Transformed errors are Gauss-Markov

Problem

Random sample \(\left( y_i,x_i \right)\) for \(i=1, \ldots ,n\)
Correctly specified linear regression model, \(y_i=x'_iβ+ε_i\) where \(E\left( ε_i \mid x_i \right)=0\) .
\(Var\left( ε_i \mid x_i \right)=σ^2σ_i^2\) where \(σ^2\) is unknown and \(σ_i^2\) is known or in matrix form, \(Var\left( ε \mid X \right)=σ^2Ψ\) where \(Ψ=diag\left( σ_1^2, \ldots ,σ_n^2 \right)\) .
Transformed variables, vector form: \(y^*=Ψ^{-1/2}y\) , \(X^*=Ψ^{-1/2}X\) , \(ε^*=Ψ^{-1/2}ε\)
Transformed model: \(y^*=X^*β+ε^*\)

Show that \(ε^*=Ψ^{-1/2 }ε\) are GM errors

Solution

\[E\left( Ψ^{-1/2 }ε \mid X \right)=Ψ^{-1/2 }E\left( ε \mid X \right)=Ψ^{-1/2 }⋅0=0\]

\[Var\left( Ψ^{-1/2}ε \mid X \right)=Ψ^{-1/2}Var\left( ε \mid X \right)Ψ^{-1/2}=Ψ^{-1/2}σ^2ΨΨ^{-1/2}=σ^2Ψ^{-1/2}ΨΨ^{-1/2}\]

Since \(Ψ=Ψ^{1/2}Ψ^{1/2}\) we have

\[σ^2Ψ^{-1/2}ΨΨ^{-1/2}=σ^2Ψ^{-1/2}Ψ^{1/2}Ψ^{1/2}Ψ^{-1/2}=σ^2I⋅I=σ^2I\]