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.