  In this paper we develop valid inference for high-dimensional time series. We
extend the desparsified lasso to a time series setting under Near-Epoch
Dependence (NED) assumptions allowing for non-Gaussian, serially correlated and
heteroskedastic processes, where the number of regressors can possibly grow
faster than the time dimension. We first derive an error bound under weak
sparsity, which, coupled with the NED assumption, means this inequality can
also be applied to the (inherently misspecified) nodewise regressions performed
in the desparsified lasso. This allows us to establish the uniform asymptotic
normality of the desparsified lasso under general conditions, including for
inference on parameters of increasing dimensions. Additionally, we show
consistency of a long-run variance estimator, thus providing a complete set of
tools for performing inference in high-dimensional linear time series models.
Finally, we perform a simulation exercise to demonstrate the small sample
properties of the desparsified lasso in common time series settings.
