Econometrics-I-11 - Econometrics I Professor William Greene...

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Part 11: Asymptotic Distribution  Theory Econometrics I Professor William Greene Stern School of Business Department of Economics
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Part 11: Asymptotic Distribution  Theory Econometrics I Part 11 – Asymptotic  Distribution Theory ™    1/42
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Part 11: Asymptotic Distribution  Theory Preliminary This 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. ™   2/42
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Part 11: Asymptotic Distribution  Theory 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? ™    3/42 x x
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Part 11: Asymptotic Distribution  Theory Convergence Definitions, kinds of convergence as n grows large: 1. To a constant; example , the sample mean, converges to the population mean. 2. To a random variable; example , a t statistic with n -1 degrees of freedom converges to a standard normal random variable ™    4/42 x
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Part 11: Asymptotic Distribution  Theory Convergence to a Constant Sequences and limits. Sequence of constants, indexed by n (n(n+1)/2 + 3n + 5) Ordinary limit : --------------------------  ? (1/2) (n2 + 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. ™    5/42
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Part 11: Asymptotic Distribution  Theory 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. This is the “weak” law of large numbers.) Note the great generality of the preceding result. (1/n)Σig(zi) converges to E[g(zi)]. ™    6/42 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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Part 11: Asymptotic Distribution  Theory Probability Limit ™    7/42 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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