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Several methods have been proposed for detecting zero modification in the first-order integer-valued autoregressive (INAR(1)) process. A basic problem of these tests is that they rely upon asymptotic results. In this paper, an alternative test is introduced which makes direct use of the approximate distribution of the number of zeros, which can be described by a beta-binomial distribution. A hybrid estimator of the mean parameter of the marginal distribution of the Poisson INAR(1) process is given. A simulation study shows that power and size of the proposed test are competitive. Finally, real data examples are provided. [ABSTRACT FROM AUTHOR]

Recently, there has been a growing interest in integer-valued time series models, especially in multivariate models. Motivated by the diversity of the infinite-patch metapopulation models, we propose an extension to the popular bivariate INAR(1) model, whose innovation vector is assumed to be time-dependent in the sense that the mean of the innovation vector is linearly increased by the previous population size. We discuss the stationarity and ergodicity of the observed process and its subprocesses. We consider the conditional maximum likelihood estimate of the parameters of interest, and establish their large-sample properties. The finite sample performance of the estimator is assessed via simulations. Applications on crime data illustrate the model. [ABSTRACT FROM AUTHOR]

To study urban air quality, this paper proposes a novel categorical time series model, which is based on a linear combination of bounded Poisson distribution and discrete distribution to describe the dynamic and systemic features of air quality, respectively. Daily air quality level data of three major cities in China, including Beijing, Shanghai and Guangzhou, are analyzed. It is concluded that the air quality in Beijing is the worst among the three cities but is gradually improving, and its dynamics is also the most pronounced. Theoretically, the design of our model increases the flexibility of the probabilistic structure while ensuring a dynamic feedback mechanism without high computational stress. We estimate the parameters through an adaptive Bayesian Markov chain Monte Carlo sampling scheme and show the satisfactory finite sample performance of the model through simulation studies. [ABSTRACT FROM AUTHOR]

Inspired by the study of air quality level data, this article proposes a new model for the normalcy‐dominant ordinal time series. The proposed model is based on a new zero‐one‐inflated bounded Poisson distribution with an autoregressive feedback mechanism in intensity. Under certain conditions, the stationarity and maximum likelihood estimation are established for the model. Moreover, a Lagrange multiplier test is constructed to detect the inflation phenomenon in the model. Applications find that the model can adequately capture the air quality level data in 30 major cities in China. More importantly, this article uses the fitted models to make the overall and dynamic air quality rankings for these cities, and finds that both rankings are rational and informative to the public. [ABSTRACT FROM AUTHOR]

The Glosten–Jagannathan–Runkle GARCH (GJR‐GARCH) model is popular in accounting for asymmetric responses in the volatility in the analysis of continuous‐valued financial time series, but asymmetric responses in the volatility are also observed in time series of counts or ℤ‐valued time series, such as the daily number of stock transactions or the daily stock returns divided by tick price (1 cent). Two different integer‐valued GARCH models based on Poisson distribution have been proposed for these two types of discrete data respectively. Shifted geometric distribution is more flexible than Poisson distribution, whose variance is greater than its mean. In this article, we propose a GJR‐GARCH model based on shifted geometric distribution for ℤ‐valued time series exhibiting asymmetric volatility. Basic probabilistic properties of the new model are given, and the maximum likelihood method is used to estimate unknown parameters and the asymptotic normality of corresponding estimators is established. A simulation study is presented to illustrate the estimation method. An empirical application to a real data concerning the daily stock returns divided by tick price is considered to show the proposed model's superiority compared with existing models. [ABSTRACT FROM AUTHOR]

The bivariate integer-valued autoregressive model of order 1 (BINAR(1)) is popular in fitting bivariate time series of counts, and the bivariate negative binomial (BNB) distribution can be chosen as its innovation's distribution, which is more flexible than the traditional bivariate Poisson distribution. It is well known that BNB distributions can be constructed in different ways, and these distributions will be reviewed in this paper. Performances of BINAR(1) models based on these BNB distributions with explanatory variables being included in the survival probability are compared. To estimate unknown parameters, the conditional maximum likelihood method is considered and evaluated by Monte Carlo simulations. Two sales counts are used to compare performances of the above models, and some interesting conclusions are also given. [ABSTRACT FROM AUTHOR]

Abstract: Overdispersion in time series of counts is very common and has been well studied by many authors, but the opposite phenomenon of underdispersion may also be encountered in real applications and receives little attention. Based on popularity of the generalized Poisson distribution in regression count models and of Poisson INGARCH models in time series analysis, we introduce a generalized Poisson INGARCH model, which can account for both overdispersion and underdispersion. Compared with the double Poisson INGARCH model, conditions for the existence and ergodicity of such a process are easily given. We analyze the autocorrelation structure and also derive expressions for moments of order 1 and 2. We consider the maximum likelihood estimators for the parameters and establish their consistency and asymptotic normality. We apply the proposed model to one overdispersed real example and one underdispersed real example, respectively, which indicates that the proposed methodology performs better than other conventional model-based methods in the literature. [Copyright &y& Elsevier]

A Poisson distribution is commonly used as the innovation distribution for integer-valued autoregressive models, but its mean is equal to its variance, which limits flexibility, so a flexible, one-parameter, infinitely divisible Bell distribution may be a good alternative. In addition, for a parameter with a small value, the Bell distribution approaches the Poisson distribution. In this paper, we introduce a new first-order, non-negative, integer-valued autoregressive model with Bell innovations based on the binomial thinning operator. Compared with other models, the new model is not only simple but also particularly suitable for time series of counts exhibiting overdispersion. Some properties of the model are established here, such as the mean, variance, joint distribution functions, and multi-step-ahead conditional measures. Conditional least squares, Yule–Walker, and conditional maximum likelihood are used for estimating the parameters. Some simulation results are presented to access these estimates' performances. Real data examples are provided. [ABSTRACT FROM AUTHOR]

• Heavy-tailedness is not formally considered for time series of counts. • The paper is the first attempt for modelling heavy-tailedness. • The proposed INAR model can capture other common features of dependent counts. • The considered model outperforms other models. • Numbers of cases, NSF fundings and transactions are analyzed, respectively. Count data frequently exhibit overdispersion, zero inflation and even heavy-tailedness (the tail probabilities are non-negligible or decrease very slowly) in practical applications. Many models have been proposed for modelling count data with overdispersion and zero inflation, but heavy-tailedness is less considered. The proposed model, a new integer-valued autoregressive process with generalized Poisson-inverse Gaussian innovations, is capable of capturing these features. The generalized Poisson-inverse Gaussian family is very flexible, which includes Poisson distribution, Poisson inverse Gaussian distribution, discrete stable distribution and so on. Stationarity and ergodicity of this model are investigated and the expressions of marginal mean and variance are provided. Conditional maximum likelihood is used for estimating the parameters, and consistency and asymptotic normality for the estimators are presented. Further, we consider the h -step forecast and diagnostics for the proposed model. The proposed model is applied to three real data examples. In the first example, we consider the monthly number of cases of Polio, which validates that the proposed model can take into account count data with excessive zeros. Then, we illustrate the use of the proposed model through an application to the numbers of National Science Foundation fundings. Finally, we apply the proposed model to the numbers of transactions in 5-min intervals for the stock traded at Empire District Electric Company. The second and third examples show that the proposed model has a good performance in modelling heavy-tailed count data. [ABSTRACT FROM AUTHOR]