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Analytical nonlinear shrinkage of large-dimensional covariance matrices

Author

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  • Olivier Ledoit
  • Michael Wolf

Abstract

This paper establishes the first analytical formula for optimal nonlinear shrinkage of large-dimensional covariance matrices. We achieve this by identifying and mathematically exploiting a deep connection between nonlinear shrinkage and nonparametric estimation of the Hilbert transform of the sample spectral density. Previous nonlinear shrinkage methods were numerical: QuEST requires numerical inversion of a complex equation from random matrix theory whereas NERCOME is based on a sample-splitting scheme. The new analytical approach is more elegant and also has more potential to accommodate future variations or extensions. Immediate benefits are that it is typically 1,000 times faster with the same accuracy, and accommodates covariance matrices of dimension up to 10, 000. The difficult case where the matrix dimension exceeds the sample size is also covered.

Suggested Citation

  • 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.
  • Handle: RePEc:zur:econwp:264
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    File URL: http://www.econ.uzh.ch/static/wp/econwp264.pdf
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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.
    2. 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.
    3. Engle, Robert & Colacito, Riccardo, 2006. "Testing and Valuing Dynamic Correlations for Asset Allocation," Journal of Business & Economic Statistics, American Statistical Association, vol. 24, pages 238-253, April.
    4. Bala Rajaratnam & Dario Vincenzi, 2016. "A theoretical study of Stein's covariance estimator," Biometrika, Biometrika Trust, vol. 103(3), pages 653-666.
    5. Harry Markowitz, 1952. "Portfolio Selection," Journal of Finance, American Finance Association, vol. 7(1), pages 77-91, March.
    6. Silverstein, J. W., 1995. "Strong Convergence of the Empirical Distribution of Eigenvalues of Large Dimensional Random Matrices," Journal of Multivariate Analysis, Elsevier, vol. 55(2), pages 331-339, November.
    7. Silverstein, J. W. & Choi, S. I., 1995. "Analysis of the Limiting Spectral Distribution of Large Dimensional Random Matrices," Journal of Multivariate Analysis, Elsevier, vol. 54(2), pages 295-309, August.
    8. Abadir, Karim M. & Distaso, Walter & Žikeš, Filip, 2014. "Design-free estimation of variance matrices," Journal of Econometrics, Elsevier, vol. 181(2), pages 165-180.
    9. 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.
    10. Silverstein, J. W. & Bai, Z. D., 1995. "On the Empirical Distribution of Eigenvalues of a Class of Large Dimensional Random Matrices," Journal of Multivariate Analysis, Elsevier, vol. 54(2), pages 175-192, August.
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    Citations

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    Cited by:

    1. 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.
    2. 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.
    3. 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.

    More about this item

    Keywords

    Kernel estimation; Hilbert transform; large-dimensional asymptotics; nonlinear shrinkage; rotation equivariance;

    JEL classification:

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

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