Parameter estimation in continuous time Markov switching models: a semi-continuous Markov chain Monte Carlo approach

Abstract

We want to estimate the parameters governing a continuous time Markov switching model given observations at discrete times only. For parameter estimation in a setting with continuous time and a latent state process, using MCMC methods two approaches are common: Using time-discretization and augmenting the unknowns with the (then discrete) state process, or working in continuous time and augmenting with the full state process. In this paper, we combine useful aspects of both approaches. On the one hand, we are inspired by the discretization, where filtering for the state process is possible, on the other hand, we catch attractive features of the continuous time method, like exact estimation (i.e. no discretization error) and direct estimation of the generator matrix rather than the transition matrix. This is achieved by taking not the whole state process for data augmentation but only the states at observation times. Using results on the distribution of occupation times in Markov processes, it is possible to compute the complete data likelihood exactly. We obtain a sampler that works more robustly and more accurately especially for fast switching in the state process.