Sign of Estimate (Coefficient) of Interaction Terms

Question

I create a linear regression model with interaction term in the model say: $$y=a_0+a_1x_1+a_2x_2+a_3x_1*x_2+e$$ where $x_1$ is continuous and $x_2$ is binary. Now, I have couple of questions:

1. If $x_1$ and $x_2$ both are significant and have positive coefficients, that means that if $x_1$ and $x_2$ increase, then $y$ increases. What can I say if $x_1*x_2$ is statistically insignificant? Does it mean that increase in. Does it mean that for $x_2=1$, the effect of $x_1$ on $y$ is not changed, i.e. increase in $x_1$ still leads to increase in $y$? What if the interaction is statistically significant?

2. What can I say if the interaction term is statistically insignificant and negative? Does it indicate that if $x_2=1$, increase in $x_1$ actually leads to decrease in $y$?

Thanks.

Answer 1

When you discuss the interaction you don't discuss it in terms of what happens to y but what happens to a1 and a2. If it's positive then as a1 increases a2 increases. If it's negative then as a1 increases a2 decreases. Interactions are about differences in effects. The slopes in regression are the effects and so it's most efficiently discussed in those terms.

Once you consider the foregoing you can understand why many say that main effects don't mean anything when there is an interaction. I don't agree with that as a generalization but certainly there are many cases where it's true.