Multivariate Hidden Markov model: An application to study correlations among cryptocurrency log-returns

We provide an analysis of the market data of the major cryptocurrencies by summing a multivariate hidden Markov process also known as the latent Markov process. We model jointly the daily log-returns of BTC, ETH, XRP, LTC, and BCH. The observed log-returns are assumed to be correlated according to a variance-covariance matrix conditionally on a latent Markov process of first-order having a discrete number of latent states. In order to compare states according to their volatility, we estimate the specific variance-covariance matrix of each state. Maximum likelihood estimation of the model parameters is carried out by the Expectation-Maximization algorithm. The latent states can be ordered according to expected average values of the log-returns and their estimated volatility. We consider different model specifications in terms of number of latent states, which are identified in terms of expected log-returns and level of volatility. Under each considered scenario we also predict the latent state by the maximum a posteriori rule.