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Modelling multivariate time series of counts in a parsimonious way is a popular topic. In this paper, we consider an integer‐valued network autoregressive model with a non‐random neighbourhood structure, which uses negative binomial distribution as the conditional marginal distribution and the softplus function as the link function. The new model generalizes existing ones in the literature and has a great flexibility in modelling. Stationary conditions in cases of fixed dimension and increasing dimension are given. Parameters are estimated by maximizing the quasi‐likelihood function, and related asymptotic properties of the estimators are established. A simulation study is conducted to assess performances of the estimators, and a real data example is analysed to show superior performances of the proposed model compared with existing ones. [ABSTRACT FROM AUTHOR]

This note examines the conditional least squares (CLS) estimators for integer-valued autoregressive (INAR) models, with a focus on the INAR(1) model. Our investigation reveals that the joint limiting distribution of the CLS estimators for the autoregressive coefficient and the mean of the model errors is degenerate in the case of mildly unstable INAR(1) models with moderate deviations from unity. This degeneracy may pose challenges for statistical inference. We provide commenting notes based on the asymptotic results to address this issue. Additionally, we conduct simulation studies and analyze a real-world dataset to validate the theoretical findings. [ABSTRACT FROM AUTHOR]

Zero inflation, zero deflation, overdispersion, and underdispersion are commonly encountered in count time series. To better describe these characteristics of counts, this article introduces a zero‐modified geometric first‐order integer‐valued autoregressive (INAR(1)) model based on the generalized negative binomial thinning operator, which contains dependent zero‐inflated geometric counting series. The new model contains the NGINAR(1) model, ZMGINAR(1) model, and GNBINAR(1) model with geometric marginals as special cases. Some statistical properties are studied, and estimates of the model parameters are derived by the Yule–Walker, conditional least squares, and maximum likelihood methods. Asymptotic properties and numerical results of the estimators are also studied. In addition, some test and forecasting problems are addressed. Three real‐data examples are given to show the flexibility and practicability of the new model. [ABSTRACT FROM AUTHOR]

In the past, most threshold models considered a single threshold variable. However, for some practical applications, models with two threshold variables may be needed. In this paper, we propose a two-threshold-variable integer-valued autoregressive model based on the binomial thinning operator and discuss some of its basic properties, including the mean, variance, strict stationarity, and ergodicity. We consider the conditional least squares (CLS) estimation and discuss the asymptotic normality of the CLS estimator under the known and unknown threshold values. The performances of the CLS estimator are compared via simulation studies. In addition, two real data sets are considered to underline the superior performance of the proposed model. [ABSTRACT FROM AUTHOR]

A very attractive property of the mixed integer-valued autoregressive model is that it can capture structural changes. Inspired by this, we propose a mixed generalized Poisson integer-valued autoregressive model with a mixture of the quasi-binomial and generalized Poisson thinning operators. The first-order and higher-order mixed models are introduced, respectively. Some properties of the mixed models are presented. Conditional maximum likelihood estimation is considered for the parameters of the proposed model. The mean, median and mode of the conditional distribution are used to predict the one-step-ahead values. Furthermore, an approximate Bayesian model averaging prediction with the Bayesian information criterion is introduced to the higher-order models in the application to real data. Some Monte Carlo simulation results are presented for assessing the performance of parameter estimation and forecasting methods. At last, the applications of proposed models to real data are given for illustrating purposes. [ABSTRACT FROM AUTHOR]

The temporal aggregation (TA) and systematic sampling (SS) in continuous-valued time series were widely studied, while the TA and SS in integer-valued time series obtain very little attention. In this paper, we thoroughly discuss them based on the integer-valued generalized autoregressive conditional heteroskedasticity (INGARCH) process. On one hand, the TA and SS for INGARCH processes are studied, and related predictors are discussed. We also illustrate that the TA or SS provides approaches to estimate INGARCH processes with different frequencies. On the other hand, definitions of the strong and weak INGARCH processes are given. The aggregated INGARCH and sampled INGARCH processes are shown to be weak, that is, they satisfy a weak characterization of INGARCH processes in terms of linear projection. Quasi maximum likelihood and nonlinear least squares estimation methods are considered for the weak INGARCH processes. Related asymptotic properties and simulation results are discussed. Finally, an empirical example illustrates the applicability of our results. • Define strong and weak INGARCH processes for temporal aggregation for the first time. • Provide new methods for estimating and predicting the aggregated count time series. • Analyze an example to illustrate the applicability of our results. [ABSTRACT FROM AUTHOR]

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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]

Integer-valued time series models have been widely used, especially integer-valued autoregressive models and integer-valued generalized autoregressive conditional heteroscedastic (INGARCH) models. Recently, there has been a growing interest in multivariate count time series. However, existing models restrict the dependence structures imposed by the way they constructed. In this paper, we consider a class of flexible bivariate Poisson INGARCH(1,1) model whose dependence is established by a special multiplicative factor. Stationarity and ergodicity of the process are discussed. The maximization by parts algorithm and its modified version together with the alternative method by using R package Template Model Builder are employed to estimate the parameters of interest. The consistency and asymptotic normality for estimates are obtained, and the finite sample performance of estimators is given via simulations. A real data example is also provided to illustrate the model. [ABSTRACT FROM AUTHOR]

This article studies an observation-driven model for time series of counts, which allows for overdispersion and negative serial dependence in the observations. The observations are supposed to follow a negative binomial distribution conditioned on past information with the form of thresh old models, which generates a two-regime structure on the basis of the magnitude of the lagged observations. We use the weak dependence approach to establish the stationarity and ergodicity, and the inference for regression parameters are obtained by the quasi-likelihood. Moreover, asymptotic properties of both quasi-maximum likelihood estimators and the threshold estimator are established, respectively. Simulation studies are considered and so are two applications, one of which is the trading volume of a stock and another is the number of major earthquakes. [ABSTRACT FROM AUTHOR]

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]

The thinning operators play an important role in the analysis of integer-valued autoregressive models, and the most widely used is the binomial thinning. Inspired by the theory about extended Pascal triangles, a new thinning operator named extended binomial is introduced, which is a general case of the binomial thinning. Compared to the binomial thinning operator, the extended binomial thinning operator has two parameters and is more flexible in modeling. Based on the proposed operator, a new integer-valued autoregressive model is introduced, which can accurately and flexibly capture the dispersed features of counting time series. Two-step conditional least squares (CLS) estimation is investigated for the innovation-free case and the conditional maximum likelihood estimation is also discussed. We have also obtained the asymptotic property of the two-step CLS estimator. Finally, three overdispersed or underdispersed real data sets are considered to illustrate a superior performance of the proposed model. [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]