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

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  • Ledoit, Olivier
  • Wolf, Michael

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

  • Ledoit, Olivier & Wolf, Michael, 2015. "Spectrum estimation: A unified framework for covariance matrix estimation and PCA in large dimensions," Journal of Multivariate Analysis, Elsevier, vol. 139(C), pages 360-384.
  • Handle: RePEc:eee:jmvana:v:139:y:2015:i:c:p:360-384
    DOI: 10.1016/j.jmva.2015.04.006
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    References listed on IDEAS

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    Cited by:

    1. Zhao Zhao & Olivier Ledoit & Hui Jiang, 2019. "Risk reduction and efficiency increase in large portfolios: leverage and shrinkage," ECON - Working Papers 328, Department of Economics - University of Zurich.
    2. Olivier Ledoit & Michael Wolf, 2019. "The power of (non-)linear shrinking: a review and guide to covariance matrix estimation," ECON - Working Papers 323, Department of Economics - University of Zurich.
    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. repec:eee:csdana:v:139:y:2019:i:c:p:82-98 is not listed on IDEAS
    5. Andries C. van Vlodrop & Andre (A.) Lucas, 2018. "Estimation Risk and Shrinkage in Vast-Dimensional Fundamental Factor Models," Tinbergen Institute Discussion Papers 18-099/III, Tinbergen Institute.
    6. Sven Husmann & Antoniya Shivarova & Rick Steinert, 2019. "Data-driven covariance estimators for high-dimensional minimum-variance portfolios," Papers 1910.13960, arXiv.org, revised Dec 2019.
    7. Gianluca De Nard & Olivier Ledoit & Michael Wolf, 2018. "Factor models for portfolio selection in large dimensions: the good, the better and the ugly," ECON - Working Papers 290, Department of Economics - University of Zurich, revised Dec 2018.
    8. repec:eee:jmvana:v:168:y:2018:i:c:p:1-29 is not listed on IDEAS
    9. repec:eee:csdana:v:115:y:2017:i:c:p:199-223 is not listed on IDEAS
    10. 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.
    11. Ledoit, Olivier & Wolf, Michael, 2017. "Numerical implementation of the QuEST function," Computational Statistics & Data Analysis, Elsevier, vol. 115(C), pages 199-223.
    12. 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.
    13. Olivier Ledoit & Michael Wolf, 2019. "Shrinkage estimation of large covariance matrices: keep it simple, statistician?," ECON - Working Papers 327, Department of Economics - University of Zurich.
    14. Robert F. Engle & Olivier Ledoit & Michael Wolf, 2019. "Large Dynamic Covariance Matrices," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 37(2), pages 363-375, April.
    15. 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.
    16. Huang, Na & Fryzlewicz, Piotr, 2018. "NOVELIST estimator of large correlation and covariance matrices and their inverses," LSE Research Online Documents on Economics 89055, London School of Economics and Political Science, LSE Library.
    17. Olivier Ledoit & Michael Wolf, 2017. "Analytical nonlinear shrinkage of large-dimensional covariance matrices," ECON - Working Papers 264, Department of Economics - University of Zurich, revised Nov 2018.
    18. 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.
    19. Olivier Ledoit & Michael Wolf, 2019. "Quadratic shrinkage for large covariance matrices," ECON - Working Papers 335, Department of Economics - University of Zurich.
    20. Sven Husmann & Antoniya Shivarova & Rick Steinert, 2019. "Sparsity and Stability for Minimum-Variance Portfolios," Papers 1910.11840, arXiv.org.
    21. Joel Bun & Jean-Philippe Bouchaud & Marc Potters, 2016. "Cleaning large correlation matrices: tools from random matrix theory," Papers 1610.08104, arXiv.org.
    22. 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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