IDEAS home Printed from
   My bibliography  Save this paper

Shrinkage estimation of large covariance matrices: keep it simple, statistician?


  • Olivier Ledoit
  • Michael Wolf


Under rotation-equivariant decision theory, sample covariance matrix eigenvalues can be optimally shrunk by recombining sample eigenvectors with a (potentially nonlinear) function of the unobservable population covariance matrix. The optimal shape of this function reflects the loss/risk that is to be minimized. We introduce a broad family of covariance matrix estimators that can handle all regular functional transformations of the population covariance matrix under large-dimensional asymptotics. We solve the problem of optimal covariance matrix estimation under a variety of loss functions motivated by statistical precedent, probability theory, and differential geometry. The key statistical ingredient of our nonlinear shrinkage methodology is a new estimator of the angle between sample and population eigenvectors, without making strong assumptions on the population eigenvalues. We also compare our methodology to two simpler ones from the literature, linear shrinkage and shrinkage based on the spiked covariance model, via both Monte Carlo simulations and an empirical application.

Suggested Citation

  • 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.
  • Handle: RePEc:zur:econwp:327

    Download full text from publisher

    File URL:
    Download Restriction: no

    References listed on IDEAS

    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. Dowson, D. C. & Landau, B. V., 1982. "The Fréchet distance between multivariate normal distributions," Journal of Multivariate Analysis, Elsevier, vol. 12(3), pages 450-455, September.
    3. Haff, L. R., 1979. "An identity for the Wishart distribution with applications," Journal of Multivariate Analysis, Elsevier, vol. 9(4), pages 531-544, December.
    4. 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.
    5. 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.
    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. Tatsuya Kubokawa & Yoshihiko Konno, 1990. "Estimating the covariance matrix and the generalized variance under a symmetric loss," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 42(2), pages 331-343, June.
    8. 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.
    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. Ravi Jagannathan & Tongshu Ma, 2003. "Risk Reduction in Large Portfolios: Why Imposing the Wrong Constraints Helps," Journal of Finance, American Finance Association, vol. 58(4), pages 1651-1684, August.
    Full references (including those not matched with items on IDEAS)

    More about this item


    Large-dimensional asymptotics; random matrix theory; rotation equivariance;

    JEL classification:

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

    NEP fields

    This paper has been announced in the following NEP Reports:


    Access and download statistics


    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:zur:econwp:327. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Marita Kieser). General contact details of provider: .

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.