Need help in understanding continuous-dummy interaction in OLS regression

Question

I am having some conceptual difficulties in understanding and interpreting interaction terms (between a dummy and a continuous variable) in OLS regressions. I was hoping someone could help me out. I know there are similar questions that have been posted, but I wasn't quite sure I understood the answers there.

Let’s say we have an equation where an individual’s years of schooling is the outcome, and we have the individual’s parental income (a continuous variable) and the individual’s gender (=1 if male) as controls. And we want to see whether parental income has a differential effect by gender. So the equation would be as follows:

YearsofSchool = β0 + β1 ParentIncome+ β2 Male+ β3 ParentIncome * Male + ε

I understand that β1 denotes the effect of parental income on schooling for females, and β1+ β3 for males. So, for example, if β1 is significant, we can say that parent’s income is significantly associated with schooling of girls. Similarly, we test for the significance of β1+ β3, and conclude whether or not income has a significant association with schooling for boys.

However, what I don’t understand is how to interpret the coefficient on the interaction term, β3. What does β3 denote in this case and how should we interpret it? E.g., if we find that parental income is significantly associated with schooling for girls only, but the interaction term is insignificant, what would that mean?

I look forward to your help.

Answer 1

In this case, the interaction variable $\beta_3$, when significant, indicates that there is an interaction between income and gender. In other words, if you find the term to be statistically significant, it indicates that the linear effect of parental income on years of schooling (assuming this is modeled as a continuous variable in standard regression and not what might be the more appropriate Poisson regression), is different for boys than it is for girls.

To understand how to interpret $\beta_3$, consider the response function:

$E(Y)$ = $\beta_0+\beta_1X_1+\beta_2X_2+\beta_3X_1X_2$

When examining the response for boys, $X_2=1$, so the response function becomes:

$E(Y)$ = $\beta_0+\beta_1X_1+\beta_2+\beta_3X_1$, or equivalently by grouping terms:

$E(Y_{boys})$ = ($\beta_0+\beta_2)+(\beta_1+\beta_3)X_1$ for boys.

When examining the response for girls, $X_2=0$, so through similar algebra as we used above, the response function becomes:

$E(Y_{girls})$ = $\beta_0+\beta_1X_1$ for girls.

Taking the difference in the response functions between boys and girls we get:

$E(Y_{boys})$-$E(Y_{girls})$ = $\beta_0+\beta_1X_1+\beta_2+\beta_3X_1-\beta_0+\beta_1X_1$

=$\beta_2+\beta_3X_1$

So, you can see that $\beta_2$ represents how much greater (or smaller) the y-intercept or expected years of schooling is for boys than girls when there is no parental income. Similarly, $\beta_3$ can be interpreted as how much greater (or smaller if the sign is negative) the slope of the response function is for boys than girls.

In situations like this, I recommend graphing the two response functions on the same graph. Once you do this, the relationship between gender and incomes usually becomes obvious.

Lastly, if years of schooling is a count variable, you may want to consider using Poisson regression as Poisson tends to perform better than standard linear regression.

I hope this helps and best of luck to you!