Stability and the Lyapounov exponent of threshold AR-ARCH Models
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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. Institute of Mathematical Statistics, 2004.
The Annals of Applied Probability
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Cline, Daren BH||Pu, Huay-min H