This paper proposes a new multi-linear projection method for denoising and estimation of high-dimensional matrix-variate factor time series. It assumes that a $p_1\times p_2$ matrix-variate time series consists of a dynamically dependent, lower-dimensional matrix-variate factor process and a $p_1\times p_2$ matrix idiosyncratic series. In addition, the latter series assumes a matrix-variate factor structure such that its row and column covariances may have diverging/spiked eigenvalues to accommodate the case of low signal-to-noise ratio often encountered in applications. We use an iterative projection procedure to reduce the dimensions and noise effects in estimating front and back loading matrices and to obtain faster convergence rates than those of the traditional methods available in the literature. We further introduce a two-way projected Principal Component Analysis to mitigate the diverging noise effects, and implement a high-dimensional white-noise testing procedure to estimate the dimension of the matrix factor process. Asymptotic properties of the proposed method are established if the dimensions and sample size go to infinity. We also use simulations and real examples to assess the performance of the proposed method in finite samples and to compare its forecasting ability with some existing ones in the literature. The proposed method fares well in out-of-sample forecasting. In a supplement, we demonstrate the efficacy of the proposed approach even when the idiosyncratic terms exhibit serial correlations with or without a diverging white noise effect.

翻译：本文提出了一种新的多线性投影方法，用于高维矩阵变量因子时间序列的去噪与估计。该方法假设一个 $p_1\times p_2$ 矩阵变量时间序列由一个动态依赖的低维矩阵变量因子过程和一个 $p_1\times p_2$ 矩阵特异性序列组成。此外，该特异性序列被假定具有矩阵变量因子结构，使得其行与列协方差矩阵可能具有发散/尖峰特征值，以适应应用中常遇到的低信噪比情形。我们采用一种迭代投影程序来降低维度并减弱噪声效应，以估计前、后载荷矩阵，并获得比文献中现有传统方法更快的收敛速率。我们进一步引入了一种双向投影主成分分析以缓解发散噪声效应，并实施了一种高维白噪声检验程序来估计矩阵因子过程的维度。当维度与样本量趋于无穷时，我们建立了所提方法的渐近性质。我们还通过模拟和实际案例评估了所提方法在有限样本下的性能，并将其预测能力与文献中已有的一些方法进行了比较。所提方法在样本外预测中表现良好。在一份补充材料中，我们证明了即使特异性项表现出具有或不具有发散白噪声效应的序列相关性时，所提方法依然有效。