Beta–Negative Binomial Auto-Regressions for Modelling Integer-Valued Time Series with Extreme Observations

Abstract

Summary 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.

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“…See for instance Davis and Wu [23], Zhu [24], and Chen et al [25]. Examples of other studies that have modelled count data but with different distributions are for instance Zhu [26], where the Conway-Maxwell-Poisson distribution is used to handle over-and under-dispersion, and Gorgi [27], where a beta-negative binomial distribution is used to increase the model robustness to extreme observations. The score driven beta-negative binomial model in Gorgi [27] can be seen as a more flexible version of our model, with one extra parameter to capture the effects of extreme events.…”

“…Examples of other studies that have modelled count data but with different distributions are for instance Zhu [26], where the Conway-Maxwell-Poisson distribution is used to handle over-and under-dispersion, and Gorgi [27], where a beta-negative binomial distribution is used to increase the model robustness to extreme observations. The score driven beta-negative binomial model in Gorgi [27] can be seen as a more flexible version of our model, with one extra parameter to capture the effects of extreme events. These studies are part of the much broader framework of integer-valued generalized autoregressive conditional heteroscedastic models (INGARCH, see Weiß [20,Chapter 4] for an overview).…”

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“…( 2016 ), Xu et al. ( 2020 ) and Gorgi ( 2020 ). The daily air quality level data for three major cities in China, including Beijing, Shanghai and Guangzhou, are analyzed.…”

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“…For integer-valued data the literature is more recent, although still flourishing. Suitable smoothing conditions on the nonlinear part of the model in this case are studied by Davis et al (2003), Fokianos et al (2009), Fokianos and Tjøstheim (2012), with Poisson data, Davis and Wu (2009), Christou and Fokianos (2014) for the Negative Binomial case, and Gorgi (2020) for the Beta Negative Binomial distribution. See Wang et al (2014) who introduced a threshold autoregressive model with Poisson distribution; for more general frameworks, Woodard et al (2011), Douc et al (2013), Ahmad and Francq (2016), Davis and Liu (2016), Douc et al (2017) and Aknouche and Francq (2021), to mention a few.…”