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Mixing and moment properties of various GARCH and stochastic volatility models. (English) Zbl 1181.62125
Summary: This paper first provides some useful results on a generalized random coefficient autoregressive model and a generalized hidden Markov model. These results simultaneously imply strict stationarity, existence of higher order moments, geometric ergodicity, and $$\beta$$-mixing with exponential decay rates, which are important properties for statistical inference. As applications, we then provide easy-to-verify sufficient conditions to ensure $$\beta$$-mixing and finite higher order moments for various linear and nonlinear GARCH(1,1), linear and power GARCH$$(p,q)$$, stochastic volatility, and autoregressive conditional duration models. For many of these models, our sufficient conditions for existence of second moments and exponential $$\beta$$-mixing are also necessary. For several GARCH(1,1) models, our sufficient conditions for existence of higher order moments again coincide with the necessary ones of He and Terasvirta [J. Econom. 92, 173–192 (1999)].

##### MSC:
 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) 62P05 Applications of statistics to actuarial sciences and financial mathematics
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