Multi-Matrix Autoregressive Models with an Application to Multi-Modal Network

Matrix-valued time-series data have become increasingly prevalent across diverse fields, including economics, finance, computer science, engineering, and signal processing. However, mixed matrix-vector structures in real-world data present challenges for existing methods. To fill in this gap, we propose a multi-matrix autoregressive (MMAR) model designed to jointly model matrix time series with varying structures. The key idea is to treat matrices as elements and to model their contemporaneous and intertemporal relationships within a simultaneous equation system. In particular, the matrix-valued autoregressive model and the three-order tensor autoregressive model are special cases of the proposed model. We present three distinct estimation methods for the MMAR model, investigate their statistical properties, and provide numerical simulations to corroborate them. Furthermore, we integrate the MMAR model with connectedness network analysis, proposing a multi-modal connectedness framework to estimate the interconnectedness between the rows and columns of matrices. Finally, we provide an empirical application to concurrently model the macroeconomic matrix time series of China’s 31 provinces and a vector time series comprising the economic policy uncertainty, trade policy uncertainty, and geopolitical risk. The findings of the empirical study provide valuable insights for further research and policymaking in the relevant domains.