Why Granger causality test gives same result for restricted and unrestricted VAR models

Question

I applied granger causality test 1st in unrestricted 2 dimensional VAR(1) model and then restricted model (t>2). Both are giving the same result (the result of unrestricted VAR model). Actually unrestricted VAR model was showing that the coefficient of 1 variable is not significant so did granger causality test. But after restricting the model we got that this variable turned significant. Therefore, we proceeded for granger causality test expecting that the result will change and now it will show 'causality'. But, unfortunately, it did not. Rather, it showed the same result as previous (in case of unrestricted VAR model.

Questions are: 1) How does 'R' restrict the model using restrict command as the entire result gets changed instead of just considering the significant variables and dropping significant ones? 2) If restricted model shows only significant variables, why granger causality test ('causality' command in R) exhibits entirely different result (shows the result of the unrestricted case)?

Answer 1

1) 'Restrict' command in R re-estimates all the coefficient one-by-one based on its significance. It is nothing do with original VAR model. But manual restriction can be provided. Some variables might not be significant in the presence of other variables but might be significant if considered alone. That's why it can be found that some of the insignificant variables in unrestricted VAR model converts into significant.

2) Restricted VAR model and Granger causality are two altogether different concepts. Restricted VAR model works on marginal t-test. Whereas, Granger causality works on general linear F-test.