  A hierarchical Bayesian framework is introduced for developing rich mixture
models for real-valued time series, partly motivated by important applications
in financial time series analysis. At the top level, meaningful discrete states
are identified as appropriately quantised values of some of the most recent
samples. These observable states are described as a discrete context-tree
model. At the bottom level, a different, arbitrary model for real-valued time
series -- a base model -- is associated with each state. This defines a very
general framework that can be used in conjunction with any existing model class
to build flexible and interpretable mixture models. We call this the Bayesian
Context Trees State Space Model, or the BCT-X framework. Efficient algorithms
are introduced that allow for effective, exact Bayesian inference and learning
in this setting; in particular, the maximum a posteriori probability (MAP)
context-tree model can be identified. These algorithms can be updated
sequentially, facilitating efficient online forecasting. The utility of the
general framework is illustrated in two particular instances: When
autoregressive (AR) models are used as base models, resulting in a nonlinear AR
mixture model, and when conditional heteroscedastic (ARCH) models are used,
resulting in a mixture model that offers a powerful and systematic way of
modelling the well-known volatility asymmetries in financial data. In
forecasting, the BCT-X methods are found to outperform state-of-the-art
techniques on simulated and real-world data, both in terms of accuracy and
computational requirements. In modelling, the BCT-X structure finds natural
structure present in the data. In particular, the BCT-ARCH model reveals a
novel, important feature of stock market index data, in the form of an enhanced
leverage effect.
