High-Dimensional Covariance Matrix Estimation: Shrinkage Toward a Diagonal Target

IMF Working Papers, December 8, 2023

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Proposal: A novel shrinkage estimator for high-dimensional covariance matrices that extends the Oracle Approximating Shrinkage (OAS) of Chen et al. (2009) to target the diagonal elements of the sample covariance matrix.
Analytical result: Derivation of the closed-form solution of the shrinkage parameter.
Simulation results: When the diagonal elements of the true covariance matrix exhibit substantial variation, the proposed method reduces the Mean Squared Error compared with the OAS that targets an average variance.
Additional outcome: The improvement is larger when the true covariance matrix is sparser.
Inverse matrix result: The proposed method also reduces the Mean Squared Error for the inverse of the covariance matrix.

Extension: Builds on Oracle Approximating Shrinkage (OAS) of Chen et al. (2009).
Target: Shrinkage toward the diagonal elements of the sample covariance matrix rather than toward an average variance.
Closed-form solution: Provides an explicit closed-form expression for the shrinkage parameter.
Focus: High-dimensional covariance matrix estimation and implications for matrix inverse estimation.

Mean Squared Error (MSE) reduction:
- Against OAS targeting average variance: The new estimator yields lower MSE when diagonal elements of the true covariance matrix exhibit substantial variation.
- Effect of sparsity: The improvement in MSE is larger when the true covariance matrix is sparser.
- Inverse covariance: The estimator reduces MSE for the inverse of the covariance matrix as well.
Sensitivity: Performance advantages are tied to heterogeneity in diagonal variances and sparsity structure of the true covariance matrix.

Authors: Sakai Ando, Mingmei Xiao
Date: December 8, 2023
Series: Working Paper No. 2023/257
Issue: 257
Volume: 2023
Pages: 32
DOI: https://doi.org/10.5089/9798400260780.001
Stock No: WPIEA2023257
ISBN: 9798400260780
ISSN: 1018-5941
Subject: Econometric analysis, Estimation techniques
Keywords: Covariance Matrix, Diagonal Target, Estimation techniques, High-Dimension, matrix estimation, novel shrinkage estimator, sample correlation matrix, Shrinkage, shrinkage parameter

IMF Working Papers — High-Dimensional Covariance Matrix Estimation: Shrinkage Toward a Diagonal Target