Abstract: This paper gives necessary and sufficient conditions for stationarity and existence of second moments in mixtures of linear vector autoregressive models with autoregressive conditional heteroskedasticity. Sufficient conditions are also provided for a more general model in which the mixture components are permitted to exhibit limited forms of nonlinearity. When specialized to the corresponding non-mixture case these sufficient conditions improve on their previous counterparts obtained for nonlinear autoregressions with nonlinear conditional heteroskedasticity. In this context, a previous conjecture is also disproved. The results of the paper are proved by using the stability theory of Markov chains. Stationarity, existence of second moments of the stationary distribution, and-mixing are obtained by establishing an appropriate version of geometric ergodicity.
Key words and phrases: Autoregressive conditional heteroskedasticity, β-mixing, geometric ergodicity, Markov chain, mixture autoregressive model, nonlinear vector autoregressive model, stability.