Inference in High-Dimensional Linear Projections: Multi-Horizon Granger Causality and Network Connectedness

This paper presents a Wald test for multi-horizon Granger causality within a high-dimensional sparse Vector Autoregression (VAR) framework. The null hypothesis focuses on the causal coefficients of interest in a local projection (LP) at a given horizon. Nevertheless, the post-double-selection method on LP may not be applicable in this context, as a sparse VAR model does not necessarily imply a sparse LP for horizon h>1. To validate the proposed test, we develop two types of de-biased estimators for the causal coefficients of interest, both relying on first-step machine learning estimators of the VAR slope parameters. The first estimator is derived from the Least Squares method, while the second is obtained through a two-stage approach that offers potential efficiency gains. We further derive heteroskedasticity- and autocorrelation-consistent (HAC) inference for each estimator. Additionally, we propose a robust inference method for the two-stage estimator, eliminating the need to correct for serial correlation in the projection residuals. Monte Carlo simulations show that the two-stage estimator with robust inference outperforms the Least Squares method in terms of the Wald test size, particularly for longer projection horizons. We apply our methodology to analyze the interconnectedness of policy-related economic uncertainty among a large set of countries in both the short and long run. Specifically, we construct a causal network to visualize how economic uncertainty spreads across countries over time. Our empirical findings reveal, among other insights, that in the short run (1 and 3 months), the U.S. influences China, while in the long run (9 and 12 months), China influences the U.S. Identifying these connections can help anticipate a country's potential vulnerabilities and propose proactive solutions to mitigate the transmission of economic uncertainty.