Robust standard errors

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 where \(Var\left( ε_i \mid x_i \right)=σ_i^2\) is unknown.
• The OLS estimator is unbiased and consistent but \(Var\left( b \mid X \right)=σ^2{\left( X'X \right)}^{-1}\) no longer holds.
• The variance of the OLS estimator under heteroscedasticity:

\[Var\left( b|X \right)={\left( X'X \right)}^{-1}X'ΨX{\left( X'X \right)}^{-1}\]

• Problem: \(Ψ\) contains \(n\) unknown parameters: no hope of finding a consistent estimator of \(Ψ\) without making assumptions about \(σ_i^2\) (See FGLS).
• Definition

\[{\hat{Ψ}}_W=diag\left( e_1^2, \ldots ,e_n^2 \right)\]

• Result

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

• is a consistent estimator of \(Var\left( b|X \right)\) . The square roots of the diagonal elements of \(\widehat{Var}\left( b|X \right)\) are called White’s standard errors or robust standard errors.