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A well-conditioned estimator for large-dimensional covariance matrices

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  • Ledoit, Olivier
  • Wolf, Michael

Abstract

Many applied problems require a covariance matrix estimator that is not only invertible, but also well-conditioned (that is, inverting it does not amplify estimation error). For large-dimensional covariance matrices, the usual estimator--the sample covariance matrix--is typically not well-conditioned and may not even be invertible. This paper introduces an estimator that is both well-conditioned and more accurate than the sample covariance matrix asymptotically. This estimator is distribution-free and has a simple explicit formula that is easy to compute and interpret. It is the asymptotically optimal convex linear combination of the sample covariance matrix with the identity matrix. Optimality is meant with respect to a quadratic loss function, asymptotically as the number of observations and the number of variables go to infinity together. Extensive Monte Carlo confirm that the asymptotic results tend to hold well in finite sample.

Suggested Citation

  • 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.
  • Handle: RePEc:eee:jmvana:v:88:y:2004:i:2:p:365-411
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    References listed on IDEAS

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    5. Vinod, H. D., 1982. "Maximum entropy measurement error estimates of singular covariance matrices in undersized samples," Journal of Econometrics, Elsevier, vol. 20(2), pages 163-174, November.
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    7. Kandel, Shmuel & Stambaugh, Robert F, 1995. " Portfolio Inefficiency and the Cross-Section of Expected Returns," Journal of Finance, American Finance Association, vol. 50(1), pages 157-184, March.
    8. Brown, Stephen J, 1989. " The Number of Factors in Security Returns," Journal of Finance, American Finance Association, vol. 44(5), pages 1247-1262, December.
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