Then an observation in a random sample with constant

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Then,   =    = an observation in a random sample, with constant variance matrix and mean vector 0. 1  converges to its expectat n x z x z ion by the law of large numbers.
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Part 12: Asymptotics for the Regression Model Consistency of s2 ™  7/38 = = - - - - = - ÷ ÷ ÷ 2 2 1 1 n 1 s n K n K n K n n 1 n K 1 1 1 plims plim plim ( ) n n n 1 1 1 1 plim  plim plim ( ) plim n n n n 1 plim n What must be a -1 -1 -1 e'e 'M 'M 'M ' 'X X'X X' ' 'X X'X X' ' 0'Q 0 ε ε = ε ε ε ε = ε ε - ε ε = ε ε ε ε = ε ε - ε = σ 2 2 1 ssumed to claim plim =  E[ ] ? n ' ε ε
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Part 12: Asymptotics for the Regression Model Asymptotic Distribution ™  8/38 - = = + × ε ÷ ÷ 1 n i i i 1 1 1 n n The limiting behavior of   is the same as  that of the statistic that results when the  moment matrix is replaced by its limit. We examine the behavior of the modified  b X'X x b β - = + × ε ÷ n 1 i i i 1 sum 1 n Q x β
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Part 12: Asymptotics for the Regression Model Asymptotics ™  9/38 - = + × ε ÷ n 1 i i i 1 1 n What is the mean of this random vector? What is its variance? Do they 'converge' to something?  We use     this method to find the probability limit. What is the asymptotic distribu Q x β tion?
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Part 12: Asymptotics for the Regression Model Asymptotic Distributions p Finding the asymptotic distribution p b β in probability. How to describe the distribution? n Has no ‘limiting’ distribution p Variance  0; it is O(1/n) p Stabilize the variance? Var[n b ] ~ σ2Q-1 is O(1) p But, E[n b ]= n β which diverges n n ( b - β )  a random variable with finite mean and variance. (stabilizing transformation) n b apx. β +1/ n times that random variable ™  10/38
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Part 12: Asymptotics for the Regression Model Limiting Distribution n ( b - β ) = n ( X’X )-1 X’ε =  n ( X’X /n)-1( X’ε /n) Limiting behavior is the same as that of  n Q -1( X’ε /n) Q is a fixed matrix. Behavior depends on the random vector  n ( X’ε /n) ™  11/38
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Part 12: Asymptotics for the Regression Model Limiting Normality ™  12/38 n n i i i i 1 i 1 n i i 1 1 1 1 n n n n 1 Mean of a sample.  n    Independent observations.    Mean converges to zero (plim (1/n)  already assumed n a candidate for the Lindberg-Feller Central L = = = = ε = = = X' x w w ε w X' = 0 ε w =  2 2 i i i 2 2 imit Theorem.    Variance of each term (| ) is  '.  Variance of   /n.    Var n Based on the CLT,  n N[ , ] σ → σ → σ → σ d x x x Q w Q w 0 Q
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Part 12: Asymptotics for the Regression Model Asymptotic Distribution ™  13/38 1 -1 2 -1 -1 2 -1 2 -1 2 -1 Limiting distribution of  ' n( ) n n n is the same as that of  .
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