Interaction term flip sign when adding the main effect of time-invariant regressor in system GMM

[karamat khan]

Hello everyone,
I have the following system GMM regression where “X1” is time-varying and “X2” is time-invariant and my main interest is the interaction term of “X1*X2.”

xtabond2 Y l.Y X1 X2 X1*X2 CONTROLS, gmm(l.Y, lag(2 2) collapse) iv(X1 X2 X1*X2  CONTROLS, equation(level)) twostep nodiffsargan  orthogonal small robust

I am confused about the sign on interaction term, which is opposite to what I expect based on literature. I performed several different models, even changing the controls, measures, sample and lags but the interaction term only takes the desired sign when I remove the main effect of the time-invariant regressor. Can someone please explain to me what is possibly wrong with my model, or the opposite sign is the TRUE sign (because the sign is consistent even with alternate models)? or is it possible to use the model without the main effect of the time-invariant regressor, would it be a correct model? And what would be the possible explanation in this case?

Note: I have also read a paper published in reputable journal where the authors used system GMM and omitted the main effect of time-invariant regressor (their main focus was also the interaction term) saying that “the country-fixed effects in the equations make the separate inclusion of time-invariant regressor redundant.” However, they showed the country-fixed effects in equations but did not include the country fixed-effects in their GMM models (results show “year fixed effects” in tables but not country fixed effects). I do not understand how they claim "separate inclusion of time-invariant regressor as redundant" when they do not add the country fixed-effects in their GMM models? Can someone also please provide explanation to this?


Thanks in advance

[George Ford]

Country fixed effects may be excluded from presentation if there are many (even if included). Is a coefficient produced for X2 when you estimate it?

[karamat khan]

Thanks for reply sir, country-fixed effects (country dummies) drops coefficient for X2 (and i don't include country dummies in my model), but my main concern is the change of sign on interaction term when include/exclude the main effect of X2 with interaction term. would it be a correct model if i drop the main effect of X2 (this is the case where i get the desired sign on coefficient)? and why the sign changes on interaction term when X2 is added with interaction term (and these results remain same even when change controls, measures etc)?