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.