Modèles de Markov à variables latentes : matrice de transition non-homogène et reformulation hiérarchique
This master’s thesis is centered on the Hidden Markov Models, a family of models in which an unobserved Markov chain dictactes the behaviour of an observable stochastic process through which a noisy version of the latent chain is observed. These bivariate stochastic processes that can be seen as a natural generalization of mixture models have shown their ability to capture the varying dynamics of many time series and, more specifically in finance, to reproduce the stylized facts of financial returns. In particular, we are interested in discrete-time Markov chains with finite state spaces, with the objective of studying the contribution of their hierarchical formulations and the relaxation of the homogeneity hypothesis for the transition matrix to the quality of the fit and predictions, as well as the capacity to reproduce the stylized facts. We therefore present two hierarchical structures, the first allowing for new interpretations of the relationships between states of the chain, and the second allowing for a more parsimonious parameterization of the transition matrix. We also present three non-homogeneous models, two of which have transition probabilities dependent on observed explanatory variables, and the third in which the probabilities depend on another latent variable.  We first analyze the goodness of fit and the predictive power of our models on the series of log returns of the S&P 500 and the exchange rate between canadian and american currencies (CADUSD). We also illustrate their capacity to reproduce the stylized facts, and present interpretations of the estimated parameters for the hierarchical and non-homogeneous models. In general, our results seem to confirm the contribution of hierarchical and non-homogeneous models to these measures of performance. In particular, these results seem to suggest that the incorporation of non-homogeneous dynamics to a hierarchical structure may allow for a more faithful reproduction of the stylized facts—even the slow decay of the autocorrelation functions of squared and absolute returns—and better predictive power, while still allowing for the interpretation of the estimated parameters.
Variables explicatives Hidden Markov Model Stylized facts of financial returns Hierarchical structure Homogeneity Explanatory variables Generalized autoregressive score model Duration dependence Interpretability