BNB autoregressions for modeling integer-valued time series with extreme observations

Abstract

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.