We investigate the identification and the estimation for matrix time series CP-factor models. Unlike the generalized eigenanalysis-based method of Chang et al. (2023) which requires the two factor loading matrices to be full-ranked, the newly proposed estimation can handle rank-deficient factor loading matrices. The estimation procedure consists of the spectral decomposition of several matrices and a matrix joint diagonalization algorithm, resulting in low computational cost. The theoretical guarantee established without the stationarity assumption shows that the proposed estimation exhibits a faster convergence rate than that of Chang et al. (2023). In fact the new estimator is free from the adverse impact of any eigen-gaps, unlike most eigenanalysis-based methods such as that of Chang et al. (2023). Furthermore, in terms of the error rates of the estimation, the proposed procedure is equivalent to handling a vector time series of dimension $\max(p,q)$ instead of $p \times q$, where $(p, q)$ are the dimensions of the matrix time series concerned. We have achieved this without assuming the "near orthogonality" of the loadings under various incoherence conditions often imposed in the CP-decomposition literature, see Han and Zhang (2022), Han et al. (2024) and the references within. Illustration with both simulated and real matrix time series data shows the usefulness of the proposed approach.

翻译：本文研究矩阵时间序列CP因子模型的识别与估计问题。与Chang等人（2023）要求两个因子载荷矩阵满秩的广义特征分析方法不同，新提出的估计方法能够处理秩亏缺的因子载荷矩阵。该估计流程包含多个矩阵的谱分解和一个矩阵联合对角化算法，计算成本较低。在不依赖平稳性假设的理论保证下，所提估计方法展现出比Chang等人（2023）更快的收敛速度。事实上，新估计量不受任何特征值间隙的负面影响，这与Chang等人（2023）等大多数基于特征分析的方法不同。此外，在估计误差率方面，所提方法等价于处理维度为$\max(p,q)$的向量时间序列，而非原始$p \times q$维矩阵时间序列（其中$(p, q)$为所研究矩阵时间序列的维度）。这一成果的取得无需依赖CP分解文献中常设的各种非相干性条件下的"近正交性"假设（参见Han与Zhang（2022）、Han等人（2024）及相关文献）。通过模拟和真实矩阵时间序列数据的实证分析，验证了所提方法的有效性。