Classes of Similar Regions and Their Power Properties for Some Econometric Testing Problems

In an hypothesis testing problem involving nuisance parameters for which boundedly complete sufficient statistics exist under the null hypothesis, the class of all similar regions for the problem is characterized by the conditional distribution of the data given these sufficient statistics. If there exists a one-to-one transformationy→ (t, u) of the data,y, to the sufficient statistic,t, and a second vector of statistics,u, that is independent oftunder the null hypothesis, then the statisticuitself characterizes the class of similar regions. This paper applies this idea to five testing problems of interest in econometrics. In each case we obtain the density of the relevant statistic under the null hypothesis, when it is free of nuisance parameters, and under the alternative. Using the density under the alternative, we discuss the power properties of the class of similar tests for each problem. Other applications are also suggested.