Beta–negative binomial auto‐regressions for modelling integer‐valued time series with extreme observations

The paper introduces a general class of heavy‐tailed auto‐regressions for modelling integer‐valued time series with outliers. The specification proposed 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 stationary and ergodic solution for the class of auto‐regressive processes is shown under general conditions. 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 studied. The score‐driven specification is shown to be particularly appealing as it delivers a robust filtering method that attenuates the effect of outliers. Empirical applications to the series of narcotics trafficking reports in Sydney and the euro–pound sterling exchange rate illustrate the effectiveness of the method in handling extreme observations.