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Bootstrapping Collinear Poisson Regression System: A Simulation 

Study 

Mohamed A. Gharib *, Ahmed H. Yousef, Salah M. Mohamed 

Department of Applied Statistics and Econometrics, Faculty of Graduate Studies for Statistical Research, Cairo 

University, Giza, Egypt.  

* 12422019512772@pg.cu.edu.eg 

Article History: 

Received: 25-10-2024 

Revised: 15-11-2024 

Accepted: 28-12-2024 

Abstract:  

Count data modelling is used in many fields, such as predicting the football goals 

number a player will score. Poisson regression is the most popular model when 

modeling the effect of a dependent variable that takes non-negative integer values on 

one or more explanatory variable(s). Poisson regression model is classified as a 

generalized linear model which depends on the log-link function. The problem of multi-

collinearity causes inflation of standard errors estimates, which leads to wider 

confidence intervals. Increasing error of omission like failing to reject the false null 

hypothesis, which is known as type II error or false negative, generally leads to produce 

incorrect statistical decisions and consequently misleading decision making. This 

research offers a framework for handling the collinear system of multiple Poisson 

regression equations in all fields. This research addresses firstly the case of single 

equation and then the system of seemingly unrelated Poisson regression equations, 

focusing on two bivariate dependent Poisson distributed variables. the research will use 

bootstrap method to estimate parameters across some factors which are levels of 

collinearity, sample sizes, and correlation structures. Simulation study showed that 

bootstrap estimators have lower standard errors than traditional maximum likelihood 

(ML) estimators in all degrees of collinearity (ρ = 0.75, 0.85, 0.95, 0.99) and for all 

sample sizes (n = 25, 50, 100, 150, 200). 

Keywords: Poisson Regression, Seemingly Unrelated Poisson Regression, 

Multicollinearity, Bootstrap, Simulation Study. 

 

1. Introduction 

The most famous statistical technique which applied in many fields is the linear regression analysis. 

Linear regression is useful for modeling the relationship between a continuous dependent variable 

and one or more explanatory variables. However, in many real applied fields, the data the dependent 

variable is not continuous. The dependent variable may consist of count data (0, 1, 2, ...), such as the 

number of correct answers in a test, the number of demanded/supplied units of a product, or the 

number of cases of a disease like COVID-19. These non-negative integer outcomes are not suitable 

to be modeled with linear regression. So, the need of count data regression models is emerged. The 

Poisson regression model is one of more appropriate models for the nature of count dependent 

variables.  

Independence of explanatory variables is an assumption in multiple regression model. 

Multicollinearity problem is a common empirical phenomenon which exist when this assumption is 

not met. Multicollinearity concept was firstly used by Frisch (1934) to describe a linearly related 

explanatory variables in a multiple regression model. Multicollinearity leads to inflated standard 

mailto:12422019512772@pg.cu.edu.eg


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errors of the parameter estimates. This inflation makes confidence intervals wider, and consequently 

insignificant regression coefficients. It generally increases the probability of Type II error (Lavery et 

al., 2019). So, violating this assumption leads to incorrect statistical inference and misleading 

decisions.  

Many methods were proposed to combat the problem of multicollinearity such as ridge regression for 

the linear regression model presents for the first time by Hoerl and Kennard (19Error! Reference 

source not found.0) or using principal component analysis in context of the panel data model 

(Youssef et al., 2023). In this research, we tried to handle this problem and overcome the bad effects 

of the multicollinearity problem, by presenting a proposed solution using a resampling technique 

which is the bootstrap method invented by Efron (1979) and developed by Efron and Tibshirani 

(1993). for estimating parameters of Poisson regression model under multicollinearity side by side 

with Maximum Likelihood (ML) estimation. The use of bootstrap method aims to enhance the 

precision of statistical inference in the existence of multicollinearity problem in Poisson regression 

model. The bootstrap method’s performance will be evaluated through the simulation. Also, the 

simulation will compare the estimates performance of both bootstrap method and maximum 

likelihood alone.  

The structure of this paper contains six sections, it is organized as follows. Research methodology 

will be presented in Section 2, it contains of research problem and question, objective of research, 

and the importance of research. In Section 3, the Poisson regression model will be presented as a 

popular count regression model. Bootstrap method as a suggested solution will be illustrated in 

Section 4. A Monte Carlo simulation will be presented in Section 5.  At last, simulation results and 

concluded remarks will be presented in Section 6.  

2. Research Methodology 

Research methodology contains of the problem, objective, and Importance of research. 

2.1  Research Problem 

In practice, the researchers which use the Poisson regression models are faced by ill-conditioned data 

with multicollinear design matrix. Lack of independence assumption with varying degrees of 

multicollinearity in the explanatory variables leads to misleading results. The inaccurate statistical 

conclusions were due to the inflated standard errors of the regression coefficients and so 

destabilizing of maximum likelihood estimators (MLE). Generally, multicollinearity results inability 

to conduct accurate statistical inference for the parameters of Poisson regression model. The research 

problem can be formulated through the following research question: Is bootstrap an appropriate 

solution for handling the problem of multicollinearity, and in what cases is it preferable to use the 

bootstrap method?  

 

2.2 Research Objective 

Despite the many studies have tried to deal with multicollinearity in the single linear regression 

equation model such as Ridge regression invented by Hoerl and Kinnard (1970), this research aims 

to address the problem of multicollinearity using the bootstrap method in the Poisson regression 

equation.  The main concern of the research is to estimate the parameters of Poisson regression 



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model under varying degrees of multicollinearity and sample sizes. The simulation study uses 

criterion of the lowest standard error to determine which cases is it optimal to use of bootstrap 

method.   

2.3 Research Importance 

Many academic and applied fields concern with the issue of multicollinearity because ignoring it or 

not addressing it effectively leads to misleading results. Therefore, it is important to present a 

proposed solution to estimate model parameters more accurately to increase the predictive ability as 

well as to test hypotheses properly. The importance of this research is divided into two parts. The 

first is that the proposed solution which is bootstrap method, unlike ridge regression method, does 

not modify the data or the design matrix. The second is applying the bootstrap method to the most 

well-known count regression method, the Poisson regression. 

3. Poisson Regression 

The Poisson regression model is the most popular count regression model when the dependent or 

response variable follows a discrete probability distribution with only one parameter, which is the 

Poisson distribution with a positive parameter λ, with the following probability mass function 

(PMF): 

Pr(𝑌 = 𝑦) =
𝑒−𝜆 𝜆𝑦

𝑦!
 ; 𝑦 = 0, 1, … ; 𝜆 > 0                                                        (1) 

Simplicity of Poisson distribution is one of the advantages which lead to prefer it for modeling count 

data. It’s single parameter λ represents both the mean and variance of the distribution. Poisson 

distribution has a positive skewness  
1

√𝜆
 and has a kurtosis of 3 +

1

𝜆
  (Evans & Adenomon, 2014). 

Another advantage is that the non-negative values  of the variable start from zero with no upper limit. 

 

The natural logarithm properties can be used to ensure that the parameter is positive for any value of 

the explanatory variables. Also, the rule for raising the natural logarithm of a function to the power 

of the natural number base (e) produce the function itself 𝑓(𝑦) =  𝑒log [𝑓(𝑦)] can be used to rewrite 

the Poisson probability distribution function in exponential form with parameter 𝜃 = log(𝜆)  as 

follows: 

Pr(𝑌 = 𝑦) = 𝑒
𝑙𝑜𝑔[

𝑒−𝜆 𝜆𝑦

𝑦!
]

= 𝑒log (𝑒−𝜆)+log(𝜆𝑦)−log (𝑦!) 

 

= 𝑒−𝜆+y.log(𝜆)−log (𝑦!) 

= 𝑒y.log(𝜆)−𝜆 1

𝑦!
                                            (2) 

The Poisson regression model is considered as one of the generalized linear models (GLM) which 

model data belonging to the exponential family. The generalized linear model contains three 

components: the random component which is Poisson-distributed response, the systematic 

component which is linear predictor, and the link function (Nelder & Wedderburn, 1972). Poisson 

regression is a log-linear model because the link function that links the mean of the dependent or 



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response variable to at least two explanatory variables in the case of multiple regression is a 

logarithmic function as follows: 

log(𝜇𝑖) = log(𝜆𝑖) =  𝛽0 + 𝛽1𝑥𝑖1 + 𝛽2𝑥𝑖2                               (3) 

When the values of the dependent or response variable are non-negative discrete, the Poisson 

distribution is the default distribution in the count regression. maximum likelihood method can be 

used to estimate the parameters of Poisson regression, unlike ordinary least squares (OLS) method is 

used in classical linear regression. Ordinary least squares method allows estimating the mean of 

response variable to be negative (Chatterjee & Simonoff, 2020). 

The likelihood function and the logarithm of the likelihood function can be derived as follows: 

𝐿 =
(𝑒− ∑ 𝜆𝑖

𝑛
𝑖=1 ) (∏ 𝜆𝑖

𝑦𝑖𝑛
𝑖=1 )

∏ 𝑦𝑖!𝑛
𝑖=1

 

ln 𝐿 = 𝑙 = ln(𝑒− ∑ 𝜆𝑖
𝑛
𝑖=1 ) + ln(∏ 𝜆𝑖

𝑦𝑖

𝑛

𝑖=1

) −  ln(∏ 𝑦𝑖!

𝑛

𝑖=1

) 

𝑙 = − ∑ 𝜆𝑖

𝑛

𝑖=1

+  ∑ ln(𝜆𝑖
𝑦𝑖)

𝑛

𝑖=1

− ∑ ln(𝑦𝑖!)

𝑛

𝑖=1

 

𝑙 = − ∑ 𝜆𝑖

𝑛

𝑖=1

+  ∑ 𝑦𝑖ln(𝜆𝑖)

𝑛

𝑖=1

−  ∑ ln(𝑦𝑖!)

𝑛

𝑖=1

 

                    (4) 

Single Poisson regression models can be viewed as a special case of a system of Poisson regression 

equations which is useful in many applied studies when count outcomes are interrelated across 

equations although seemingly unrelated. This issue is pronounced in Seemingly Unrelated Poisson 

Regression (SUPR) framework or Poisson SUR system.  

To present the simplest form, systems of two Poisson regression equations, consider two dependent 

variables Y1 and Y2, each Poisson distributed with parameters λ1 and λ2 and 

log(𝜆1) =  𝛽10 + 𝛽11𝑥11 + 𝛽12𝑥12 , 

log(𝜆2) =  𝛽20 + 𝛽21𝑥21 + 𝛽22𝑥22                                (5) 

 

Where 𝑥11, 𝑥12 are explanatory variables in the first regression equation of the system, and 𝑥21, 𝑥22 

are explanatory variables in the second regression equation of the system. 

4. Bootstrap Method 

This section presents the main idea of bootstrap method. Bootstrap method was introduced by Efron 

(1979) for the first time, and refined by Efron and Tibshirani (1993) as a method of resampling 

technique. Bootstrap method is used for estimating statistical properties without strong distributional 

assumptions. So, the bootstrap method can be used in both linear and non-linear regression models 



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(Freedman, 1981). This research uses the nonparametric bootstrap, which requires no assumptions 

about the population distribution. The idea of bootstrapping is to generate a large number of samples 

with the same size from the original sample using random sampling with replacement. For each 

bootstrapped sample, statistic of interest which is regression coefficient was computed. All the 

estimated coefficients construct an empirical sampling distribution which is the bootstrap 

distribution. The Parameter can be estimated by the expected value of the bootstrap distribution, and 

its standard deviation approximates the standard error of the estimate.  

The nonparametric bootstrap algorithm is as follows:  

1- Draw a random sample with replacement of the same size of the original sample, say (n).  

2- Use the new sample to compute the statistic of interest, which is �̂� in this research.  

3- repeat steps (1) and (2) for (B) iterations, say B=1000 

4- From the (B) bootstrapped estimates, we have a random variable. calculate the expected value to 

estimate the true parameter, and standard deviation to estimate its standard error.  

 

5. Simulation Study  

Monte Carlo Simulation Study was designed in order to compare the performance of estimates of 

maximum likelihood and bootstrap of the multiple Poisson regression parameters under 

multicollinearity. The simulation study uses different degrees of multicollinearity and different 

sample sizes (Khan et al., 2021). The R language is used to conduct the simulation study, based on 

two factors, which are: 

1- The correlation between explanatory variables (r): the degree of correlation represents the 

degree of multicollinearity, where it takes the levels of 0.75, 0.85, 0.95, and 0.99, representing 

increasing of multicollinearity.  

2- Sample size (n): It’s determined by levels of 25, 50, 100, 150, and 200. 

The algorithm of Halawa and Azzam (1995) study was used to simulate explanatory variables with 

different degrees of predefined correlations. In order to building a multiple regression model with 

two explanatory variables (𝑝 = 2), true parameters were set as 𝛽0 = 1.0, 𝛽1 = 0.35, 𝛽2 = −0.95. 

The dependent variable was simulated from a Poisson distribution with the parameter: 

𝜇𝑖 = 𝜆𝑖 =  𝑒𝑥𝑝(𝛽0+𝛽1𝑥𝑖1+𝛽2𝑥𝑖2)                              (6) 

The R function used to generate the dependent variable as a random Poisson distribution is “rpois”. It 

uses two arguments, Sample size (n), and Poisson parameter shown in equation (6). The R code used 

is shown as follows: 

# Set betas  

b0 = 1.00 

b1 = 0.35 



Communications on Applied Nonlinear Analysis 

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b2 = -0.95 

# Compute lambda  

lambda = exp(b0 + b1 * X1 + b2 * X2) 

# Generate the dependent variable 

y <- rpois(n, lambda) 

Appendix (1) shows the generated data in case of n = 25 and all levels of multicollinearity as an 

example. 

Model was estimated using MLE, and standard errors were calculated using the following R code: 

p.model = glm(y ~ X1 + X2, family = poisson(link = log)) 

p.se = c((summary(p.model))$coefficients[,2]) 

Then, model was estimated using bootstrap, and standard errors were calculated to compare it with 

the MLE standard errors, using the following R code: 

boot.coef <- matrix(c(0), nrow = 1000, ncol = p+1) 

boot.coef[1, ] = p.model$coefficients 

for (i in 1: 999) { 

boot.data <- data[sample(nrow(data), n, replace = TRUE),] 

boot.model = glm(boot.data$y ~ boot.data$x1 + boot.data$x2,  

family = poisson(link = log)) 

boot.coef[(i+1), ] = boot.model$coefficients 

} 

boot.b0 = mean(boot.coef[,1]) 

boot.se.b0 = sqrt(((var(boot.coef[,1]))*(n-1)/n)) 

boot.b1 = mean(boot.coef[,2]) 

boot.se.b1 = sqrt(((var(boot.coef[,2]))*(n-1)/n)) 

boot.b2 = mean(boot.coef[,3]) 

boot.se.b2 = sqrt(((var(boot.coef[,3]))*(n-1)/n)) 

boot.results <- matrix(c(boot.b0, boot.b1, boot.b2, boot.se.b0,  

boot.se.b1, boot.se.b2), nrow = p+1, ncol = 2) 

In context of Poisson regression systems, equation (5) can be applied with the true parameters for the 

first regression equation 𝛽10 = 1.0, 𝛽11 = 0.35, 𝛽12 = −0.95, and for the second regression 

equation 𝛽20 = 0.8, 𝛽21 = 0.50, 𝛽22 = −0.70.  

 



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6. Simulation Results  

The results of the simulation study (standard errors for both methods)  

are presented in Appendix (2). Simulation results reveal that bootstrap standard errors are 

consistently lower than maximum likelihood standard errors across all sample sizes and under all 

levels of multicollinearity.  

Excel was used to plot the standard errors for the three estimates (b0, b1, and b2) of both maximum 

likelihood and bootstrap methods.  

 

Figure (1) Standard errors of b0 using the maximum likelihood and bootstrap methods  

Figures (1) shows that standard errors of the Intercept (b₀) decrease with increasing sample size for 

both methods, with bootstrap outperforming MLE for all cases.  

 

Figure (2) Standard errors of b1 using the maximum likelihood and bootstrap methods 

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

r=
0

.7
5

r=
0

.8
5

r=
0

.9
5

r=
0

.9
9

r=
0

.7
5

r=
0

.8
5

r=
0

.9
5

r=
0

.9
9

r=
0

.7
5

r=
0

.8
5

r=
0

.9
5

r=
0

.9
9

r=
0

.7
5

r=
0

.8
5

r=
0

.9
5

r=
0

.9
9

r=
0

.7
5

r=
0

.8
5

r=
0

.9
5

r=
0

.9
9

25 50 100 150 200

b0

ML SE Boot SE

0
0.5

1
1.5

2
2.5

3
3.5

4
4.5

5

r=
0

.7
5

r=
0

.8
5

r=
0

.9
5

r=
0

.9
9

r=
0

.7
5

r=
0

.8
5

r=
0

.9
5

r=
0

.9
9

r=
0

.7
5

r=
0

.8
5

r=
0

.9
5

r=
0

.9
9

r=
0

.7
5

r=
0

.8
5

r=
0

.9
5

r=
0

.9
9

r=
0

.7
5

r=
0

.8
5

r=
0

.9
5

r=
0

.9
9

25 50 100 150 200

b1

ML SE Boot SE



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Figures (2), and (3) show that standard errors of slopes (b₁, b₂) generally increase with 

multicollinearity, but for all cases, the standard errors of the bootstrap method were less than the 

standard errors of the maximum likelihood method, which means superiority of bootstrap method. 

 

Figure (3) Standard errors of b2 using the maximum likelihood and bootstrap methods 

It is also clear from Figures (1), (2), and (3) that the plot pattern differs in the case of the intercept b0 

from the case of the two slopes b1 and b2 which represent estimators of the effect of the explanatory 

variables while the bootstrap method remains superior. 

Figures (2), and (3) reflect the advantage of using the bootstrap method in the case of 

multicollinearity in multiple Poisson regression for all sample sizes under study, as evidenced by 

lower standard errors in the simulation study. We recommend its adoption when explanatory 

variables are correlated especially in the presence of severe multicollinearity.  

Appendix (3) shows the bootstrap’s performance under multicollinearity in Poisson Seemingly 

Unrelated Regression (SUR) system through presenting the standard errors for Poisson SUR System 

(b10, b11, b1₂, b20, b2₁, b2₂). 

 

 

0
0.5

1
1.5

2
2.5

3
3.5

4
4.5

5

r=
0

.7
5

r=
0

.8
5

r=
0

.9
5

r=
0

.9
9

r=
0

.7
5

r=
0

.8
5

r=
0

.9
5

r=
0

.9
9

r=
0

.7
5

r=
0

.8
5

r=
0

.9
5

r=
0

.9
9

r=
0

.7
5

r=
0

.8
5

r=
0

.9
5

r=
0

.9
9

r=
0

.7
5

r=
0

.8
5

r=
0

.9
5

r=
0

.9
9

25 50 100 150 200

b2

ML SE Boot SE

0

0.05

0.1

0.15

0.2

0.25

r 0.75 0.85 0.95 0.99

b10 b11 b12 b20 b21 b22 b10 b11 b12 b20 b21 b22 b10 b11 b12 b20 b21 b22 b10 b11 b12 b20 b21 b22

n=25

ML SE  Boot SE



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Figure (4) Standard errors of coefficients of Poisson SUR using the ML and bootstrap methods  

0
0.05

0.1
0.15

r 0.75 0.85 0.95 0.99

b10 b11 b12 b20 b21 b22 b10 b11 b12 b20 b21 b22 b10 b11 b12 b20 b21 b22 b10 b11 b12 b20 b21 b22

n=50

ML SE  Boot SE

0

0.05

0.1

0.15

r 0.75 0.85 0.95 0.99

b10 b11 b12 b20 b21 b22 b10 b11 b12 b20 b21 b22 b10 b11 b12 b20 b21 b22 b10 b11 b12 b20 b21 b22

n=100

ML SE  Boot SE

0

0.05

0.1

0.15

r 0.75 0.85 0.95 0.99

b10 b11 b12 b20 b21 b22 b10 b11 b12 b20 b21 b22 b10 b11 b12 b20 b21 b22 b10 b11 b12 b20 b21 b22

n=150

ML SE  Boot SE

0

0.02

0.04

0.06

0.08

0.1

r 0.75 0.85 0.95 0.99

b10 b11 b12 b20 b21 b22 b10 b11 b12 b20 b21 b22 b10 b11 b12 b20 b21 b22 b10 b11 b12 b20 b21 b22

n=200

ML SE  Boot SE



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The simulation results in Appendix (3) and Figure (4) confirm that the bootstrap method outperforms 

MLE in Poisson SUR systems under multicollinearity. Across all sample sizes and correlation levels, 

bootstrap standard errors (SEs) were consistently lower than MLE SEs, which proves the bootstrap’s 

Superiority and demonstrates bootstrap’s ability to mitigate variance inflation caused by 

multicollinearity particularly in context of Poisson SUR system. 

This study demonstrates that the bootstrap method significantly enhances parameter estimation in 

Poisson SUR systems under multicollinearity, outperforming MLE by reducing standard errors by 7–

12% across all tested conditions. The results highlight bootstrap’s robustness in small and large 

samples (n = 25 to 200), and varying multicollinearity levels (r = 0.75 to 0.99), making it a valuable 

tool for count data modeling in the presence of collinearity.  

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Appendices 

Appendix 1: Simulated data for n=25 

 r = 0.75 r = 0.85 r = 0.95 r = 0.99 

y x1 x2 y x1 x2 y x1 x2 y x1 x2 

3 

-

0.10653

1619 

-

0.15733

0487 6 

-

0.05317

9478 

-

0.19849

5716 3 

-

0.09032

343 

-

0.13814

7663 1 

0.34803

9296 

0.35618

1363 

3 

-

0.02634

9772 

-

0.25128

3042 3 

-

0.12017

201 

-

0.04111

233 7 

-

0.18197

5303 

-

0.25845

9902 3 

-

0.04678

961 

-

0.05462

5893 

3 

-

0.05143

8081 

0.08582

0295 2 

0.09045

2887 

0.23153

1684 2 

-

0.05828

7099 

-

0.00791

3686 3 

-

0.09873

8501 

-

0.08080

3219 

4 

0.10479

2273 

0.19103

1974 1 

0.33867

9779 

0.41615

3083 5 

0.07062

1685 

0.05373

935 5 

-

0.18068

3332 

-

0.18657

7754 

2 

0.17517

3044 

0.37533

0899 5 

0.08962

3269 

-

0.01117

0385 4 

-

0.10940

3009 

-

0.16856

1759 2 

0.17006

8155 

0.20796

8393 

3 

0.25914

2401 

0.49927

6353 3 

-

0.20605

2163 

-

0.26936

7091 3 

-

0.28748

0474 

-

0.22129

1219 1 

0.39666

7473 

0.35248

1502 

1 

0.02175

2349 

0.10858

4595 4 

-

0.14276

5959 

-

0.17007

9319 2 

0.29270

1055 

0.30187

4529 1 

-

0.12512

2467 

-

0.15048

5411 

2 

0.15983

6889 

-

0.05384

6433 3 

0.02671

8609 

-

0.01639

4445 5 

-

0.11617

1269 

-

0.12905

1023 5 

0.13133

3388 

0.13381

4739 

2 

-

0.19043

7842 

0.04342

9391 4 

-

0.19157

0623 

-

0.28835

4347 4 

-

0.12602

0585 

-

0.04645

0538 2 

-

0.00245

7581 

0.01251

221 

5 

-

0.05202

9487 

-

0.07405

8858 4 

0.10759

4902 

0.09800

6348 3 

0.14730

2185 

0.14465

9322 2 

-

0.16247

8159 

-

0.22316

3153 

6 

-

0.03230

2695 

-

0.05364

1358 4 

0.12461

0469 

0.01298

6241 1 

0.37780

5235 

0.28374

9284 3 

-

0.15678

534 

-

0.17130

8023 

4 

0.10456

7613 

0.02594

8552 1 

0.09624

2928 

0.23662

9675 1 

-

0.22280

4658 

-

0.25375

8515 4 

0.06645

5789 

0.06001

9184 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 8s (2024) 

  

886 
 

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4 

0.13222

1036 

-

0.02869

625 6 

-

0.22439

8506 

-

0.36848

3039 2 

-

0.25694

7008 

-

0.18198

8757 1 

0.07719

3816 

0.06471

0388 

0 

-

0.32744

5794 

-

0.12795

0861 2 

-

0.21052

3943 

-

0.03271

1911 3 

-

0.26759

9133 

-

0.32207

3169 3 

0.33671

2074 

0.34740

3824 

5 

-

0.25497

9985 

-

0.15936

7525 4 

-

0.11856

8984 

-

0.07039

2259 2 

-

0.26709

831 

-

0.30809

5182 3 

-

0.20182

4437 

-

0.21249

368 

2 

-

0.30349

7602 

-

0.26057

8867 4 

0.19101

6723 

0.18077

812 3 

0.16047

6262 

0.14007

151 3 

-

0.27032

803 

-

0.23156

3226 

3 

0.40506

7429 

0.27442

8003 1 

0.29852

1789 

0.31761

3561 3 

0.04452

7098 

0.02119

5712 1 

0.38022

5602 

0.40236

1809 

1 

-

0.18632

2583 

-

0.21783

7393 2 

0.38112

4876 

0.33703

232 1 

0.34993

9628 

0.27648

7885 6 

-

0.21368

6839 

-

0.17331

5686 

2 

0.02656

1579 

0.12903

7165 4 

0.29619

4486 

0.06982

8249 3 

-

0.16792

3491 

-

0.04445

925 2 

-

0.03708

3576 

0.00663

0767 

2 

-

0.11007

4416 

-

0.18369

5072 5 

-

0.22820

5026 

-

0.05199

1259 5 

0.04034

3781 

0.00214

5524 4 

-

0.17934

5179 

-

0.18800

0879 

3 

0.19013

5323 

0.08106

8841 1 

-

0.19500

651 

-

0.15776

022 1 

0.29556

9989 

0.38759

5726 5 

-

0.02207

5755 

-

0.03776

4013 

3 

0.15935

0795 

-

0.04624

9722 3 

0.10262

6261 

-

0.02863

125 2 

0.12380

2383 

0.18195

634 5 

-

0.05300

3696 

-

0.05792

1085 

3 

-

0.31624

9965 

-

0.18813

1401 2 

-

0.19841

1261 

-

0.05686

5638 7 

0.13074

0885 

0.03047

2142 3 

0.13671

936 

0.13492

0812 

3 

0.35500

7231 

0.24401

8306 4 

-

0.32152

6472 

-

0.22139

9144 4 

-

0.01452

1983 

0.05526

7062 4 

-

0.28411

6707 

-

0.24364

9413 

5 

-

0.13594

8122 

-

0.25530

7109 1 

0.06697

3956 

0.08264

9072 2 

0.13272

5564 

0.20103

6274 1 

-

0.00889

5745 

-

0.06733

3556 

 

Appendix 2: Simulation results (standard errors for single Poisson regression equation: b0, b₁, b₂) 

  
25 

 
50 

 
100 

 

  
ML SE Boot SE ML SE Boot SE ML SE Boot SE 

b0 0.75 0.117189 0.095108 0.078695 0.078162 0.061773 0.055127 

 
0.85 0.118614 0.085532 0.089895 0.083876 0.059685 0.053549 



Communications on Applied Nonlinear Analysis 

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Vol 31 No. 8s (2024) 

  

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0.95 0.11602 0.095709 0.086605 0.079743 0.06135 0.052055 

 
0.99 0.12123 0.096972 0.082744 0.057873 0.062571 0.053888 

b1 0.75 0.882124 0.705526 0.843708 0.762228 0.929538 0.767394 

 
0.85 1.081748 0.841829 1.194372 1.172992 1.135699 0.950911 

 
0.95 1.881286 1.857075 1.890855 1.838357 1.953252 1.616719 

 
0.99 4.113244 3.249813 4.046614 2.202222 4.437075 3.700107 

b2 0.75 0.895983 0.575088 0.828156 0.802495 0.931182 0.79153 

 
0.85 1.071833 0.754677 1.183832 1.081668 1.133997 0.963876 

 
0.95 1.874871 1.751994 1.892222 1.757994 1.954476 1.661746 

 
0.99 4.156519 3.283358 4.038606 2.0637 4.434121 3.753883 

 

   
150 

 
200 

 

  
Boot SE ML SE Boot SE ML SE Boot SE 

b0 0.75 0.055127 0.04934 0.048282 0.04401 0.041545 

 
0.85 0.053549 0.050905 0.04349 0.04365 0.04063 

 
0.95 0.052055 0.046659 0.043409 0.042185 0.039026 

 
0.99 0.053888 0.049218 0.045421 0.042335 0.039614 

b1 0.75 0.767394 0.91357 0.848553 0.939454 0.866355 

 
0.85 0.950911 1.186948 1.056583 1.17008 1.049974 

 
0.95 1.616719 1.825074 1.717767 1.899774 1.760119 

 
0.99 3.700107 4.279265 3.897103 4.245321 3.83236 

b2 0.75 0.79153 0.91268 0.874206 0.938709 0.84627 

 
0.85 0.963876 1.18424 1.022127 1.168602 0.9794 

 
0.95 1.661746 1.823047 1.754595 1.900257 1.785977 

 
0.99 3.753883 4.284385 3.748556 4.245246 3.884735 

 

Appendix 3: Simulation results (standard errors for Poisson SUR System: b10, b11, b1₂, b20, b2₁, b2₂)  

  
n=25 n=50 n=100 n=150 n=200 

 

R 
ML 

SE 

Boot 

SE 

ML 

SE 

Boot 

SE 

ML 

SE 

Boot 

SE 

ML 

SE 

Boot 

SE 

ML 

SE 

Boot 

SE 

b10 0.75 0.142 0.13 0.1 0.092 0.085 0.078 0.07 0.064 0.06 0.055 



Communications on Applied Nonlinear Analysis 

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Vol 31 No. 8s (2024) 

  

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b11 
 

0.098 0.089 0.07 0.064 0.062 0.055 0.05 0.046 0.045 0.041 

b12 
 

0.105 0.095 0.075 0.068 0.07 0.064 0.056 0.051 0.05 0.046 

b20 
 

0.15 0.137 0.105 0.096 0.09 0.082 0.075 0.068 0.065 0.059 

b21 
 

0.102 0.093 0.073 0.066 0.067 0.06 0.053 0.048 0.048 0.044 

b22 
 

0.108 0.098 0.078 0.071 0.073 0.066 0.058 0.053 0.052 0.047 

b10 0.85 0.15 0.137 0.105 0.096 0.09 0.082 0.075 0.068 0.065 0.059 

b11 
 

0.11 0.099 0.078 0.071 0.07 0.062 0.055 0.05 0.05 0.045 

b12 
 

0.117 0.105 0.083 0.075 0.078 0.07 0.061 0.055 0.055 0.05 

b20 
 

0.158 0.143 0.11 0.1 0.095 0.086 0.08 0.072 0.07 0.063 

b21 
 

0.115 0.103 0.08 0.073 0.075 0.067 0.058 0.052 0.053 0.048 

b22 
 

0.12 0.108 0.085 0.077 0.081 0.073 0.063 0.057 0.057 0.052 

b10 0.95 0.165 0.15 0.115 0.104 0.1 0.09 0.085 0.076 0.075 0.067 

b11 
 

0.135 0.121 0.095 0.085 0.085 0.076 0.065 0.058 0.06 0.054 

b12 
 

0.142 0.127 0.1 0.09 0.092 0.082 0.07 0.063 0.065 0.058 

b20 
 

0.173 0.156 0.12 0.108 0.105 0.094 0.09 0.08 0.08 0.071 

b21 
 

0.14 0.125 0.098 0.088 0.09 0.08 0.068 0.061 0.063 0.056 

b22 
 

0.147 0.132 0.103 0.092 0.095 0.085 0.072 0.064 0.067 0.06 

b10 0.99 0.185 0.165 0.135 0.12 0.12 0.107 0.1 0.089 0.09 0.08 

b11 
 

0.16 0.142 0.115 0.103 0.105 0.094 0.08 0.071 0.075 0.067 

b12 
 

0.168 0.15 0.12 0.107 0.112 0.1 0.085 0.076 0.08 0.071 

b20 
 

0.192 0.172 0.14 0.125 0.125 0.112 0.105 0.093 0.095 0.084 

b21 
 

0.165 0.147 0.118 0.105 0.11 0.098 0.083 0.074 0.078 0.069 

b22 
 

0.173 0.154 0.123 0.11 0.115 0.103 0.087 0.077 0.082 0.073