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Econometrics-I-11

# The clt does not state that means of samples have

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Unformatted text preview: The CLT does not state that means of samples have normal distributions. &#152;&#152;&#152;™™ ™ 19/42 Part 11: Asymptotic Distribution A Central Limit Theorem &#152;&#152;&#152;™™ ™ 20/42 μ σ- μ → σ 1 n 2 d Lindeberg-Levy CLT (the simplest version of the CLT) If x ,..., x are a random sample from a population with finite mean and finite variance , then n(x ) N(0,1) Note, not the limitin = σ- μ → n d n g distribution of the mean, since the mean, itself, converges to a constant. A useful corollary: if plim s , and the other conditions are met, then n(x ) N(0,1) s Part 11: Asymptotic Distribution Lindeberg-Levy vs. Lindeberg-Feller Lindeberg-Levy assumes random sampling – observations have the same mean and same variance. Lindeberg-Feller allows variances to differ across observations, with some necessary assumptions about how they vary. Most econometric estimators require Lindeberg- Feller (and extensions such as Lyapunov). &#152;&#152;&#152;™™ ™ 21/42 Part 11: Asymptotic Distribution Order of a Sequence Order of a sequence ‘Little oh’ o(.). Sequence hn is o(n) (order less than n) iff n- hn 0. Example: hn = n1.4 is o(n1.5) since n-1.5 hn = 1 /n.1 0. ‘Big oh’ O(.). Sequence hn is O(n) iff n- hn a finite nonzero constant. Example 1: hn = (n2 + 2n + 1) is O(n2). Example 2: ixi2 is usually O(n1) since this is nthe mean of xi2 and the mean of xi2 generally converges to E[xi2], a finite constant. What if the sequence is a random variable? The order is in terms of the variance. Example: What is the order of the sequence in random sampling? Var[ ] = σ2/n which is O(1/n) &#152;&#152;&#152;™™ ™ 22/42 n x n x Part 11: Asymptotic Distribution Asymptotic Distribution An asymptotic distribution is a finite sample approximation to the true distribution of a random variable that is good for large samples, but not necessarily for small samples. Stabilizing transformation to obtain a limiting distribution. Multiply random variable xn by some power, a, of n such that the limiting distribution of naxn has a finite, nonzero variance. Example, has a limiting variance of zero, since the variance is σ2/n. But, the variance of √n is σ2. However, this does not stabilize the distribution because E[ ] = √ nμ. The stabilizing transformation would be &#152;&#152;&#152;™™ ™ 23/42 n x n x n n x n(x )- μ Part 11: Asymptotic Distribution Asymptotic Distribution Obtaining an asymptotic distribution from a limiting distribution Obtain the limiting distribution via a stabilizing transformation Assume the limiting distribution applies reasonably well in finite samples Invert the stabilizing transformation to obtain the asymptotic distribution Asymptotic normality of a distribution....
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The CLT does not state that means of samples have normal...

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