Copula hidden Markov model with unknown number of states

In this study, we investigate the construction of hidden Markov models (HMMs) with copulas serving as the emission distributions. Additionally, we relax the traditional assumption that the number of hidden states must be predetermined before model fitting. Instead, in our approach, the number of states is estimated simultaneously with other model parameters of the copula-HMM when datasets are applied. This is achieved by incorporating the hierarchical Dirichlet process as a prior during the Bayesian inference procedure. We provide a comprehensive algorithm for this methodology, including a detailed implementable example using the t-copula, which is a novel contribution not previously available in the copula literature. The proposed estimator was validated through simulation studies, which demonstrated its superiority over traditional BIC-based approaches for model selection in HMMs. Furthermore, we applied this methodology to real data to examine the dependence structure among stock markets.