Granger Causality coefficient comparison

Question

I am working with a time series data and found that $a_t$ Granger causes $b_{t+1}$ and $b_t$ granger causes $a_{t+1}$. The results were obtained through Stata.

1. With the coefficients obtained, is it possible to determine which Granger causal direction is stronger? If it can't be determined by just looking at the coefficients, is there a method to do so?

   Also, Granger causality was tested on 2 conditions. In one subset, a policy was adopted. In the other subset, the policy was not adopted. The results was that a granger causes $b_{t+1}$ in both cases.

2. Is there a way to test whether the Granger causality coefficient in condition one was significantly different in condition two? One idea is to introduce an interaction term, but is this possible with Granger causality tests?

Answer 1

In both cases, the last step in Granger Causality entails testing whether the square of the residuals from your two different regressions come from the same population. One set of residuals come from your base model (a simple autoregressive model), the other set of residuals come from adding the independent variable that you are testing to the base model. So far this is just describing part of Granger Causality.

So, let's say when you figured that A Granger causes B, and the related comparison of the squared residuals were associated with either a P value or a Significance of F of 0.20; and when you ran Granger Causality in the other direction, B Granger causes A, you got the respective P value or Significance of F of 0.01... In this case, you would say that B Granger causes A a lot more than the reverse because the related P value of 0.01 is a lot more statistically significant than the 0.2 in the opposite direction.

When I mention P value, this entails that you either use an Unpaired t test or Mann-Whitney test to test how different the square of the residuals are between the base autoregressive model and the extended model when you add the other variable to be tested. Meanwhile, the Significance of F is if you use an F test to test the same thing. Clive Granger, the inventor of Granger Causality, did develop his methodology using an F test. However, in most cases an Unpaired t test or Mann-Whitney test would be just as if not more appropriate. The results directionally should be the same either test you use.