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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...

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    $\begingroup$ Hint: what would the columns in the design matrix look like for the models? $\endgroup$
    – mdewey
    Oct 22, 2023 at 15:16

1 Answer 1

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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 I assume that there are only male and female (somewhat outdated to be honest).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.

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