Envelope Matrix Autoregressive Models

Matrix-valued data is commonly collected over time in many scientific fields. However, existing methods for handling such data are limited and often suffer from overparameterization. In response, Chen, Xiao, and Yang introduced the matrix autoregressive (MAR) model as an alternative to traditional time series analysis, which relies on vectorization and vector autoregression frameworks. By preserving the original structure of matrices, the MAR model avoids the loss of valuable column and row information. This approach offers a significant reduction in dimensions and enables explicit interpretations of the data. However, when applied to high-dimensional matrix time series, the MAR model faces challenges due to the large size of the coefficient matrices involved. It struggles to differentiate between relevant and irrelevant information, making it inefficient in extracting relevant information from complex data. To address these limitations, we propose envelope-based MAR (EMAR) models that effectively identify and eliminate irrelevant information. Our proposed EMAR approach achieves substantial efficiency gains in estimation and forecasting by reducing parameters and constructing a link between the mean function and covariance structure. This is achieved by using the minimal reducing subspaces of covariance matrices. We establish the asymptotic properties of our proposed estimators and compare their efficiency and accuracy to existing methods through simulation studies under both normality and non-normality conditions. Furthermore, we provide two real-world applications in economics and business to demonstrate the effectiveness of our approach.