Hidden Markov models and their extensions with applications in finance and gambling

Abstract

Hidden Markov models (HMMs) are time series models which incorporate serial dependence via a latent (hidden) discrete-time Markov chain (DTMC). Consequently, standard HMMs assume that the distribution which generates an observation at a particular point in time depends on the chosen state of the latent DTMC at that point in time. HMMs also motivate several important extensions. One such extension is the hidden semi-Markov model (HSMM). HSMMs generalize HMMs by allowing dwell-time distributions in states to be modelled explicitly instead of relying on the geometric distribution assumption imposed by the HMM setup. Thus, state changes and state persistence are now controlled by what is called a semi-Markov chain. The application which follows sees the implementation of HMMs and HSMMs with normal state-dependent distributions to model daily returns of the S&P 500 Index and the BTC/USD exchange rate. The aim is that of identifying market regimes mainly bull and bear market phases. Another important extension to the standard HMM is to allow for the inclusion of time-varying covariates in the state-dependent parameters. Through appropriate link functions, the state-dependent means can be allowed to change, not only according to the state, but also according to covariate information. This framework leads to the HMM-GLM hybrid, where HMM Regression (HMMR) is a special case when the state-dependent distributions are assumed normal. The application which follows involves modelling problem gambling behaviour through the use of HMMs with seasonal adjustments. The aim is that of identifying inactive, moderately active, and highly active periods by monitoring players’ gambling history.