Gaussian Mixture Autoregressive Conditional Heteroscedastic Model

Mixture time series models have received continuous attention because of their flexibilities in modeling.This paper proposes a kind of mixture time series models based on autoregressive conditional heteroscedastic models called gaussian mixture autoregressive-autoregressive conditional heteroscedastic,short for GMAR-ARCH.In the previous mixture AR-ARCH model,the mixture weights are time-independent variables.Despite this model has achieved good results,but the previous mixture AR-ARCH models only have stationary and conditional likelihood functions with imposing some additional restrictions on the parameters and do not give the stationary distribution or the proof of ergodicity.In this paper,the weights are defined in a special way that is related to time.Because of the time varying weights,we can model the multimodal date in practical and get the density of stationary distribution in theoretically.The ergodicity is proved by the theory of Markov chain and the consistency can be proved easily.We use the maximum likelihood estimation of the parameters and the numerical simulation are given.