  Independent or i.i.d. innovations is an essential assumption in the
literature for analyzing a vector time series. However, this assumption is
either too restrictive for a real-life time series to satisfy or is hard to
verify through a hypothesis test. This paper performs statistical inference on
a sparse high-dimensional vector autoregressive time series, allowing its white
noise innovations to be dependent, even non-stationary. To achieve this goal,
it adopts a post-selection estimator to fit the vector autoregressive model and
derives the asymptotic distribution of the post-selection estimator. The
innovations in the autoregressive time series are not assumed to be
independent, thus making the covariance matrices of the autoregressive
coefficient estimators complex and difficult to estimate. Our work develops a
bootstrap algorithm to facilitate practitioners in performing statistical
inference without having to engage in sophisticated calculations. Simulations
and real-life data experiments reveal the validity of the proposed methods and
theoretical results.
  Real-life data is rarely considered to exactly satisfy an autoregressive
model with independent or i.i.d. innovations, so our work should better reflect
the reality compared to the literature that assumes i.i.d. innovations.
