  This paper deals with a parametrized family of partially observed bivariate
Markov chains. We establish that, under very mild assumptions, the limit of the
normalized log-likelihood function is maximized when the parameters belong to
the equivalence class of the true parameter, which is a key feature for
obtaining the consistency of the maximum likelihood estimators (MLEs) in
well-specified models. This result is obtained in the general framework of
partially dominated models. We examine two specific cases of interest, namely,
hidden Markov models (HMMs) and observation-driven time series models. In
contrast with previous approaches, the identifiability is addressed by relying
on the uniqueness of the invariant distribution of the Markov chain associated
to the complete data, regardless its rate of convergence to the equilibrium.
