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.

翻译：我们用正规回归法研究高维时间序列的Ganger因果关系测试。 为了进行正确的推断, 我们依赖对无症状差异进行恒定估计, 并开发高维环境中的推论理论。 为了确认时间序列数据结构, 我们侧重于小群体 LASSO 估测器, 其中包括LASSO 和 LASSO 组的特殊案例 。 我们为低维回归系数组设定了偏差中央限值, 并研究了 HAC 依据稀有群体 LASSO 残留物得出的长期差异估计值 。 这导致个人回归系数和群体的有效时间序列推断, 包括Granger因果性测试。 治疗依赖于一个新的Fuk- Nagaev 不平等性, 用于比高斯尾部更重的 $tau$- tagev 混合过程, 这是一种独立的兴趣。 在一次实验应用中, 我们研究了VIX 和 金融新闻 之间的重因果关系。