General framework and model building in the class of Hidden Mixture Transition Distribution models

Abstract

Modeling time series that present non-Gaussian features plays as central role in many fields, including finance, seismology, psychological, and life course studies. The Hidden Mixture Transition Distribution model is an answer to the complexity of such series. The observed heterogeneity can be induced by one or several latent factors, and each level of these factors is related to a different component of the observed process. The time series is then treated as a mixture and the relation between the components is governed by a Markovian latent transition process. This framework generalizes several specifications that appear separately in related literature. Both the expectation and the standard deviation of each component are allowed to be functions of the past of the process. The latent process can be of any order, and can be modeled using a discrete Mixture Transition Distribution. The effects of covariates at the visible and hidden levels are also investigated. One of the main difficulties lies in correctly specifying the structure of the model. Therefore, we propose a hierarchical model selection procedure that exploits the multilevel structure of our approach. Finally, we illustrate the model and the model selection procedure through a real application in social science.