Under h n r n a rv n r 1 n r n a d q 2 and so the q 2

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Under H 0 , 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
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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
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To test a single restriction we can use an asymptotic t statistic. If the null is H 0 : r 1 1 ... r p p a , or H 0 : 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
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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
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