<p>Regime-switching multivariate time series models tend to involve a large number of parameters, which complicates their estimation and inference. We propose a new class of parsimonious regime-switching multivariate time series models such that there is a stationary Markov process for each regime with non-Gaussian marginal distributions. A simple serial dependence construction is suggested for transitions between regimes. The stochastic process in each regime has the property that all lower-dimensional sub-processes follow a regime-switching process sharing the same latent regime sequence and having the same Markov order as the multivariate process. This results in a model that is closed under margins, a property that allows inference on the latent regimes to be based on chosen lower-dimensional subprocesses and enables the use of a more computationally convenient multi-stage estimation procedure. We conduct a simulation study to evaluate the finite sample performance of the proposed estimation procedure. Then, we apply the model to a macroeconomic dataset to infer the latent business cycle and compare it to a relevant benchmark.</p>