Integer-valued Autoregressive Model Research Articles

Modeling Z-valued time series based on new versions of the Skellam INGARCH model

Recently, there has been a growing interest in integer-valued time series models, including integer-valued autoregressive (INAR) models and integer-valued generalized autoregressive conditional heteroscedastic (INGARCH) models, but only a few of them can deal with data on the full set of integers, that is, Z={…,−2,−1,0,1,2,…}. Although some attempts have been made to deal with Z-valued time series, these models do not provide enough flexibility in modeling some specific integers (e.g., 0, ±1). A symmetric Skellam INGARCH(1,1) model was proposed in the literature, but it only considered zero-mean processes, which limits its application. We first extend the symmetric Skellam INGARCH model to an asymmetric version, which can deal with non-zero-mean processes. Then we propose a modified Skellam model which adopts a careful treatment on integers 0 and ±1 to satisfy a special feature of the data. Our models are easy-to-use and flexible. The maximum likelihood method is used to estimate unknown parameters and the log-likelihood ratio test statistic is provided for testing the asymmetric model against the modified one. Simulation studies are given to evaluate performances of the parametric estimation and log-likelihood ratio test. A real data example is also presented to demonstrate good performances of newly proposed models.

Forecasting overdispersed INAR(1) count time series with negative binomial marginal

This paper addresses the coherent forecasting problem for overdispersed integer-valued autoregressive (INAR) model of order one having negative binomial marginal distribution. INAR models with Poisson or geometric marginal distribution have been used by several researchers to tackle the forecasting and related issues in low count time series. However, when the process results in relatively higher counts with overdispersion, these models do not provide satisfactory fit and good forecasts. We use negative binomial INAR(1) (NBINAR(1)) model for forecasting the count time series by deriving its exact forecast distribution. Extensive simulation study has been carried out to assess the performance of the forecasts obtained using NBINAR(1) with its INAR(1) counterparts. Two real data sets have been analyzed using the proposed methodology.

Recent progress in parameter change test for integer-valued time series models

In this study, we review a recent progress regarding the change point test for integer-valued time series models, specifically concentrating on the CUSUM test for integer-valued autoregressive (INAR) and generalized autoregressive conditional heteroscedastic (INGARCH) models. Because time series often experience changes in underlying models, the change point test has been a fundamental issue in time series analysis during the past decades. We first introduce the CUSUM test in a general set-up and then construct estimate-, score vector- and residual-based CUSUM tests in INAR and INGARCH models and state their limiting null distributions. Finally, the residual-based CUSUM of squares test and the robust change point test based on the density power divergence are addressed.

Forecasting the Number of Prisoners in Nganjuk With Integer-Valued Pth-Order Autoregressive (INAR(P))

The most used time series model is the time series model which assumes the variables being tested are continuous in a discrete time period and produces continuous values. Whereas in many applications, such as number of monthly accidents, number of doctor visits per year a person makes, etc., needed a discrete time series model to handle discrete variables and produce discrete values as well. Time series model that handles count or non-negative integer data is the Integer-valued Autoregressive model with the pth-order or INAR(p). INAR(p) is the counterpart of AR(p) model for integer data. To get integer results, this model uses binomial thinning operator which implements probabilistic operations with discrete distribution that are suitable to model count data such as Poisson and Binomial, also use median forecasting method. Model parameters will be estimated using the Yule-Walker method. In this research, INAR(p) time series model will be applied to number of prisoners in Nganjuk from April 2013 until July 2016 to help tackle overcapacity problems in prisons that led to many negative impacts. Through model specification, the best model for forecast the case is INAR(2). In this data, based on the measure of AIC, BIC, and AICc, the INAR(2) model achieved better performance than its AR(2) counterpart.

Interval Estimation of Random Coefficient Integer-Valued Autoregressive Model Based on Mean Empirical Likelihood Method

In this paper, we study the use of the mean empirical likelihood (MEL) method in a first-order random coefficient integer-valued autoregressive model. The MEL ratio statistic is established, its limiting properties are discussed, and the confidence regions for the parameter of interest are derived. Furthermore, a simulation study is presented to demonstrate the performance of the proposed method. Finally, a real data analysis of dengue fever is performed.

Scale-, time- and asset-dependence of Hawkes process estimates on high frequency price changes

The statistical estimate of the branching ratio η of the Hawkes model, when fitted to windows of mid-price changes, has been reported to approach criticality () as the fitting window becomes large. In this study – using price changes from the EUR/USD currency pair traded on the Electronic Broking Services (EBS) interbank trading platform and the S&P 500 E-mini futures contract traded at the Chicago Mercantile Exchange (CME) – it is shown that the estimated branching ratio depends little upon window size and is usually far from criticality. This is done by controlling for exogenous non-stationarities/heterogeneities at inter- and intraday scales, accomplished by using information criteria to select the degree of flexibility of the Hawkes immigration intensity, either piecewise constant or adaptive logspline, estimated using an expectation maximization (EM) algorithm. The (positive) bias incurred by keeping the immigration intensity constant is small for time scales up to two hours, but can become as high as 0.3 for windows spanning days. This emphasizes the importance of choosing an appropriate model for the immigration intensity in the application of Hawkes processes to financial data and elsewhere. The branching ratio is also found to have an intraday seasonality, where it appears to be higher during times where market activity is dominated by supposedly reflexive automated decisions and a lack of fundamental news and trading. The insights into the microstructure of the two considered markets derived from our Hawkes process fits suggest that equity futures exhibit more complex non-stationary features, are more endogenous, persistent and traded at higher speed than spot foreign exchange. We complement our point process study with EM-estimates of integer-valued autoregressive (INAR) time series models at even longer scales of months. Transferring our methodologies to the aggregate bin-count setting confirms that, even at these very long scales, criticality can be rejected.

Comparison of BINAR(1) models with bivariate negative binomial innovations and explanatory variables

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.

Modeling Digital Camera Monitoring Count Data with Intermittent Zeros for Short-Term Prediction

Digital camera monitoring has revolutionised survey designs in many fields, as an important source of information. The extended sampling coverage offered by this monitoring scheme makes it preferable compared to other traditional methods of survey. However, data obtained from digital camera monitoring are often highly variable, and characterized by sparse periods of zero counts, interspersed with missing observations due to outages. In practice, missing data of relatively shorter duration are mostly observed and are often imputed using interpolation techniques, ignoring long-term trends leading to inherent estimation biases. In this study, we investigated time series forecasting methods that adequately handle intermittency and produce plausible estimates for imputation and forecasting purposes. The study utilised a yearlong digital camera monitoring data set of hourly counts of powerboat launches at three boat ramps in Western Australia. Several time series forecasting methods were evaluated and the accuracies of their point estimates of forecasts for various lead times in hours of up to one week were assessed using cross-validation techniques. Intermittent demand forecasting techniques, including Croston’s method and Syntetos-Boylan Approximation (SBA) models, and count data forecasting methods including autoregressive conditional Poisson (ACP) models, integer-valued moving average (INMA) models, and integer-valued autoregressive (INAR) models were evaluated. ACP and INAR models performed better than intermittent demand forecasting techniques for short forecast horizons and provided some evidence of their sufficiency in predicting the dynamics in recreational boating activities. This result established that, in as much as intermittency may be a key feature for a given dataset, it should not override the systemic characteristics of data in the application of forecasting techniques. Our results provide plausible estimates for short-term missing data and forecasts for monitoring events, with applications in supporting proper tracking of usage of facilities, guiding resource allocations and providing insightful perspectives for management decisions.

The approximation of GSTAR model for discrete cases through INAR model

Modelling non-negative discrete valued for time series analysis was introduced by previous researchers, that is Integer-Valued Autoregressive (INAR) model. The model involves both the parameter index and state space are discrete. Various phenomena within the scope of discrete cases are not only affected by previous discrete observations, but also affected by the discrete observations at surrounding locations in previous time. As a result, this model involves two parameter indices namely time and space, and also state space which are discrete. In continuous cases, a model has been introduced to overcome this problem, namely Generalized Space Time Autoregressive (GSTAR) model (discrete parameter index and continuous state space). Using this model to solve the discrete case produces a continuous forecast value, so it needs a discretization process. In the time series model for discrete cases, thinning operators have been introduced as a substitute for discretization in a continuous time series. The thinning operator paradigm in the discrete time series will be developed in the space-time model proposed in this paper. This paper will also introduce a new model, namely a space-time model for cases related to discrete cases. This means that the index of time and location parameters, as well as the state space used is a discrete state space. The properties that apply to the model are adopted from the INAR model and the GSTAR model for continuous cases. The main focus of this paper is the development of thinning operators from time series to space-time models. This paper also provides a modeling procedure using the GSTAR model for discrete cases. The result of this paper is the formation of a GSTAR model with a discrete state space.

Model Averaging of Integer-Valued Autoregressive Model With Covariates

This article considers model averaging in the class of the integer-valued autoregressive (INAR) processes. The INAR process is a class of structural models that can be used to model dependent count data in fields such as medicine, economics, finance and marketing. It specifies the data generating process (DGP) via the distributions of certain thinning and arrival processes instead of an approximate multiplicative and additive relationship, which means that maximum likelihood (ML) is a natural candidate for estimation. Forecasts can not only be evaluated in terms of points and intervals but also through the entire forecast distribution. Here, we propose a K-fold cross-validation model averaging (KCVMA) estimator based on ML and prove that the KCVMA estimator is asymptotically optimal in a metric equivalent to a squared loss criterion. The KCVMA estimators are consistent, provided at least one candidate model is not underfitted. The probabilistic nature of the INAR process also enables us to conduct probabilistic forecasts based on the KCVMA estimator. Monte Carlo simulations and empirical analysis show that the KCVMA estimator can improve forecast accuracy over existing model averaging and model selection methods.

Forecasting the number of confirmed new cases of COVID-19 in Italy for the period from 19 May to 2 June 2020.

In this paper we forecast the spread of the coronavirus disease 2019 outbreak in Italy in the time window from May 19 to June 2, 2020. In particular, we consider the forecast of the number of new daily confirmed cases. A forecast procedure combining a log-polynomial model together with a first-order integer-valued autoregressive model is proposed. An out-of-sample comparison with forecasts from an autoregressive integrated moving average (ARIMA) model is considered. This comparison indicates that our procedure outperforms the ARIMA model. The Root Mean Square Error (RMSE) of the ARIMA is always greater than that of the our procedure and generally more than twice as high as the our procedure RMSE. We have also conducted Diebold and Mariano (1995) tests of equal mean square error (MSE). The tests results confirm that forecasts from our procedure are significantly more accurate at all horizons. We think that the advantage of our approach comes from the fact that it explicitly takes into account the number of swabs.

A New Extension of Thinning-Based Integer-Valued Autoregressive Models for Count Data.

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.

A geometric minification integer-valued autoregressive model

In this manuscript we introduce a new minification integer-valued autoregressive model of the first-order motivated to solve the problem which can arise when the binomial thinning or the negative binomial thinning operator are used. Namely, if one of these thinning operators is used in construction of the minification model, then it is possible that the model becomes zero constantly over time. As a solution for this problem, we construct a minification model by using a modification of the negative binomial thinning operator. Many important properties of the introduced model are derived and these properties are applied for the estimation of the unknown parameters and used to show the applicability and adequacy of the model. Three estimation methods are considered and the performances of the obtained estimates by these methods are checked through some simulations for different true values of the parameters. The performances of one-step-ahead predictions based on our model and four competitive models are checked through Monte-Carlo simulations. We have shown that our model with geometric marginal distributions can provide better fit in comparison with these competitive models applied to the well-known polio real data set. Finally, the adequacy of the model on this real data set is proved by using the parametric bootstrap approach.

Bayesian generalizations of the integer-valued autoregressive model

We develop two Bayesian generalizations of the Poisson integer-valued autoregressive model. The AdINAR(1) model accounts for overdispersed data by means of an innovation process whose marginal distributions are finite mixtures, while the DP-INAR(1) model is a hierarchical extension involving a Dirichlet process, which is capable of modeling a latent pattern of heterogeneity in the distribution of the innovations rates. The probabilistic forecasting capabilities of both models are put to test in the analysis of crime data in Pittsburgh, with favorable results.

First-order integer-valued autoregressive process with Markov-switching coefficients

A class of integer-valued autoregressive models is considered, which are based on a latten process e.g., a Markov chain. Some statistical properties of this class of models are discussed. Moreover, parameter estimation and forecasting are addressed. A Monte Carlo simulation study is conducted to examine the finite sample performance of the given estimation procedure. Finally, the performance of these models is illustrated by empirical applications to two sets of real counts.

A new one-parameter discrete distribution with associated regression and integer-valued autoregressive models

Abstract This study introduces the Poisson-Bilal distribution and its associated two models for modeling the over-dispersed count data sets. The Poisson-Bilal distribution has tractable properties and explicit forms for its statistical properties. A new over-dispersed count regression model and integer-valued autoregressive process with flexible innovation distribution are defined and studied comprehensively. Two real data sets are analyzed to prove empirically the importance of proposed models. Empirical findings show that the Poisson-Bilal distribution has important application fields in time series and regression modeling.

Efficient estimation in periodic [formula omitted] model: Nonparametric innovation distributions case

The efficient estimation problem of a semi-parametric periodic integer-valued autoregressive PINARp model of arbitrary order is considered. The unspecified distribution of the innovation process of this model is supposed to satisfy only some mild technical assumptions.We therefore provide efficient estimates for both parameters of the model, namely a periodic autoregressive parameter and a periodic probability law of the innovation non-negative integer values process which is seen as an infinite dimensional parameter. The performances of these efficient estimations are shown through an intensive simulation study and an application on real data set.

A dependent counting INAR model with serially dependent innovation

ABSTRACT To provide a more flexible model of count data, we extend the first-order integer-valued autoregressive model with serially dependent innovations based on the dependent thinning operator. This model is appropriate for modelling the number of dependent random events affecting each other when the number of new cases depend on the previous count through a linear functional relationship. Several statistical properties of the model are determined, parameters are estimated by some methods and their properties are studied via simulations. This study was carried out to investigate the efficiency of the new model by two real count data sets, the number of contagious diseases and robbery.

Scale-, time- and asset-dependence of Hawkes process estimates on high frequency price changes

The statistical estimate of the branching ratio η of the Hawkes model, when fitted to windows of mid-price changes, has been reported to approach criticality (η = 1) as the fitting window becomes large. In this study -- using price changes from the EUR/USD currency pair traded on the Electronic Broking Services (EBS) interbank trading platform and the S&P 500 E-mini futures contract traded at the Chicago Mercantile Exchange (CME) -- it is shown that the estimated branching ratio depends little upon window size and is usually far from criticality. This is done by controlling for exogenous non-stationarities/heterogeneities at inter- and intraday scales, accomplished by using information criteria to select the degree of flexibility of the Hawkes immigration intensity, either piecewise constant or adaptive logspline, estimated using an expectation maximization (EM) algorithm. The bias incurred by keeping the immigration intensity constant is small for time scales up to two hours, but can become as high as 0.3 for windows spanning days. This emphasizes the importance of choosing an appropriate model for the immigration intensity in the application of Hawkes processes to financial data and elsewhere. The branching ratio is also found to have an intraday seasonality, where it appears to be higher during times where market activity is dominated by supposedly reflexive automated decisions and a lack of fundamental news and trading. The insights into the microstructure of the two considered markets derived from our Hawkes process fits suggest that equity futures exhibit more complex non-stationary features, are more endogenous, persistent and traded at higher speed than spot foreign exchange. We complement our point process study with EM-estimates of integer-valued autoregressive (INAR) time series models at even longer scales of months. Transferring our methodologies to the aggregate bin-count setting confirms that even at these very long scales, criticality can be rejected.

Random environment binomial thinning integer-valued autoregressive process with Poisson or geometric marginal

To predict time series of counts with small values and remarkable fluctuations, an available model is the $r$ states random environment process based on the negative binomial thinning operator and the geometric marginal. However, we argue that the aforementioned model may suffer from the following two drawbacks. First, under the condition of no prior information, the overdispersed property of the geometric distribution may cause the predictions fluctuate greatly. Second, because of the constraints on the model parameters, some estimated parameters are close to zero in real-data examples, which may not objectively reveal the correlation relationship. For the first drawback, an $r$ states random environment process based on the binomial thinning operator and the Poisson marginal is introduced. For the second drawback, we propose a generalized $r$ states random environment integer-valued autoregressive model based on the binomial thinning operator to model fluctuations of data. Yule–Walker and conditional maximum likelihood estimates are considered and their performances are assessed via simulation studies. Two real-data sets are conducted to illustrate the better performances of the proposed models compared with some existing models.