Prediction with Mixture Autoregressive Models

Mixture autoregressive (MAR) models have the attractive property that the shape of the conditional distribution of a forecast depends on the recent history of the process. In particular, it may have a varying number of modes over time. We show that the distributions of the multi-step predictors in MAR models are also mixtures and specify them analytically. In the important case when the original MAR model is a mixture of normal or, more generally, α-stable distributions, the multi-step distributions are also mixtures of normal (respectively α-stable) distributions. The latter provide a framework for introducing (possibly skewed) multi-modal components with heavy tails. It is valuable to know that these important classes of mixtures are “closed” in respect to prediction since this allows for deriving the predictive distributions by simple arithmetic manipulation of the parameters of the original distributions. Our approach is applicable to other models as well.