Weighted least squares, regression

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 \(Var\left( ε_i|x_i \right)=x_i^2σ^2\) . If you want to estimate the model using WLS, which regression should you run?

  1. \(y_i/x_i\) on \(x_i\) with intercept
  2. \(y_i/x_i\) on \(x_i\) without intercept
  3. \(y_i/x_i\) on \(1/x_i\) with intercept
  4. \(y_i/x_i\) on \(1/x_i\) without intercept
  5. \(y_i/x_i^2\) on \(1/x_i^2\) och \(1/x_i\) without intercept

Solution

You need to divide by the square root of \(x_i^2\) or \(x_i\) (assuming \(x_i≥0\) ). The transformed model is

\[y_i/x_i=β_1/x_i+β_2x_i/x_i+ε_i/x_i\]

or

\[y_i/x_i=β_1/x_i+β_2+ε_i/x_i\]

The correct answer is therefore c.