Flipping of coefficients signs after introducing interaction effects

Question

My question as the title suggests is about flipping of coefficient signs when we introduce interaction variables based on two independent variables. More precisely,
My baseline regression is

Y_it=α_0+〖β_1 X1〗_it+〖β_2 X2〗_it

I obtain a positive sign for X1 as expected in my hypothesis. Now when I introduce interaction in the next regression to see whether X2 positively moderates effects of X1 over Y(i.e., a positive coefficient for interaction term) in below regression

Y_it=α_0+〖β_1 X1〗_it+〖β_2 X2〗_it+〖β_3 (X1*X2)〗_it I obtain positive sign for interaction term (β_3) as expected in my second hypothesis but sign for X1 coeeficient (β_1) turns negative and significant. as opposed to the baseline regression (Hyp 1).

Is it a problem if sign of the two or any of the variables used in interaction changes as my main variable of interest is the interaction term and not the two independent variables here? Thanks for your help.

Answer 1

You have encountered a case of "Simpson's Paradox" --- whether this is a problem or not really depends on what you are trying to determine. Usually in an analysis there is some objective reason to prefer the presence or absence of an interaction effect, depending on the research question and the model fit. That will tell you which is the appropriate model and therefore what output is relevant to your inferences.

Answer 2

No, that is not a problem. Generally speaking adding any additional variable can change the coefficient of the variables that have been in the model before.