# Interpreting interaction term in a regression if interaction term is statistically insignificant

Can you please help me with the following. I have a regression with an interaction term:

Y = A + B + A*B


where A is a continuous variable and B is a dummy variable. I have the following coefficient of a regression:

Y = -0.3A + 0.01B + 0.02A*B


Coefficient on interaction maybe significant or insignificant.

So, the way I interpret the effect of A in the presence of B (B==1) on Y is (-0.3+0.02 = -0.1). If (-0.1) is statistically different from zero, then there is no effect of A (in the presence of B) on Y. Am I correct? What if the interaction term is not statistically significant, but A is stat-ly significant with (A+A*B) not different from zero? No effect?

Let's think about what adding an interaction term does to a model. You've left the intercept out but I'm going to assume you do have one: $$Y_i=\beta_0+\beta_1A_i+\beta_2B_i+\beta_3A_iB_i$$ where $$\beta_0$$ is the intercept, $$\beta_1,\beta_2$$ and $$\beta_3$$ are the coefficients for $$A, B$$ and their interaction respectively, $$A$$ is a continuous variable, and $$B$$ is a dummy variable.

Now, this is the model we get when $$B = 0$$: $$Y_i=\beta_0+\beta_1A_i$$ And this is the model we get when $$B = 1$$: $$Y_i=\beta_0+\beta_1A_i+\beta_2+\beta_3A_i\\=(\beta_0+\beta_2)+(\beta_1+\beta_3)A_i$$ As you can see, there are two things going on:

1. Adding the dummy variable $$B$$ lets the model fit different intercepts for the two levels of $$B$$. Notice that when $$B=0$$ the intercept is $$\beta_0$$ but when $$B=1$$ the intercept is $$\beta_0+\beta_2$$.
2. Adding the interaction term lets the model fit different slopes of $$A$$ for the two levels of $$B$$. Notice that when $$B=0$$ the slope of $$A$$ is $$\beta_1$$ but when $$B=1$$ the slope of $$A$$ is $$\beta_1+\beta_3$$.

Therefore, to answer your question, what does the statistical significance of the interaction term tell us? If the interaction term is statistically significant, then we know that there is strong evidence to suggest that the effect of $$A$$ on $$Y$$ is different for different levels of $$B$$ (the slope of $$A$$ changes).

PS: I once wrote a blog post about this very topic, you can find it here if you're interested.

• Thank you for your reply! Very helpful! I fully get your point. But the thing that confuses me is once I run a regression, I get the coefficient on interaction term being insignificant. But if I test whether coefficients of A + A*B are different from zero, I fail to reject this. (coeff. on A is stat-ly significant). Why this might be the case? Should I get the results different from zero? Commented Sep 28, 2021 at 16:40
• @AlbertoAlvarez If you get a statistically insignificant interaction term, all it means is that you don't need different slopes for the two levels of $B$. You don't need to do any further tests - in fact, I'm not even sure about how you're testing whether the coefficients of $A+A\times B$ are different from zero. Commented Sep 28, 2021 at 16:49
• Ok, I see your point, thank you! I test it using stata: test _b[A]+_b[A*B]=0 Commented Sep 28, 2021 at 16:51