2. [Derivation/Conceptual] Consider the multiple linear regression model Y = XBFE where Y is the n x 1 column vector of responses, X is the n x (p + 1) matrix for the predictors (with intercept), and"e ~ MVN(0, o' Inx). Recall that we have the estimator "8 = (XTX)-1XTY. We showed that B is unbiased since E(B) = B, and that Var(B) 02 ( XT X ) -1.(a) The covariance matrix of the predictors Var(A) - o'(X"X)-1 shows that the estimates of the regression parameters are often correlated. Using this fact, let's look back at the simple linear regression case, where we can write X - What is Cou(Bo, 1) in simple linear regression in terms of $1, ..., I,? [ Recall that we showed how to compute Var(fo) and Var(81) from first principles; so this part of the problem completes the story for the SLR parameters. | (b) Recall that for MLR, the diagonal elements of Var(A) - o'(XX)-1 give us the variances Var(Bo), Var(Bi),..., Var(Bp), which are then the basis of constructing confidence intervals for the parameters. In this part, we will show that 8 - (XTX)-1XTY in fact minimizes those variances among all linear unbiased estimators (and thus gives us the most precise CIs for the By's). Follow these steps: . Consider constructing an alternative linear estimator for the parameters, let's write it as (* - AY for a matrix of constants A. What condition must hold on A in order for 8* to be unbiased? . Define D - A - (X"X) 1X". If f* is unbiased, compute Var(#*) in terms of the matrices D and X. . In order to minimize the diagonal elements of Var((*), what condition must hold on D? Recall that a positive semidefinite matrix must have all diagonal elements 2 0. Finally, what do we conclude about 8 - (XX]-1XTY