Interaction Term F-Test

[Sophia Meyer]

Dear all,
i have a question regarding the usage of interaction terms in my regression.
I have estimated a model Y =b1 X1 +b2 X2 + b3 (X1 × X2) + u
The main effect of X1 is statistically significant, also is the coefficient of the interaction term, but the coefficient of X2 isn´t signficant.
As far as i know, i cant interpret the main effects independently if tthere is a signficant interaction term because they only represent the effect if the ther main variable is zero.
Now i conducted an F-test for testing if X1 and the interaction term are jointly signficant. I did this because i am interested in the effect of X1.
The F-test implicates that x1 and the interaction term are not jointly signficant different from zero.
Now i am very unsure how to proceed. Can i conclude that there is an signifcant interaction effect between X1 and X2 because the coefficient of the interaction is significant? or can i conclude that there is no interaction effect between x1 and x2 because the f-test of X1 and X3 implicates that there is no jointly singicance?
i also want to apologize for not including my stata codes, but i am using data of a company i am not allowed to share.
Can you help me with my problem? thank you very much!
kind regards,
Sophia

[Maarten Buis]

By including an interaction term you gave up on the concept of one effect of x1. There are now multiple effects of x1, namely one per value of x2. So testing whether the effect is zero no longer makes sense, also a joint test of the main effect and the interaction term is meaningless. What you need to do is substantively think what the hypothesis is that you want to test, and than translate that to a statistical test.

If your substantive hypothesis is about the effect of x1, then you cannot include an interaction term. If you need the interaction term, then you will probably end up with something graphical, as there is no longer a simple yes/no decision as there is no longer one effect.

[Sophia Meyer]

my hypothesis is that is x1 has no effect on Y.
so i should not look at the p-Values of the interaction term?
for example, the partial effect of x1 is now b1 + b3X2.
How can i test the hypothesis that there is no signficant effect of X1 on Y? and what woul i do if the interaction term wasnt signficant? should i then intepret the main effects or also create a graphic?
thank you mr Buis!
kind regards,
Sophia

[Maarten Buis]

By adding the interaction term the effect of x1 changes when x2 changes. That is the purpose of an interaction term. This means that there are now multiple effects for x1. So an hypothesis that the effect of x1 is zero no longer makes sense, as the effect no longer exists. So you need to rethink your hypothesis. There can be no technical solution for a conceptual problem like yours. You need to solve it on the conceptual level.

[Phil Bromiley]

To offer a slightly different slant on Maarten's answer, if the model includes x1, x2, and x1*x2, dy/dx1 for x1 depends on x2. It is possible to test whether the values (note, values, not value) of dy/dx1 are significantly different from zero across the range of observed x2 (or any other range for x2) using the margins command.