Sparsity concepts and estimation procedures for high‐dimensional vector autoregressive models

High‐dimensional‐20 vector autoregressive (VAR) models are important tools for the analysis of multi‐variate time series. This article 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. Furthermore, it is shown that, under this sparsity setting, thresholding extends 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 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.