In this paper, we consider the nonstationary matrix-valued time series with common stochastic trends. Unlike the traditional factor analysis which flattens matrix observations into vectors, we adopt a matrix factor model in order to fully explore the intrinsic matrix structure in the data, allowing interaction between the row and column stochastic trends, and subsequently improving the estimation convergence. It also reduces the computation complexity in estimation. The main estimation methodology is built on the eigenanalysis of sample row and column covariance matrices when the nonstationary matrix factors are of full rank and the idiosyncratic components are temporally stationary, and is further extended to tackle a more flexible setting when the matrix factors are cointegrated and the idiosyncratic components may be nonstationary. Under some mild conditions which allow the existence of weak factors, we derive the convergence theory for the estimated factor loading matrices and nonstationary factor matrices. In particular, the developed methodology and theory are applicable to the general case of heterogeneous strengths over weak factors. An easy-to-implement ratio criterion is adopted to consistently estimate the size of latent factor matrix. Both simulation and empirical studies are conducted to examine the numerical performance of the developed model and methodology in finite samples.

翻译：本文研究具有共同随机趋势的非平稳矩阵值时间序列。与传统因子分析将矩阵观测值展平为向量的做法不同，我们采用矩阵因子模型以充分挖掘数据内在的矩阵结构，允许行与列随机趋势之间的交互作用，从而提升估计收敛速度并降低计算复杂度。当非平稳矩阵因子满秩且特质分量具有时间平稳性时，主要估计方法建立在样本行协方差矩阵与列协方差矩阵的特征分析基础上；该方法进一步扩展至更灵活的设定场景——当矩阵因子存在协整关系且特质分量可能非平稳时。在允许弱因子存在的温和条件下，我们推导了因子载荷矩阵与非平稳因子矩阵估计量的收敛理论。特别地，所发展的方法与理论适用于弱因子强度存在异质性的普遍情形。采用易于实现的比率准则可一致估计潜在因子矩阵的维度。通过模拟研究与实证分析，检验了所建模型与方法在有限样本中的数值表现。