Heteroscedasticity

Problem

In the LRM, \(y_i=β_1+β_2x_i+ε_i\) , suppose that the errors are heteroscedastic, \(Var\left( ε_i \mid x_i \right)=σ^2x_i^2\) for \(i=1, \ldots ,n\) . Define \(ν_i=ε_i/x_i\) for \(i=1, \ldots ,n\) . Show that \(ν_i\) are homoscedastic and \(Var\left( ν_i \mid x_i \right)=σ^2\) for \(i=1, \ldots ,n\) .

Hint: \(Var\left( a+bY|X \right)=b^2Var(Y|X)\)