The Simulation Study to Test the Performance of Quantile Regression Method With Heteroscedastic Error Variance


CAUCHY –Jurnal Matematika Murni dan Aplikasi 
Volume 5(1)(2017), Pages 36-41 
p-ISSN: 2086-0382; e-ISSN: 2477-3344 
 

Submitted: 15 May 2017 Reviewed: 24 July 2017 Accepted: 2 November 2017 
DOI: http://dx.doi.org/10.18860/ca.v5i1.4209 

The Simulation Study to Test the Performance of Quantile 
Regression Method With Heteroscedastic Error Variance 

Ferra Yanuar1*, Laila Hasnah2, Dodi Devianto3  

1,2,3Jurusan Matematika FMIPA Universitas Andalas 
Kampus Limau Manis 25163 Padang 

* Corresponding Author. E-mail: ferrayanuar@yahoo.co.id 

ABSTRACT  

Least square estimator has many limitations. This estimator will not be a Best Linear Unbiased 
Estimator (BLUE) in the condition of the variance error term have heteroscedasticity problem. Quantile 
regression is a robust approach in situations where the limitation addressed above present for least square 
estimator. The purpose of this study is  to describe the performance of quantile regression method in 
modeling a data set which contain the heteroscedasticity problem. To achieve the goal, a data set is 
generated and statistical framework quantile method then applied to the data.  The consistency of the 
proposed model is then checked by doing a simulation study. This study proves that the quantile regression 
method is able to produce acceptable parameter model since the proposed models have large Pseudo R2 and 
small mean square error (MSE) for all parameter estimated. It could be conclude here that quantile 
regression method is an unbiased estimator method and able to result acceptable model althought in the 
present of heteroscedasticity problem of variance error. 

 
Keywords: Heteroscedasticity, quantile regression, simulation study, Pseudo R2, mean square error (MSE). 

INTRODUCTION  

  In modeling the relationship between covariates and responses, it need estimator method 
to estimate the parameter model. To estimate the value of parameters, it usually use the Ordinary 
Least Squares (OLS). The principle of this method is to minimize the sum of the squares of the 
error. This OLS is applied if all model assumptions are met (independent observations, linearity 
of conditional means, normality of response variable and homogeneity of error variance). In all 
model assumptions are met, the estimator method is called as BLUE (Best Linear Unbiased 
Estimator). However, if one or more of the assumptions are not met, the results could be 
misleading [1]. 
  In regression, an error is how far a point deviates from the regression line. Ideally, our 
data should be homoscedastic (i.e. the variance of the errors should be constant). In many real 
world applications, this situation rarely happens. Most data is heteroscedastic by nature. Due to 
these limitatios, an alternative approach to classical linear regression is in demand. Quantile 
regression is a robust approach in situations where the limitations addressed above [2] present 
for ordinary least square estimator.  
  The quantile method is one of the regression modeling methods by dividing a batch of data 
into the same parts after the data is sorted from the smallest or the largest [1, 3]. Quantile 
regression is an approach in regression analysis introduced by Koenker and Basset [4]. Quantile 

mailto:ferrayanuar@yahoo.co.id


The Simulation Study to Test the Performance of Quantile Regression Method With Heteroscedastic 
Error Variance 

Ferra Yanuar 37 

regression in his theory is able to overcome the violation of normality assumptions, 
heteroscedasticity, multicollinearity problems and so on. This method uses the parameter 
estimation approach by separating or dividing the data into quantities, by assuming the 
conditional quantization function on a distribution of data and minimizing the absolute 
asymmetry of unsymmetric weighted error and presupposes a conditional quantile function on a 
distribution of data [5]. 
  In this paper, we adopt the quantile regression approach to modeling groups of data with 
non homogeneity of error variance. Section 2 of the paper, describes the theoretical framework of 
quantile regression and its indicators to determine the goodness of fit of the proposed model. In 
section 3, we illustrate the implementation of quantile regression through a simulated case-study. 
We choose two covariates in our model hypothesis.as the predictors to the response variable. We 
end with a short discussion in Section 4. 
 
FUNDAMENTAL THEORIES AND RELATED WORKS  
   
 In stricly linear models, a simple approach to estimating the conditional quantiles is 
suggested in Koenker and Basset [2].  Based on the classical regression model, we have: 

𝑦𝑖 = 𝒙𝒊′𝜷 + 𝜀𝑖 , 𝑖 = 1,2, … , 𝑛                      (1) 
where 𝑦𝑖  are dependent variable for each data , 𝑥𝑖  are independent matrix 𝑛𝑥𝑝, with  (𝑦𝑖 ) =
𝒙𝒊′𝜷, 𝜷 is vector of parameter model size 𝑝𝑥1, 𝜀𝑖  are error for each data . 

The parameter estimate by classical regression, by minimizing the sum of the error 
squares, is written as  

𝑚𝑖𝑛 ∑ (𝑦𝑖 − 𝑦�̂�)
2𝑛

𝑖=1                                                      (2) 

where  are estimated value for each data . 

2.1.     Quantile Regression Method 
The prediction based on the median, that is, by minimizing the absolute number of errors 

can be written using following equation : 
𝑚𝑖𝑛 ∑ |𝑦𝑖 − 𝑦�̂�|

2𝑛
𝑖=1                                (3) 

 Furthermore, the linear equations for -th quantil can be written as follows : 
𝑦𝑖 = 𝒙𝒊′𝜷𝝉 + 𝜀𝑖 ,   𝑖 = 1,2, . . , 𝑛           (4) 
Noted that 𝑦�̂� = 𝒙𝒊′𝜷𝝉, the parameter estimate for th quantile is to minimize the absolute 

value of the error by weighting τ for the positive and weighted error (1- τ) for the negative error 
[4]. The 𝜏th (0 < 𝜏 < 1) quantile of 𝜀𝑖  is the value, 𝑄𝜏, for which 𝑃 (𝜀𝑖 < 𝑄𝑌(𝜏)) = 𝜏. The 𝜏th 

conditional quantile of 𝑦𝑖  given 𝒙𝒊 is then simply [6,7] : 
𝑄𝜏(𝑦𝑖 |𝒙𝒊) = 𝒙𝒊′𝜷𝝉, 

where 𝜷𝝉, is a vector of coefficients dependent of  𝜏. 
 The 𝜏th regression quantile is defined as any solution, �̂�𝝉, to the quantile regression 
minimasation problem : 

 𝑚𝑖𝑛𝛽𝜖ℛ ∑ 𝜌𝜏
𝑛
𝑖=1 (𝑦𝑖 − 𝒙𝒊′𝜷𝝉),                       (5) 

where the loss function : 

 𝜌𝜏(𝑢) = 𝑢(𝜏 − 𝐼(𝑢 < 0)).               (6) 

Equivalently, we may rewrite (5) as :  
𝑚𝑖𝑛𝛽𝜖ℛ {∑ 𝜏|𝑦𝑖 − 𝒙𝒊′𝜷𝝉| + ∑ (1 − 𝜏)|𝑦𝑖 − 𝒙𝒊′𝜷𝝉|

𝑛
𝑖=1

𝑛
𝑖=1 }                                     (7) 

              
 The meaning which can be added to explain the equation (7), namely that all observations 
greater than the quantile value, multiplied by the weighting τ and the observations whose value 
is less than the quantile multiplied by 1 − 𝜏.  
 
2.2  Goodness of Fit using  Pseudo R2 

Simple quantile regression model with n independent variables can be formed as follows:
   

𝑄𝜏(�̂�|𝒙) = �̂�0(𝜏) + �̂�1(𝜏)𝒙 + ⋯ + �̂�𝑛 (𝜏)𝒙            (8) 

i

i

ˆ
iy i







The Simulation Study to Test the Performance of Quantile Regression Method With Heteroscedastic 
Error Variance 

Ferra Yanuar 38 

The indicator of the goodness of fit for the model can be predicted with Pseudo R2 as defined below 
[2]: 

   Pseudo 𝑅𝜏
2 = 1 −

𝑅𝐴𝑊𝑆𝜏

𝑇𝐴𝑆𝑊𝜏
              (9)  

where : 

𝑅𝐴𝑊𝑆𝜏 = ∑ 𝜏|𝑦𝑖 − �̂�0(𝜏) − �̂�1(𝜏)𝒙𝒊 − ⋯ − �̂�𝑛 (𝜏)𝒙𝒊|
𝑦𝑖≥�̂�0(𝜏)+�̂�1(𝜏)𝒙𝒊+⋯+�̂�𝑛(𝜏)𝒙

 

      + ∑ (1 − 𝜏)|𝑦𝑖 − �̂�0(𝜏) − �̂�1(𝜏)𝒙𝒊  − �̂�𝑛(𝜏)𝒙𝒊|𝑦𝑖<�̂�0(𝜏)+�̂�1(𝜏)𝒙𝒊+⋯+�̂�𝑛(𝜏)𝒙   

                               (10) 
and 
 𝑇𝐴𝑆𝑊𝜏 = ∑ 𝜏|𝑦𝑖 − �̂�| +𝑦𝑖≥𝜏 ∑ (1 − 𝜏)|𝑦𝑖 − �̂�|𝑦𝑖<𝜏               (11) 

 
The value of 𝑅𝐴𝑊𝑆𝜏  (Residual Absolute Sum of Weighted) is always less than the value of 

𝑇𝐴𝑆𝑊𝜏 (Total Absolute Sum of Weighted) so that the Pseudo 𝑅𝜏
2 will be in the range 0 to 1. The 

closer the Pseudo R2 value to one the model will be better. However, the virtues of Pseudo R2 can 
not be used to test the overall goodness of fit for the model, it can only be used to test the merits 
of the selected quantile [1,2]. 

 
2.3 Mean Square Error (MSE) 

The parameter estimate obtained is said to be good if it has a small bias and small variance. 
Therefore, to see the goodness of estimating the parameters based on the bias and variance values 
simultaneously, represented in the value of Mean Square Error (MSE) [8, 9, 10], formulated as 
follows : 

𝑀𝑆𝐸(�̂�𝑗 (𝜏𝑞 )) = 𝑉𝑎𝑟 ((�̂�𝑗 (𝜏𝑞 )) + 𝐵𝑖𝑎𝑠 (�̂�𝑗 (𝜏𝑞 ))
2                          (12) 

where : 

𝑀𝑆𝐸(�̂�𝑗 (𝜏𝑞 ))  : value of MSE for 𝑗 = 1,2, … 𝑝  

     𝜏𝑞 = 𝑞𝑢𝑎𝑛𝑡𝑖𝑙 𝑞 = 1,2, … , 𝑘   

𝑉𝑎𝑟 ((�̂�𝑗 (𝜏𝑞 ))  : variance for selected quantile 

𝑉𝑎𝑟 ((�̂�𝑗 (𝜏𝑞 )) =
𝑛 ∑ (�̂�𝑗(𝜏𝑞))

2𝑛
𝑖=1 −(∑ (�̂�𝑗(𝜏𝑞))

𝑛
𝑖=1 )

2

𝑛(𝑛−1)
  

𝐵𝑖𝑎𝑠 (�̂�𝑗 (𝜏𝑞 ))            : the value of bias for selected quantile is obtained from the mean of the 

difference of the expected value and the estimated value, or : 

𝐵𝑖𝑎𝑠 (�̂�𝑗 (𝜏𝑞 )) =
1

𝑛
∑ ((�̂�𝑗 (𝜏𝑞 )) −

𝑛
𝑖=1 (𝛽𝑗 (𝜏𝑞 ))     

 
RESULT AND DISCUSSION 
  We describe our approach to quantile regression by conducting the simulation study. In 
this research, we design two covariates each measuring 100 samples. The response variable, 𝑦𝑖  is 
generated from the model : 

𝑦𝑖 =  𝛽0 + 𝛽1𝑥𝑖1 + 𝛽2𝑥𝑖2 + 𝜀𝑖 ,            𝑖 = 1, … ,100    (13) 
where covariate 𝑥𝑖1 is generated from a standard normal distribution and 𝑥𝑖2 is generated from 
exponential with one degrees of freedom. The parameter 𝛽0, 𝛽1 and 𝛽2 are set to 1.2, 1, and 1.7 
respectively. The data for error is also generated by taking the mean at zero and its variances have 

heteroscedasticity problems. We consider the heteroscedastic normal, 𝑁(0, √0.01 x (𝑿𝜷)2) for 

distribution of error term.  
 In this example, we choose 𝜏 = 0.10, 0.25, 0.50, 0.75 and 0.90 as the quantile points for 
estimated. Table 1 below shows the parameter estimated and its corresponding standard error.
  
  



The Simulation Study to Test the Performance of Quantile Regression Method With Heteroscedastic 
Error Variance 

Ferra Yanuar 39 

Table 1. Quantile Regression Estimates 
 

𝜏th Quantile Parameter Estimates Standard error 

0.10 
b0 1.0251* 0.0308 
b1 0.8004* 0.0220 
b2 1.4783* 0.0219 

0.25 
b0 1.0891* 0.0310 
b1 0.8438* 0.0222 
b2 1.5931* 0.0220 

0.50 
b0 1.1840* 0.0417 
b1 0.9955* 0.0298 
b2 1.7640* 0.0296 

0.75 
b0 1.2829* 0.0435 
b1 1.0871* 0.0312 
b2 1.8561* 0.0309 

0.90 
b0 1.3215* 0.0287 
b1 1.1514* 0.0205 
b2 2.0225* 0.0204 

      (* significant at α = 0,05) 

Table 1 informs us that estimated coefficient (�̂�0, �̂�1 and �̂�2) for all quantile points are close 
to initial value, 𝑏0 = 1.2  𝑏1 = 1 and 𝑏2 = 1.7. For example, consider at 0.50th quantile, proposed 

parameter estimated are  �̂�0 = 1.1840, �̂�1 = 0.9955 and �̂�2= 1.7640. 
The next analysis in the quantile regression is a consistency test of the proposed model to  

reveal the performance of the quantile approach and its associated algorithm in recovering the 
true parameters of the quantile regression analysis. Consistency test is done by doing simulation 
study. Simulation study does so by generating a set of new data set by sampling with replacement 
from the original data set, and fitting the model to each new data set [11, 12]. To compute standard 
errors for calculating the 95% confidence interval of all parameters in this study, roughly 25 
model fits are determined. The goodness of fit of each model are also calculated. Table 2 presents 
the result taken from the simulation study.  

 
Table 2.  Simulation Results of 25 Data Sets Using Quantile Regression Approach  

 
-th 

Quantile 
Parameter 

Paramter Estimated  
(Standard Error) 

95% Interval 
Confidence 

Pseudo 
R2 

0.10 
b0 1.0317* (0.0439) (1.0233 ; 1.0401) 

0.8104 b1 0.8974* (0.0312) (0.8898 ; 0.9049) 
b2 1.5091* (0.0321) (1.4973 ; 1.5208) 

0.25 
b0 1.0992* (0.0322) (1.0928 ; 1.0992) 

0.8288 b1 0.9334* (0.0229) (0.9270 ; 0.9398) 
b2 1.1613* (0.0229) (1.6000 ; 1.6226) 

0.50 
b0 1.1912* (0.0301) (1.1861 ; 1.1972) 

0.8477 b1 0.9949* (0.0220) (0.9906 ; 0.9992) 
b2 1.6973* (0.0218) (1.6897 ; 1.7050) 

0.75 
b0 1.2837* (0.0341) (1.2747 ; 1.2928) 

0.8655 b1 1.0533* (0.0243) (1.0478 ; 1.0589) 
b2 1.8074* (0.0244) (1.7925; 1.8223) 

0.90 
b0 1.3693* (0.0414) (1.3550 ; 1.3837) 

0.8854 b1 1.1064* (0.0299) (1.0961 ; 1.1168) 
b2 1.9164* (0.0295) (1.8976 ; 1.9352) 

      (* significant at α = 0,05) 

Table 2 informs us that the estimated value of the model parameter coefficients (�̂�0, �̂�1 and 

�̂�2) on each quantile are close to the initial value (𝑏0 = 1.2  𝑏1 = 1 and 𝑏2 = 1.7). For example, 





The Simulation Study to Test the Performance of Quantile Regression Method With Heteroscedastic 
Error Variance 

Ferra Yanuar 40 

consider the 50th quantile, the estimated value for �̂�0, �̂�1 and �̂�2 are 1.1916, 0.9949, and 1.6973 
respectively. Based on Table 2, we also know that all parameter estimated values fall within 95% 
confidence intervals obtained from the simulation study. It means quantile 95% confidence 
interval seem to work well here and parameter estimated are acceptable.  

The goodness of fit for the each quantile regression model is presented by the value of 
Pseudo R2, shown in the last column in Table 2. All Pseudo R2 values obtained here are more than 
80%, indicating that all proposed model are adequate and could be accepted.  

In this study we also determine the value of of MSE (Mean Square Error) to ensure that 
parameter estimated have small bias and small variance. Table 3 below presents the MSE value of 
the quantile regression method for all three parameter estimated at corresponding quantile 
points. 

Table  3.   The Value of MSE For Any Quantile Points  
 

Parameter 
MSE (Mean Square Error) 

0.10 0.25 0.50 0.75 0.90 
b0 0.0011 0.0011 0.0008 0.0022 0.0054 
b1 0.0015 0.0012 0.0005 0.0008 0.0028 
b2 0.0037 0.0033 0.0015 0.0058 0.0093 

 
Table 3 above  gives information that all parameter estimated have small MSE. These 

results indicate that quantile regression method is able to produce unbiased parameter model 
since it has small bias and small variance.  

Based on any results here, we could believe that the power of our quantile regression 
method result the best fit for the model althought in the existence of heteroscedastic error 
variance.  
 
CONCLUSIONS 
  This present study purposes to describe the performace of quantile regression method in 
modeling the data containing non-uniform variance problem (heteroscedasticity). A data set with 
two covariates is generated, each measuring 100 samples. The distribution for error term is 

heteroscedastic normal, 𝑁(0, √0.01 x (𝑿𝜷)2) is designed to generate a response variable. Each 

parameter model are also set as initial values. A simulation study is done to check the power of 
quantile regression algorithm.  
 This study resulted that all parameter estimated are close to the initial values. The value of 
Pseudo R2 for all proposed model at any selected quantile points are quite large, more than 80%. 
Based on simulation study, the value of parameter estimated are within 95% confidence intervals 
indicating that parameter estimated could be accepted. This study also result that quantile 
regression method is able to produce small value of MSE. Therefore, it could be concluded here 
that quantile regression methods is unbiased estimator method and could result the acceptable 
model although in the due of heteroscedasticity problem of error variance. 
 
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