ReferencesFINAL EXAM ECON 7801 SPRING 2001ECONOMICS DEPARTMENT, UNIVERSITY OF UTAHThis is a closed book exam but you may bring one sheet with formulas withyou; write your name on the formula sheet and submit it together with your exam.General formulas, please, not the answers to the questions in the class notes. (Thissame ruling will also apply to the field exam.)The first two questions are repeats from the Midterm, and they are obligatory foreveryone. Other than that you should pick and choose and do as many questions asyou can in the 90 minutes time.Problem 16. 2 points We are in the multiple regression model y = Xβ + εεε withintercept, i.e., X is such that there is a vector a with ι = Xa. Define the row vector¯x>=1nι>X, i.e., it has as its jth component the sample mean of the jth independentDate of exam Tuesday, May 1st, 9–10:30 am.12 U OF U ECONOMICSvariable. Using the normal equations X>y = X>Xˆβ, show that ¯y =¯x>ˆβ (i.e., theregression plane goes through the center of gravity of all data points).Answer. Premultiply the normal equation by a>to get ι>y − ι>Xˆβ = 0. Premultiply by 1/n toget the result. Problem 17. Assumeˆˆβ is the constrained least squares estimator subject to theconstraint Rβ = o, andˆβ is the unconstrained least squares estimator.• a. 1 point With the usual notation ˆy = Xˆβ andˆˆy = Xˆˆβ, show that(1) y =ˆˆy + (ˆy −ˆˆy) + ˆεPoint out these vectors in the reggeom simulation.Answer. In the reggeom-simulation, y is the pu rple line; Xˆˆβ is the red line starting at the origin,one could also call itˆˆy; X(ˆβ −ˆˆβ) = ˆy −ˆˆy is the light blue line, and ˆε is the green line which doesnot start at the origin. In other words: if one projects y on a plane, and also on a line in that plane,and then connects the footpoints of these two projections, one obtains a zig-zag line with two rightangles. • b. 4 points Show that in (1) the three vectorsˆˆy, ˆy −ˆˆy, and ˆε are orthogonal. Youare allowed to use, without proof, the following formula forˆˆβ:(2)ˆˆβ =ˆβ − (X>X)−1R>R(X>X)−1R>−1(Rˆβ − u).FINAL EXAM ECON 7801 SPRING 2001 3Answer. One has to verify that the scalar products of the three vectors on the right hand side of(1) are zero.ˆˆy>ˆε =ˆˆβ>X>ˆε = 0 and (ˆy −ˆˆy)>ˆε = (ˆβ −ˆˆβ)>X>ˆε = 0 follow from X>ˆε = o;geometrically on can simply say that ˆy andˆˆy are in the space spanned by the columns of X, andˆε is orthogonal to that space. Finally, using (2) forˆβ −ˆˆβ,ˆˆy>(ˆy −ˆˆy) =ˆˆβ>X>X(ˆβ −ˆˆβ) ==ˆˆβ>X>X(X>X)−1R>R(X>X)−1R>−1Rˆβ ==ˆˆβ>R>R(X>X)−1R>−1Rˆβ = 0becauseˆˆβ satisfies the constraint Rˆˆβ = o, henceˆˆβ>R>= o>. Problem 18. 3 points Explain the meanings of all the terms in t he following identity(3) ˆyi= (1 − hii)ˆyi(i) + hiiyiand use that equation to explain why hiiis called the “leverage” of the ith observation.Is every observation with high leverage also “influential” (in the sense that its removalwould greatly change the regression estimates)?Answer. ˆyiis the fitted value for the ith obser vation, i.e., it is the BLUE of ηi, of the expectedvalue of the ith observation. It is a weighted average of two quantities: the actual observationyi(which has ηias expec ted value), and ˆyi(i), which is the BLUE of ηibased on all the otherobservatio ns except the ith. The weight of the ith observation in this weighted average is called the4 U OF U ECONOMICS“leverage” of the ith observation. The sum of all leverages is always k, the number of parametersin the regression. If the leverage of one individual point is much greater than k/n, then th is pointhas much more influence on its own fitted value than one should expect just based on the numberof observations,Leverage is not the same as influence; if an observation has high leverage, but by accidentthe observed value yiis very close to ˆyi(i), then removal of this observation will not change theregression results much. Leverage is potential influence. Leverage does not depend on any of theobservatio ns, one only needs the X matrix to compute it. Problem 19. Prove the following facts about the diagonal elements of the so-called“hat matrix” H = X(X>X)−1X>, which has its name because Hy = ˆy, i.e., itputs the hat on y.• a. 1 point H is a projection matrix, i.e., it is symmetric and idempotent.Answer. Symmetry follows from the laws for the transposes of products: H>= (ABC)>=C>B>A>= H where A = X, B = (X>X)−1which is symmetric, and C = X>. IdempotencyX(X>X)−1X>X(X>X)−1X>= X(X>X)−1X>. • b. 1 point Prove that a symmetric idempoten t matrix is nonnegative definite.Answer. If H is symmetric and idempotent, then for arbitrary g, g>Hg = g>H>Hg = kHgk2≥0. But g>Hg ≥ 0 for all g is the criterion which makes H nonnegative definite. • c. 2 points Show that(4) 0 ≤ hii≤ 1FINAL EXAM ECON 7801 SPRING 2001 5Answer. If eiis the vector with a 1 on the ith place and zeros e verywhere else, then e>iHei= hii.From H nonnegative definite follows therefore that hii≥ 0. hii≤ 1 follows because I − H issymmetric and idempotent (and therefore nonnegative definite) as well: it is the projection on theorthogonal complement. • d. 2 points Show: the average value of the hiiisPhii/n = k/n, where k is thenumber of columns of X. (Hint: for this you must compute the trace tr H.)Answer. The average can be writt en as1ntr(H) =1ntr(X(X>X)−1X>) =1ntr(X>X(X>X)−1) =1ntr(Ik) =kn.Here we used tr BC = tr CB. • e. 1 point Show that1nιι>is a project ion matrix. Here ι is the n-vector of ones.• f . 2 points Show: If the regression has a constant term, then H −1nιι>is aprojection matrix.Answer. If ι, the vector of ones, is one of the columns of X (or a linear combination of thesecolumns), this means there is a vector a with ι = Xa. From this follows Hιι>= X(X>X)−1X>Xaι>=Xaι>= ιι>. One can use this to show that H −1nιι>is idemp otent: (H −1nιι>)(H −1nιι>) =HH − H1nιι>−1nιι>H +1nιι>1nιι>= H −1nιι>−1nιι>+1nιι>= H −1nιι>. • g. 1 point Show: If the regression has a constant term, then one can sharpeninequality (4) to 1/n ≤ hii≤ 1.6 U OF U ECONOMICSAnswer. H − ιι>/n is a projection matrix, therefore nonnegative definite, therefore its diagonalelements hii− 1/n are nonnegative. Problem 20. 3 points What are the main concepts used in modern