# Interpreting Granger Causality F-test

This question is a bit basic (I reviewed the previous postings on similar subjects, but still need help with this). Thanks in advance for any answer.

The question is if A & B are two time-series and are decided to be likely Granger-Causal with a SSR F-test below with a p-value p1, and test statistic t1. Similarly C & D are two other time-series which are also likely causal with test p-values p2, and test statistic t2. Can I compare the p-values and the t-statistics to deem how likely more causal one is compared to another?

An experiment is created where the 'load' is changed, and a pair of time-series for two variables are generated for each such load. That is, a pair of time-series exists for each discrete load value point (say ten specific load values).

A Granger Causality test for two time-series using python statsmodels package (R reports similar results) reports the following for the ssr F-test statistic.

Granger Causality ('number of lags (no zero)', 1) ssr based F test: F=5.4443 , p=0.0198 , df_denom=1385, df_num=1 ssr based chi2 test: chi2=5.4561 , p=0.0195 , df=1 likelihood ratio test: chi2=5.4454 , p=0.0196 , df=1 parameter F test: F=5.4443 , p=0.0198 , df_denom=1385, df_num=1

I have the following questions:

1. The fact that the p-value << 0.05 indicates that there is 'likely' some 'Granger Causality' between the two time series for a given load. Is this just a likely/unlikely (i.e., boolean) statement, or can I read something more into the F-statistic reported? The null hypothesis being: there is no granger causality between the two series.

2. If the p-value is consistently less that << 0.05 across the loads, can I read anything into the F test statistics relative to each other (i.e., does it mean anything to compare the F-statistic for each load?) If so, can I say anything about 'how much causal', one is relative to the other. In other words, what would be and if there is a way to quantify the causality when p-value is consistently << 0.05

3. If one p-value for a given load is much >> 0.05, while << 0.05 for other, then is there a way to represent the results across all loads using a single value?

In the first case, we could use a comparison of independent R-values from regression where in each regression we obtain our correlations from $R^2=1-\dfrac{SSE}{SST}$, and then test the significance of difference of R-values. For the second case, correlated correlations, smaller differences in R-values are more easy to detect.