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Unformatted text preview: Applied Econometrics William Greene Department of Economics Stern School of Business Applied Econometrics 12. Asymptotics for the Least Squares Estimator in the Classical Regression Model Setting The least squares estimator is ( X X )1 X ′ y = ( X X )1 Σ i x i y i = β + ( X X )1 Σ i x i ε i So, it is a constant vector plus a sum of random variables. Our ‘finite sample’ results established the behavior of the sum according to the rules of statistics. The question for the present is how does this sum of random variables behave in large samples? Well Behaved Regressors A crucial assumption: Convergence of the moment matrix X ′ X /n to a positive definite matrix of finite elements, Q What kind of data will satisfy this assumption? What won’t? Does stochastic vs. nonstochastic matter? Various conditions for “well behaved X ” Probability Limit = = = + × ε × ε × ε ∑ ∑ 1 n i i i 1 1 n i i i i i 1 We use 'convergence in mean square. Adequate for almost all problems, not adequate for some time series problems. 1 1 n n 1 1 1 ( ' ' n n n b X'X x b  b  X'X x x β β29( β29 = = = ε ε ∑ ∑ ∑ 1 n i 1 1 1 n i i j j 2 i 1 1 n 1 1 1 ' n n n In E[( '  ] in the double sum, terms with unequal subscripts have expectation zero. E[( ' n j=1 X'X X'X x x X'X b  b  X b  b  = β29( β29 β29( β29 = ε σ σ = = ∑ 1 1 n 2 i j i 2 i 1 1 1 1 2 2 1 1 1  ] 'E[  ] n n n 1 1 1 1 n n n n n n X X'X x x X X'X X'X X'X X'X X'X = Mean Square Convergence E[ b  X ]= β for any X. Var[ b  X ] 0 for any specific X b converges in mean square to β Probability Limit = = = + × ε = × ε = = = = ∑ ∑ 1 n i i i 1 1 1 n i i i 1 1 1 1 1 n n 1 1 1 1 n n n n 1 1 Plim( ) plim n n 1 1 1 plim plim plim n n n b X'X x b  X'X x X'X X' b  X'X X' X'X X' X'X β β ε β ε ε = = 1 1 1 plim n 1...
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This note was uploaded on 06/20/2011 for the course ECON 803 taught by Professor Pp during the Spring '11 term at Thammasat University.
 Spring '11
 PP
 Econometrics

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