Interpreting results of panel granger test

I ran the Panel Granger Causality test from the plm package and I'm unsure how to interpret the results. I don't know what "Ztilde" represents, and I'm not sure if the p-value represents the probability of the null hypothesis, or the probability of the alternate hypothesis.

Here's what the results look like:

    Panel Granger (Non-)Causality Test (Dumitrescu/Hurlin (2012))

data:  y ~ x
Ztilde = -0.43184, p-value = 0.6659
alternative hypothesis: Granger causality for at least one individual


Replication code:

trial <- data.frame(id=c(1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2),
timeid=c(1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9),
x=c(0.003, 0.04, 0.00683, 0.0, 0.0, 0.0, 0.0, 0.00012, 0.31, 0.003, 0.04, 0.00683, 0.0, 0.0,  0.0, 0.0, 0.0, 0.1),
y=c(0.03, 1.04, 0.05683, 0.1, 0.5, 0.0, 0.0, 0.12, 0.041, 0.003, 0.04, 0.00683, 0.0, 0.0,  0.0, 0.0, 1.0, 0.0))
trial_pdf <- pdata.frame(as.data.frame(trial), index=c("id", "timeid"))
results <- pgrangertest(y ~ x, data = trial_pdf, order=1,
test="Ztilde")
results


I have a broad understanding of how the Granger test works, I'm just unfamiliar with the terminology used here and there is virtually no documentation on the R website. Thank you!

• I do not get what you mean by "virtually no documentation on the R website". Why don't you look at the function's documentation and referenced literature? See ?plm::pgangertest. Feb 28, 2021 at 22:02
• Yes, my point is that the R docs on pgrangertest just isn't very helpful. It has no explanation of what Ztilde is or whether the p-value represents the probability of the null or alternate hypothesis. The referenced paper does, but the question is how the R implementation lines up with the paper - i.e. are the variable names the same? Mar 2, 2021 at 1:59
• Information about Ztilde is actually given in the documentation. Regards p-value you seem to have found what it is supposed to mean by learning about the test in general. It is not clear to me what you meant by "are the variable names the same"; if by variable you mean the variables tested, their names depend on your data. Mar 5, 2021 at 17:29
• I meant variable names between the R implementation (i.e. the documentation says enabling the "Ztilde" or "Zbar" options computes those respective statistics), and the paper referenced by the documentation. It seems likely that they are the same but they also may not be! Mar 5, 2021 at 18:47
• Well, at some point you would either need to trust that things with the same name in a software implementation and in the paper are the same or you look at the source code and check yourself or implement your own test. Mar 5, 2021 at 19:04