Optimal Test for Stochastic Unit Root with Markov Switching

We propose a new test for fixed unit root against Markov switching unit root models. The testing problem is challenging due to nonstationarity and the presence of nuisance parameters not identified under the null. This problem is different to the existing literature on stochastic unit roots, where the dynamic structure of the random coefficients are assumed to be fully specified. We fully exploit the Markov property of the random coefficients and we derive a class of information matrix-type tests and show that they are equivalent to the likelihood ratio test. Hence, our tests are asymptotically optimal. Moreover these tests are easy to implement as they do not require the estimation of the model under the alternative. And limiting distribution of the test statistic is pivotal, so critical values could be tabulated. Then we apply our results to the macroeconomic datasets considered by Nelson and Plosser to test for the existence of Markov Switching in the unit roots