Asymptotic null distribution of the likelihood ratio test in Markov Switching Models

Markov switching (MS) models raise a problem known as testing hypotheses when a nuisance parameter is not identified under the null hypothesis. I show that the asymptotic distribution theory used for testing in presence of such a problem appears to work also for MS models, even though its validity can be questioned because of identically zero scores under the null estimates. Assuming the validity of this distribution theory, I derive the asymptotic null distribution of the likelihood ratio (LR) test for various MS models. Monte Carlo experiments show that the LR asymptotic distributions approximate the empirical distributions very well.