  This article introduces a general class of heavy-tailed autoregressions for
modeling integer-valued time series with outliers. The proposed specification
is based on a heavy-tailed mixture of negative binomial distributions that
features an observation-driven dynamic equation for the conditional
expectation. The existence of a unique stationary and ergodic solution for the
class of autoregressive processes is shown under a general contraction
condition. The estimation of the model can be easily performed by Maximum
Likelihood given the closed form of the likelihood function. The strong
consistency and the asymptotic normality of the estimator are formally derived.
Two examples of specifications illustrate the flexibility of the approach and
the relevance of the theoretical results. In particular, a linear dynamic
equation and a score-driven equation for the conditional expectation are
considered. The score-driven specification is shown to be particularly
appealing as it delivers a robust filtering method that attenuates the impact
of outliers. An empirical application to the time series of narcotics
trafficking reports in Sydney illustrates the effectiveness of the method in
handling extreme observations.
