A Gaussian Mixture autoregressive model for univariate time series

This paper presents a general formulation for the univariate nonlinear autoregressive model discussed by Glasbey [Journal of the Royal Statistical Society: Series C, 50(2001), 143-154] in the first order case, and provides a more thorough treatment of its theoretical properties and practical usefulness. The model belongs to the family of mixture autoregressive models but it differs from its previous alternatives in several advantageous ways. A major theoretical advantage is that, by the definition of the model, conditions for stationarity and ergodicity are always met and these properties are much more straightforward to establish than is common in nonlinear autoregressive models. Moreover, for a pth order model an explicit expression of the (p+1)-dimensional stationary distribution is known and given by a mixture of Gaussian distributions with constant mixing weights. Lower dimensional stationary distributions have a similar form whereas the conditional distribution given the past observations is a Gaussian mixture with time varying mixing weights that depend on p lagged values of the series in a natural way. Due to the known stationary distribution exact maximum likelihood estimation is feasible, and one can assess the applicability of the model in advance by using a nonparametric estimate of the density function. An empirical example with interest rate series illustrates the practical usefulness of the model.