Weighted least squares, properties

Problem

In the model \(y_i=β_1+β_2x_i+ε_i\) GM holds. However, you believe, incorrectly, that \(Var\left( ε_i|x_i \right)=x_iσ^2\) and you run

\[ \frac{y_i}{\sqrt{x_i}}= \frac{β_1}{\sqrt{x_i}}+β_2\sqrt{x_i}+error term\]

  1. Is the WLS estimator unbiased?
  2. Is the WLS estimator consistent?
  3. Is the WLS estimator efficient?
  4. Are the standard errors correct?

Solution

Your transformed model will now be heteroscedastic. Exogeneity will still hold so a,b are true while c,d are false.