The negative binomial (NB) regression model has become a widely utilized tool in panel data analysis across various fields, including economics, healthcare, demography, and environmental sciences (see Ahmed et al. [1], Abonazel et al. [2], and Youssef et al. [3]). Panel data analysis has emerged as a prominent area of research in econometrics literature as it involves studying data along both the temporal and sectoral dimensions to derive the maximum benefit from the available data. Panel datasets combine cross-sectional and time-series observations, enabling a more in-depth analysis of the data over time [4, 5]. By including individual-specific fixed effects, panel data can account for these differences and provide more accurate estimates of the effects of the variables of interest. Within this framework, count panel data models are particularly relevant when the dependent variable represents non-negative integer values generated from counting the occurrence of a particular event, for example, the number of accidents on roads, patents granted to countries and companies, and the number of deaths due to a specific disease (see Winkelmann [6]).

One of the key challenges in analyzing count panel data is the issue of over-dispersion; this problem appears when the variance of the dependent variable is greater than the mean. Therefore, the NB panel model has a dispersion parameter ϕ_i; this parameter allows var(y_it) > E(y_it). This violates the core assumption of the classical Poisson model and often leads to inefficient or biased estimates. To address this limitation, the NB model is a powerful statistical model that is commonly used in analyzing count data. Unlike the Poisson model, the NB model is capable of handling over-dispersion in count panel data; this characteristic makes it an important model in various fields, such as economics, finance, public health, criminology, environmental health, transportation, finance, and law, where it provides a more flexible and accurate approach for analyzing count data. Therefore, several studies have focused on the use of the NB model. For example, Cameron and Trivedi [7], Yaacob et al. [8], Hana [9], Al-Taweel and Algamal [10], Alobaidi et al. [11], Rashad et al. [12], Niang et al. [13], Liu et al. [14], and Suryadi et al. [15].

Therefore, the FENB model is great importance in statistical analysis due to its ability to address over-dispersion in count panel data and control unobserved heterogeneity among individuals or entities in panel data. The model captures individual-specific or entity-specific characteristics that are constant over time. The FENB model finds applications in various fields such as economics, public health, criminology, and social sciences. Youssef et al. [16] conducted a study on estimating the number of patents in some high-income countries using the NB model and Poisson model based on the panel data models. The research aimed to develop accurate models for predicting patents. The results indicate that the FENB model is more suitable and performs better in analyzing the data. This research offers valuable contributions to the field of patent analysis by providing reliable models for estimating patents, enhancing our understanding of the factors that influence global patent trends, and providing useful insights for researchers and policymakers in the field of estimating patents. Kumara and Chin [17] conducted a study analyzing accident data for 41 countries over 15 years. They used the FENB model to examine the effect of socioeconomic and infrastructure factors on fatal road accidents in the Asia-Pacific region. The study also employed the AIC statistic to assess the suitability of the model. Guimaraes [18] used the FENB model under a specific set of assumptions. The study showed that this model effectively addresses over-dispersion. This research emphasizes the significance of the FENB model for accurately analyzing count panel data, particularly in the presence of over-dispersion. In the FENB model, Hausman et al. [19] added the individual fixed effects α_i and the NB dispersion parameter (ϕ_i) into the model. Then, the FENB model given is

where y_it is the dependent variable and takes values of non-negative integers, i.e., y_it ≥ 0. The mean and the variance of the NB model with fixed effects are not equal. Γ(·) is the gamma function, and θi=αiϕi is the parameter of the individual effects and dispersion, which is assumed to be constant over time for each cross section, while λ_it depends on covariates by the function ln(λit)=Xk,it′β.

We can use the conditional maximum likelihood (CML) estimation method to estimate the parameters of the FENB model [19]. The NB model with fixed effects assumes that for a given cross section (i), the dependent variable (y_it) is independent over time and ∑t=1Tyit has a NB distribution with parameters θ_i and ∑t=1Tλit, where

For an individual (i), the dependent variable y_it is independent over time as follows:

The joint probability function for the i^th observation for Equation 1 is

Then, the conditional joint probability function for the i^th observation is

The CML estimation of the NB model with fixed effects can be obtained by maximizing the following log CML function:

For a more comprehensive understanding of CML estimation and its applications see Yousse et al. [16, 20]. In the context, the FENB model is particularly useful because it not only accounts for over-dispersion but also controls for unobserved, time-invariant heterogeneity among individuals. This makes the FENB model a powerful tool for analyzing panel count data with complex structures. However, one major issue that has been neglected in the literature is the effect of outliers on parameters estimation in the FENB model. Outliers can occur due to rare events or abnormal changes in specific units over time. Classical estimation techniques, such as the CML, are known to be highly sensitive to such outliers. This highlights the need for robust estimation methods that can produce accurate and stable results even in the presence of outliers. Despite the progress made in developing robust estimators for the fixed effects Poisson model by Ahmed et al. [1], Abonazel et al. [2], and Youssef et al. [3], there has been little interest in extending these robust methods to the NB model. This gap in literature provides a key motivation for the present study.

According to the work of Youssef et al. [3], who introduced robust estimation methods for the Poisson fixed effects panel model, this study extends the robustness framework to the NB case to accommodate both under-dispersion and over-dispersion in count panel data. Accordingly, this section aims to develop robust estimators for the FENB model in the presence of dispersion and outliers, based on the Huber, Hampel, and Tukey weight functions. The performance of these estimators is evaluated through Monte Carlo simulations under varying levels of outliers, dispersion levels, cross-sectional sizes, and time dimensions. Furthermore, the proposed methods are applied to real-world datasets in two empirical applications involving selected European countries, providing strong evidence of their practical effectiveness.

The NB model addresses over-dispersion that may exist in the data, where the model provides more accurate and reliable estimates; this makes it suitable for a wide range of applications in various fields. The count panel data can contain outliers, and these outliers can have a significant impact on the estimated parameters, where some studies focus on addressing outliers in some models using different estimation methods (see for example Huber [21, 22], Rousseeuw and Leroy [23], Garay et al. [24], Toka and Cetin [25], Susanti et al. [26], Maronna et al. [27], Tüzun et al. [28], Youssef et al. [3, 29], and Ke et al. [30]). On the other hand, panel data regression models may suffer from outliers that can negatively affect classical estimation methods, where there are limited studies on robust estimation methods for fixed effects panel data regression model (see Beyaztas and Bandyopadhyay [31], Amelia et al. [32], Víšek [33], and Bramati and Croux [34]).

In this context, there is not a specific robust method available for estimating parameters in the FENB regression model. Therefore, we propose a robust estimation method for NB regression to deal with count panel data when these data contain outliers. The method utilizes robust estimation techniques that are less sensitive to the influence of outliers, providing more accurate and robust parameter estimates. These methods are based on weight functions that include Huber, Hampel, and Tukey bisquare and minimize the residuals, aiming to improve the robustness and reliability of parameter estimation in the presence of outliers, see for example Hampel et al. [35] and Venables and Ripley [36]. The residuals of the i^th observation for the FENB model can be expressed as

The robust estimator for the FENB model can be obtained by minimizing the objective function (ρ) concerning all β, which can be expressed as follows:

where ξit=Rit(θ^i∑t=1Tλit)1/2, λit=exp(Xk,it′β), and σ^MAD is the estimated standard deviation, where calculated using the median absolute deviation, which is defined as follows:

where Med is the median. Based on the function ρ(ξ_it), we can obtain the robust estimator of β by differentiating the objective function (ρ) concerning the parameters β, and equate the partial derivatives to zero in Equation 2, this yields the following expression:

where ψ(u_it) is called the influence function and equal ρ′(uit). We can rewrite the Equation 3 by using the weight function _ξ(u_it), which equals ψ(uit)uit, and then, we can obtain the first-order condition for robust estimators as follows:

where ξit′=∂∂βξit and W_ξ(ξ_it) is matrix (NT × NT). Three robust estimates for β in the FENB model can be obtained by solving Equation 4 using the weight functions of Huber (W_HR(ξ_it)), Hampel (W_HM(ξ_it)), and Tukey bisquare (W_TK(ξ_it)), as introduced by Baltagi [37] and Youssef et al. [3] in the context of panel data.

We will present an algorithm for handling outliers in the NB panel data model with fixed effects to obtain stable and robust estimates as follows:

In this section, we will present an algorithm to evaluate the impact of outliers on the estimates of the FENB model using a Monte Carlo simulation procedure to conduct a comprehensive analysis to understand the flexibility of the NB model with fixed effects in the presence of outliers. The simulation was conducted using R software. This algorithm was designed specifically for the Monte Carlo simulation study of the FENB model, where it aims to assess the model's sensitivity and accuracy in various conditions. The results from this simulation are expected to contribute significantly to the existing literature, offering an accurate perspective on the applicability and reliability of the FENB model in empirical research, especially in fields where extreme values are not merely outliers but pivotal points of interest. The algorithm utilized for conducting the Monte Carlo simulation study of the FENB model is as follows:

The simulation results are presented in Tables 1–4 for small, moderate, and large sample sizes. These tables report the total MSE and total MAE values for four estimators: the classical (non-robust) estimator NOFNB and three robust estimators (FNBHM, FNBHR, and FNBTK). The simulation is conducted under various scenarios defined by cross-sectional sizes (N = 50, 100, 200, and400), time periods (T = 5, 10, and20), outlier proportions (τ = 0%, 5%, 10%, 15%, and20%), and two dispersion levels representing under-dispersion (ϕ = 0.5) and over-dispersion (ϕ = 2).

In the context of clean data free from outliers (τ = 0%), the NOFNB estimator consistently performs better than the proposed estimators (FNBHM, FNBHR, and FNBTK) across all cases of N, T, and ϕ. This suggests that under ideal data conditions, the classical CML method offers higher efficiency. For example, at N = 50, ϕ = 0.5, T = 5, and τ = 0%, the MSE of the NOFNB estimator is 0.1614, whereas the values for FNBHM, FNBHR, and FNBTK are 0.2331, 0.2758, and 0.3345, respectively.

In contrast, when the data contain outliers (τ > 0%), the FNBHM, FNBHR, and FNBTK estimators outperform the classical NOFNB estimator. The presence of outliers negatively affects the performance of NOFNB, leading to inflated error values, while the robust methods maintain greater stability and accuracy. For instance, at N = 200, ϕ = 2, T = 20, and τ = 10%, the MSE values of FNBHM, FNBHR, and FNBTK are 0.0597, 0.0663, and 0.0566, respectively, compared to 0.4512 for NOFNB.

Moreover, as the proportion of outliers increases, both MSE and MAE tend to rise; however, the rate of increase is slower for the proposed estimators than for the NOFNB. For example, when τ increases from 10 to 15% at N = 100, ϕ = 2, and T = 10, the MAE values of FNBHM, FNBHR, and FNBTK increase from 0.4100, 0.4087, and 0.3826 to 0.6216, 0.6759, and 0.5661, respectively, while the MAE of NOFNB jumps from 1.0490 to 1.4436. These results highlight the greater robustness of the proposed estimators in the presence of outliers.

As the number of cross-sectional units increases from N = 50 to N = 400, the values of MSE and MAE generally decrease for all estimators, indicating enhanced estimation accuracy due to the larger unit's size and increased information. For example, when T = 10, τ = 5%, and ϕ = 2, the MSE of the classical estimator NOFNB decreases from 0.1871 at N = 50 to 0.1565 at N = 400. Similarly, for the proposed robust estimator FNBTK, the MSE drops from 0.0507 to 0.0123 over the same increase in N. This decreasing trend is consistently observed for both robust and non-robust estimators. However, robust estimators continue to offer more reliable estimates in the presence of outliers, regardless of the value of N. Although the classical estimator benefits significantly from increased sample size under clean data conditions, it becomes less reliable when the data contain outliers.

Increasing the time dimension (T) also enhances the performance of all estimators by providing more repeated observations over time. When holding N = 200, ϕ = 2, and τ = 10% fixed, increasing T from 5 to 20 leads to a noticeable reduction in both MSE and MAE values. For instance, the MSE of the proposed estimator FNBHM decreases from ~0.0765 at T = 5 to 0.0597 at T = 20, reflecting a substantial improvement in estimation precision. This trend highlights the importance of longer time-series dimensions in reducing estimation error, even in the presence of moderate levels of outliers. We also find that, when comparing the performance across both under-dispersion (ϕ = 0.5) and over-dispersion (ϕ = 2) scenarios, the robust estimators generally maintain their superiority in the presence of outliers.

The MRE for robust estimators (FNBHR, FNBHM, and FNBTK) is illustrated in Figures 1, 2 clustered by percentages of outliers (from 5 to 20%) and time-series (from 5 to 20) for all units separately. The evaluation was performed separately for two cases; the first case expresses under-dispersion (ϕ = 0.5), while the second case expresses over-dispersion (ϕ = 2).

The results presented in Figures 1, 2 highlight the differences in estimators performance under varying dispersion levels. The efficiency of these estimators increases when increasing the size of the cross section (N), the time series (T), the percentage of the outliers (τ), and the dispersion parameter (ϕ), but the FNBTK estimator remains more efficient. Overall, the findings indicate that the MRE values of the FNBTK estimator are consistently higher than those of the FNBHR and FNBHM estimators. This suggests that the FNBTK estimator is more efficient than the other two robust estimators across all scenarios of N, T, τ, and ϕ.

The practical relevance of the proposed estimation methods is evaluated through real-life panel data applications from selected European countries, representing different types of count panel data frequently encountered in public health and economic contexts. To demonstrate the flexibility and robustness of these estimators under real-world conditions, two empirical case studies are presented. The first application utilizes COVID-19 data, while the second focuses on patent data, providing insights into the dynamics of innovation and economic development over time. Both datasets exhibit the panel structure necessary to evaluate the performance of classical and robust estimators under varying degrees of heterogeneity, over-dispersion, and the presence of outliers. Through these applications, the practical value and adaptability of the proposed methods are highlighted across real-world scenarios.

In this research study, we have presented proposed robust estimators for the FENB panel model. To evaluate the performance of these estimators, we conducted the simulation study and applied them to real panel datasets. By combining Monte Carlo simulation and the applications of real data, we can conduct a comprehensive evaluation of the proposed estimators, ensuring their effectiveness and reliability. This dual approach allows us to assess the estimators' performance across different situations, providing a comprehensive understanding of their robustness and practical utility under diverse conditions.

The simulation results indicate that in the absence of outliers, the NOFNB estimator outperforms the robust estimators, but, when outliers are present, the proposed robust estimators (FNBHM, FNBHR, and FNBTK) exhibit higher effectiveness compared to the non-robust estimator. Furthermore, the results of the empirical studies in European countries indicate that the proposed robust estimators are more efficient than other estimators when outliers are present in the datasets. In addition, the FNBTK robust estimator demonstrates greater efficiency compared to FNBHM and FNBHR robust estimators.

The results of this study are expected to have practical implications for researchers and policymakers in the field of scientific research and development. In addition, it contributes to the advancement of statistical methodology for analyzing count panel data in the presence of outliers and provides a better understanding of the factors influencing count panel data in different fields by emphasizing the importance of the NB panel model. In future work, we plan to propose new robust estimators for the FENB model or for one of the other count panel data models.