Two-way Threshold Matrix Autoregression

Matrix-valued time series data are widely available in various applications, attracting increasing attention in the literature. However, while nonlinearity has been recognized, the literature has so far neglected a deeper and more intricate level of nonlinearity, namely the {\it row-level} nonlinear dynamics and the {\it column-level} nonlinear dynamics, which are often observed in economic and financial data. In this paper, we propose a novel two-way threshold matrix autoregression (TWTMAR) model. This model is designed to effectively characterize the threshold structure in both rows and columns of matrix-valued time series. Unlike existing models that consider a single threshold variable or assume a uniform structure change across the matrix, the TWTMAR model allows for distinct threshold effects for rows and columns using two threshold variables. This approach achieves greater dimension reduction and yields better interpretation compared to existing methods. Moreover, we propose a parameter estimation procedure leveraging the intrinsic matrix structure and investigate the asymptotic properties. The efficacy and flexibility of the model are demonstrated through both simulation studies and an empirical analysis of the Fama-French Portfolio dataset.