  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.
