 # Quadratic shrinkage for large covariance matrices

Ledoit, Olivier; Wolf, Michael (2020). Quadratic shrinkage for large covariance matrices. Working paper series / Department of Economics 335, University of Zurich.

## Abstract

This paper constructs a new estimator for large covariance matrices by drawing a bridge between the classic Stein (1975) estimator in finite samples and recent progress under large-dimensional asymptotics. The estimator keeps the eigenvectors of the sample covariance matrix and applies shrinkage to the inverse sample eigenvalues. The corresponding formula is quadratic: it has two shrinkage targets weighted by quadratic functions of the concentration (that is, matrix dimension divided by sample size). The first target dominates mid-level concentrations and the second one higher levels. This extra degree of freedom enables us to outperform linear shrinkage when optimal shrinkage is not linear (which is the general case). Both of our targets are based on what we term the “Stein shrinker”, a local attraction operator that pulls sample covariance matrix eigenvalues towards their nearest neighbors, but whose force diminishes with distance, like gravitation. We prove that no cubic or higher-order nonlinearities beat quadratic with respect to Frobenius loss under large-dimensional asymptotics. Non-normality and the case where the matrix dimension exceeds the sample size are accommodated. Monte Carlo simulations confirm state-of-the-art performance in terms of accuracy, speed, and scalability.

## Abstract

This paper constructs a new estimator for large covariance matrices by drawing a bridge between the classic Stein (1975) estimator in finite samples and recent progress under large-dimensional asymptotics. The estimator keeps the eigenvectors of the sample covariance matrix and applies shrinkage to the inverse sample eigenvalues. The corresponding formula is quadratic: it has two shrinkage targets weighted by quadratic functions of the concentration (that is, matrix dimension divided by sample size). The first target dominates mid-level concentrations and the second one higher levels. This extra degree of freedom enables us to outperform linear shrinkage when optimal shrinkage is not linear (which is the general case). Both of our targets are based on what we term the “Stein shrinker”, a local attraction operator that pulls sample covariance matrix eigenvalues towards their nearest neighbors, but whose force diminishes with distance, like gravitation. We prove that no cubic or higher-order nonlinearities beat quadratic with respect to Frobenius loss under large-dimensional asymptotics. Non-normality and the case where the matrix dimension exceeds the sample size are accommodated. Monte Carlo simulations confirm state-of-the-art performance in terms of accuracy, speed, and scalability.

## Statistics

Detailed statistics

Item Type: Working Paper 03 Faculty of Economics > Department of Economics Working Paper Series > Department of Economics 330 Economics C13 Inverse shrinkage, Hilbert transform, large-dimensional asymptotics, signal amplitude, Stein shrinkage, Kovarianzmatrix, Schätzfunktion, Monte-Carlo-Simulation English December 2020 13 Nov 2019 15:34 04 Dec 2020 11:36 Working paper series / Department of Economics 73 1664-705X Revised version Green https://www.econ.uzh.ch/static/release/workingpapers.php?id=1014

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