Tree-structured generalized autoregressive conditional heteroscedastic models

We propose a new generalized autoregressive conditional heteroscedastic (GARCH) model with tree-structured multiple thresholds for the estimation of volatility in financial time series. The approach relies on the idea of a binary tree where every terminal node parameterizes a (local) GARCH model for a partition cell of the predictor space. The fitting of such trees is constructed within the likelihood framework for non-Gaussian observations: it is very different from the well-known regression tree procedure which is based on residual sums of squares. Our strategy includes the classical GARCH model as a special case and allows us to increase model complexity in a systematic and flexible way. We derive a consistency result and conclude from simulation and real data analysis that the new method has better predictive potential than other approaches.     

Testing the Martingale Hypothesis

We propose new tests of the martingale hypothesis based on generalized versions of the Kolmogorov–Smirnov and Cramér–von Mises tests. The tests are distribution-free and allow for a weak drift in the null model. The methods do not require either smoothing parameters or bootstrap resampling for their implementation and so are well suited to practical work. The article develops limit theory for the tests under the null and shows that the tests are consistent against a wide class of nonlinear, nonmartingale processes. Simulations show that the tests have good finite sample properties in comparison with other tests particularly under conditional heteroscedasticity and mildly explosive alternatives. An empirical application to major exchange rate data finds strong evidence in favor of the martingale hypothesis, confirming much earlier research.     

Hidden Markov chains in generalized linear models

We show how the concept of hidden Markov model may be accommodated in a setting involving multiple sequences of observations. The resulting class of models allows for both interrelationships between different sequences and serial dependence within sequences. Missing values in the observation sequences may be handled in a straightforward manner. We also examine a group of methods, based upon the observed Fisher Information matrix, for estimating the covariance matrix of the parameter estimates. We illustrate the methods with both real and simulated data sets.     

A semiparametric Bayesian model for repeatedly repeated binary outcomes

Summary.  We discuss the analysis of data from single-nucleotide polymorphism arrays comparing tumour and normal tissues. The data consist of sequences of indicators for loss of heterozygosity (LOH) and involve three nested levels of repetition: chromosomes for a given patient, regions within chromosomes and single-nucleotide polymorphisms nested within regions. We propose to analyse these data by using a semiparametric model for multilevel repeated binary data. At the top level of the hierarchy we assume a sampling model for the observed binary LOH sequences that arises from a partial exchangeability argument. This implies a mixture of Markov chains model. The mixture is defined with respect to the Markov transition probabilities. We assume a non-parametric prior for the random-mixing measure. The resulting model takes the form of a semiparametric random-effects model with the matrix of transition probabilities being the random effects. The model includes appropriate dependence assumptions for the two remaining levels of the hierarchy, i.e. for regions within chromosomes and for chromosomes within patient. We use the model to identify regions of increased LOH in a data set coming from a study of treatment-related leukaemia in children with an initial cancer diagnostic. The model successfully identifies the desired regions and performs well compared with other available alternatives.