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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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