slides_10_asymptotic

# The important conclusion is that n is more robust but

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The important conclusion is that ̃ n is more robust but, if the Poisson distribution is correct, it can be much less efficient (asymptotically) than ̂ n . Generally in econometrics there is a tradeoff between robustness (consistency) and asymptotic efficiency. 69

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The fact that ̃ n has tractable and desirable finite-sample properties – for example, it is the BLUE of when we consider only linear functions of Y i : i 1,..., n – is particular to this example. In more complex environments, no estimators will have tractable finite-sample properties. 70
DEFINITION : Let Θ p be a parameter vector, and let E be a class of n -asymptotically normal estimators of . In other words, each member of E is a sequence ̂ n : n 1,2,... such that n ̂ n is asymptotically normal for all . (Sometimes E is called a CAN class, short for consistent and asymptotically normal ). 71

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If n : n 1,2,..., E is such that for any other ̂ n : n 1,2,... E , Avar n ̂ n  Avar n n  is psd , all Θ , then n : n 1,2,..., is asymptotically efficient in E . As a shorthand, n is often said to be best asymptotically normal ( BAN ) in in E . 72
Unlike in the case of a best unbiased estimator, an asymptotically efficient estimator is never unique because we can always change the estimator in ways that do not affect its asymptotic distribution. An important result is that if we have an asymptotically efficiency estimator of Θ then we automatically have an asymptotically efficient estimator of any smooth function of . 73

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THEOREM : Let ̂ n : n 1,2,... and ̃ n : n 1,2,... be two n -asymptotically normal estimators with asymptotic variances C 1 and C 2 , respectively, such that C 2 C 1 is psd. Let g : Θ m be continuously differentiable on the open set Θ p and define g . Consider two estimators of , ̂ n g ̂ n and ̃ n g ̃ n . Then ̂ n is asymptotically efficient relative to ̃ n . 74
Proof : We know from Slutsky’s theorem that ̂ n and ̃ n are consistent , and we know from the delta method that each estimator is n -asymptotically normal with Avar n ̂ n  G C 1 G Avar n ̃ n  G C 2 G where G g is the m p Jacobian. 75

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Now, we have to show that Avar n ̃ n  Avar n ̂ n  is psd.
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