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Wooldridge IE AISE SSM app e - APPENDIX E SOLUTIONS TO...

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This edition is intended for use outside of the U.S. only, with content that may be different from the U.S. Edition. This may not be resold, copied, or distributed without the prior consent of the publisher. 123 APPENDIX E SOLUTIONS TO PROBLEMS E.1 This follows directly from partitioned matrix multiplication in Appendix D. Write X = 1 2 n x x x # , X = ( 1 x 2 x n x ), and y = 1 2 n y y y # Therefore, X X = 1 n t t t = x x and X y = 1 n t t t = x y . An equivalent expression for ˆ β is 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 ˆ β = 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.
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