Matrix-valued time series are ubiquitous in modern economics and finance, yet modeling them requires navigating a trade-off between flexibility and parsimony. We propose the Matrix Autoregressive model with Common Factors (MARCF), a unified framework for high-dimensional matrix time series that bridges the structural gap between the Matrix Autoregression (MAR) and Matrix Factor Model (MFM). While MAR typically assumes distinct predictor and response subspaces and MFM enforces identical ones, MARCF explicitly characterizes the intersection of these subspaces. By decomposing the coefficient matrices into common, predictor-specific, and response-specific components, the framework accommodates distinct input and output structures while exploiting their overlap for dimension reduction. We develop a regularized gradient descent estimator that is scalable for high-dimensional data and can efficiently handle the non-convex parameter space. Theoretical analysis establishes local linear convergence of the algorithm and statistical consistency of the estimator under high-dimensional scaling. The estimation efficiency and interpretability of the proposed methods are demonstrated through simulations and an application to global macroeconomic forecasting.

翻译：矩阵值时间序列在现代经济学与金融学中普遍存在，然而对其建模需要在灵活性与简约性之间权衡。我们提出了带公共因子的矩阵自回归模型（MARCF），这是一个面向高维矩阵时间序列的统一框架，旨在弥合矩阵自回归（MAR）与矩阵因子模型（MFM）之间的结构差异。MAR通常假设预测变量与响应变量处于不同的子空间，而MFM则强制要求两者处于同一子空间；MARCF则显式刻画了这些子空间的交集。通过将系数矩阵分解为公共成分、预测变量特定成分与响应变量特定成分，该框架既能适应不同的输入与输出结构，又能利用其重叠部分实现降维。我们开发了一种正则化梯度下降估计器，该估计器可扩展至高维数据，并能高效处理非凸参数空间。理论分析证明了算法在局部具有线性收敛性，且估计器在高维尺度下具有统计一致性。通过仿真实验及在全球宏观经济预测中的应用，验证了所提方法的估计效率与可解释性。