Consider the Simple Regression Model Yi = β₀ + β₁X_i $\begin{array} { l } 2 \\i\end{array}$

Essay

Consider the simple regression model Yi = β₀ + β₁X_i + u_i where X_i > 0 for all i, and the conditional variance is var(u_i
| X_i)= θX $\begin{array} { l } 2 \\i\end{array}$ where θ is a known constant with θ > 0.
(a)Write the weighted regression as $\bar Y$ _i = β₀
$\bar X$ _0i + β₁
$\bar X$ _1i + $\bar u$ _i. How would you construct $\bar Y$ _i, $\bar X$ _0i and $\bar X$ _1i?
(b)Prove that the variance of is $\bar u$ _i homoskedastic.
(c)Which coefficient is the intercept in the modified regression model? Which is the slope?
(d)When interpreting the regression results, which of the two equations should you use, the original or the modified model?