Understanding interaction terms in a regression

Question

To my understanding interaction terms help if you believe the relationship between some terms undergoes a fundamental change.

I am working on this exercise in my book:

And

I computed a linear regression on the given data to produce the following table of information:

I see that the estimated value for the $x_2$ term is negative and relatively large. So this term has a significant impact on my model.

Now my trouble is to understand why we should or should not include an interaction term for this scenario. I need a nudge in the right direction on how to think about this.

Answer 1

It essentially allows for the effect of age to be different depending on gender. Definitely try adding one and see if the term is statistically significant. It may help to see the regression equation to understand why an interaction term is useful.

---

Without Interaction Term $$y = \beta_1 x_1 + \beta_2 x_2$$ Let $x_2$ (gender) = 0 $$y = \beta_1 x_1$$ We have every unit increase in age leads to a $\beta_1$ increase in y

Let $x_2$ (gender) = 1 $$y = \beta_1 x_1 + \beta_2$$ We have every unit increase in age leads to a $\beta_1$ increase in y

$\therefore$ age has same effect for both genders

---

With Interaction Term $$y = \beta_1 x_1 + \beta_2 x_2 + \beta_3 x_1 x_2$$ Let $x_2$ (gender) = 0 $$y = \beta_1 x_1$$ We have every unit increase in age leads to a $\beta_1$ increase in y

Let $x_2$ (gender) = 1 $$y = \beta_1 x_1 + \beta_2 + \beta_3 x_1$$ $$y = (\beta_1 + \beta_3) x_1 + \beta_2$$ We have every unit increase in age leads to a $\beta_1 + \beta_3$ increase in y

$\therefore$ age has different effect for each gender