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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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 tt t = xx 1 n t = xy . An equivalent expression for is ˆ β = ˆ β 1 1 1 n t n = 1 1 n t ny = 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 n = 1 1 n t nu = 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 ) -1 A X y = A -1 ( X X ) -1 X y = A -1 . β ± ˆ β (ii) By definition of the fitted values, ˆ t y = and ˆ t x β t y ± = . Plugging z t and into the second equation gives t z β ± β ± t y ± = ( x t A )( A -1 ) = = ˆ β ˆ β t x
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 05/03/2009 for the course ECON 418 taught by Professor Breman during the Spring '08 term at University of Arizona- Tucson.

Page1 / 2

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online