(Solved): 11 June 2022, 2 hours 1. Consider the regression model Y = Bo + BX + , i = 1, ..., n ...

11 June 2022, 2 hours 1. Consider the regression model Y? = Bo + B?X? + €¡, i = 1, ..., n. n n Let X = n ¹X?, Y = n-¹Yi, Bo and 3? be the least squares estimators of Bo and i=1 i=1 3?, respectively, Y? = Bo + B?X; be the fitted values, and e; = Y; - ?; be the residuals. We have here that 72 B?=Sxy/Sxx = a;Y?, Yis i=1 n where Sxy = (X? – X)(Y? –? ?), Sxx = Ê(X¡ – X)², and a¡ = (X¡ — Ñ)/Sxx. - i=1 i=1 n Note that ??; = 0 and ¹2a;X; = 1. i=1 i=1 n 1.a. Show that ?e = 0. (15 pts) i=1 1.b. Show that the estimators and ? are unbiased. (15 pts) n 1.c. Show that S² = (n ? 2)-¹?e? is an unbiased estimator of o². (15 pts) i=1 1.d. Calculate Var(B) and Var(8?). (10 pts)