Count Time Series Models Research Articles

Modelling ℤ-valued time series with Skellam thinning-based INAR(1) process

ABSTRACT Non-stationary count time series, which are small in value and show a trend having relatively large fluctuation, are commonly encountered in real-world applications. Differencing is a commonly employed method to transform a non-stationary time series into a stationary one. However, the differenced count time series is defined on Z and cannot be fitted by the mainstream count time series models, which are defined on N 0 . Although some integer-valued autoregressive (INAR) models based on the signed thinning operator have been proposed for fitting Z -valued time series, these models still encounter difficulties in terms of data generation mechanism, model selection, and parameter estimation. These difficulties may pose challenges for practitioners in practical applications. Therefore, this article utilizes the reparameterized version of the Skellam (Poisson difference) distribution to construct a new thinning operator, thereby introducing a class of Z -valued time series models. Strict stationarity, ergodicity, statistical properties, and parameter estimation methods of the processes are detailedly discussed. An application to the monthly counts of newly listed companies illustrates that the new model is superior in data analysis of finance and economics.

A Copula Discretization of Time Series-Type Model for Examining Climate Data

The study presents a comparative analysis of climate data under two scenarios: a Gaussian copula marginal regression model for count time series data and a copula-based bivariate count time series model. These models, built after comprehensive simulations, offer adaptable autocorrelation structures considering the daily average temperature and humidity data observed at a regional airport in Mobile, AL.

What drives cryptocurrency pump and dump schemes: Coin versus market factors?

This paper investigates both coin-specific and market-based factors that drive cryptocurrency pump-and-dump schemes. It analyzes a data set comprising 1457 pump events that occurred from January 3, 2018, to January 2, 2022. Empirical findings, derived from binary cross-sectional regression models, reveal several characteristics that increase the likelihood of cryptocurrencies being pumped. These include lower market capitalization, lower trading volume, greater social media popularity, increased developer activity, and fewer exchanges trading them. Furthermore, the study employs count time-series models to examine market-based factors. The results indicate that periods of higher volatility or uncertainty are associated with an increase in pre-announced pump-and-dump activities. Additionally, the analysis shows that macroeconomic factors and specific time-related effects – such as Sundays, certain months, and the COVID-19 period – are significant in explaining the frequency of pump occurrences. Based on these findings, the article discusses several targeted recommendations.

Specifications tests for count time series models with covariates

AbstractWe propose a goodness-of-fit test for a class of count time series models with covariates which includes the Poisson autoregressive model with covariates (PARX) as a special case. The test criteria are derived from a specific characterization for the conditional probability generating function, and the test statistic is formulated as a $$L_2$$ L 2 weighting norm of the corresponding sample counterpart. The asymptotic properties of the proposed test statistic are provided under the null hypothesis as well as under specific alternatives. A bootstrap version of the test is explored in a Monte–Carlo study and illustrated on a real data set on road safety.

Comparison of estimation and prediction methods for a zero-inflated geometric INAR(1) process with random coefficients

This study explores zero-inflated count time series models used to analyze data sets with characteristics such as overdispersion, excess zeros, and autocorrelation. Specifically, we investigate the ZIGINAR RC ( 1 ) process, a first-order stationary integer-valued autoregressive model with random coefficients and a zero-inflated geometric marginal distribution. Our focus is on examining various estimation and prediction techniques for this model. We employ estimation methods, including Whittle, Taper Spectral Whittle, Maximum Empirical Likelihood, and Sieve Bootstrap estimators for parameter estimation. Additionally, we propose forecasting approaches, such as median, Bayesian, and Sieve Bootstrap methods, to predict future values of the series. We assess the performance of these methods through simulation studies and real-world data analysis, finding that all methods perform well, providing 95% highest predicted probability intervals that encompass the observed data. While Bayesian and Bootstrap methods require more time for execution, their superior predictive accuracy justifies their use in forecasting.

Determining economic factors for sex trafficking in the United States using count time series regression

The article presents a robust quantitative approach for determining significant economic factors for sex trafficking in the United States. The aim is to study monthly counts of sex trafficking-related convictions, and use a wide range of economic variables as covariates to investigate their effect on conviction counts. A count time series model is considered along with a regression setup to include economic time series as covariates (economic factors) to explain the counts on sex trafficking-related convictions. The statistical significance of these economic factors is investigated and the significant factors are ranked based on appropriate model selection methods. The inclusion of time-lagged versions of the economic factor time series in the regression model is also explored. Our findings indicate that economic factors relating to immigration policy, consumer price index and labor market regulations are the most significant in explaining sex trafficking convictions.

Integer-Valued Split-BREAK Process with a General Family of Innovations and Application to Accident Count Data Modeling

This paper presents a novel count time-series model, named integer-valued Split-BREAK process of the first order, abbr. INSB(1) model. This process is examined in terms of its basic stochastic properties, such as stationarity, mean, variance and correlation structure. In addition, the marginal distribution, over-dispersion and zero-inflation properties of the INSB(1) process are also examined. To estimate the unknown parameters of the INSB(1) process, an estimation procedure based on probability generating functions (PGFs) is proposed. For the obtained estimators, their asymptotic properties, as well as the appropriate simulation study, are examined. Finally, the INSB(1) process is applied in the dynamic analysis of some real-world series, namely, the numbers of serious traffic accidents in Serbia and forest fires in Greece.

A multivariate heavy-tailed integer-valued GARCH process with EM algorithm-based inference

A new multivariate integer-valued Generalized AutoRegressive Conditional Heteroscedastic (GARCH) process based on a multivariate Poisson generalized inverse Gaussian distribution is proposed. The estimation of parameters of the proposed multivariate heavy-tailed count time series model via maximum likelihood method is challenging since the likelihood function involves a Bessel function that depends on the multivariate counts and its dimension. As a consequence, numerical instability is often experienced in optimization procedures. To overcome this computational problem, two feasible variants of the expectation-maximization (EM) algorithm are proposed for estimating the parameters of our model under low and high-dimensional settings. These EM algorithm variants provide computational benefits and help avoid the difficult direct optimization of the likelihood function from the proposed process. Our model and proposed estimation procedures can handle multiple features such as modeling of multivariate counts, heavy-tailedness, overdispersion, accommodation of outliers, allowances for both positive and negative autocorrelations, estimation of cross/contemporaneous-correlation, and the efficient estimation of parameters from both statistical and computational points of view. Extensive Monte Carlo simulation studies are presented to assess the performance of the proposed EM algorithms. Two empirical applications of our approach are provided. The first application concerns modeling bivariate count time series data on cannabis possession-related offenses in Australia, while the second one involves modeling intraday high-frequency financial transactions data from multiple holdings in the U.S. financial market.

Doubly-Inflated Poisson INGARCH Models for Count Time Series

Doubly-Inflated Poisson INGARCH Models for Count Time Series

Conditional-mean multiplicative operator models for count time series

Multiplicative error models (MEMs) are commonly used for real-valued time series, but they cannot be applied to discrete-valued count time series as the involved multiplication would not preserve the integer nature of the data. Thus, the concept of a multiplicative operator for counts is proposed (as well as several specific instances thereof), which are then used to develop a kind of MEMs for count time series (CMEMs). If equipped with a linear conditional mean, the resulting CMEMs are closely related to the class of so-called integer-valued generalized autoregressive conditional heteroscedasticity (INGARCH) models and might be used as a semi-parametric extension thereof. Important stochastic properties of different types of INGARCH-CMEM as well as relevant estimation approaches are derived, namely types of quasi-maximum likelihood and weighted least squares estimation. The performance and application are demonstrated with simulations as well as with two real-world data examples.

Multivariate epidemic count time series model.

An infectious disease spreads not only over a single population or community but also across multiple and heterogeneous communities. Moreover, its transmissibility varies over time because of various factors such as seasonality and epidemic control, which results in strongly nonstationary behavior. In conventional methods for assessing transmissibility trends or changes, univariate time-varying reproduction numbers are calculated without taking into account transmission across multiple communities. In this paper, we propose a multivariate-count time series model for epidemics. We also propose a statistical method for estimating the transmission of infections across multiple communities and the time-varying reproduction numbers of each community simultaneously from a multivariate time series of case counts. We apply our method to incidence data for the novel coronavirus disease 2019 (COVID-19) pandemic to reveal the spatiotemporal heterogeneity of the epidemic process.

Investigation of Factors Affecting Housing Sales in Turkey in Terms of Covid-19 Period: Count Time Series Model

Türkiye’de ekonomik büyümenin önemli bileşenlerinden biri konut sektörüdür. Konut arz ve talebinin dengesinin sağlanmasında ise konut satış tahminlerinin yeni nesil tekniklerle doğru bir şekilde yapılmasına bağlıdır. Bununla birlikte, konut satış miktarına ait tahminler yapılırken incelenen dönemdeki ekonomik ve finansal krizler, salgın hastalıklar, doğal afetler gibi olağanüstü olaylar dikkate alınmalıdır. Bu çalışmanın amacı, Türkiye’de konut talebi ile makroekonomik değişkenler arasındaki ilişkinin kovid salgınının etkisini de dikkate alarak sayma zaman serisi analizi ile tahmin edilmesidir. Elde edilen bulgular, tüketici fiyat indeksi ve konut kredi faizlerinin konut satış miktarları üzerinde negatif bir etkiye sahip olduğunu; dolar kuru ve konut fiyat indeksinin ise konut satış miktarlarını pozitif etkilediğini ortaya çıkarmıştır. Kovid sonrası dönemde ise tüketici fiyat indeksindeki artışların konut satış miktarı üzerinde daha yüksek bir azaltıcı etkisi söz konusu iken; dolar kuru, konut fiyat indeksi ve konut kredi faizlerindeki artışların konut satış miktarı üzerinde pozitif bir etki yarattığı görülmüştür.

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>

Modelling Clusters of Corporate Defaults: Regime-Switching Models Significantly Reduce the Contagion Source

Abstract In this paper, we report robust evidence that the process of corporate defaults is time-dependent and can be modelled by extending an autoregressive count time series model class via the introduction of regime-switching. That is, some of the parameters of the model depend on the regime of an unobserved Markov chain, capturing the model changes during clusters observed for count time series in corporate defaults. Thus, the process of corporate defaults is more dynamic than previously believed. Moreover, the contagion effect—that current defaults affect the probability of other firms defaulting in the future—is reduced compared to models without regime-switching, and is only present in one regime. A two-regime model drives the counts of monthly corporate defaults in the United States. To estimate the model, we introduce a novel quasi-maximum likelihood estimator by adapting the extended Hamilton–Gray algorithm for the Poisson autoregressive model.

Multivariate Score-Driven Models for Count Time Series To Assess Financial Contagion

Multivariate Score-Driven Models for Count Time Series To Assess Financial Contagion

Climate-Based Modeling and Prediction of Rice Gall Midge Populations Using Count Time Series and Machine Learning Approaches

The Asian rice gall midge (Orseolia oryzae (Wood-Mason)) is a major insect pest in rice cultivation. Therefore, development of a reliable system for the timely prediction of this insect would be a valuable tool in pest management. In this study, occurring between the period from 2013–2018: (i) gall midge populations were recorded using a light trap with an incandescent bulb, and (ii) climatological parameters (air temperature, air relative humidity, rainfall and insulations) were measured at four intensive rice cropping agroecosystems that are endemic for gall midge incidence in India. In addition, weekly cumulative trapped gall midge populations and weekly averages of climatological data were subjected to count time series (Integer-valued Generalized Autoregressive Conditional Heteroscedastic—INGARCH) and machine learning (Artificial Neural Network—ANN, and Support Vector Regression—SVR) models. The empirical results revealed that the ANN with exogenous variable (ANNX) model outperformed INGRACH with exogenous variable (INGRCHX) and SVR with exogenous variable (SVRX) models in the prediction of gall midge populations in both training and testing data sets. Moreover, the Diebold–Mariano (DM) test confirmed the significant superiority of the ANNX model over INGARCHX and SVRX models in modeling and predicting rice gall midge populations. Utilizing the presented efficient early warning system based on a robust statistical model to predict the build-up of gall midge population could greatly contribute to the design and implementation of both proactive and more sustainable site-specific pest management strategies to avoid significant rice yield losses.

Model Selection for Time Series Count Data with Over-Dispersion

Time series of count with over-dispersion is the reality often encountered in many biomedical and public health applications. Statistical modelling of this type of series has been a great challenge. Rottenly, the Poisson and negative binomial distributions have been widely used in practice for discrete count time series data, their forms are too simplistic to accommodate features such as over-dispersion. Unable to account for these associated features while analyzing such data may result in incorrect and sometimes misleading inferences as well as detection of spurious associations. Therefore, the need for further investigation of count time series models suitable to fit count time series with over-dispersion of different level. The study therefore proposed a best model that can fit and forecast time series count data with different levels of over-dispersion and sample sizes Simulation studies were conducted using R statistical package, to investigate the performances of Autoregressiove Conditional Poisson (ACP) and Poisson Autoregressive (PAR) models. The predictive ability of the models were observed at different steps ahead. The relative performance of the models were examined using Akaike Information criteria (AIC) and Hannan-Quinn Information Criteria (HQIC). Conclusively, the best model to fit was ACP at different sample sizes. The predictive abilities of the four fitted models increased as sample size and number of steps ahead were increased

A multivariate statistical approach to predict COVID-19 count data with epidemiological interpretation and uncertainty quantification.

For the analysis of COVID‐19 pandemic data, we propose Bayesian multinomial and Dirichlet‐multinomial autoregressive models for time‐series of counts of patients in mutually exclusive and exhaustive observational categories, defined according to the severity of the patient status and the required treatment. Categories include hospitalized in regular wards (H) and in intensive care units (ICU), together with deceased (D) and recovered (R). These models explicitly formulate assumptions on the transition probabilities between these categories across time, thanks to a flexible formulation based on parameters that a priori follow normal distributions, possibly truncated to incorporate specific hypotheses having an epidemiological interpretation. The posterior distribution of model parameters and the transition matrices are estimated by a Markov chain Monte Carlo algorithm that also provides predictions and allows us to compute the reproduction number Rt. All estimates and predictions are endowed with an accuracy measure obtained thanks to the Bayesian approach. We present results concerning data collected during the first wave of the pandemic in Italy and Lombardy and study the effect of nonpharmaceutical interventions. Suitable discrepancy measures defined to check and compare models show that the Dirichlet‐multinomial model has an adequate fit and provides good predictive performance in particular for H and ICU patients.

On robust estimation of negative binomial INARCH models

We discuss robust estimation of INARCH models for count time series, where each observation conditionally on its past follows a negative binomial distribution with a constant scale parameter, and the conditional mean depends linearly on previous observations. We develop several robust estimators, some of them being computationally fast modifications of methods of moments, and some rather efficient modifications of conditional maximum likelihood. These estimators are compared to related recent proposals using simulations. The usefulness of the proposed methods is illustrated by a real data example.

Analysis and Forecasting of Risk in Count Processes

Risk measures are commonly used to prepare for a prospective occurrence of an adverse event. If we are concerned with discrete risk phenomena such as counts of natural disasters, counts of infections by a serious disease, or counts of certain economic events, then the required risk forecasts are to be computed for an underlying count process. In practice, however, the discrete nature of count data is sometimes ignored and risk forecasts are calculated based on Gaussian time series models. But even if methods from count time series analysis are used in an adequate manner, the performance of risk forecasting is affected by estimation uncertainty as well as certain discreteness phenomena. To get a thorough overview of the aforementioned issues in risk forecasting of count processes, a comprehensive simulation study was done considering a broad variety of risk measures and count time series models. It becomes clear that Gaussian approximate risk forecasts substantially distort risk assessment and, thus, should be avoided. In order to account for the apparent estimation uncertainty in risk forecasting, we use bootstrap approaches for count time series. The relevance and the application of the proposed approaches are illustrated by real data examples about counts of storm surges and counts of financial transactions.