For what reason, why is $x_i^{\frac{1}{2}}$ applied to transform the usual regression to transform Weighted Least Regression?

Question

In the image below, the linear regression with heteroscedasticity got transformed into homoscedasticity by dividing the usual linear regression with $x_i^{\frac{1}{2}}$. As a result of transformation, it becomes a homoscedastic linear regression. I am wondering for what reason we can apply $x_i^{\frac{1}{2}}$ to the equation. Is that because it is just the way to transform the a heteroscedastic regression into a homoscedastic regression?

Answer 1

It's cause they've assumed that the conditional variance is a linear function of $x.$ This is expressed in the equation $\mathrm{Var}(\epsilon_i|x_i) = \sigma^2x_i.$ (And note it's a huge assumption for the purposes of example... not generic at all).

Then they divide by $\sqrt{x_i}$ so that the new noise $\epsilon_i'= \epsilon_i/\sqrt{x_i}$ has variance $\mathrm{Var}(\epsilon_i'|x_i) = \sigma^2.$ In other words, it has constant variance.

You could do the same thing if, say, the variance were quadratic in $x$: $\mathrm{Var}(\epsilon_i|x_i) = \sigma^2x_i^2.$ Then you'd divide through by $x_i$ rather than $\sqrt{x_i}.$