Econometrics-I-12 - Econometrics I Professor William Greene...

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Part 12: Asymptotics for the Regression Model Econometrics I Professor William Greene Stern School of Business Department of Economics
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Part 12: Asymptotics for the Regression Model Econometrics I Part 12 – Asymptotics for  the Regression Model ™  1/38
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Part 12: Asymptotics for the Regression Model Setting The least squares estimator is ( XX )-1 Xy = ( XX )-1i x iyi = + ( XX )-1i x iεi 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? ™  2/38
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Part 12: Asymptotics for the Regression Model 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 ™  3/38
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Part 12: Asymptotics for the Regression Model Probability Limit ™  4/38 - = - = = + × ε ÷ ÷ × ε × ε ÷ ÷ 1 n i i i 1 1 n i i i i i 1 i 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 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 =
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Part 12: Asymptotics for the Regression Model Mean Square Convergence E[ b | X ]= β for any X. Var[ b | X ]0 for any specific well behaved X b converges in mean square to β ™  5/38
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Part 12: Asymptotics for the Regression Model Crucial Assumption of the Model ™  6/38 = ÷ ε ε i i 1 What must be assumed to get plim ? n (1)   = a random vector with finite means and variance and identical distributions. (2)   =  a random variable with a constant distribution with finite mean X' 0 x = ε ε ε i i i i i i n i i 1  and variance and E[ ]= 0 (3)   and   statistically independent.
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