High-Dimensional Multivariate VAR Estimation with Spatio-Temporal Structure

Abstract

High-dimensional vector autoregressive (VAR) models provide a flexible framework for characterizing dynamic dependence in multivariate spatio-temporal systems, but their unrestricted estimation becomes infeasible when multiple variables are observed over many spatial locations. This paper develops a structured estimation procedure for high-dimensional multivariate VAR processes that explicitly incorporates spatial information. We decompose each block transition matrix into a cross-variable dependence coefficient and a spatial transition matrix, and constrain the spatial transition matrices through a pre-specified spatial graph. The resulting estimator is formulated as a weighted $\ell_1$-regularized least-squares problem, where the weights encode spatial proximity or topological similarity and induce stronger shrinkage on spatially implausible interactions. Since the objective function is bi-convex, we estimate the cross-variable dependence matrix and the spatial transition matrices through an alternating convex-search algorithm implemented with ADMM. Under stability and restricted-eigenvalue-type conditions for high-dimensional VAR processes, we establish convergence to a blockwise stationary point in the subgradient sense and derive high-probability estimation error bounds for both components of the model. Simulation studies demonstrate that the proposed estimator accurately recovers sparse transition structures and improves over existing two-step $\ell_1$-regularized methods in support recovery and estimation accuracy. An application to North American climate data illustrates that the method recovers interpretable variable-dependence networks and spatial interaction patterns across different climate regions.