Now suppose that n 0 then n x n n n x n n n n d

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Now, suppose that n 0. Then n X ̄ n n n X ̄ n n n n d Normal 0,1 . If we assume n o / n then n X ̄ n o n X ̄ n n d Normal / ,1 47
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We can apply this result immediately to the statistic T n 1 in the Poisson example. For the other two statistics, we use another general result. 48
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If W n p – for example, if we can show W n n p 0, along with n – then n X ̄ n n W n n X ̄ n n 1 W n 1 n X ̄ n n n X ̄ n n o p 1   O p 1 n X ̄ n n o p 1 . 49
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We cannot use local power analysis to choose among these three statistics when the Poisson distribution holds. Recall that T n 3 is valid more generally (because it is just the usual asymptotic t statistic that can be used for any population distribution). In the vast majority of econometric applications, how one estimates the asymptotic variance has no effect on asymptotic size or local power if the variance estimator is consistent under the null (and hence under local alternatives). 50
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