On mixture autoregressive conditional heteroskedasticity

We consider mixture univariate autoregressive conditional heteroskedastic models, both with Gaussian or Student t-distributions, which were proposed in the literature for modeling nonlinear time series. We derive sufficient conditions for second order stationarity of these processes. Then we propose an algorithm in matrix form for the estimation of model parameters, and derive a formula in closed form for the asymptotic Fisher information matrix. Our results are proved by using the theory of time series models with Markov changes in regime. An illustrative example of the theoretical results and a real application on financial data complete the paper.