On a mixture autoregressive conditional heteroscedastic model

We propose a mixture autoregressive conditional heteroscedustic (MAR-ARCH) model for modeling nonlinear time series. The models consist of a mixture o f K autoregressive components with autoregressive conditional heteroscedast icity that is, the conditional mean of the progress variable follows a mixt ure AR (MAR) process, whereas the conditional variance of the process varia ble follows a mixture ARCH process. In addition to the advantage of better description of the conditional distributions from the MAR model, the MAR-AR CH model allows a more flexible squared autocorrelation structure. The stat ionarity conditions, autocorrelation function, and squared autocorrelation function are derived. Construction of multiple step predictive distribution s is discussed. The estimation can be easily done through a simple EM algor ithm, and the model selection problem is addressed. The shape-changing feat ure of die conditional distributions makes these models capable of modeling time series with multimodal conditional distributions and with heterosceda sticity. The models are applied to two real datasets and compared to other competing models. The MAR-ARCH models appear to capture features of the dat a better than the competing models.