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 \(E\left( ε_i \mid x_i \right)=0\) .
• Heteroscedasticity so \(Var\left( ε_i \mid x_i \right)=σ^2\) no longer holds
• We say that we have a simple form of heteroscedasticity if

\[Var\left( ε_i \mid x_i \right)=σ^2σ_i^2\]

• where \(σ^2>0\) is an unknown constant and \(σ_i^2>0\) is known (conditionally on \(X\) ).
• Example,

\[Var\left( ε_i \mid x_i \right)=σ^2x_{i,2}^2\]

• Note: with heteroscedasticity, \(σ^2\) is no longer the common \(Var\left( ε_i \mid x_i \right)\) . \(σ^2\) is now an arbitrary constant.