  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.
