An Efficient Iterative Least Squares Algorithm for Large-dimensional Matrix Factor Model via Random Projection

TL;DR

This paper introduces a simple iterative least squares algorithm for large-dimensional matrix factor models, offering an alternative to PCA-based methods, with proven convergence and practical effectiveness demonstrated through simulations and real data.

Contribution

The paper proposes a novel iterative least squares approach for matrix factor models, improving efficiency and robustness over existing PCA-based methods, with theoretical convergence guarantees.

Findings

Algorithm converges at a known rate

Estimates remain valid even with overestimated factor numbers

Effective in financial and macroeconomic data analysis

Abstract

The matrix factor model has drawn growing attention for its advantage in achieving two-directional dimension reduction simultaneously for matrix-structured observations. In this paper, we propose a simple iterative least squares algorithm for matrix factor models, in contrast to the Principal Component Analysis (PCA)-based methods in the literature. In detail, we first propose to estimate the latent factor matrices by projecting the observations with two deterministic weight matrices, which are chosen to diversify away the idiosyncratic components. We show that the inferences on factors are still asymptotically valid even if we overestimate both the row/column factor numbers. We then estimate the row/column loading matrices by minimizing the squared loss function under certain identifiability conditions. The resultant estimators of the loading matrices are treated as the new weight/projection matrices and thus the above update procedure can be iteratively performed until convergence. Theoretically, given the true dimensions of the factor matrices, we derive the convergence rates of the estimators for loading matrices and common components at any $s$-th step iteration. Additionally, we propose an eigenvalue-ratio method to estimate the pair of factor numbers consistently. Thorough numerical simulations are conducted to investigate the finite-sample performance of the proposed methods and two real datasets associated with financial portfolios and multinational macroeconomic indices are used to illustrate our algorithm's practical usefulness.