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271 APPENDIX E

SOLUTIONS TO PROBLEMS

E.1 This follows directly from partitioned matrix multiplication in Appendix D. Write

X = 12n ?? ? ? ? ? ???x x x , X ' = (1'x 2'x n 'x ), and y = 12n ?? ? ? ? ? ???

y y y

Therefore, X 'X = 1n t t t ='∑x x and X 'y = 1n

t t t ='∑x y . An equivalent expression for ?β is

?β = 1

11n t t t n --=??' ???∑x x 11n t t t n y -=??' ???

∑x which, when we plug in y t = x t β + u t for each t and do some algebra, can be written as

?β= β + 111n t t t n --=??' ???∑x x 11n t t t n u -=??' ???

∑x . As shown in Section E.4, this expression is the basis for the asymptotic analysis of OLS using matrices.

E.2 (i) Following the hint, we have SSR(b ) = (y – Xb )'(y – Xb ) = [?u

+ X (?β – b )]'[ ?u + X (?β – b )] = ?u '?u + ?u 'X (?β – b ) + (?β – b )'X '?u + (?β – b )'X 'X (?β – b ). But by the first order conditions for OLS, X '?u

= 0, and so (X '?u )' = ?u 'X = 0. But then SSR(b ) = ?u '?u + (?β – b )'X 'X (?β – b ), which is what we wanted to show.

(ii) If X has a rank k then X 'X is positive definite, which implies that (?β

– b ) 'X 'X (?β – b ) > 0 for all b ≠ ?β

. The term ?u '?u does not depend on b , and so SSR(b ) – SSR(?β) = (?β– b ) 'X 'X (?β

– b ) > 0 for b ≠?β.

E.3 (i) We use the placeholder feature of the OLS formulas. By definition, β = (Z 'Z )-1Z 'y =

[(XA )' (XA )]-1(XA )'y = [A '(X 'X )A ]-1A 'X 'y = A -1(X 'X )-1(A ')-1A 'X 'y = A -1(X 'X )-1X 'y = A -1?β

.

(ii) By definition of the fitted values, ?t y = ?t x β and t y = t

z β. Plugging z t and β into the second equation gives t y = (x t A )(A -1?β) = ?t x β = ?t

y .

(iii) The estimated variance matrix from the regression of y and Z is 2σ(Z 'Z )-1 where 2σ is the error variance estimate from this regression. From part (ii), the fitted values from the two

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