Bubble modeling by mixed causal-noncausal autoregressive processes

Some financial time series exhibit short periods of explosive local trends followed by an abrupt decline. Such trends can be a result of speculative bubble phenomena. A bubble is formed when investors’ future profits expectations influence the present market value of securities. Mixed causal-noncausal autoregressive processes (MAR) are able to better capture such behavior in comparison to standard causal ARIMA models. In the first part of this work we propose an alternative distribution (Voigt) to model the disturbances in the MAR processes. The Voigt, a convolution of Gaussian and Cauchy distributions, is used in atomic and molecular spectroscopy, and is more flexible than other heavy-tail distributions. The second part of this work extends the MAR models to Markov switching mixed causal-noncausal autoregressive processes (MSMAR) with Cauchy distributed errors to account for changes in regime at different times. Parameter estimation of both models MAR with Voigt errors and MSMAR is performed in a Bayesian framework via MCMC algorithms. The models are tested for performance with a simulation study and then applied to Bitcoin/USD exchange rate data