Development and Application of Hidden Markov Models in the Bayesian Framework

This thesis develops new hidden Markov models and applies them to financial market\nand macroeconomic time series.\nChapter 1 proposes a probabilistic model of the return distribution with rich and\nheterogeneous intra-regime dynamics. It focuses on the characteristics and dynamics of bear market rallies and bull market corrections, including, for example, the probability of transition from a bear market rally into a bull market versus back to the primary bear state. A Bayesian estimation approach accounts for parameter and regime uncertainty and provides probability statements regarding future regimes and returns. A Value-at-Risk example illustrates the economic value of our approach.\nChapter 2 develops a new efficient approach to model and forecast time series data\nwith an unknown number of change-points. The key is assuming a conjugate prior for the time-varying parameters which characterize each regime and treating the regime duration as a state variable. Conditional on this prior and the time-invariant parameters,\nthe predictive density and the posterior of the change-points have closed forms. The conjugate prior is further modeled as hierarchical to exploit the information across regimes. This framework allows breaks in the variance, the regression coefficients or both. In addition to the time-invariant structural change probability, one extension assumes the regime duration has a Poisson distribution. A new Markov Chain Monte Carlo sampler draws the parameters from the posterior distribution efficiently. The model is applied to Canadian inflation time series.\nChapter 3 proposes an infinite dimension Markov switching model to accommodate\nregime switching and structural break dynamics or a combination of both in a Bayesian framework. Two parallel hierarchical structures, one governing the transition probabilities and another governing the parameters of the conditional data density, keep the model parsimonious and improve forecasts. This nonparametric approach allows for regime persistence and estimates the number of states automatically. A global identification algorithm for structural changes versus regime switching is presented. Applications\nto U.S. real interest rates and inflation compare the new model to existing parametric alternatives. Besides identifying episodes of regime switching and structural breaks,\nthe hierarchical distribution governing the parameters of the conditional data density\nprovides significant gains to forecasting precision.

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