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ERGODICITY, MIXING, AND EXISTENCE OF MOMENTS OF A CLASS OF MARKOV MODELS WITH APPLICATIONS TO GARCH AND ACD MODELS

Published online by Cambridge University Press:  11 June 2008

Mika Meitz*
Affiliation:
University of Oxford
Pentti Saikkonen*
Affiliation:
University of Helsinki
*
Address correspondence to Mika Meitz, Department of Economics, University of Oxford, Manor Road Building, Manor Road, Oxford, OX1 3UQ, United Kingdom; e-mail: mika.meitz@economics.ox.ac.uk.
Pentti Saikkonen, Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68, FIN-00014 University of Helsinki, Finland; e-mail: pentti.saikkonen@helsinki.fi.

Abstract

This paper studies a class of Markov models that consist of two components. Typically, one of the components is observable and the other is unobservable or “hidden.” Conditions under which geometric ergodicity of the unobservable component is inherited by the joint process formed of the two components are given. This implies existence of initial values such that the joint process is strictly stationary and β-mixing. In addition to this, conditions for the existence of moments are also obtained, and extensions to the case of nonstationary initial values are provided. All these results are applied to a general model that includes as special cases various first-order generalized autoregressive conditional heteroskedasticity (GARCH) and autoregressive conditional duration (ACD) models with possibly complicated nonlinear structures. The results only require mild moment assumptions and in some cases provide necessary and sufficient conditions for geometric ergodicity.

Type
Research Article
Copyright
Copyright © Cambridge University Press 2008

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ERGODICITY, MIXING, AND EXISTENCE OF MOMENTS OF A CLASS OF MARKOV MODELS WITH APPLICATIONS TO GARCH AND ACD MODELS
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