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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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  • Mika Meitz
  • Pentti Saikkonen
  • University of Helsinki

Abstract

This paper studies a class of Markov models which 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 â-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 which includes as special cases various first order generalized autoregressive conditional heteroskedasticity (GARCH) and autoregressive conditional duration (ACD) models with possibly complicated non-linear structures. The results only require mild moment assumptions and in some cases provide necessary and sufficient conditions for geometric ergodicity.

Suggested Citation

  • Mika Meitz & Pentti Saikkonen & University of Helsinki, 2007. "Ergodicity, mixing, and existence of moments of a class of Markov models with applications to GARCH and ACD models," Economics Series Working Papers 327, University of Oxford, Department of Economics.
    • Handle: RePEc:oxf:wpaper:327
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    More about this item

    Keywords

    Generalized Autoregressive Conditional Heteroskedasticity; Autoregressive Conditional Duration; GARCH-in-mean; Nonlinear Time Series Models; Geometric Ergodicity; Mixing; Strict Stationarity; Existence of Moments; Markov Models;
    All these keywords.

    JEL classification:

    • C10 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - General
    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes

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