A Regime switching Student-t copula model for the analysis of cryptocurrencies data

Flexible statistical models have an important role in explaining the joint distribution of financial returns. In these analyses, it is necessary to consider abrupt switches in the market conditions, especially if the focus is on cryptoassets, the market of which is characterized by high instabilities. Regime switching (RS) copula models represent a powerful tool to formulate the joint distribution of time-series accurately: they are based on a copula distribution with parameters governed by a hidden Markov process of first-order so as to account for the correlation patterns between series. The hidden states represent different market regimes, each described by a state-specific vector of copula parameters. We propose RS copula models as a valuable instrument for describing the joint behavior of log- returns. We choose a Student-t copula function to consider extreme dependent values appropriately as they are often observed in financial returns. We split the modeling process into two steps: the first one consists in fitting the marginal distribution of each univariate time-series, while the second one deals with the estimation of the joint distribution of the log-returns described by a RS copula model. Maximum likelihood estimation of the model parameters is carried out by the expectation-maximization (EM) algorithm, which alternates two steps until convergence: at the E-step, we compute the expectation of the log-likelihood evaluated using the current values for the parameters and, at the M-step, parameters estimates are updated by maximizing the expected complete-data log-likelihood computed at the previous step. The main computational burdens deal with estimating the correlation matrix (R) and the number of degrees of freedom (v) of the Student t-copula. At this aim, we propose performing the M-step by computing R given v using a closed form solution obtained from a constrained optimization of the log-likelihood using Lagrange multipliers. Then, we numerically maximize the log-likelihood with respect to v given the previous update of R. The proposal is validated through a simulation study showing that the estimators have good finite sample properties. We consider data on daily log-returns over four years of five cryptos Bitcoin, Bitcoin Cash, Ethereum, Litecoin, and Ripple as an application.