Matrix Autoregressive Time Series With Reduced‐Rank and Sparse Structural Constraints

Matrix‐ and tensor‐valued time series models have emerged as effective tools to address the challenges posed by high‐dimensional time series data. These models utilize the multi‐classification structures inherent in data variables to decompose large interaction networks into smaller, more manageable sub‐networks. To further reduce dimensionality, recent research has explored regularized matrix‐valued time series models. This study builds upon this line of work by proposing the RR‐S‐MAR model—a matrix autoregressive (MAR) model that incorporates a reduced‐rank structure on one side and a sparse structure on the other. We address key challenges related to the estimation, inference, and selection of the proposed model. For regularized estimation, we develop an alternating least‐squares algorithm, while statistical inference is conducted using a bootstrapping method. To optimize the selection of rank and sparsity level, we introduce an extended Bayesian information criterion (EBIC). Simulation studies demonstrate the convergence of the estimation algorithm and validate the effectiveness of the proposed model selection criterion. Finally, we apply the RR‐S‐MAR model to economic data, showcasing its practical utility and providing insights through real‐world analysis and interpretation.