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. 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 practical usefulness.
翻译:矩阵要素模型在同时为矩阵结构观测实现双向减少方面的好处日益引起人们的注意。在本文件中,我们建议对矩阵要素模型采用简单的迭代最小方方程式算法,与文献中基于本组成部分分析(PCA)的方法相对照。详细而言,我们首先提议用两个确定性重量矩阵来估计潜在要素矩阵,通过两个确定性重量矩阵来预测这些观测,选择这些矩阵是为了使特异性组成部分多样化。我们表明,即使我们高估行/柱要素数字,对各种因素的推论仍然毫无实际意义。我们然后通过在某些可识别性条件下尽量减少平方损失函数来估计行/栏装载矩阵。因此,我们首先提议用新的加权/预测性矩阵来估计潜在要素矩阵,从而在趋同之前可以反复进行上述更新程序。理论上,考虑到要素矩阵的真正层面,我们得出在任何一步的指数上装载矩阵和共同组成部分的估算值的趋同率。我们随后通过在某些可识别性条件下尽量减少平方位损失功能的行/负载量矩阵矩阵,然后用两种宏观经济模型模拟方法来调查所拟议的宏观经济实用性指数。