Solved – Combining two linear regression model into a single linear model using covariates

Question

Let's say we have two very similar models:

$Y_{i}$ =
$\beta_{0}+$
$\beta_{1}X_{1,i}
+\beta_{2}X_{2,i}+e_{i}$

and

$Y_{i}$ =
$\alpha_{0}$ +
$\alpha_{1}X_{1,i}
+\alpha_{2}X_{2,i}+e_{i}$

Let's say the first model is for men, and the second for women.

Mathematically, how do I combine the two linear regression models together? Do I multiply or add?

Additional information: The data sets are derived from Hamermesh & Biddle's (1994) paper of "Beauty and the labor market". And we divide the dummary variables into

$X_{1,i} = {above average looking}$

$X_{2,i} = {below average looking}$

$X_{3,i} = {female}$

Answer 1

It sounds like you want a fully interacted model.

In which case, you interact all the terms with the dummy variable:

$Y_i = \beta_0 + \beta_1 X_{1,i} + \beta_2 X_{2,i} + \beta_3 X_{3,i} + \beta_4 (X_{1,i}*X_{3,i}) + \beta_5 (X_{2,i}*X_{3,i}) + \epsilon_i$

$\beta_1$ and $\beta_2$ are the effects of $X_1$ and $X_2$ when $X_{3,i} = 0$

$\beta_1 + \beta_4$ is the effect of $X_1$ when $X_{3,i} = 1$

$\beta_2 + \beta_5$ is the effect of $X_2$ when $X_{3,i} = 1$