Stability and the Lyapounov exponent of threshold AR-ARCH Models
  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. 
Daren B. H. Cline and Huay-min H. Pu