Stability and the Lyapounov exponent of threshold AR-ARCH Models

Details Stability and the Lyapounov exponent of threshold AR-ARCH Models Stability and the Lyapounov exponent of threshold AR-ARCH Models

2005-03-24

arxiv.org/abs/math/0503547v1

Annals of Applied Probability 2004, Vol. 14, No. 4, 1920-1949

Mathematics

Probability

Published at http://dx.doi.org/10.1214/105051604000000431 in the Annals of Applied Probability (http://www.imstat.org/aap/) by

Scientific paper

10.1214/105051604000000431

The Lyapounov exponent and sharp conditions for geometric ergodicity are determined of a time series model with both a threshold autoregression term and threshold autoregressive conditional heteroscedastic (ARCH) errors. The conditions require studying or simulating the behavior of a bounded, ergodic Markov chain. The method of proof is based on a new approach, called the piggyback method, that exploits the relationship between the time series and the bounded chain. The piggyback method also provides a means for evaluating the Lyapounov exponent by simulation and provides a new perspective on moments, illuminating recent results for the distribution tails of GARCH models.

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