Integer‐Valued Autoregressive Models With Survival Probability Driven By A Stochastic Recurrence Equation

Abstract

This paper proposes a new class of integer‐valued autoregressive models with a dynamic survival probability. The peculiarity of this class of models lies in the specification of the survival probability through a stochastic recurrence equation. The proposed models can effectively capture changing dependence over time and enhance both the in‐sample and out‐of‐sample performance of integer‐valued autoregressive models. This point is illustrated through an empirical application to a real‐time series of crime reports. Additionally, this paper discusses the reliability of likelihood‐based inference for the class of models. In particular, this study proves the consistency of the maximum likelihood estimator and a plug‐in estimator for the conditional probability mass function in a misspecified model setting.

On a flexible construction of a negative binomial model

“…(13) Table 5 shows the results in terms of point and density forecasting at different horizons h. We note that our model performs well, when compared to the different models proposed in Gorgi (2018), in particular when compared to the GAS-NBINAR model. Overall, we can conclude that the negative-binomial continuous-time Markov process performs well for estimation and forecasting purposes.…”

“…This section is devoted to the study of the monthly number of offensive conduct reported in the city of Blacktown, Australia, from January 1995 to December 2014. Following Gorgi (2018), we employ our Gorgi (2018), this indicates over-dispersion, and thus a negative binomial distribution for the error term is a suitable assumption for the data. In Table 4, we compare the estimation values of (r, q, c) and the computational times of our exact transition probability (Exact), Joe's transition probability by computing numerically the integral in Matlab (Numerical Inversion 1) and Joe's transition probability by using an arbitrary precision integral (Numerical Inversion 2).…”

“…In particular, we give a closed expression for the transition probability corresponding to Zhu and Joe (2010b) model. We further illustrate the convenience of having a simple expression for transition probabilities via a simulation study and a real data analysis based on a crime reports dataset studied by Gorgi (2018). The performance of our approach is compared, in terms of computational time, with the method of Zhu and Joe (2010b) when the transition is evaluated with numerical methods.…”

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

“…BNB–GAS can approximate arbitrarily well some existing models that have been proposed in the literature. As α →∞, BNB–GAS becomes a score‐driven model with negative binomial distribution; see Gorgi (2018) for an application of the GAS framework with negative binomials. As additionally r →∞, the model becomes a Poisson auto‐regressive model, which belongs to the class of models of Davis et al .…”

Zero-Inflated Autoregressive Conditional Duration Model for Discrete Trade Durations With Excessive Zeros

“…Koopman and Lit (2017) used the bivariate Poisson distribution for a number of goals in football matches and the Skellam distribution for a score difference. Gorgi (2018) used the Poisson distribution as well as the negative binomial distribution for offensive conduct reports.…”