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.

翻译：矩阵因子模型因其在同时实现矩阵结构观测数据的双向降维方面具有优势而日益受到关注。本文提出了一种用于矩阵因子模型的简单迭代最小二乘算法，这与文献中基于主成分分析（PCA）的方法形成对比。具体而言，我们首先提出通过使用两个确定性权重矩阵对观测数据进行投影来估计潜在因子矩阵，这两个矩阵被选择用于分散异质成分。我们证明，即使高估了行/列因子数量，对因子的推断仍具有渐近有效性。然后，我们在特定可识别性条件下通过最小化平方损失函数来估计行/列载荷矩阵。由此得到的载荷矩阵估计量被视为新的权重/投影矩阵，因此上述更新过程可以迭代进行直至收敛。理论上，给定因子矩阵的真实维度，我们推导了载荷矩阵和共同成分的估计量在任何第$s$步迭代时的收敛速度。此外，我们提出了一种特征值比率方法来一致地估计这对因子数量。通过大量数值模拟研究了所提方法的有限样本性能，并利用两个与金融投资组合和跨国宏观经济指标相关的真实数据集展示了我们算法的实际应用价值。