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This paper considers hypothesis tests for nonlinear econometric models when the parameter space is restricted under the alternative hypothesis. Multivariate one-sided tests are a leading example. Optimal tests, called directed tests, are derived using a weighted average power criterion. The likelihood ratio test is shown to be admissible and to maximize power against alternatives that are arbitrarily distant from the null hypothesis. Exact results are established ﬁrst for Gaussian linear regression models with known variance. Asymptotic analogues are then established for dynamic nonlinear models. Simulation is used to compare the tests discussed in the paper. The D–W ∞ directed test is found to perform best in an overall sense for multivariate one-sided alternatives. The he D–W ∞ and LR tests are found to perform likewise for mixed one- and two-sided alternatives.
Andrews, Donald W.K., "Hypothesis Testing with a Restricted Parameter Space" (1993). Cowles Foundation Discussion Papers. 1303.