Robust Covariance Estimation for Approximate Factor Models

TL;DR

This paper introduces a robust method for covariance estimation in approximate factor models, utilizing Huber loss minimization to handle various distributions and providing theoretical guarantees and empirical validation.

Contribution

It proposes a novel two-step framework for robust covariance estimation that works under broader distributional assumptions, including elliptical distributions.

Findings

Method achieves element-wise optimal rates.

Applicable to sub-Gaussian and elliptical distributions.

Validated through simulations and real data.

Abstract

In this paper, we study robust covariance estimation under the approximate factor model with observed factors. We propose a novel framework to first estimate the initial joint covariance matrix of the observed data and the factors, and then use it to recover the covariance matrix of the observed data. We prove that once the initial matrix estimator is good enough to maintain the element-wise optimal rate, the whole procedure will generate an estimated covariance with desired properties. For data with only bounded fourth moments, we propose to use Huber loss minimization to give the initial joint covariance estimation. This approach is applicable to a much wider range of distributions, including sub-Gaussian and elliptical distributions. We also present an asymptotic result for Huber's M-estimator with a diverging parameter. The conclusions are demonstrated by extensive simulations and real data analysis.