  We consider the estimation and inference in a system of high-dimensional
regression equations allowing for temporal and cross-sectional dependency in
covariates and error processes, covering rather general forms of weak temporal
dependence. A sequence of regressions with many regressors using LASSO (Least
Absolute Shrinkage and Selection Operator) is applied for variable selection
purpose, and an overall penalty level is carefully chosen by a block multiplier
bootstrap procedure to account for multiplicity of the equations and
dependencies in the data. Correspondingly, oracle properties with a jointly
selected tuning parameter are derived. We further provide high-quality
de-biased simultaneous inference on the many target parameters of the system.
We provide bootstrap consistency results of the test procedure, which are based
on a general Bahadur representation for the $Z$-estimators with dependent data.
Simulations demonstrate good performance of the proposed inference procedure.
Finally, we apply the method to quantify spillover effects of textual sentiment
indices in a financial market and to test the connectedness among sectors.
