Analytical Nonlinear Shrinkage of Large-Dimensional Covariance Matrices

University of Zurich, Department of Economics, Working Paper No. 264, Revised version

56 Pages Posted: 4 Oct 2017 Last revised: 12 Nov 2018

See all articles by Olivier Ledoit

Olivier Ledoit

University of Zurich - Department of Economics

Michael Wolf

University of Zurich - Department of Economics

Date Written: November 2018

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.

Keywords: Kernel estimation, Hilbert transform, large-dimensional asymptotics, nonlinear shrinkage, rotation equivariance

JEL Classification: C13

Suggested Citation

Ledoit, Olivier and Wolf, Michael, Analytical Nonlinear Shrinkage of Large-Dimensional Covariance Matrices (November 2018). University of Zurich, Department of Economics, Working Paper No. 264, Revised version. Available at SSRN: https://ssrn.com/abstract=3047302 or http://dx.doi.org/10.2139/ssrn.3047302

Olivier Ledoit (Contact Author)

University of Zurich - Department of Economics ( email )

Wilfriedstrasse 6
Z├╝rich, 8032
Switzerland

Michael Wolf

University of Zurich - Department of Economics ( email )

Wilfriedstrasse 6
Zurich, 8032
Switzerland

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