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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

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

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

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

Mataunited17
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1 Answers1

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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 $X_{3,i} = 1$

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paqmo
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    @pagma thanks! Could you explain me more how you derived that model? How did you set it up to end up with that? – Mataunited17 Oct 31 '16 at 18:05
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    For $X_3 = 0$, $Y_i = \beta_0 + \beta_1 X_{1,i} + \beta_1 X_{2,i}$. For $X_3 = 1$, $Y_i = (\beta_0 * \beta_3) + (\beta_1 + \beta_4) * X_{1,i} + (\beta_2 + \beta_5) * X_{2,i}$. So in this case, $a_0 = (\beta_0 * \beta_3)$, $a_1 X_{1,i} = (\beta_1 + \beta_4) * X_{1,i}$, and $a_2 X_{2,i} = (\beta_2 + \beta_5) * X_{2,i}$. – paqmo Oct 31 '16 at 18:23
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    @paqma Hey again! What does is "the effect of" mean? (referring back to your first post) – Mataunited17 Nov 04 '16 at 16:45
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    The unique variation of X associated with $Y$ (net of other variables). Standard interpretation of regression coefficients. How much of an increase in $Y$ is associated with a unit increase in $X$. – paqmo Nov 04 '16 at 17:01