Modeling flat stretches, bursts, and outliers in time series using mixture transition distribution models

The class of mixture transition distribution (MTD) time series models is extended to general non-Gaussian time series. In these models the conditional distribution of the current observation given the past is a mixture of conditional distributions given each one of the last p observations. They can capture non-Gaussian and nonlinear features such as flat stretches, bursts of activity, outliers, and changepoints in a single unified model class. They can also represent time series defined on arbitrary state spaces, univariate or multivariate, continuous, discrete or mixed, which need not even be Euclidean. They perform well in the usual case of Gaussian time series without obvious nonstandard behaviors. The models are simple. analytically tractable, easy to simulate, and readily estimated. The stationarity and autocorrelation properties of the models are derived. A simple EM algorithm is given and shown to work well for estimation. The models are applied to several real and simulated datasets with satisfactory results. They appear to capture the features of the data better than the best competing autoregressive integrated moving average (ARIMA) models.