Inference when a nuisance parameter is weakly identified under the null hypothesis

When a nuisance parameter is weakly identified under the null hypothesis, the usual asymptotic theory breaks down and standard tests may exhibit significant size distortions. We provide asymptotic approximations under a drifting parameter DGP for distributions of classical tests and of those designed for the case of complete nonidentification. Simulations with a simple SETAR model show that the usual asymptotic theory does fail, although actual sizes of the classical Likelihood Ratio test display surprising robustness to the degree of identification. (C) 2004 Elsevier B.V. All rights reserved.