Suppose you have a dependent variable Y. Suppose someone runs 2 regressions: one by interacting a variable A with B and another by interacting A with C on Y. The person doing these regressions concludes that there is no significant evidence for a relationship between the interaction variables and the dependent variable.

Someone else claims that the effects of the first regression cancel out the effects of the second regression (i.e. they work against each other). This person claims that this is the source of the lack of evidence of a relationship.

How are you supposed to re-work the original models/regressions to accomodate and test for the second person's claims?

I am not too gifted in econometrics to make a clear judgement on this. I was thinking of doing a difference-in-difference method to somehow separate the two effects (A with B and A with C) and see their effects separately. Am I heading in the right direction or am I totally off?

Thank you very much.