Envelope Matrix Autoregressive Models for High-Dimensional Matrix-Valued Time Series

Abstract

Matrix-valued time series arise in many fields when observations are naturally organized by two dimensions, such as geographic regions and economic indicators, assets and financial characteristics, or brain regions and frequency bands. Traditional vector autoregressive approaches require vectorizing the observations, which ignores their inherent row and column structures and may lead to severe overparameterization. Matrix autoregressive models preserve this structure and provide a more parsimonious and interpretable framework. However, in high-dimensional settings, they may still involve large coefficient matrices and do not distinguish between information that is relevant and irrelevant to the temporal dynamics. This talk presents an envelope matrix autoregressive model that identifies and removes irrelevant variation through reducing subspaces of the covariance matrices. By linking the mean and covariance structures, the proposed approach reduces the number of parameters and improves estimation and forecasting efficiency. The asymptotic properties of the estimators are established, and their finite-sample performance is evaluated through simulation studies under both normal and non-normal distributions. Two applications involving economic and business data illustrate the practical advantages of the proposed method.