Statistical Estimation of High-Dimensional Vector Autoregressive Models

TL;DR

This paper develops a new sparsity scheme for high-dimensional VAR models, demonstrating how thresholding extends estimator consistency across various matrix norms, with extensive simulations validating the approach.

Contribution

It introduces a more suitable sparsity assumption for high-dimensional VAR models and shows how thresholding enhances estimator consistency for diverse inference tasks.

Findings

Proposed a new sparsity scheme for high-dimensional VAR models.

Thresholding extends estimator consistency to various matrix norms.

Simulation results validate the effectiveness of the proposed estimators.

Abstract

High-dimensional vector autoregressive (VAR) models are important tools for the analysis of multivariate time series. This paper focuses on high-dimensional time series and on the different regularized estimation procedures proposed for fitting sparse VAR models to such time series. Attention is paid to the different sparsity assumptions imposed on the VAR parameters and how these sparsity assumptions are related to the particular consistency properties of the estimators established. A sparsity scheme for high-dimensional VAR models is proposed which is found to be more appropriate for the time series setting considered. Furthermore, it is shown that, under this sparsity setting, threholding extents the consistency properties of regularized estimators to a wide range of matrix norms. Among other things, this enables application of the VAR parameters estimators to different inference problems, like forecasting or estimating the second-order characteristics of the underlying VAR process. Extensive simulations compare the finite sample behavior of the different regularized estimators proposed using a variety of performance criteria.