  We study Granger causality testing for high-dimensional time series using
regularized regressions. To perform proper inference, we rely on
heteroskedasticity and autocorrelation consistent (HAC) estimation of the
asymptotic variance and develop the inferential theory in the high-dimensional
setting. To recognize the time series data structures we focus on the
sparse-group LASSO estimator, which includes the LASSO and the group LASSO as
special cases. We establish the debiased central limit theorem for low
dimensional groups of regression coefficients and study the HAC estimator of
the long-run variance based on the sparse-group LASSO residuals. This leads to
valid time series inference for individual regression coefficients as well as
groups, including Granger causality tests. The treatment relies on a new
Fuk-Nagaev inequality for a class of $\tau$-mixing processes with heavier than
Gaussian tails, which is of independent interest. In an empirical application,
we study the Granger causal relationship between the VIX and financial news.
