Econometrics-I-11 - 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 11. Asymptotic Distribution Theory
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Preliminary     This and our class presentation will be a  moderately detailed sketch of these results.   More complete presentations appear in Chapter  4 of your text. Please read this chapter  thoroughly. We will develop the results that we  need as we proceed.  Also, (I believe) that this  topic is the most difficult conceptually in this  course, so do feel free to ask questions in  class.
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Asymptotics: Setting         Most modeling situations involve stochastic  regressors, nonlinear models or nonlinear estimation  techniques.  The number of exact statistical results,  such as expected value or true distribution, that can  be obtained in these cases is very low.  We rely,  instead, on approximate results that are based on  what we know about the behavior of certain statistics  in large samples.  Example from basic statistics:   What can we say about 1/     We know a lot about    .   What do we know about its reciprocal? x x
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Convergence Definitions, kinds of convergence as  n  grows large: 1.  To a constant;  example , the sample mean,  2.  To a random variable;  example , a  t  statistic  with  -1 degrees of freedom  x
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Convergence to a Constant Sequences and limits. Sequence of constants, indexed by n                           (n(n+1)/2 + 3n + 5) Ordinary limit : --------------------------    ?                                (n 2  + 2n + 1) (The use of the “leading term”) Convergence of a random variable .  What does it mean  for a random variable to converge to a constant?   Convergence of the variance to zero.  The random  variable converges to something that is not random.
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Convergence Results Convergence of a sequence of random variables to a constant -  convergence in mean square: Mean converges to a constant,  variance converges to zero.   (Far from the most general, but  definitely sufficient for our purposes.) A convergence theorem for sample moments.   Sample moments  converge in probability to their population counterparts. Generally the form of  The Law of Large Numbers . (Many forms; see  Appendix D in your text.)  Note the great generality of the preceding result.  (1/n) Σ i g(z i ) converges  to E[g(z i )]. 2 1 1 , [ ] , Var[ ]= / 0 n n i i n n n x x E x x n = = Σ = μ → μ σ
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Probability Limit θ ε → ∞ - θ ε = = θ θ n n n Let   be a constant,   be any positive value,  and n index the sequence.
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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.

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Econometrics-I-11 - Applied Econometrics William Greene...

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