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Tyler's M-estimator, random matrix theory, and generalized elliptical distributions with applications to finance

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  • Frahm, Gabriel
  • Jaekel, Uwe

Abstract

In recent publications standard methods of random matrix theory were applied to principal components analysis of high-dimensional financial data. We discuss the fundamental results and potential shortcomings of random matrix theory in the light of the stylized facts of empirical finance. Especially, our arguments are based on the impact of nonlinear dependencies such as tail dependence. After a brief discussion of the stylized facts we present the class of multivariate generalized elliptical distributions. This class allows for the modeling of various anomalies frequently observed in financial data. Thus it will serve as a general model for the investigation of standard methods of random matrix theory. It is shown that the Marčenko-Pastur law generally fails when analyzing the empirical distribution function of the eigenvalues given by the sample covariance matrix of generalized elliptically distributed data. As an alternative we derive a random matrix referred to as the spectral estimator which is distribution-free within the class of generalized elliptical distributions. Moreover, we show that the spectral estimator corresponds to Tyler's M-estimator and many important properties of the spectral estimator can be obtained from the corresponding literature. Substituting the sample covariance matrix by the spectral estimator resolves the problems which are due to the stylized facts and the Marčenko-Pastur law remains valid. This holds even if the data are not generalized elliptically distributed but mutually independent.

Suggested Citation

  • Frahm, Gabriel & Jaekel, Uwe, 2007. "Tyler's M-estimator, random matrix theory, and generalized elliptical distributions with applications to finance," Discussion Papers in Econometrics and Statistics 2/07, University of Cologne, Institute of Econometrics and Statistics.
  • Handle: RePEc:zbw:ucdpse:207
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    References listed on IDEAS

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    Cited by:

    1. Frahm, Gabriel & Jaekel, Uwe, 2010. "A generalization of Tyler's M-estimators to the case of incomplete data," Computational Statistics & Data Analysis, Elsevier, vol. 54(2), pages 374-393, February.
    2. Zhang, Teng & Cheng, Xiuyuan & Singer, Amit, 2016. "Marčenko–Pastur law for Tyler’s M-estimator," Journal of Multivariate Analysis, Elsevier, vol. 149(C), pages 114-123.
    3. Fan, Jianqing & Han, Fang & Liu, Han & Vickers, Byron, 2016. "Robust inference of risks of large portfolios," Journal of Econometrics, Elsevier, vol. 194(2), pages 298-308.
    4. Soloveychik, I. & Trushin, D., 2016. "Gaussian and robust Kronecker product covariance estimation: Existence and uniqueness," Journal of Multivariate Analysis, Elsevier, vol. 149(C), pages 92-113.
    5. Alexander Bade & Gabriel Frahm & Uwe Jaekel, 2009. "A general approach to Bayesian portfolio optimization," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 70(2), pages 337-356, October.
    6. Bade, Alexander & Frahm, Gabriel & Jaekel, Uwe, 2008. "A general approach to Bayesian portfolio optimization," Discussion Papers in Econometrics and Statistics 1/08, University of Cologne, Institute of Econometrics and Statistics.

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