(a) The covariance matrix of the predictors Var(B) = 02(X"X)- 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 X2 X = What is Cov(Bo, B1) in simple linear regression in terms of x1, ..., In? [ Recall that we showed how to compute Var(Bo) and Var(B1) 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(B) = 02(X X)- give us the variances Var(Bo), Var(B1), ..., Var(Bp), which are then the basis of constructing confidence intervals for the parameters. In this part, we will show that B = (XTX)-1XTY in fact minimizes these variances among all linear unbiased estimators (and thus gives us the most precise CIs for the Bj'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 (* to be unbiased? . Define D = A - (XTX)-1XT. If * is unbiased, compute Var(3*) in terms of the matrices D and X. . In order to minimize the diagonal elements of Var(8*), 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 3 = (XTX)-1XTY