Q:

In Granger causality test, what does very low p-value means when you know there is little correlation

I am testing a comparison between two time series to see if/how a variable Xs interacts/affects variable Ys.
I have come across some cases where I use cross-correlation where I verify there is little correlation and anti-correlation but the p-values of these cases are like 0.00001.
According to what I understood the lower the p-value is the higher the chance of the Null hypothesis being false. However since the cross-correlation algorithm says there is little correlation between them, I find odd the p-value being so low.
So what I want to know is, in what situations this can happen or if low p-value can mean something else, or even if all my approach/logic is rubbish since I am self-learning and doing this for the first time so there a decent chance that might happen.

A:

One explanation could be that partial correlation is higher than regular correlation, i.e. the regression 
$$ y_t=\beta_0+\beta_1 x_s+u_t $$
yields an insignificant $\beta_1$ while
$$ y_t=\beta_0+\beta'_1 x_s+\alpha_1 y_s+v_t $$
yields a significant $\beta'_1$. That could happen. For Granger causality test it is the partial correlation that matters.

ould you tell me though how could I very that on R. Since I don't think there is a function to verify the partial correlation between two time series

Partial correlation depends on what other variables you are considering. Run a regression model and check the coefficient significance – that will tell whether the partial correlation is significant. That is what the Granger causality test is doing. (So you already know the answer.)

