Econometrics-I-11

# Note not the limitin σ μ n d n g distribution of

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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

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Part 11: Asymptotic Distribution  Theory 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). ™  21/42
Part 11: Asymptotic Distribution  Theory 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 nthe 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) ™  22/42 n x n x

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Part 11: Asymptotic Distribution  Theory 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 ™  23/42 n x n x n n x n(x ) - μ
Part 11: Asymptotic Distribution  Theory 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. ™  24/42 d a 2 a 2 a 2 2 n(x ) / N[0,1] Assume holds in finite samples.  Then, n(x ) N[0, ]     (x ) N[0, / n]            x N[ , / n] Asymptotic distribution. / n  the asymptotic variance.   - μ σ → - μ → σ - μ → σ → μ σ σ =

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Part 11: Asymptotic Distribution  Theory Asymptotic Efficiency p Comparison of asymptotic variances p How to compare consistent estimators? If both converge to constants, both variances go to zero.
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