We consider a multidimensional, continuous-time model where the observation process is a diffusion with drift and volatility coefficients being modeled as continuous-time, finite-state Markov chains with a common state process. For the econometric estimation of the states for drift and volatility and the rate matrix of the underlying Markov chain, we develop both an exact continuous time and an approximate discrete-time Markov chain Monte Carlo (MCMC) sampler and compare these approaches with maximum likelihood (ML) estimation. For simulated data, MCMC outperforms ML estimation for difficult cases like high rates. Finally, for daily stock index quotes from Argentina, Brazil, Mexico, and the USA we identify four states differing not only in the volatility of the various assets but also in their correlation. Copyright The Author 2009. Published by Oxford University Press. All rights reserved. For Permissions, please e-mail: email@example.com, Oxford University Press.