  A regime-switching multivariate time series model which is closed under
margins is built. The model imposes a restriction on all lower-dimensional
sub-processes to follow a regime-switching process sharing the same latent
regime sequence and having the same Markov order as the original process. The
margin-closed regime-switching model is constructed by considering the
multivariate margin-closed Gaussian VAR($k$) dependence as a copula within each
regime, and builds dependence between observations in different regimes by
requiring the first observation in the new regime to depend on the last
observation in the previous regime. The property of closure under margins
allows inference on the latent regimes based on lower-dimensional selected
sub-processes and estimation of univariate parameters from univariate
sub-processes, and enables the use of multi-stage estimation procedure for the
model. The parsimonious dependence structure of the model also avoids a large
number of parameters under the regime-switching setting. The proposed model is
applied to a macroeconomic data set to infer the latent business cycle and
compared with the relevant benchmark.
