Large Covariance Estimation by Thresholding Principal Orthogonal Complements

TL;DR

This paper introduces the POET method for high-dimensional covariance estimation, combining factor models and thresholding to handle sparsity and diverging eigenvalues, with theoretical guarantees and practical applications.

Contribution

The paper proposes the POET estimator that integrates factor analysis and thresholding, providing convergence rates and robustness in high-dimensional covariance estimation.

Findings

POET effectively estimates covariance matrices in high dimensions.

The impact of factor estimation diminishes as data dimensionality increases.

Simulation studies confirm the theoretical convergence results.

Abstract

This paper deals with the estimation of a high-dimensional covariance with a conditional sparsity structure and fast-diverging eigenvalues. By assuming sparse error covariance matrix in an approximate factor model, we allow for the presence of some cross-sectional correlation even after taking out common but unobservable factors. We introduce the Principal Orthogonal complEment Thresholding (POET) method to explore such an approximate factor structure with sparsity. The POET estimator includes the sample covariance matrix, the factor-based covariance matrix (Fan, Fan, and Lv, 2008), the thresholding estimator (Bickel and Levina, 2008) and the adaptive thresholding estimator (Cai and Liu, 2011) as specific examples. We provide mathematical insights when the factor analysis is approximately the same as the principal component analysis for high-dimensional data. The rates of convergence of the sparse residual covariance matrix and the conditional sparse covariance matrix are studied under various norms. It is shown that the impact of estimating the unknown factors vanishes as the dimensionality increases. The uniform rates of convergence for the unobserved factors and their factor loadings are derived. The asymptotic results are also verified by extensive simulation studies. Finally, a real data application on portfolio allocation is presented.