2 A Uni ed Framework for Asymptotically Optimal Tests

This paper puts forward a uni ed framework for asymptotically optimal tests in parametric models and applies the new theory to two tests. The rst test is a classical test in locally asymptotically normal (LAN) models, but the assumptions are weakened and a lacuna in the literature is lled. The second test concerns the location of the threshold point. The main result here is that the optimal test in the weighted average power sense is based on the posterior odds which depends on the prior on the local parameter space and is not unique. Furthermore, the likelihood ratio test is not asymptotically equivalent to the posterior odds and there is a discrete component in its asymptotic distribution. Since the asymptotic distribution of the posterior odds is not pivotal to the true value under the null hypothesis, a parametric bootstrap is used to nd asymptotic critical values. The results in the second test are very di¤erent from those in classical LAN models. Keywords: optimal test, threshold regression, posterior odds, Bayes factor, nonstandard testing problem, integrated likelihood ratio, limits of experiments JEL-Classification: C11, C12, C21. Email: p.yu@auckland.ac. nz. I would like to thank professor Adrian Raftery for introducing me the optimality property of Bayes factors. I also want to thank Professor Van der Vaart for helping me solving a technical di¢ culty. This paper bene ts heavily from helpful comments of Bruce Hansen and careful modi cations of Peter Phillips for which I want to express my gratitude. All remaining errors are mine. 1 Introduction Optimal testing is part of classical statistical theory and has important applications throughout econometrics. In the frequentist literature, the Neyman-Pearson lemma is the cornerstone for considering optimality, but only applies for simple hypotheses. When the hypotheses are composite, the weighted average power (WAP) criterion is often used. For example, Andrews and Ploberger (1994) consider parametric optimal tests using WAP when a nuisance parameter is present only under the alternative. In the Bayes literature, Bayes factors are the usual choice for testing. It is well known that Bayes factors have some optimality properties. For example, Je¤reys (1961) shows the Bayes factor minimizes the sum of the Type I and Type II error probabilities averaged over the prior. It is standard to apply the decision theory to the testing problem in nite samples; see, e.g., Section 8.1 of Chamberlain (2007). But the decision theory is not applied to optimal testing in large samples yet. This paper tries to nd the optimal tests in the asymptotic WAP sense by using the decision theory framework. Such a uni ed framework is given in Section 2. This framework is very powerful. It helps us to sort out the key problems we need to answer when we try to nd a test with the maximum WAP. The basic result is that the posterior odds procedure is optimal in the WAP sense as long as it is asymptotically pivotal to the local parameters under the null hypothesis. There is considerable literature related to the topic discussed in this paper. Andrews (1994) builds the large sample correspondence between classical hypothesis tests and Bayesian posterior odds tests in locally asymptotically normal (LAN) models, but no optimality results are covered. See also his Section 7 for the interaction of frequentist and Bayesian literature in hypothesis testing. Andrews and Ploberger (1994) discuss optimal tests when a nuisance parameter can not be identi ed under the null, but their Theorem 2 actually claims that the optimal tests there have the smallest weighted average type I and type II error probabilities, not the maximum WAP. A classical test in LAN models is reconsidered in Section 3, but the assumptions are weakened and the lacuna in Andrews and Ploberger (1994) is lled. A more interesting test about the location of the threshold point in threshold regression is discussed in Section 4. The threshold regression model is speci ed as y = ( x0 1 + 1e; x0 2 + 2e; q ; q > : (1) where q is the threshold variable used to split the sample, x 2 R, ( 1; 2) 2 R and ( 1; 2) are threshold parameters on mean and variance in the two regimes, E[ejx; q] = 0, E[e] = 1 is a normalization of the error variance and adopts conditional heteroskedasticity, and all the other variables are understood as in the linear regression framework. In the parametric model, the density of e conditional on (x; q) is fejx;q (ejx; q; ), 2 R is some nuisance parameter a¤ecting the shape of this error distribution only, the joint distribution of (x; q) is fx;q (x; q), the marginal density of q is fq (q), and the unknown parameter is = ; 1; 0 2; 1; 2; 0 0 ( ; ). Threshold regression models of the type (1) have many applications in economics; see the introduction of Yu (2007) or Lee and Seo (2008) for a summary. The hypotheses of interest are H0 : = 0; H1 : 6= 0: (2) This test is never considered independently although it appears as an intermediate step in the con dence interval construction literature such as Hansen (2000). The optimal tests depend on the weighting scheme used, so are not unique and depend on user preferences. Furthermore, the likelihood ratio test is not