Contributions to the Theory of Optimal Tests

This paper considers tests which maximize the weighted average power (WAP). The focus is on determining WAP tests subject to an uncountable number of equalities and/or inequalities. The unifying theory allows us to obtain tests with correct size, similar tests, and unbiased tests, among others. A WAP test may be randomized and its characterization is not always possible. We show how to approximate the power of the optimal test by sequences of nonrandomized tests. Two alternative approximations are con-sidered. The first approach considers a sequence of similar tests for an in-creasing number of boundary conditions. This discretization allows us to implement the WAP tests in practice. The second method finds a sequence of tests which approximate the WAP test uniformly. This approximation allows us to show that WAP similar tests are admissible. The theoretical framework is readily applicable to several econometric models, including the important class of the curved-exponential family. In this paper, we consider the instrumental variable model with heteroskedas-tic and autocorrelated errors (HAC-IV) and the nearly integrated regressor model. In both models, we find WAP similar and (locally) unbiased tests which dominate other available tests.