Time series of matrix-valued data are increasingly available in various areas including economics, finance, social science, among others. These data may shed light on the inter-dynamical relationships between two sets of attributes, for instance, countries and economic indices. The matrix autoregressive (MAR) model provides a parsimonious approach for analyzing such data. However, the MAR model, being a linear model with parametric constraints, cannot capture the nonlinear patterns in the data, such as regime shifts in the dynamics. We propose a mixture matrix autoregressive (MMAR) model for analyzing potential regime shifts in the dynamics between two attributes, for instance, due to recession versus expansion, or stable period versus pandemic. We propose an EM algorithm for maximum likelihood estimation. We derive some theoretical properties of the proposed method including consistency and asymptotic distribution, and illustrate its performance via simulations and real applications.

翻译：矩阵值数据的时间序列在经济学、金融学、社会科学等多个领域日益普及。这类数据能够揭示两组属性（例如国家与经济指数）之间的相互动态关系。矩阵自回归（MAR）模型为分析此类数据提供了一种简约的方法。然而，作为具有参数约束的线性模型，MAR模型无法捕捉数据中的非线性模式，如动态过程中的机制转换。我们提出了一种混合矩阵自回归（MMAR）模型，用于分析两种属性之间动态过程中可能存在的机制转换——例如由经济衰退与扩张、或稳定期与疫情期引发的转换。我们提出了用于最大似然估计的EM算法。推导了所提方法的一些理论性质，包括一致性和渐近分布，并通过模拟和实际应用验证了其性能。