Weighted least squares

Summary

Setup

• The LRM with random sampling

\[y_i=β_1+β_2x_{i,2}+β_3x_{i,3}+…+β_kx_{i,k}+ε_i\]

• 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

\[y_i=β_1+β_2x_{i,2}+β_3x_{i,3}+ε_i\]

where

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

Divide both sides by \(x_{i,2}\) gives us the transformed model :

\[ \frac{y_i}{x_{i,2}}=β_1 \frac{1}{x_{i,2}}+β_2+β_3 \frac{x_{3i}}{x_{i,2}}+ \frac{ε_i}{x_{i,2}} \]

The errors in the transformed model, \( \frac{ε_i}{x_{i,2}}\) , satisfy all GM assumptions. In particular

\[Var\left(\frac{ε_i}{x_{i,2}}| x_i \right)= \frac{1}{x_{i,2}^2}Var\left( ε_i|x_i \right)= \frac{1}{x_{i,2}^2}σ^2x_{i,2}^2=σ^2\]

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