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

Mika Meitz · Pentti Saikkonen

135 citations0/0 files storedEconometric Theory

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Financial Risk and Volatility Modeling Probability and Statistical Research Mathematical Dynamics and Fractals

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.

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References

1986Generalized autoregressive conditional heteroskedasticity 1982Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation 1993On the Relation between the Expected Value and the Volatility of the Nominal Excess Return on Stocks 1998Autoregressive Conditional Duration: A New Model for Irregularly Spaced Transaction Data 1990Stationarity and Persistence in the GARCH(1,1) Model 2003ASYMPTOTIC THEORY FOR A VECTOR ARMA-GARCH MODEL 1996Modeling volatility persistence of speculative returns: A new approach 1995All in the family Nesting symmetric and asymmetric GARCH models

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2006MIXING PROPERTIES OF A GENERAL CLASS OF GARCH(1,1) MODELS WITHOUT MOMENT ASSUMPTIONS ON THE OBSERVED PROCESS 2008Nonparametric estimation of conditional VaR and expected shortfall 2009On asymptotic theory for multivariate GARCH models 2012Nonparametric Copula-Based Test for Conditional Independence with Applications to Granger Causality 2008Stability of nonlinear AR‐GARCH models

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