Problem 2. (Working through a simple linear model with heteroscedasticity, i.e. Var(e) is not constant.) Most of the results for linear regression we have learnt so far assumes homoscedasticity, i.e. the errors 6,- have a constant variance 02. In this exercise, we will work with a heteroscedastic simple linear model, i.e. a model where the error variance Var(e) changes with 3:. Consider the linear model: Y; = g + 313:,- + 67;, i: 1, . . . , n, where 6,- ~ N(0, 02m?) and 61, . . . ,6\" are independent. (a) Show that even for the above model, the least square estimates 61 and 30 are still given by: W and 30 = l7 315;, (thus, the same form you have seen before). i=1 i _ 31: (b) Show rigorously the following three properties regarding the sampling distribution of 5'1: i \" = ii at\