An Interpretable and Efficient Infinite-Order Vector Autoregressive Model for High-Dimensional Time Series

TL;DR

This paper introduces a sparse, interpretable infinite-order VAR model for high-dimensional time series that captures complex temporal patterns efficiently, overcoming key limitations of traditional VARMA models.

Contribution

It proposes a novel sparse infinite-order VAR model with separate interpretability of temporal and cross-sectional structures, along with efficient estimation and model selection methods.

Findings

Model effectively captures rich temporal patterns.

Estimation methods have provable error bounds.

Application to macroeconomic data demonstrates practical utility.

Abstract

As a special infinite-order vector autoregressive (VAR) model, the vector autoregressive moving average (VARMA) model can capture much richer temporal patterns than the widely used finite-order VAR model. However, its practicality has long been hindered by its non-identifiability, computational intractability, and difficulty of interpretation, especially for high-dimensional time series. This paper proposes a novel sparse infinite-order VAR model for high-dimensional time series, which avoids all above drawbacks while inheriting essential temporal patterns of the VARMA model. As another attractive feature, the temporal and cross-sectional structures of the VARMA-type dynamics captured by this model can be interpreted separately, since they are characterized by different sets of parameters. This separation naturally motivates the sparsity assumption on the parameters determining the cross-sectional dependence. As a result, greater statistical efficiency and interpretability can be achieved with little loss of temporal information. We introduce two $\ell_1$-regularized estimation methods for the proposed model, which can be efficiently implemented via block coordinate descent algorithms, and derive the corresponding nonasymptotic error bounds. A consistent model order selection method based on the Bayesian information criteria is also developed. The merit of the proposed approach is supported by simulation studies and a real-world macroeconomic data analysis.