High-dimensional covariance matrix estimation in approximate factor models

TL;DR

This paper develops a method for estimating high-dimensional covariance matrices in factor models by assuming sparse error covariance, allowing for cross-sectional correlation, and using adaptive thresholding techniques.

Contribution

It introduces a new estimation approach that relaxes the independence assumption of idiosyncratic components in factor models, suitable for high-dimensional financial data.

Findings

Effective covariance estimation in high dimensions

Allows for correlated idiosyncratic errors

Demonstrates improved estimation accuracy

Abstract

The variance--covariance matrix plays a central role in the inferential theories of high-dimensional factor models in finance and economics. Popular regularization methods of directly exploiting sparsity are not directly applicable to many financial problems. Classical methods of estimating the covariance matrices are based on the strict factor models, assuming independent idiosyncratic components. This assumption, however, is restrictive in practical applications. By assuming sparse error covariance matrix, we allow the presence of the cross-sectional correlation even after taking out common factors, and it enables us to combine the merits of both methods. We estimate the sparse covariance using the adaptive thresholding technique as in Cai and Liu [J. Amer. Statist. Assoc. 106 (2011) 672--684], taking into account the fact that direct observations of the idiosyncratic components are unavailable. The impact of high dimensionality on the covariance matrix estimation based on the factor structure is then studied.