenmodes misses most of the dynamics, and that a covariance estimator correctly evaluates both volatilities and correlations.

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    File URL: http://arxiv.org/pdf/0903.1525
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    Paper provided by arXiv.org in its series Papers with number 0903.1525.

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    Date of creation: Mar 2009
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    Handle: RePEc:arx:papers:0903.1525

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    1. M. Potters & J. P. Bouchaud & L. Laloux, 2005. "Financial Applications of Random Matrix Theory: Old Laces and New Pieces," Papers physics/0507111, arXiv.org.
    2. Gilles Zumbach, 2004. "Volatility processes and volatility forecast with long memory," Quantitative Finance, Taylor & Francis Journals, Taylor & Francis Journals, vol. 4(1), pages 70-86.
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    Cited by:
    1. Caporin, Massimiliano & McAleer, Michael, 2014. "Robust ranking of multivariate GARCH models by problem dimension," Computational Statistics & Data Analysis, Elsevier, Elsevier, vol. 76(C), pages 172-185.
    2. Michael McAleer & Massimiliano Caporin, 2011. "Ranking Multivariate GARCH Models by Problem Dimension:An Empirical Evaluation," KIER Working Papers, Kyoto University, Institute of Economic Research 778, Kyoto University, Institute of Economic Research.
    3. Nikolaus Hautsch & Lada M. Kyj & Roel C.A. Oomen, 2009. "A blocking and regularization approach to high dimensional realized covariance estimation," SFB 649 Discussion Papers SFB649DP2009-049, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
    4. Romain Allez & Jean-Philippe Bouchaud, 2012. "Eigenvector dynamics: general theory and some applications," Papers 1203.6228, arXiv.org, revised Jul 2012.

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