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Spectrum estimation: a unified framework for covariance matrix estimation and PCA in large dimensions

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  • Olivier Ledoit
  • Michael Wolf

Abstract

Covariance matrix estimation and principal component analysis (PCA) are two cornerstones of multivariate analysis. Classic textbook solutions perform poorly when the dimension of the data is of a magnitude similar to the sample size, or even larger. In such settings, there is a common remedy for both statistical problems: nonlinear shrinkage of the eigenvalues of the sample covariance matrix. The optimal nonlinear shrinkage formula depends on unknown population quantities and is thus not available. It is, however, possible to consistently estimate an oracle nonlinear shrinkage, which is motivated on asymptotic grounds. A key tool to this end is consistent estimation of the set of eigenvalues of the population covariance matrix (also known as the spectrum), an interesting and challenging problem in its own right. Extensive Monte Carlo simulations demonstrate that our methods have desirable finite-sample properties and outperform previous proposals.

Suggested Citation

  • Olivier Ledoit & Michael Wolf, 2013. "Spectrum estimation: a unified framework for covariance matrix estimation and PCA in large dimensions," ECON - Working Papers 105, Department of Economics - University of Zurich, revised Jul 2013.
  • Handle: RePEc:zur:econwp:105
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    References listed on IDEAS

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    1. Ledoit, Olivier & Wolf, Michael, 2004. "A well-conditioned estimator for large-dimensional covariance matrices," Journal of Multivariate Analysis, Elsevier, vol. 88(2), pages 365-411, February.
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    Citations

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    Cited by:

    1. Ikeda, Yuki & Kubokawa, Tatsuya, 2016. "Linear shrinkage estimation of large covariance matrices using factor models," Journal of Multivariate Analysis, Elsevier, vol. 152(C), pages 61-81.
    2. Olivier Ledoit & Michael Wolf, 2016. "Numerical implementation of the QuEST function," ECON - Working Papers 215, Department of Economics - University of Zurich, revised Jan 2017.
    3. Joongyeub Yeo & George Papanicolaou, 2016. "Random matrix approach to estimation of high-dimensional factor models," Papers 1611.05571, arXiv.org, revised Nov 2017.
    4. Tsukuma, Hisayuki, 2016. "Estimation of a high-dimensional covariance matrix with the Stein loss," Journal of Multivariate Analysis, Elsevier, vol. 148(C), pages 1-17.
    5. Robert F. Engle & Olivier Ledoit & Michael Wolf, 2016. "Large dynamic covariance matrices," ECON - Working Papers 231, Department of Economics - University of Zurich, revised Apr 2017.
    6. Ikeda, Yuki & Kubokawa, Tatsuya & Srivastava, Muni S., 2016. "Comparison of linear shrinkage estimators of a large covariance matrix in normal and non-normal distributions," Computational Statistics & Data Analysis, Elsevier, vol. 95(C), pages 95-108.
    7. Olivier Ledoit & Michael Wolf, 2017. "Direct nonlinear shrinkage estimation of large-dimensional covariance matrices," ECON - Working Papers 264, Department of Economics - University of Zurich.
    8. Joel Bun & Jean-Philippe Bouchaud & Marc Potters, 2016. "Cleaning large correlation matrices: tools from random matrix theory," Papers 1610.08104, arXiv.org.
    9. Tsubasa Ito & Tatsuya Kubokawa, 2015. "Linear Ridge Estimator of High-Dimensional Precision Matrix Using Random Matrix Theory ," CIRJE F-Series CIRJE-F-995, CIRJE, Faculty of Economics, University of Tokyo.
    10. repec:eee:csdana:v:115:y:2017:i:c:p:199-223 is not listed on IDEAS
    11. Olivier Ledoit & Michael Wolf, 2014. "Nonlinear shrinkage of the covariance matrix for portfolio selection: Markowitz meets Goldilocks," ECON - Working Papers 137, Department of Economics - University of Zurich, revised Feb 2017.

    More about this item

    Keywords

    Large-dimensional asymptotics; covariance matrix eigenvalues; nonlinear shrinkage; principal component analysis;

    JEL classification:

    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General

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