Consider the Simple Regression Model $Y _ { i } = \beta _ { 0 } + \beta _ { 1 } X _ { i } + u _ { i }$

Essay

Consider the simple regression model $Y _ { i } = \beta _ { 0 } + \beta _ { 1 } X _ { i } + u _ { i }$ where $X _ { \mathrm { i } } > 0$ for all $i$ , and the conditional variance is $\operatorname { var } \left( u _ { i } \mid X _ { i } \right) = \theta X _ { i } ^ { 2 }$ where $\theta$ is a known constant with $\theta > 0$ . (a) Write the weighted regression as $\tilde { Y } _ { i } = \beta _ { 0 } \tilde { X } _ { 0 i } + \beta _ { 1 } \tilde { X } _ { 1 i } + \tilde { u } _ { i }$ . How would you construct $\tilde { Y } _ { i }$ , $\tilde { X } _ { 0 i }$ and $\tilde { X } _ { 1 i } ?$