OLS variance under heteroscedasticity

Problem

Random sample \(\left( y_i,x_i \right)\) for \(i=1, \ldots ,n\)
Correctly specified linear regression model, \(y_i=x'_iβ+ε_i\) or \(y=Xβ+ε\) 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 \(Ψ=diag\left( σ_1^2, \ldots ,σ_n^2 \right)\) .
\(b_{OLS}={\left( X'X \right)}^{-1}X'y\)

Show that so

\[Var\left( b_{OLS} \mid X \right)=σ^2{\left( X'X \right)}^{-1}X'ΨX{\left( X'X \right)}^{-1}\]

Solution

\(b_{OLS}=β+{\left( X'X \right)}^{-1}X'ε\) so

\[Var\left( b_{OLS} \mid X \right)=Var\left( {\left( X'X \right)}^{-1}X'ε \mid X \right)\]

The transpose of \({\left( X'X \right)}^{-1}X'\) is \({X\left( X'X \right)}^{-1}\) ,

\[Var\left( {\left( X'X \right)}^{-1}X'ε \mid X \right)={\left( X'X \right)}^{-1}X'Var\left( ε \mid X \right){X\left( X'X \right)}^{-1}=\]

\[{\left( X'X \right)}^{-1}X'σ^2Ψ{X\left( X'X \right)}^{-1}=σ^2{\left( X'X \right)}^{-1}X'Ψ{X\left( X'X \right)}^{-1}\]