Interpreting Granger Causality F-test

Question

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?

Answer 1

One method would be to look at correlations, and, there are two cases to consider. That, in turn, depends on whether or not we can assume independence. The worst case scenario is correlation of independent processes. In that case, we could be comparing for A & B, dollar values on the US Dow Jones market and the Canadian exchange in Toronto, to C & D representing temperatures in Osaka and Tokyo, Japan. The second case would be correlated correlations, where we could be comparing values on the the US and Canadian exchanges to the values on the Osaka and Tokyo exchanges.

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.

Discussion So what if the correlation from Granger Causality is better for the North American markets than for the temperatures in Japan? There are two problems with this. First is that we are left with no real valid conclusions because we are comparing "apples to oranges" in the figurative sense. Second is because the prevailing winds in Osaka are sometimes from the direction of Tokyo and sometimes to Tokyo, so that Granger Causality could underestimate the actual relationship if seasonal variations in temperature and prevailing winds are ignored. In the second case, correlated correlations, one could find that the changes in the the larger markets (New York and Tokyo) precede changes in the smaller markets (Toronto and Osaka), but, why would we compare correlations anyway? Why wouldn't we, for example, do a round robin and Granger Causality compare New York and Tokyo markets directly, if that is what we want to know?