Bounds for Inference with Nuisance Parameters Present only under the Alternative.

In hypothesis testing with nuisance parameters present only under the alternative two issues typically arise: (i) critical values are date dependent and so cannot be tabulated; (ii) we need to choose a functional over the nuisance parameter space. We address the first issue by providing easily computable bounds for the case of dependent and heterogeneous observations. We tackle the second issue by suggesting a weighted average statistic with weights given by the (quasi) likelihood over the nuisance parameter space. The small sample behavior of our procedure is analyzed via few Monte Carlo simulations; we consider conditional moment tests and tests for nonlinearities in the SETAR model. For samples of 200-400 observations, the behavior of the suggested weighted average statistic is rather satisfactory. An empirical illustration using date on U.S. male unemployment is provided.