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slides_13_inferasymptotic

# Log 2 x ̄ and this clearly differs from the previous

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Unformatted text preview: log 2 X ̄ and this clearly differs from the previous t statistic. ∙ Both converge in distribution to Normal 0,1 under the null, and they have exactly the sample local power function. 68 ∙ An uncomfortable fact about large-sample inference is that there are many different statistics that are asymptotically equivalent in the sense that the have the same local power function. One could conceivably reject H using one of the statistics but not the other. ∙ At a minimum, the asymptotic p-values will usually differ. Suppose with n 400 we obtain x ̄ .8. and so ̂ exp − .8 ≈ .45. Rounding to two decimal places, 20 exp − .8 − .5 exp − .8 .8 ≈ − 2.52 20 .8 − log 2 .8 ≈ 2.39 69 (Note that if the alternative is H 1 : .5 this is equivalent to H 1 : log 2 , which is why the signs are different.) ∙ So, the asymptotic p-values, gotten in both cases from the Normal 0,1 distribution, are different. (The one-sided p-values are about .0059 in the first case and about .0084 in the second case.) ∙ In practice, this has not been viewed as much of a problem, as there is often a “natural” way to compute the statistic. 70 ∙ The key to testing a single hypothesis using asymptotic methods is to just remember the general formula for a t statistic. If is a parameter and n ̂ − d → Normal 0, 2 , so that 2 is the asymptotic variance of n ̂ − (which depends on the parameter vector ), then the asymptotic standard error of ̂ is se ̂ 2 ̂ n ≡ ̂ n 71 ∙ The asymptotic t statistic for testing H : is ̂ − se ̂ n ̂ − ̂ , and this has a limiting standard normal distribution under H . Local power analysis can also be carried out. 72 Testing Multiple Restrictions ∙ Let be a p 1 vector in the open set Θ ⊂ p , and let c : Θ → q be a continuously differentiable function. Assume the q ≤ p restrictions to be tested are H : c ∙ For example, suppose p 3 and we want to test H : 1 2 1, 3 so there are q 2 restrictions. Then c c 1 c 2 1 2 − 1 3 73 ∙ Given n ̂ n − n d → Normal , V we use the delta method to obtain the asymptotic variance of n c ̂ n under the null hypothesis c : Avar n c ̂ n C VC ′ and the Wald test statistic is n c ̂ n ′ C ̂ n VC ̂ n ′ − 1 n c ̂ n c ̂ n ′ C ̂ n VC ̂ n ′ / n − 1 c ̂ n 74...
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