The transformed model under simple form of heteroscedasticity in marix form

Summary

• Setup
• 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
• Definition:

\[Ψ=diag\left( σ_1^2, \ldots ,σ_n^2 \right)\]

• \(Ψ\) is a diagonal matrix with elements \(σ_1^2, \ldots ,σ_n^2\) on the diagonal.
• Result:

\[Var\left( ε \mid X \right)=σ^2Ψ\]

• where \(σ^2\) is unknown while \(Ψ\) is known (conditionally on \(X\) ).
• Definition:

\[Ψ^{1/2}=diag\left( σ_1, \ldots ,σ_n \right)\]

• where \(σ_i=\sqrt{σ_i^2}\) , the standard deviation of the known part of the error term \(ε_i\) .
• Result:

\[Ψ^{1/2}Ψ^{1/2}=Ψ\]

• \(Ψ^{1/2}\) has full rank (invertible)
• Definition: the transformed regression in vector form \(\)

\[Ψ^{-1/2}y=Ψ^{-1/2}Xβ+Ψ^{-1/2}ε\]

• where \(Ψ^{-1/2}\) is the inverse of \(Ψ^{1/2}, Ψ^{-1/2}Ψ^{1/2}=I\) .
• Definition: transformed variables and transformed error terms in vector form

\[y^*=Ψ^{-1/2}y\]

\[X^*=Ψ^{-1/2}X\]

\[ε^*=Ψ^{-1/2}ε\]

• Result: the transformed regression model can be written as

\[y^*=X^*β+ε^*\]