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Stability of nonlinear stochastic recursions with application to nonlinear AR-GARCH models

Part of: Markov processes Mathematical economics Inference from stochastic processes Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Daren B. H. Cline
Daren B. H. Cline*
Affiliation:
Texas A&M University
*
∗ Postal address: Department of Statistics, Texas A&M University, College Station, TX 77843-3143, USA. Email address: dcline@stat.tamu.edu
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Abstract

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We characterize the Lyapunov exponent and ergodicity of nonlinear stochastic recursion models, including nonlinear AR-GARCH models, in terms of an easily defined, uniformly ergodic process. Properties of this latter process, known as the collapsed process, also determine the existence of moments for the stochastic recursion when it is stationary. As a result, both the stability of a given model and the existence of its moments may be evaluated with relative ease. The method of proof involves piggybacking a Foster-Lyapunov drift condition on certain characteristic behavior of the collapsed process.

Keywords

Stochastic recursionGARCH modelergodicitystationary distribution

MSC classification

Primary: 60G10: Stationary processes 60J05: Discrete-time Markov processes on general state spaces
Secondary: 62M10: Time series, auto-correlation, regression, etc. 91B84: Economic time series analysis
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 39 , Issue 2 , June 2007 , pp. 462 - 491
Copyright
Copyright © Applied Probability Trust 2007 

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