Non-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 \(E\left( ε_i \mid x_i \right)=0\) .
• We say that we have a non-simple form of heteroscedasticity if

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

• where \(σ_i^2\) is unknown .
• In some cases, it is possible to estimate each \(σ_i^2\) for \(i=1,..,n\) . \({\hat{σ}}_i^2\) is then an estimator of \(σ_i^2\) .
• Definition, \(\hat{Ψ}\)

\[\hat{Ψ}=diag\left(  {\hat{σ}}_1^2, \ldots , {\hat{σ}}_n^2 \right)\]

• Definition: the Feasible Weighted Least Squares (FGLS) estimator is defined as

\[b_{FWLS}={\left( X'{\hat{Ψ}}^{-1}X \right)}^{-1}X'{\hat{Ψ}}^{-1}y\]

• This is also called the EGLS estimator, where E stands for estimated.
• The properties of the FGLS estimator is more complicated and, in general, \(E\left( b_{FWLS} \right)\) and \(Var\left( b_{WLS}|X \right)\) cannot be found. The FGLS estimator is in general not unbiased and not BLUE.
• However, the FWLS estimator is often consistent and as an estimator if the variance of \(b_{FWLS}\) we use

\[\widehat{Var }\left( b_{FWLS}|X^* \right)={\left( X'{\hat{Ψ}}^{-1}X \right)}^{-1}\]

• Example.
• Suppose that

\[Var\left( ε_i \mid x_i \right)=σ_i^2=exp \left( x'_iγ \right)\]

• \(γ\) is a \(k×1\) vector of unknown parameters.
• \(γ\) can be estimated by running the auxiliary regression

\[log e_i^2=x'_iγ+error term\]

• We then use

\[{\hat{σ}}_i^2=exp \left( x'_i\hat{γ} \right)\]

• Under mild conditions, \(\hat{γ}\) will be a consistent estimator of \(γ\) , \(b_{FWLS}\) a consistent estimator of \(β\) which is asymptotically more efficient than the OLS estimator and the formula for the estimated variance will be asymptotically correct.