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appendixe_s - APPENDIX E SOLUTIONS TO PROBLEMS E.1 This...

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APPENDIX E SOLUTIONS TO PROBLEMS E.1 This follows directly from partitioned matrix multiplication in Appendix D. Write X = , X = ( 1 2 n x x x # 1 x 2 x n x ), and y = 1 2 n y y y # Therefore, X X = and X y = 1 n t t t = x x 1 n t t t = x y . An equivalent expression for is ˆ β = ˆ β 1 1 1 n t t t n = x x 1 1 n 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 + ˆ β 1 1 1 n t t t n = x x 1 1 n 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.3 (i) We use the placeholder feature of the OLS formulas. By definition, = ( Z Z ) -1 Z y = [( XA ) ( XA )] -1 ( XA ) y = [ A ( X X ) A ] -1 A X y = A -1 ( X X ) -1 ( A
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