Weighted least squares

Summary

Setup

  • The LRM with random sampling

yi=β1+β2xi,2+β3xi,3++βkxi,k+εiyi=β1+β2xi,2+β3xi,3++βkxi,k+εi

  • All explanatory variables are exogenous.
  • The error terms are heteroscedastic.
  • The OLS estimator is unbiased and consistent but the OLS standard errors are inconsistent.

Example

yi=β1+β2xi,2+β3xi,3+εiyi=β1+β2xi,2+β3xi,3+εi

where

Var(εi|xi)=σ2x2i,2Var(εi|xi)=σ2x2i,2

Divide both sides by xi,2xi,2 gives us the transformed model :

yixi,2=β11xi,2+β2+β3x3ixi,2+εixi,2yixi,2=β11xi,2+β2+β3x3ixi,2+εixi,2

The errors in the transformed model, εixi,2εixi,2 , satisfy all GM assumptions. In particular

Var(εixi,2|xi)=1x2i,2Var(εi|xi)=1x2i,2σ2x2i,2=σ2Var(εixi,2|xi)=1x2i,2Var(εi|xi)=1x2i,2σ2x2i,2=σ2

Estimating the transformed model using OLS will give us efficient estimates of all parameters as well as consistent standard errors.