Closed-form likelihood function of Markov-switching models

For a class of Markov-switching models, the likelihood function and inferred state distributions for a given sample are shown to have closed-form representations under a set of sufficient conditions. Based on these results, it is demonstrated that the closed-form partial derivatives (when exist) of the likelihood function can be readily found. These results may be used to improve the efficiency of numerical optimization techniques used for estimating the Markov-switching models.