Square of a dummy

Problem

You consider the model

\(y_i=β_1+β_2x_{i,2}+γd_i+ε_i\)

but you suspect that the real model is non-linear in data. A friend suggest that you instead consider

\[y_i=β_1+β_2x_{i,2}+β_3x_{i,2}^2+γ_1d_i+γ_2d_i^2+ε_i\]

(adding squares of variables). Will it be possible to estimate this model using OLS?

Solution

No. Since \(d_i\) is zero or one, \(d_i^2=d_i\) for all observations. We cannot include two variables that are the same, we get perfect multicollinearity.