Many or few variables when testing for Granger causality?

Question

I am reading Levendis "Time Series Econometrics: Learning Through Replication" (2018) and there are two statements about Granger causality that kinda confuses me. The statements themselves are not confusing, but the implications.

1. "Granger causality is much harder to establish with more than 2 variables". So I guess few variables are better, and using just two variables are even more better.
2. "Tests of Granger causality are sensitive to omitted variables. Researchers should include all relevant variables in their analysis". So here, more relevant variables are better, I guess.

Can we add more relevant variables without jeopardizing Granger causality?

Answer 1

The implication is, the model should be adequately specified, without omitting relevant variables. Once the model is adequately specified, Granger causality will be easier to establish if the model has fewer variables than if it has more. (I am not taking a stance regarding how reasonable the quoted statements are but rather just rephrasing/interpreting them.)

This characterizes the systems you may be modelling (few or many variables) and the modelling choices (avoid omitting variables). In applications of time series econometrics (e.g. in macroeconomics or finance), usually you cannot change the underlying system (make it contain fewer variables). Then you do not have much choice over the number of variables in your model -- unless you are willing to violate the suggestion to avoid omitting relevant ones and risk working with an inadequate model and consequently obtaining invalid results.