The conditional expectation of squared residuals with homoscedasticity

Problem

Consider \(y_i=β_1+β_2x_i+ε_i\) where GM holds. Show that the conditional expectation of squared error terms does not depend on \(x_i\) .

Solution

In this case \(E\left( ε_i^2|x_i \right)=Var\left( ε_i|x_i \right)=σ^2\) which does not depend on \(x_i\) . Running a regression of \(e_i^2\) on \(x_i^2\) should result in an insignificant effect of \(x_i^2\) .