# Interaction with dummies - 2 distinct models

What exactly is the difference between those two models:

model 1:

$$Income_i = \beta_0 + \beta_1 \text{female}_i + \beta_2 \text{experience}_i + \beta_3 \text{female}_i \cdot \text{experience}_i + u_i$$

vs.

model 2:

$$Income_i = \beta_0 + \beta_1 \text{female}_i + \beta_2 \text{male}_i \cdot \text{experience}_i + \beta_3 \text{female}_i \cdot \text{experience}_i + u_i$$

where for simplicity, let's suppose experience is continuous and male/female are indicating the gender (dummy variables). I am familiar with model 1. This is basically what some statistical software are doing as default (e.g. R with *-operator). But how should we think about model 2? I first thought it was some sort of dummy variable trap, because both male and female dummies enter the regression, but it turned out to be a completely fine model that is quite common. Can you explain the intuition behind model 2? I think it somehow tries to split the effect of experience into its gender-components...

• Hint: what would the columns in the design matrix look like for the models? Oct 22, 2023 at 15:16

Let $$Y$$ be income, $$X_1$$ be the indicator for female and $$X_2$$ be experience. The indicator for male is $$1-X_2$$. I assume that there are only male and female (somewhat outdated to be honest). Then model 2 is $$Y=\beta_0+\beta_1X_1+\beta_2(1-X_1)X_2+\beta_3X_1X_2+U=\beta_0+\beta_1X_1+\beta_2X_2+(\beta_3-\beta_2)X_1X_2+U.$$ This is the same as model 1, only what is called $$\beta_3$$ in model 1 is $$\beta_3^*=\beta_3-\beta_2$$ in model 2. The two models are mathematically equivalent. The only difference regards the interpretation of the parameters. Model 2 separates the impact of experience for male and female. Model 1 takes male as baseline and $$\beta_3^*$$ is about how experience plays out differently for female compared to male.