A Truncated Mixture Transition Model for Interval-valued Time Series

We propose a model for interval-valued time series (ITS), e.g. the collection of daily intervals of high/low stock returns over time, that specifies the conditional joint distribution of the upper and lower bounds of the interval as a mixture of truncated bivariate normal distribution. This specification guarantees that the natural order of the interval (upper bound not smaller than lower bound) is preserved. The model also captures the potential conditional heteroscedasticity and non-Gaussian features in ITS. The standard EM algorithm, when applied to the estimation of mixture models with truncated distribution, does not provide a closed-form solution in M step. We propose a new EM algorithm that solves this problem. We establish the consistency of the maximum likelihood estimator. Monte Carlo simulations show the new EM algorithm has good convergence properties. We apply the model to the interval-valued IBM daily stock returns and it exhibits superior performance over competing methods.