  Vector autoregression (VAR) models are widely used to analyze the
interrelationship between multiple variables over time. Estimation and
inference for the transition matrices of VAR models are crucial for
practitioners to make decisions in fields such as economics and finance.
However, when the number of variables is larger than the sample size, it
remains a challenge to perform statistical inference of the model parameters.
In this article, we propose the de-biased Lasso and two bootstrap de-biased
Lasso methods to construct confidence intervals for the elements of the
transition matrices of high-dimensional VAR models. We show that the proposed
methods are asymptotically valid under appropriate sparsity and other
regularity conditions. To implement our methods, we develop feasible and
parallelizable algorithms, which save a large amount of computation required by
the nodewise Lasso and bootstrap. A simulation study illustrates that our
methods perform well in finite samples. Finally, we apply our methods to
analyze the price data of stocks in the S&P 500 index in 2019. We find that
some stocks, such as the largest producer of gold in the world, Newmont
Corporation, have significant predictive power over the most stocks.
