High-Dimensional Vector Autoregressive Time Series Modeling via Tensor Decomposition

The classical vector autoregressive model is a fundamental tool for multivariate time series analysis. However, it involves too many parameters when the number of time series and lag order are even moderately large. This paper proposes to rearrange the coefficient matrices of the model into a tensor form such that the parameter space can be restricted in three directions simultaneously via tensor decomposition. The proposed method substantially expands the capacity of vector autoregressive modeling for a large number of time series. In contrast, the widely used reduced-rank regression method can restrict the parameter space in only one direction. Moreover, to handle high-dimensional time series, this paper considers imposing sparsity on factor matrices to improve the interpretability and estimation efficiency, which leads to a sparsity-inducing estimator. For the low-dimensional case, we derive asymptotic properties of the proposed least squares estimator and introduce an alternating least squares algorithm. For the high-dimensional case, we establish non-asymptotic properties of the sparsity-inducing estimator and propose an ADMM-based algorithm for regularized estimation. Simulation experiments and a real data example demonstrate the advantages of the proposed approach over various existing methods