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
Date Written: November 2018
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
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