Sampling distribution for single-regression Granger causality estimators

The single-regression Granger-Geweke causality estimator has previously been shown to solve known problems associated with the more conventional likelihood-ratio estimator; however, its sampling distribution has remained unknown. We show that, under the null hypothesis of vanishing Granger causality, the single-regression estimator converges to a generalized ?2 distribution, which is well approximated by a G distribution. We show that this holds too for Geweke's spectral causality averaged over a given frequency band, and derive explicit expressions for the generalized ?2 and G-approximation parameters in both cases. We present a Neyman–Pearson test based on the single-regression estimators, and discuss how it may be deployed in empirical scenarios. We outline how our analysis may be extended to the conditional case, point-frequency spectral Granger causality, and the important case of state-space Granger causality.