Decomposing Co-Movements in Matrix-Valued Time Series: A Pseudo-Structural Reduced-Rank Approach

Abstract

A pseudo-structural framework is proposed for analyzing contemporaneous co-movements in stationary reduced-rank matrix autoregressive (RRMAR) models. Unlike conventional vector autoregressive (VAR) models that discard the matrix structure, the formulation preserves it, enabling a decomposition of co-movements into three interpretable components: row-specific, column-specific, and joint (row--column) interactions across the matrix-valued time series. The estimator admits standard asymptotic inference and a BIC-type criterion is proposed for the joint selection of the reduced ranks and the autoregressive lag order. The method's finite-sample performance in terms of estimation accuracy, coverage, and rank selection is validated through simulation experiments, including cases of rank misspecification. Practical usefulness is illustrated through an application to labor market data from nine Midwestern U.S. states, revealing distinct row-, column-, and joint co-movement patterns.