Asymptotic Null Distribution of the Likelihood Ratio Test in Markov Switching Models

The Markov Switching Model, introduced by Hamilton (1988Hamilton ( , 1989)), has been used in various economic and financial applications where changes in regime play potentially an important role.While estimation methods for these models are by now well established, such is not the case for the corresponding testing procedures.The Markov switching models raise a special problem known in the statistics literature as testing hypotheses in models where a nuisance parameter is not identified under the null hypothesis.In these circumstances, the asymptotic distributions of the usual tests (likelihood ratio, Lagrange multiplier, Wald tests) are non-standard.In this paper, we show that, if we treat the transition probabilities as nuisance parameters in a Markov switching model and set the null hypothesis in terms uniquely of the parameters governed by the Markov variable, the distributional theory proposed by Hansen (1991a) is applicable to Markov switching models under certain assumptions.Based on this framework, we derive analytically, in the context of two-state Markov switching models, the asymptotic null distribution of the likelihood ratio test (which is shown to be also valid for the Lagrange multiplier and Wald tests under certain conditions) and the related covariance functions.Monte Carlo simulations show that the asymptotic distributions offer a very good approximation to the corresponding empirical distributions.This specification encompasses the specifications most frequently used in the literature for two-state Markov switching models.Two exceptions are noteworthy: the state-dependent autoregressive specification used in Garcia and Perron (1995) or the time-varying transition probability model used in Diebold, Lee, and Weinbach (1993) and Filardo (1992).This point is clear when looking at the element of the score vector corresponding to derived in Lemma !