Simulation Study of the Using of Bayesian Quantile Regression in Non-normal Error


CAUCHY –Jurnal Matematika Murni dan Aplikasi 
Volume 5(3) (2018), Pages 121-126 
p-ISSN: 2086-0382; e-ISSN: 2477-3344 

Submitted: 04 October 2018 Reviewed: 08 October 2018 Accepted: 30 November 2018 
DOI: http://dx.doi.org/10.18860/ca.v5i3.5633  

Simulation Study of the Using of Bayesian Quantile Regression in Non-
normal Error 

Catrin Muharisa1, Ferra Yanuar2*, Dodi Devianto3  

1,2,3 Department of Mathematics, Faculty of Mathematics and Natural Science, Andalas 
University, Kampus Limau Manis, 25163, Padang - Indonesia 

Email: catrinmuharisa@gmail.com, ferrayanuar@sci.unand.ac.id, 
ddevianto@sci.unand.ac.id  

*Corresponding Author Email: ferrayanuar@sci.unand.ac.id 

ABSTRACT  

The purposes of this paper are to introduce the ability of the Bayesian quantile regression method in 
overcoming the problem of the non-normal errors. In this research we do simulation study to apply the 
proposed method. We generate data and assume error by asymmetric Laplace distribution. In this research, 
we solve the non-normal problem using quantile regression method and Bayesian quantile regression 
method and then we compare. The quantile regression approach we divide the data into any quantiles, then 
estimated the conditional quantile function and minimize absolute error that is asymmetrical. Bayesian 
regression method used the asymmetric Laplace distribution in likelihood function. Markov Chain Monte 
Carlo method using Gibbs sampling algorithm is applied then to estimate the parameter in Bayesian 
regression method. Convergence and confidence interval of parameter estimated are also checked. Bayesian 
quantile regression method results has more significance parameter and smaller confidence interval than 
quantile regression method. The best regression equation model is Bayesian quantile method used quantile 
value 0.50. This study proves that Bayesian quantile regression method can produce acceptable parameter 
estimate for non-normal error. 

 

Keywords: Quantile regression; Asymmetric Laplace distribution; Gibbs sampling; Markov Chain 
Monte Carlo; Bayesian quantile regression 

INTRODUCTION 

Regression analysis is a tool in experienced statistics development and used in 
many areas of life. The analysis has a purpose to estimate the relationship between 
dependent variable with independent variable [1]. There are several methods used to 
estimate parameters in the regression equation, one of the most commonly used is the 
Ordinary Least Square (OLS) method. The used of OLS method is based on several 
assumptions, one of which is the assumption of normality. Furthermore, developed 
Quantile regression method which is generally used in case of econometrics. In this 
quantile regression approach, we divide the data into any quantiles, then estimated the 
conditional quantile function and minimize absolute error that is asymmetrical. In the 
method of quantile regression usually requires large data size.  

Bayes introduced a method to estimate parameters by utilizing initial information 
called prior distribution. This method is known as the Bayesian method. Prior distribution 
can be derived from the prior research data or based on the researcher's intuition[2]. 
Prior information from the distribution of parameters is then combined with information 
from data obtained from sampling or so-called likelihood function so that posterior 

mailto:catrinmuharisa@gmail.com
mailto:ferrayanuar@sci.unand.ac.id
mailto:ddevianto@sci.unand.ac.id
mailto:ferrayanuar@sci.unand.ac.id


Simulation Study of the Using of Bayesian Quantile Regression in Non-normal Error 

Catrin Muharisa 122 

distribution is obtained. The mean and variance of this posterior distribution or the 
posterior mean and posterior variance then can be estimated.  

Bayesian Method uses  MCMC algorithm (Markov Chain Monte Carlo). MCMC can 
be easily used to obtain posterior distribution even in that situation complex [3]. In this 
research will be estimated model parameters with combining the quantile regression 
method and the Bayesian regression method called the Bayesian quantile regression 
method. 

 

METHODS  

In this section, we will introduce two models, quantile regression and Bayesian 
quantile regression. Here, we denote Y as the dependent variable, X is the independent 
variable. 

1. Quantile Regression 

 Quantile function denoted by 𝑄𝜃 where 𝜃, 0 ≤ 𝜃 ≤ 1. Given 𝑌 be a random 
variable with a cumulative distribution function 𝐹𝑌 = 𝑃(𝑌 ≤ 𝑦).  The Quantile regression  
𝜃𝑡ℎ of  𝑌 can be simply written as follows [4]: 
 

𝑄𝜃(𝑌) ≔ 𝐹𝑌
−1(𝜃) = inf⁡{𝑦:𝐹𝑌(𝑦) ≥ 𝜃}                    (1) 

 
 Given 𝑌 is dependent variable, and 𝑋 is independent variable have 𝑝 dimension. 
Let 𝐹𝑌(𝑦|𝑋 = 𝑥) = 𝑃(𝑌 ≤ 𝑦|𝑋 = 𝑥) conditional cumulative function notation of 𝑌 given 
𝑋 = 𝑥. The conditional quantile 𝜃𝑡ℎ of 𝑌 defined as bellows [4]: 
 

𝑄𝜃(𝑌|𝑋 = 𝑥) = inf⁡{𝑦:𝐹𝑌(𝑦|𝑥) ≥ 𝜃}.            (2) 
 
 Based on the median concept of estimate for 𝛽 from the quantile regression 𝜃𝑡ℎ 
obtained by minimizing the absolute number of errors by weighting 𝜃 for positive error 
and weighting 1 − 𝜃 for negative error [5] : 
 

𝑎𝑟𝑔𝑚𝑖𝑛
𝑥𝑖
𝑇𝛽0∈𝑅

𝜃 ∑ 𝜌𝜃
𝑛
𝑖=1 ⁡(𝑦𝑖 − 𝑄𝜃(𝑌|𝑋))⁡           (3) 

 
where 𝜽 is quantile indeks ∈ (𝟎,𝟏) and 𝝆𝜽 is asymmetrice loss function for 𝑸𝜽(𝒀|𝑿) =
𝑿𝑻𝜷. 

2. Bayesian Quantile Regression 

The Bayesian method uses Markov Monte Carlo chain (MCMC) to estimate the 
posterior distribution. Given 𝒚 = (𝑦1,𝑦2,…,𝑦𝑛), where the prior distribution of 𝛽 is 𝑝(𝛽). 
The prior distribution taken in this research is prior informative those originating from 
previous research [6]. Determination of prior distribution parameters are very subjective, 
depending on the researcher's intuition. A variable 𝑌 is said to follows Asymmetric 
Laplace Distribution with the density function of the probability as follows [2]: 

 
𝑓𝑝(𝑦) = 𝑝(1 − 𝑝)exp⁡{𝜌𝜃(𝑦𝑖 − 𝜇)}    (4) 

 
and likelihood function as follows : 
 

𝐿(𝑦|𝛽) = 𝑝𝑛(1 − 𝑝)𝑛exp⁡{−∑ 𝜌𝜃𝑖 (𝑦𝑖 − 𝑥𝑖
𝑇𝛽)} .  (5) 



Simulation Study of the Using of Bayesian Quantile Regression in Non-normal Error 

Catrin Muharisa 123 

 
 Then the posterior distribution of 𝛽, 𝑓(𝛽|𝑦) is given by 

𝑓(𝛽|𝑦) ∝ 𝐿(𝑦|𝛽)⁡𝑝(𝛽)         
  

∝ 𝑝𝑛(1 − 𝑝)𝑛exp⁡{−∑ 𝜌𝜃𝑖 (𝑦𝑖 − 𝑥𝑖
𝑇𝛽)}⁡𝑝(𝛽)                                    (6) 

 
 
 

RESULTS AND DISCUSSION  

 In this study data generated by software R that consists of two independent 
variables (𝑋1,𝑋2) and one response variable (𝑌), each independent variable (𝑋1,𝑋2) 
spreads according to the normal distribution (𝑋1⁡~⁡𝑁(0,1)) and (𝑋2⁡~⁡𝑁(0,1)). While the 
response variable (𝑌), is set value 𝑦 = 0,7⁡𝑥1 +⁡𝑥2 + 𝜀, where 𝜀~𝐴𝐿𝐷(0,1,0.75). Each 
variable measured 150 sample data. 
 Estimation results of each model parameter for each quantile using the quantile 
regression method can be seen in the following Table 1 
 

Table 1. Estimated parameter model of quantile regression method 

Quantile (𝜽𝒕𝒉) 𝜷𝟏 Se 𝜷𝟐 Se 

0.05 

0.25 

0.50 

0.75 

0.95 

0,97237 (0,30104) 

0,7057 (0,11758) 

1,00287 (0,00031**) 

1,20465 (0,0000**) 

1,12467 (0,1943) 

0,9369 

0,44829 

0,27177 

0,19633 

0,86254 

4,43896 (0,000055**) 

0,92898 (0,06935) 

1,25697 (0,00007**) 

1,22291 (0,0000**) 

1,27023 (0,1956) 

1,06125 

0,50779 

0,30784 

0,30784 

0,97702 

**Significant on level 𝛼=0.05  Se=Standart error 

 Based on Table 1, the parameter of 𝛽1 significant in quantities to 0.50 and 0.75. 
While the parameter 𝛽2 is significant in quantiles to 0.05, 0.50 and 0.75. Hence, the 
coefficient parameter of each quantile satisfied when used quantile 0.50. 

 Estimation results for each quantile using the Bayesian quantile regression 
method can be seen in the Table 2 

Table 2. Estimated parameter model of  Bayesian quantile regression method 

Quantile (𝜽𝒕𝒉) 𝜷𝟏 𝜷𝟐 

0.05 0,527 3,499 

0.25 0,749 1,037 

0.50 1,04 1,22 



Simulation Study of the Using of Bayesian Quantile Regression in Non-normal Error 

Catrin Muharisa 124 

0.75 1,077 1,195 

0.95 1,08 1,48 

 

 Based on the coefficient parameter of each quantile satisfied when used quantile 
0.25. Based on the point estimated value of the Bayesian quantile regression closer to the 
beta value than the quantile regression. Estimation of model parameters with quantile 
regression method and Bayesian quantile regression method as previously obtained. Next 
will be compared by using the confidence interval. The comparison results are shows in 
the following Table 3  
 

Table 3. Confidence interval of  quantile regression and Bayesian quantile regression 

Quantile (𝜽𝒕𝒉) 𝜷𝟏 𝜷𝟐 
QR BQR QR BQR 

0.05 
 
 

0.25 
 
 

0.50 
 
 

0.75 
 
 

0.95 

Lower Limit 
Upper Limit 
Difference 

Lower Limit 
Upper Limit 
Difference 

Lower Limit 
Upper Limit 
Difference 

Lower Limit 
Upper Limit 
Difference 

Lower Limit 
Upper Limit 
Difference 

-1,84973 
2,54512 
4,39485 
-0,07067 
1,51635 
1,58702 
0,7346 

1,53139 
0,79679 
0,49631 
1,46396 
0,96765 
0,6571 

1,64607 
0,98897 

-0,819 
1,17 

1,989 
0,189 
1,33 

1,141 
0,706 
1,41 

0,704 
0,601 

1,5283 
0,9273 
0,369 
1,72 

1,351 

-2,8179 
4,63321 
7,45111 
0,34168 
1,84061 
1,49893 
0,69714 
1,54094 
0.8438 

0,87919 
1,4644 

0,58521 
0,58451 
2,69848 
2,11397 

1,663 
4,77 

3,107 
0,462 
1,63 

1,168 
0,801 
1,61 

0,809 
0,795 

1,5898 
0,7948 
0,337 

2,7 
2,363 

         QR=Quantile Regression BQR=Bayesian Quantile Regression 
 
Based on Table 3, it can be seen in the quantile 0.05 by using the quantile 

regression method on 𝛽1 parameter the result is not significant, where Bayesian quantile 
regression method is significant. On the parameter of 𝛽2 to the quantile 0.05 using the 
quantile regression method and Bayesian quantile regression method not significant. In 
quantiles 0.25, 0.50, 0.75, and 0.95 it can be seen that using the Bayesian quantile and 
quantile regression methods has been significant. From the analysis, the best regression 
equation model is obtained when using Bayesian quantile regression analysis with 0.50 
quantile value. 

 
The next step in Bayesian quantile regression approach is convergence test of 

convergence of model parameters that have been estimated parameter. Test is using 
history trace plot and density plot [3]. Figure 1 and Figure 2 presents a trace plot and 
density plot for some selected parameters. 



Simulation Study of the Using of Bayesian Quantile Regression in Non-normal Error 

Catrin Muharisa 125 

 

 Figure 1. Trace Plot of parameter 𝛽1 for 𝜃 = 0.50 

 

 Figure 2. Trace Plot of parameter 𝛽2 for 𝜃 = 0.50 

Figure 1 and Figure 2 it can be concluded that the assumption of convergence is 
related. Data distribution has been stable as it is between two parallel horizontal lines. 

 

Figure 3. Density plot of parameter 𝛽1 for 𝜃 = 0.50 

 

Figure 4. Density plot of parameter 𝛽2 for 𝜃 = 0.50 

In Figure 3 and Figure 4, the density plot for some parameters show the normal 
distributed curve. This result informs us that the selected parameters model is 
convergent.  Based on the convergence examination of the trace plot and density plot, it 
can be concluded that the alleged model has satisfied the criterion of convergence. 
 
CONCLUSIONS 
 In this research, we use the analysis quantile and Bayesian quantile regression 
to analyze simulation study in non-normal error. Obtained the best regression equation 
model is Bayesian quantile method used quantile value 0.50. Bayesian quantile regression 
method using Gibbs algorithm sampling better estimator than quantile regression 
method. Because the Bayesian method has more parameter significance and confidence 
interval values smaller. 
 
 



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Catrin Muharisa 126 

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