Integer-valued GARCH Model Research Articles

Bayesian modeling of spatial integer-valued time series

Many infectious diseases spread through person-to-person contact, either directly or indirectly. The proposal incorporates spatial-temporal patterns into multivariate integer-valued GARCH models with either a generalized Poisson distribution or zero-inflated generalized Poisson distribution in order to describe these features of the data. By considering the neighboring locations of the target series, the set-up incorporates a flexible and continuous conceptualization of distance to present the spatial components, thereby highlighting for the non-separability of space and time. Such an approach eliminates the need to pre-assign a spatial weight matrix. Newly designed models are utilized to investigate time-series counts of infectious diseases, enabling inference, prediction, and model selection within a Bayesian framework through Markov chain Monte Carlo (MCMC) algorithms. As an illustration, design simulation studies and multivariate weekly dengue cases are scrutinized for the performance of the Bayesian methods. The proposed models successfully capture the characteristics of spatial dependency, over-dispersion, and a large portion of zeros, providing a comprehensive model for the observed phenomena in the data.

A covariate-driven beta-binomial integer-valued GARCH model for bounded counts with an application

A covariate-driven beta-binomial integer-valued GARCH model for bounded counts with an application

An alternative hyper-Poisson integer-valued GARCH model with application to polio, internet protocol and COVID-19 data

<abstract> <p>Time series of counts are observed widely in actuarial science, finance, epidemiology and biology. These time series may exhibit over-, equi- and under-dispersion. The Poisson distribution is commonly used in count time series models, but it is restricted by the equality of mean and variance. Other distributions such as the generalized Poisson, double Poisson, hyper-Poisson, and COM-Poisson distributions have been proposed to replace the Poisson distribution to model the different levels of dispersion in time series of counts. These models have certain limitations such as complex expressions for the mean and variance which complicate the formulation as GARCH models. In this study, we propose an alternative hyper-Poisson (AHP) distribution, with simple forms of conditional mean and variance, for an integer-valued GARCH (INGARCH) model for time series of counts that also exhibit the different levels of dispersion. We demonstrate that the AHP-INGARCH model is comparable to some existing INGARCH models. Additionally, the model can cover a wider range of dispersion. The maximum likelihood estimation can be used to estimate the parameters of the proposed model. Applications to three real-life data sets related to polio, internet protocol and daily COVID-19 new deaths underscore the usefulness of the proposed model in studying both over-dispersed and under-dispersed time series of counts.</p> </abstract>

Robust estimation for the one-parameter exponential family integer-valued GARCH(1,1) models based on a modified Tukey’s biweight function

Robust estimation for the one-parameter exponential family integer-valued GARCH(1,1) models based on a modified Tukey’s biweight function

Periodic stationarity conditions for mixture periodic INGARCH models

<abstract><p>This paper proposes strict periodic stationarity and periodic ergodicity conditions for a finite mixture integer-valued GARCH model with $ S $-periodic time-varying parameters that depend on the state of an independent and periodically distributed regime sequence. In this model, the past conditional mean values depend on the past of the regime variable in the same order, so the model is characterized by path-regime dependence. We also propose sufficient conditions for periodic stationarity when the conditional means are nonlinear of past observations. The results are applied to various discrete conditional distributions.</p></abstract>

Forecasting transaction counts with integer-valued GARCH models

Abstract Using numerous transaction data on the number of stock trades, we conduct a forecasting exercise with INGARCH models, governed by various conditional distributions; the Poisson, the linear and quadratic negative binomial, the double Poisson and the generalized Poisson. The model parameters are estimated with efficient Markov Chain Monte Carlo methods, while forecast evaluation is done by calculating point and density forecasts.

A new class of integer-valued GARCH models for time series of bounded counts with extra-binomial variation

A new class of integer-valued GARCH models for time series of bounded counts with extra-binomial variation

USING OBSERVATION DRIVEN MODELS TO FIT COUNT DAILY NEW CONFIRMED CASES OF “COVID 19” IN EGYPT

On 11 of March, the World Health Organization declared corona virus as a pandemic. This virus has spread in at least 218 countries with at least one case reported until the end of 2020. In Egypt, the first case was reported on 14/02/2020. At the end of 2020, it confirms its higher daily rise of new This paper suggests to use the autoregressive conditional Poisson (ACP) models, also known as integer-valued GARCH (briefly: INGARCH) models which belong to the observation driven models (ODMs) category to model the count daily new confirmed cases of COVID 19 in Egypt. We compare between the robust and the non-robust estimators for the Poisson INARCH(1) model as a simple class of the INGARCH (p, q) models by simulations. The application results indicate that the robust version of the Poisson INARCH(1) model is more reasonable to fit the number of new confirmed cases of COVID 19.

Robust Estimation for Poisson Integer-Valued GARCH Models Using a New Hybrid Loss

The Poisson integer-valued GARCH model is a popular tool in modeling time series of counts. The commonly used maximum likelihood estimator is strongly influenced by outliers, so there is a need to develop a robust M-estimator for this model. This paper has three aims. First, the authors propose a new loss function, which is a hybrid of the tri-weight loss for relatively small errors and the exponential squared loss for relatively large ones. Second, Mallows’ quasi-likelihood estimator (MQLE) is proposed as an M-estimator and its existence, uniqueness, consistency and asymptotic normality are established. In addition, a data-adaptive algorithm for computing MQLE is given based on a data-driven selection of tuning parameters in the loss function. Third, simulation studies and analysis of a real example are conducted to illustrate the performance of the new estimator, and a comparison with existing estimators is made.

Bayesian inference of nonlinear hysteretic integer-valued GARCH models for disease counts

This study proposes a class of nonlinear hysteretic integer-valued GARCH models in order to describe the occurrence of weekly dengue hemorrhagic fever cases via three meteorological covariates: precipitation, average temperature, and relative humidity. The proposed model adopts the hysteretic three-regime switching mechanism with a buffer zone that are able to explain various characteristics. This allows for having consecutive zeros in the lower regime and large counts to appear up in the upper regime. These nonlinear hysteretic integer-valued GARCH models include Poisson, negative binomial, and log-linked forms. We utilize adaptive Markov chain Monte Carlo simulations for making inferences and prediction and employ two Bayesian criteria for model comparisons and the relative root mean squared prediction error for evaluation. Simulation and analytic results emphasize that the hysteretic negative binomial integer-valued GARCH model is superior to other models and successfully offers an alternative nonlinear integer-valued GARCH model to better describe larger values of counts.

A generalized mixture integer-valued GARCH model

We propose a generalized mixture integer-valued generalized autoregressive conditional heteroscedastic model to provide a more flexible modeling framework. This model includes many mixture integer-valued models with different distributions already studied in the literature. The conditional and unconditional moments are discussed and the necessary and sufficient first- and second-order stationary conditions are derived. We also investigate the theoretical properties such as strict stationarity and ergodicity for the mixture process. The conditional maximum likelihood estimators via the EM algorithm are derived and the performances of the estimators are studied via simulation. The model can be selected in terms of both the number of mixture regimes and the number of orders in each regime by several different criteria. A real-life data example is also given to assess the performance of the model.

Residual-based CUSUM of squares test for Poisson integer-valued GARCH models

ABSTRACTThis study considers the problem of testing for a parameter change in integer-valued time series models in which the conditional density of current observations is assumed to follow a Poisson distribution. As a test, we consider the CUSUM of the squares test based on the residuals from INGARCH models and find that the test converges weakly to the supremum of a Brownian bridge. A simulation study demonstrates its superiority to the residual and standardized residual-based CUSUM tests of Kang and Lee [Parameter change test for Poisson autoregressive models. Scand J Statist. 2014;41:1136–1152] and Lee and Lee [CUSUM tests for general nonlinear inter-valued GARCH models: comparison study. Ann Inst Stat Math. 2019;71:1033–1057.] as well as the CUSUM of squares test based on standardized residuals.

Monitoring of count data time series: Cumulative sum change detection in Poisson integer valued GARCH models

This article presents a cumulative sum (CUSUM) monitoring approach for count-data time series. A seasonal integer-valued generalized autoregressive conditional heteroscedasticity (INGARCH(1,1)) time series model with Poisson deviates is used to develop a likelihood ratio test formulation to detect changes in the process accounting for temporal correlations and seasonality. Simulation studies show that the proposed CUSUM monitoring approach can provide significantly improved performance in applications where serial correlation or seasonality is prevalent. A case study with real traffic crash counts is presented to illustrate the application of the proposed methodology for roadway safety improvement.

A negative binomial mixture integer-valued GARCH model

This paper generalizes the negative binomial integer-valued GARCH model (NBINGARCH) to a negative binomial mixture integer-valued GARCH (NBMINGARCH) for modeling time series of counts with presence of overdispersion. This class of models consists of a mixture of K stationary or non-stationary negative binomial integer-valued GARCH components. The advantage of these models over the NBINGARCH models includes the ability to handle multimodality and nonstationary components. Compared to the MINGARCH models, this class of models is more flexible to describe the greater degrees of overdispersion. The necessary and sufficient first and second order stationarity conditions are investigated. The estimation of parameters is done through an EM algorithm and the model is selected by some information criterions. Some simulation results and real data application are provided. Keywords: Time series of counts, Mixture model, Negative binomial, Overdispersion. AMS 2010 Mathematics Subject Classification : 62F10, 62M10

A generalized regime-switching integer-valued GARCH(1, 1) model and its volatility forecasting

A generalized regime-switching integer-valued GARCH(1, 1) model and its volatility forecasting

Zero-truncated compound Poisson integer-valued GARCH models for time series

ABSTRACTStarting from the compound Poisson INGARCH models, we introduce in this paper a new family of integer-valued models suitable to describe count data without zeros that we name zero-truncated CP-INGARCH processes. For such class of models, a probabilistic study concerning moments existence, stationarity and ergodicity is developed. The conditional quasi-maximum likelihood method is introduced to consistently estimate the parameters of a wide zero-truncated compound Poisson subclass of models. The conditional maximum likelihood method is also used to estimate the parameters of ZTCP-INGARCH processes associated with well-specified conditional laws. A simulation study that compares some of those estimators and illustrates their finite distance behaviour as well as a real-data application conclude the paper.

Mean targeting estimator for the integer-valued GARCH(1, 1) model

The integer-valued GARCH model is commonly used in modeling time series of counts. Maximum likelihood estimation (MLE) is used to estimate unknown parameters, but numerical results for MLE are sensitive to the choice of initial values, which also occurs in estimating the GARCH model. To alleviate this numerical difficulty, we propose an alternative to MLE and name it as mean targeting estimation (MTE), which is an analogue to variance targeting estimation used in the GARCH model. Consistency and asymptotic normality for MTE are established. Comparisons with the standard MLE are provided and the merits of the mean targeting method are discussed. In particular, it is shown that MTE can be superior to MLE for estimating parameters or prediction when the model is well specified and misspecified. We conduct numerical studies to confirm our theoretical findings and illustrate the practical utility of our proposals.

A new bivariate integer-valued GARCH model allowing for negative cross-correlation

Univariate integer-valued time series models, including integer-valued autoregressive (INAR) models and integer-valued generalized autoregressive conditional heteroscedastic (INGARCH) models, have been well studied in the literature, but there is little progress in multivariate models. Although some multivariate INAR models were proposed, they do not provide enough flexibility in modeling count data, such as volatility of numbers of stock transactions. Then, a bivariate Poisson INGARCH model was suggested by Liu (Some models for time series of counts, Dissertations, Columbia University, 2012), but it can only deal with positive cross-correlation between two components. To remedy this defect, we propose a new bivariate Poisson INGARCH model, which is more flexible and allows for positive or negative cross-correlation. Stationarity and ergodicity of the new process are established. The maximum likelihood method is used to estimate the unknown parameters, and consistency and asymptotic normality for estimators are given. A simulation study is given to evaluate the estimators for parameters of interest. Real and artificial data examples are illustrated to demonstrate good performances of the proposed model relative to the existing model.

Monitoring parameter shift with Poisson integer-valued GARCH models

ABSTRACTThis study examines the statistical process control chart used to detect a parameter shift with Poisson integer-valued GARCH (INGARCH) models and zero-inflated Poisson INGARCH models. INGARCH models have a conditional mean structure similar to GARCH models and are well known to be appropriate to analyzing count data that feature overdispersion. Special attention is paid in this study to conditional and general likelihood ratio-based (CLR and GLR) CUSUM charts and the score function-based CUSUM (SFCUSUM) chart. The performance of each of the proposed methods is evaluated through a simulation study, by calculating their average run length. Our findings show that the proposed methods perform adequately, and that the CLR chart outperforms the GLR chart when there is an increased shift of parameters. Moreover, the use of the SFCUSUM chart in particular is found to lead to a lower false alarm rate than the use of the CLR chart.

Branching processes in generalized autoregressive conditional environments

AbstractBranching processes in random environments have been widely studied and applied to population growth systems to model the spread of epidemics, infectious diseases, cancerous tumor growth, and social network traffic. However, Ebola virus, tuberculosis infections, and avian flu grow or change at rates that vary with time—at peak rates during pandemic time periods, while at low rates when near extinction. The branching processes in generalized autoregressive conditional environments we propose provide a novel approach to branching processes that allows for such time-varying random environments and instances of peak growth and near extinction-type rates. Offspring distributions we consider to illustrate the model include the generalized Poisson, binomial, and negative binomial integer-valued GARCH models. We establish conditions on the environmental process that guarantee stationarity and ergodicity of the mean offspring number and environmental processes and provide equations from which their variances, autocorrelation, and cross-correlation functions can be deduced. Furthermore, we present results on fundamental questions of importance to these processes—the survival-extinction dichotomy, growth behavior, necessary and sufficient conditions for noncertain extinction, characterization of the phase transition between the subcritical and supercritical regimes, and survival behavior in each phase and at criticality.