Interactive dummy variables

Summary

Main points:

• In a regression model, an interactive dummy variable is a product of a dummy variable and an explanatory variable, $d_ix_{i,j}, i=1,…,n$ for any $j=2,…,k$ . $d_ix_{i,j}$ can take the value 0 or $x_{i,j}$ .
• We can add an interactive dummy variable to our regression model:

$$y_i=β_1+β_2x_{i,2}+β_3x_{i,3}+…+β_kx_{i,k}+γd_ix_{i,2}+ε_i i=1,…,n$$

• The marginal effect of $x_2$ of individuals coded as $d_i=0$ will be $β_2$ .
• The marginal effect of $x_2$ of individuals coded as $d_i=1$ will be $β_2+γ$ . $γ$ is the extra marginal effect that group 1 observations have compared to group 0 observations.
• You can have many interactive dummy variables in a LRM, for example

$$y_i=β_1+β_2x_{i,2}+β_3x_{i,3}+…+β_kx_{i,k}+γ_1d_ix_{i,2}+γ_2d_ix_{i,3}+ε_i i=1,…,n$$

• You can combine dummy variables and interactive dummy variables in the same regression, for example

$$y_i=β_1+β_2x_{i,2}+β_3x_{i,3}+…+β_kx_{i,k}+γ_1d_i+γ_2d_ix_{i,2}+ε_i i=1,…,n$$