The transformed model under simple form of heteroscedasticity

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: the transformed regression model for \(i=1, \ldots ,n \)

\[ \frac{y_i}{σ_i}= \frac{x'_iβ}{σ_i}+ \frac{ε_i}{σ_i}\]

• Definition: transformed variables and transformed error terms

\[y_i^*=y_i/σ_i\]

\[x_i^*=x_i/σ_i=\left( 1/σ_i,x_{i,2}/σ_i, \ldots ,x_{i,k}/σ_i \right)\]

\[ε_i^*=ε_i/σ_i\]

• Result: the transformed regression model can be written as

\[y_i^*={x_i^*}'β+ε_i^*\]