Econometrics-I-12

# Econometrics-I-12 - Applied Econometrics William Greene...

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Applied Econometrics William Greene Department of Economics Stern School of Business

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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?

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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 ÷ ÷ β β29( β29 = 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 - = - - =  ÷ ÷     ε ε ÷  ÷ ÷    ∑ ∑ = β29( β29 β29( β29 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 - - - = - - -    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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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
= ÷ 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. Then,   =   = an observation in a random sample, with constant variance matrix and mean vector 0. 1

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## This note was uploaded on 11/23/2011 for the course ECON B30.3351 taught by Professor Professorw.greene during the Spring '10 term at NYU.

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Econometrics-I-12 - Applied Econometrics William Greene...

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