Properties of the weighted least squares estimator

Summary

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 \(y_i^*=y_i/σ_i\) , \(x_i^*=x_i/σ_i\) and \(ε_i^*=ε_i/σ_i\)
Transformed variables, vector form: \(y^*=Ψ^{-1/2}y\) , \(X^*=Ψ^{-1/2}X\) , \(ε^*=Ψ^{-1/2}ε\)
Transformed model: \(y_i^*={x_i^*}'β+ε_i^*\) or \(y^*=X^*β+ε^*\)
WLS estimator \(b_{WLS}={\left( \sum_{i=1}^{n}{ x_i^*{x_i^*}' } \right)}^{-1}\sum_{i=1}^{n}{ x_i^*y_i^* }\) or
In matrix form: \(b_{WLS}={\left( {X^*}'X^* \right)}^{-1}{X^*}'y^*={\left( X'Ψ^{-1}X \right)}^{-1}X'Ψ^{-1}y\)

Result: the transformed model \(y_i^*={x_i^*}'β+ε_i^*\) will satisfy the GM conditions :

\[E\left( ε_i^*|x_i^* \right)=0\]

\[Var\left( ε_i^*|x_i^* \right)=σ^2\]

\(b_{WLS}=β+{\left( X'Ψ^{-1}X \right)}^{-1}X'Ψ^{-1}ε\) is the WLS statistical formula
\(b_{WLS}\) is an unbiased estimator of \(β\)
\(b_{WLS}\) is BLUE