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2. [Derivation/Conceptual] Consider the multiple linear regression model Y = XB + E 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, 2 Inxn). Recall that we have the estimator B = (XTX)-1xTY. We showed that / is unbiased since E(B) = 3, and that Var(B) = 02 (X] X)-1.(a) The covariance matrix of the predictors Var() = (72(XTX)_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 1 $1 1 {122 X = _ 1 can What is Cov(Bo,B1) in simple linear regression in terms of 3:1,. .?,a:n [ Recall that we showed how to compute Va'r([30) and Varwl) from rst principles; so this part of the problem completes the story for the SLR parameters ] (b) Recall that for MLR, theAdiagonal elements of Var(,) = 02 (X TX )_1 give us the variances Var(o), Var(,6'1), . . . , Var(p), which are then the basis of constructing condence intervals for the parameters. In this part, we will show that E = (X TX )'1X T17 in fact minimizes these variances among all linear unbiased estimators (and thus gives us the most precise 01s for the Bj's). Follow these steps: 0 Consider constructing an alternative linear estimator for the parameters, let's write it as 5* = A? for a matrix of constants A. What condition must hold on A in order for 3* to be unbiased? 0 Dene D = A (XTX)_1XT. If 3* is unbiased, compute Var(*) in terms of the matrices D and X. o In order to minimize the diagonal elements of Var(/*), what condition must hold on D? Recall that a positive semidenite matrix must have all diagonal elements 2 0. Finally, what do we conclude about 3 = (XTX)_1XT17