# Partial Cross-Correlation, and its relationship to Granger Causality?

Definitions:

Granger Causality (GC):

1. To the future of the target channel $$y(t)$$ fit past values of the target channel $$y(t-1)$$, ... $$y(t-n)$$
2. To the future of the target channel $$y(t)$$ fit past values of the target channel $$y(t-1)$$, ... $$y(t-n)$$ and the past values of the source channel $$x(t-1)$$, ... $$x(t-n)$$
3. Test if the 2nd fit is significantly better than the first one

Partial Cross-Correlation Function (PCCF):

1. To the future of the target channel $$y(t)$$ fit its past values $$y(t-1)$$, ... $$y(t-n)$$, compute residual $$\tilde{y}(t)$$
2. To the future of the source channel $$x(t)$$ fit its past values $$x(t-1)$$, ... $$x(t-n)$$, compute residual $$\tilde{x}(t)$$
3. Compute cross-correlation between residuals $$\tilde{x}(t)$$, $$\tilde{y}(t)$$, shifted by time $$\tau$$

Questions:

1. How is GC related to 1-step PCCF (using $$\tau=1$$)?
2. Is PCCF useful in practice? If yes, what is it typically used for?

I am a bit surprised by a gap in the literature. There is tons of literature on GC, and an extensive wikipedia page. There is tons of literature on the Partial Autocorrelation Function (PACF), and a small wikipedia page. There is no wikipedia page for PCCF, and there is little mention of it besides a few recent publications in neuroscience, even though extension of PACF to PCCF seems simple and natural.