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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.

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