# 17-072/III (2017-08-02)

Author(s)
Francisco (F.) Blasques, VU Amsterdam, The Netherlands; Tinbergen Institute, The Netherlands; Marc Nientker, VU Amsterdam, The Netherlands
Keywords:
Ergodicity, GARCH-type models, mixing, nonlinear time series, stationarity,stochastic recurrence equations, threshold models
JEL codes:
C50, C51, C58

This article generalises the results of Saidi and Zakoian (2006) to a considerably larger class of nonlinear ARCH models with discontinuities, leverage effects and robust news impact curves. We propose a new method of proof for the existence of a strictly stationary and phi-mixing solution. Moreover, we show that any path converges to this solution. The proof relies on stochastic recurrence equation theory and builds on the work of Bougerol (1993) and Straumann (2005). The assumptions that we need for this approach are less restrictive than those imposed in Saidi and Zakoian (2006) and typically found in Markov chain theory, as they require very little from the distribution of the underlying process. Furthermore, they can be stated in a general setting for random functions on a separable Banach space as is done in Straumann and Mikosch (2006). Finally, we state sufficient conditions for the existence of moments.