On a mixture autoregressive model

We generalize the Gaussian mixture transition distribution (GMTD) model introduced by Le and co‐workers to the mixture autoregressive (MAR) model for the modelling of non‐linear time series. The models consist of a mixture of K stationary or non‐stationary AR components. The advantages of the MAR model over the GMTD model include a more full range of shape changing predictive distributions and the ability to handle cycles and conditional heteroscedasticity in the time series. The stationarity conditions and autocorrelation function are derived. The estimation is easily done via a simple EM algorithm and the model selection problem is addressed. The shape changing feature of the conditional distributions makes these models capable of modelling time series with multimodal conditional distributions and with heteroscedasticity. The models are applied to two real data sets and compared with other competing models. The MAR models appear to capture features of the data better than other competing models do.