Weighted least squares, weighting

Problem

In the model \(y_i=β_1+β_2x_i+ε_i\) we have \(E\left( ε_i \mid x_i \right)=0\) and \(Var\left( ε_i|x_i \right)=\sqrt{x_i}σ^2\) . If you want to estimate the model using WLS you must estimate a transformed model where we divide both sides with …?

  1. \(\sqrt{x_i}\)
  2. \(\sqrt{x_i}σ^2\)
  3. \(x_i^{1/4}\)
  4. \(x_i\)
  5. \(x_i^2\)

Solution

In general, if \(Var\left( ε_i|x_i \right)=\) \(σ^2\) then you need to divide by the square root of an alien. In our case, you need to divide by the square root of \(\sqrt{x_i}\) . Using powers, you need to divide by

\[{\left( {\left( x_i \right)}^{1/2} \right)}^{1/2}=x_i^{1/4}\]