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∙ in econometrics the statistic t n is typically

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Unformatted text preview: ∙ In econometrics, the statistic T n is typically called the Wald statistic , and often denoted W n . ∙ Under H , n R ̂ n − a ′ RV ̂ n R ′ − 1 n R ̂ n − a d → q 2 and so the q 2 distribution is used to obtain critical values or to compute p-values. The derivation is a special case of the result for local alternatives. 62 ∙ A similar argument to the previous case shows that, under H n 1 , RVR ′ − 1/2 n R ̂ n − a d → Normal RVR ′ − 1/2 , I q and so T n has a limiting noncentral chi-square distribution with q df and noncentrality parameter ′ RVR ′ − 1 63 ∙ To test a single restriction we can use an asymptotic t statistic. If the null is H : r 1 1 ... r p p a , or H : r ′ a for a p 1 vector r , the asymptotic t statistic is n r ′ ̂ n − a r ′ V ̂ n r r ′ ̂ n − a se r ′ ̂ n − a We can compare this to standard normal or t n − p critical values (the latter being a finite-sample adjustment that often seems to work better). 64 ∙ Local power analysis is somewhat rare in practice, mainly because a few basic testing priniciples are applied to the most efficient estimators possible. Only rarely do we design a survey where we can try to control the power. ∙ For survey design, power analysis can be helpful in choosing sample sizes so that the null of (usually) no effect can be detected with “reasonably high” probability for alternatives “sufficiently far” from the null. But these parameters must be chosen by the researcher. 65 6 . Testing Nonlinear Restrictions Testing Single Restrictions ∙ Suppose that we have random sampling from the Poisson distribution. Define exp − P X . Suppose we want to test H : .5. Let ̂ X ̄ be the estimator of (and drop the sample size index). The MLE of is then ̂ exp − ̂ . To construct an asymptotic t statistic, ̂ − .5 se ̂ we need to find se ̂ , which is the asymptotic standard error of ̂ . 66 ∙ We use the delta method: se ̂ exp − ̂ se ̂ [because | d exp − / d | exp − ]. Of course, se ̂ se X ̄ ̂ / n (because Var X ̄ / n for the Poisson distribution). So an asymptotic t statistic based on ̂ is ̂ − .5 exp − ̂ ̂ / n n exp − X ̄ − .5 exp − X ̄ X ̄ 67 ∙ Note that there is not a unique asymptotic t statistic for testing this hypothesis. We can instead test H : − log .5 log 2 The t statistic based on the null stated this way is simply X ̄ − log 2 X ̄ / n n X ̄ − log...
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∙ In econometrics the statistic T n is typically called...

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