On a Mixture Autoregressive Conditional Heteroscedastic Model

Abstract

We propose a mixture autoregressive conditional heteroscedastic (MAR-ARCH) model for modeling nonlinear time series. The models consist of a mixture of K autoregressive components with autoregressive conditional heteroscedasticity; that is, the conditional mean of the process variable follows a mixture AR (MAR) process, whereas the conditional variance of the process variable follows a mixture ARCH process. In addition to the advantage of better description of the conditional distributions from the MAR model, the MAR-ARCH model allows a more flexible squared autocorrelation structure. The stationarity conditions, autocorrelation function, and squared autocorrelation function are derived. Construction of multiple step predictive distributions is discussed. The estimation can be easily done through a simple EM algorithm, and the model selection problem is addressed. The shape-changing feature of the conditional distributions makes these models capable of modeling time series with multimodal conditional distributions and with heteroscedasticity. The models are applied to two real datasets and compared to other competing models. The MAR-ARCH models appear to capture features of the data better than the competing models.