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

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