Matrix Factor Analysis: From Least Squares to Iterative Projection

TL;DR

This paper develops a least squares and a robust Huber loss-based iterative projection method for large-dimensional matrix factor models, providing convergence analysis and demonstrating superior robustness to heavy-tailed errors.

Contribution

It offers a least-square interpretation of Projected Estimators for matrix factor models and extends to Huber loss for robustness, with theoretical convergence rates and practical advantages.

Findings

Huber estimators outperform existing methods with heavy-tailed data

Convergence rates established for least squares and Huber estimators

Empirical application shows Huber estimator's robustness in financial data

Abstract

In this article, we study large-dimensional matrix factor models and estimate the factor loading matrices and factor score matrix by minimizing square loss function. Interestingly, the resultant estimators coincide with the Projected Estimators (PE) in Yu et al.(2022), which was proposed from the perspective of simultaneous reduction of the dimensionality and the magnitudes of the idiosyncratic error matrix. In other word, we provide a least-square interpretation of the PE for matrix factor model, which parallels to the least-square interpretation of the PCA for the vector factor model. We derive the convergence rates of the theoretical minimizers under sub-Gaussian tails. Considering the robustness to the heavy tails of the idiosyncratic errors, we extend the least squares to minimizing the Huber loss function, which leads to a weighted iterative projection approach to compute and learn the parameters. We also derive the convergence rates of the theoretical minimizers of the Huber loss function under bounded $(2+\epsilon)$th moment of the idiosyncratic errors. We conduct extensive numerical studies to investigate the empirical performance of the proposed Huber estimators relative to the state-of-the-art ones. The Huber estimators perform robustly and much better than existing ones when the data are heavy-tailed, and as a result can be used as a safe replacement in practice. An application to a Fama-French financial portfolio dataset demonstrates the empirical advantage of the Huber estimator.