  We show for the first time that, under the null hypothesis of vanishing
Granger causality, the single-regression Granger-Geweke estimator converges to
a generalised $\chi^2$ distribution, which may be well approximated by a
$\Gamma$ distribution. We show that this holds too for Geweke's spectral
causality averaged over a given frequency band, and derive explicit expressions
for the generalised $\chi^2$ and $\Gamma$-approximation parameters in both
cases. We present an asymptotically valid Neyman-Pearson test based on the
single-regression estimators, and discuss in detail how it may be usefully
employed in realistic scenarios where autoregressive model order is unknown or
infinite. We outline how our analysis may be extended to the conditional case,
point-frequency spectral Granger causality, state-space Granger causality, and
the Granger causality $F$-test statistic. Finally, we discuss approaches to
approximating the distribution of the single-regression estimator under the
alternative hypothesis.
