STAT 3008:Applied Regression Analysis 2021-22 Term 1 Assignment #2 (Problem #4c Revised) Due: October 20", 2021 (Wed) at 11:59pm This assignment covers material from Section 2.5 and 3.7 of the lecture notes. * * Please submit the hardcopy of the R-code and R-outputs for Problem #3 and Problem #4. You need to show your calculation in details order to obtain full scores. Please present your answers in at least 4 significant figures Note that the solutions will be available on Oct 22" (Fri) at 5pm, as the Mid-term exam will be on Oct 28. No late assignment will be accepted after the solutions are posted. Problem #1 [18 points]: Consider the multiple linear regression: nx1 Y = X P te , with E(e) =0,x, and Var(e) = o'I, nx(p+1) (p+1)x1 nxl (a) [10 points] Based on the fact that the OLS estimates P = (X'X) 'X' Y, show that E(Y'Y) = B'X'XP +(p+1)0- (b) [8 points] Based on the result from part (a), and the fact that E(RSS) = E(e'e) = ...=0' (n -p -1) , show that |EE( )? ) = EE(? )+ EE(@?) Problem #2 [18 points]: Consider a multiple linear regression with n=3 and p=0 (i.e. no EV!): N 3 0 X = Y = E(Y) = -1 Var( Y) = 0 (a) [6 points] Compute (X'X)" and show that (X'X) "X'Y gives y = (y, + yz + 3)/3. (b) [6 points] Compute H =X(X'X)"'X' and express Y'(I-H) Y in terms of yu, yz and y;. What is the value of E(Y'(I-H)Y)? (c) [6 points] Based on the fact that E(Y'AY)=E[tr(Y'AY)]=tr[AE(YY')], compute the value of E(Y' (I-H) Y). Problem #3 [24 points]: Consider the dataset Htwt from the alr4 library. Suppose that we want to fit a simple linear regression with wt as the response variable (i.e. y