  This paper studies theory and inference of an observation-driven model for
time series of counts. It is assumed that the observations follow a Poisson
distribution conditioned on an accompanying intensity process, which is
equipped with a two-regime structure according to the magnitude of the lagged
observations. The model remedies one of the drawbacks of the Poisson
autoregression model by allowing possibly negative correlation in the
observations. Classical Markov chain theory and Lyapunov's method are utilized
to derive the conditions under which the process has a unique invariant
probability measure and to show a strong law of large numbers of the intensity
process. Moreover the asymptotic theory of the maximum likelihood estimates of
the parameters is established. A simulation study and a real data application
are considered, where the model is applied to the number of major earthquakes
in the world.
