  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.
