The WLS estimator is unbiased

Problem

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

Show that \(b_{WLS}\) is unbiased, \(E\left( b_{WLS} \right)=β\)

Solution

First find \(E\left( b_{WLS} \mid X \right)\) using \(b_{WLS}=β+{\left( X'Ψ^{-1}X \right)}^{-1}X'Ψ^{-1}ε\)

\[E\left( β+{\left( X'Ψ^{-1}X \right)}^{-1}X'Ψ^{-1}ε \mid X \right)=β+E\left( {\left( X'Ψ^{-1}X \right)}^{-1}X'Ψ^{-1}ε \mid X \right)=\]

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

Next, \(E\left( b_{WLS} \right)=E\left( E\left( b_{WLS} \mid X \right) \right)=E\left( β \right)=β\) by FWL.