Asymptotic Properties of the Maximum Likelihood Estimator in Regime-Switching Models with Time-Varying Transition Probabilities

We prove the asymptotic properties of the maximum likelihood estimator (MLE) in time-varying transition probability (TVTP) regime-switching models. This class of models extends the constant regime transition probability in Markov-switching models to a time-varying probability by including information from observations. An important feature in this proof is the mixing rate of the regime process conditional on the observations, which is time varying owing to the time-varying transition probabilities. Consistency and asymptotic normality follow from the almost deterministic geometrically decaying bound of the mixing rate. The assumptions are verified in regime-switching autoregressive models with widely-applied TVTP specifications. A simulation study examines the finite-sample distributions of the MLE and compares the estimates of the asymptotic variance constructed from the Hessian matrix and the outer product of the score. The simulation results favour the latter. As an empirical example, we compare three leading economic indicators in terms of describing U.S. industrial production.