Title


Science and Technology Indonesia
e-ISSN:2580-4391 p-ISSN:2580-4405
Vol. 5, No. 3, July 2020

Research Paper

Simulation Study of Autocorrelated Error Using Bayesian �antile Regression
Nayla Desviona1, Ferra Yanuar2*

1,2Department of Mathematics, Andalas University, Limau Manis Campus, Padang 25163, Indonesia
*Corresponding author: ferrayanuar@sci.unand.ac.id

Abstract
The purpose of this study is to compare the ability of the Classical �antile Regression method and the Bayesian �antile
Regression method in estimating models that contain autocorrelated error problems using simulation studies. In the quantile
regression approach, the data response is divided into several pieces or quantiles condition on indicator variables. Then, The
parameter model is estimated for each selected quantiles. The parameters are estimated using conditional quantile functions
obtained by minimizing absolute asymmetric errors. In the Bayesian quantile regression method, the data error is assumed to be
asymmetric Laplace distribution. The Bayesian approach for quantile regression uses the Markov Chain Monte Carlo Method
with the Gibbs sample algorithm to produce a converging posterior mean. The best method for estimating of parameter is the
method that produces the smallest absolute value of bias and the smallest confidence interval. This study resulted that the
Bayesian �antile method produces smaller absolute bias values and confidence interval than the quantile regression method.
These results proved that the Bayesian �antile Regression method tends to produce be�er estimate values than the �antile
Regression method in the case of autocorrelation errors.

Keywords
�antile Regression Method, Bayesian �antile Regression Method, Confidence Interval, Autocorrelation.

Received: 20 May 2020, Accepted: 16 July 2020

https://doi.org/10.26554/sti.2020.5.3.70-74

1. INTRODUCTION

Parameter estimation in linear regression analysis is performed
using the Ordinary Least Squares Method. This method has
several assumptions that must be approved in order to get the
Best Linear Unbias Estimator (BLUE). In some empirical data, not
all assumptions can be ful�lled, such as autocorrelation among
errors.

Then, the quantile regression method appears to overcome
weaknesses in the Ordinary Least Square (OLS) method. This
method uses a parameter estimation approach by separating
or dividing data into quantiles by assuming conditional quan-
tile functions in data distribution and minimizing the absolute
symmetric weighted errors. This quantile regression analysis is
used to overcome assumptions that are not met, including the
existence of autocorrelation, normality assumptions, no multi-
collinearity and homogeneity of variances (Yanuar et al., 2019a)

Large data size is usually needed in the quantile regression
method. A sampling of large data requires a lot of time and a
lot of energy. Therefore it is used the Bayes method to evaluate
parameters with quantile regression. Bayesian related to variable
selection in quantile regression has received great attention in
the literature because the Bayes method is able to get good
models with small data (Oh et al., 2016), (Yanuar et al., 2013),

(Yanuar et al., 2019d).
In Bayes’s views, the unknown parameter is assumpted as

a random variable and has distribution. Distribution related
to the parameter can be obtained from corresponding previous
research or based on expert opinion, this distribution known as
the prior distribution. Then, the prior distribution is combined
with information from data obtained from sampling (known
as likelihood function). The combination of both distribution
then results in a posterior distribution of parameters. Averages
and variations of this posterior distribution are made estimators
for the regression parameters by the Bayesian method (Yanuar
et al., 2019b). In case of di�culties to identify the distribution
of posterior distribution or because of complex formulation,
Bayesian method uses the MCMC (Markov Chain Monte Carlo)
algorithm to estimate the mean posterior and variance posterior
of the parameter model.

In previous studies (Muharisa et al., 2018), the Bayesian Quan-
tile Regression Method with Abnormal Error has been discussed
in the case of Low Birth Weight (BBLR) in West Sumatra in the
data of 2016 to 2018. Furthermore (Delviyanti et al., 2018) has
been examined the application of the Quantile Regression with
the Bootstrap Method to the autocorrelated error in the case
of the interest rate on Indonesia’s in�ation rate. This article

https://doi.org/10.26554/sti.2020.5.3.70-74


Desviona et. al. Science and Technology Indonesia, 5 (2020) 70-74

will compare the ability of the Quantile Regression method and
the Bayesian quantile Regression method in overcoming the
autocorrelated error problem using a simulation study.

2. EXPERIMENTAL SECTION

2.1 Materials

2.1.1 Quantile Regression

The quantile method is one of the methods of regression mod-
eling by dividing data sets into several equal parts with data
sorted from the smallest or largest. Quantile regression, in the-
ory, is able to overcome the existence of autocorrelation, normal-
ity assumptions, heteroskedasticity, multicollinearity problems
and etc. Quantile regression minimizes asymmetric weighted
data and agrees with the data function on the data distribution,
(Muharisa et al., 2018). Linear equations for quantile � can be
written with the equation :

Y = X

′

i
�(�) + " (1)

The general estimation for it to � can be written as follows:

argmin
x
′

i
���R

n

∑

i=1

�
�
(yi − Q�(Y |X)) (2)

Where :
� : Show as quantile index �(0,1 )
�
�

: is an asymmetric loss function
Q
�
(Y |X) = X

′

i
� the quantile function to � from Y on X con-

dition, (Yanuar et al., 2019a).

2.1.2 Bayesian Methods

Bayes introduced a parameter estimation method where we need
to know the form of the initial distribution (prior) of parameters
to �nd the estimated � parameter of the population, known as
the Bayesian method. The Bayesian method uses the prior f (�)
distribution together with the likelihood function to determine
the posterior f (�|x1,x2, ...,xn) distribution (Yanuar et al., 2019c)

Given y = (y1,y2, . . .yn) where the prior distribution of � is �
(�). The prior distribution taken in this research is prior informa-
tive which is originating from previous research. Determination
of prior distribution parameters are very subjective, depending
on the researcher’s intuition.

2.1.3 Bayesian Quantile Regression

A random variable of Y is said to follow Asymmetric Laplace
Distribution (ALD)(�, �, �) with location � parameter, scale of
�>0 parameter and skewness p parameter in (0,1), the density
function of the probability of ALD as follows, (Alhamzawi and
Yu, 2012) ; (Feng and He, 2015).

f (y|�,�,�) =

p(1 − p)

�

exp(−�p(

y − �

�

)) (3)

Where �p is loss function that is de�ned as follows :

�p(u) = u(p − 1u<0) (4)

With I shows the indicator function. (Benoit and Van den
Poel, 2010) ; (Yanuar et al. (2019).

2.2. Methods

The data used in this study was data generated using R software
version 3.6.1 (R Development Core Team, 2011). The data used in
this study consisted of two variables X1 and X2 each generated
from N (0.1). While the dependent Y variable was set with the
value Yt = 0.5X1t + 2X2t + "t ."t = Sin(Seq(0.1�, 15.0�, 0.1�)) + Zt
with ∼ N(0, 0.1) to t = 1, 2, ..., 150.

3. RESULTS AND DISCUSSION

In this section, We will describe the results of parameter estima-
tions and comparisons for the Quantile regression and Bayesian
Quantile regression methods.

3.1. Durbin Watson (DW) test on Simulation Data

In this study, to see the existence of autocorrelation in error from
simulation data used The Dubin Watson test.

Based on the results of Durbin Watson (DW) using R version
3.6.1 software (R Development Core Team, 2011) the statistical
DW value was 0.124904. To �nd out whether the error of the
simulation data was free from autocorrelation, We have com-
pared the statistical DW values with DW table values. With the
number of independent K variables was 2 and the number of n
observations was 150.

Then the values for d
L
=1.7062 and du=1.7602 were obtained

from the DW Table values. In other words, Based on the Durbin
Watson test, if the DW value lies between 0 and d

L
stated that

0<DW<d
L

Or 0<0.124904<1.7062. Then H0 is rejected, the error
contains autocorrelation. Thus, in this case, the simulation data
contains autocorrelated errors.

3.2. Parameter Estimation by using the Quantile Regres-
sion and Bayesian Quantile Regression Model
At this step estimation parameters for quantile regression and
Bayesian quantile regression will be performed. The data con-
sists of two variables X1 and X2 each generated from N (0,1).
While the dependent Y variable was set with the value Yt =
0.5X1t + 2X2t + "t

The quantile regression method is done by dividing or split-
ting the data into groups that have di�erent estimations of the
results in the quantiles.

In the Bayesian quantile regression method, the results of esti-
mation are also seen in each quantiles. The Bayesian method uses
the MCMC to estimate its posterior distribution with the help of
R version 3.6.1 software (R Development Core Team, 2011). The
results of parameter estimation by the quantile method and the
Bayesian quantile method are presented in Table 1.

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Desviona et. al. Science and Technology Indonesia, 5 (2020) 70-74

Table 1. Estimated Parameters by Both Methods.

Quantile Quantile Reg Bayesian Quantile Reg
to � �1 �2 �1 �2
0.1 0.5230* 1.9899 0.4308* 1.9356*
0.25 0.4722* 1.9996* 0.4325* 1.9357*
0.5 0.3787* 1.9632 0.4340* 1.9391*
0.75 0.4282* 2.0619* 0.4329* 1.9296*
0.9 0.4779* 2.0418 0.4230* 1.9366*

Figure 1. Trace plot �1 with quantile 0.25

In Table 1, it can be seen that the parameter estimation for
�1 in quantile regression has been signi�cant for all quantiles by
selecting the real level � = 0.05. Whereas �2 is only signi�cant at
the quantiles 0.25 and 0.75. While the estimated parameters for
Bayesian quantile regression for �1 and �2 have been signi�cant
for all each quantiles with a real level � = 0.05.

The analysis continued with the Durbin Watson (DW) test,
to estimate the two methods by looking at the DW values in each
quantiles, this aimed to �nd out whether the model meets the
Best Linear Unbias Estimator (BLUE) by seeing no more assump-
tions that are violated. For the Quantile Regression Method in
the simulation data, the DW value was 2.0829 and the Bayesian
Quantile Regression had DW value 2.1132 with a p-value <0.05
. In the DW table with n=150 and K = 2 and the values, D

L
=

1.7249 and D
L

= 1.7915.
Based on the Durbin Watson test, if the DW value is located

at D
U

< 4-D
L
, then H0 is accepted.

Thus, the error does not contain autocorrelation for both
methods. Likewise for quantile 0.25, 0.5, 0.75, 0.9.

3.3 Comparison between the Quantile Regression
Method and the Bayesian Quantile Regression
In this section, We compared the estimated results of the model
parameters with the two methods by estimating the value of the
Absolute Bias and the width of the Bayes con�dence interval.

Figure 2. Plot Density �1 with quantile 0.25

Estimated results for both criteria are presented in Table 2.
Based on Table 2, it is known that the Absolute bias value

for �1 results of Bayesian quantile regression tends to produce
a smaller value than the quantile regression results. Smaller
values are marked in bold. While the Absolute Bias value for �2
results of the quantile regression tends to produce a smaller value
than the Bayesian quantile regression results. This is acceptable
because the Bayes estimator is a biased estimator so that the
results of estimated posterior mean may slightly deviate from
the true values.

Then the analysis results from Table 2, also show that the
con�dence interval is 95% for �1 and �2 with the Bayes approach
tends to produce a smaller value than the classical quantile
method. This result is acceptable because the Bayes method
tends to produce a smaller variance so that in the calculation of
the con�dence interval will also produce a smaller value than
the classical quantile method.

Thus, it could be concluded that the Bayes estimator method
tends to produce better guess values than the quantile method
in autocorrelated error cases.

Furthermore, the convergence test would be conducted for
the parameters of the model that has been obtained. The conver-
gence test used was by choosing a trace plot, density plot and
autocorelation plot of the selected parameters.

Figure 1 presents a trace plot of the �1 parameter at the
quantile 0.25. This �gure shows that the estimated values of
the parameter with the iteration process has spread around two

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Desviona et. al. Science and Technology Indonesia, 5 (2020) 70-74

Table 2. Comparison of Absolute Bias & Interval of Con�dence Method of Quantile Regression and Bayesian Quantile Regression.

Quantile Absolute Bias Bayes Credible Interval
to � QR BQR QR BQR

�1 �2 �1 �2 �1 �2 �1 �2

0.1 0.023 0.01 0.0692 0.0643 0.5578 0.4438 0.3005 0.2792
0.25 0.0578 0.046 0.0475 0.0643 0.4381 0.3744 0.3205 0.2921
0.5 0.1212 0.037 0.0666 0.0609 0.304 0.3246 0.3231 0.2746
0.75 0.0718 1.794 0.0711 0.0704 0.2996 0.2891 0.2354 0.2912
0.9 0.052 0.072 0.0772 0.0634 0.4443 0.2955 0.3111 0.2962

Figure 3. ACF plot �1 with quantile 0.25

linear horizontal lines so the parameter is said to be convergent.
The Density Plot model parameters for quantile 0.25 with

�1 can be seen in Figure 2 and for the ACF plot can be seen in
Figure 3. In Figure 3, the ACF plot shows stationary data on
average. It can be seen that the ACF plot on the lag line above
does not cross the 1.0 ACF line.

In this case, it can be said that there is no error containing
autocorrelation for the selected quartile, the quartile 0.25 with
the parameter �1.

4. CONCLUSIONS

This study used a simulation study to apply the quantile re-
gression and Bayesian quantile regression methods for auto-
correlated errors. By comparing the quantile values in each
parameters, and looking at the Absolute Bias values and the
smallest con�dence interval for each quantiles in the parameter.
The quantile regression method is done by dividing or splitting
the data into groups that have di�erent estimates of the results

in the quantiles. In the Bayesian quantile regression method
the parameter estimation results are also seen in each quantiles.
The Bayesian method uses MCMC to estimate its posterior dis-
tribution. The obtained results proved that the best method for
estimating model parameter is the Bayesian quantile regression
method. By looking at the quantile value of each parameter, the
smallest quantile 0.25 was obtained based on the absolute bias
value and the smallest con�dence interval 95%.

REFERENCES

Alhamzawi, R. and K. Yu (2012). Variable selection in quantile
regression via Gibbs sampling. Journal of Applied Statistics,
39(4); 799–813

Benoit, D. F. and D. Van den Poel (2010). Binary quantile regres-
sion: A Bayesian approach based on the asymmetric Laplace
density. Journal of Applied Econometrics

Delviyanti, O. S., F. Yanuar, and D. Devianto (2018). Simulation
Study The Implementation of Quantile Bootstrap Method on
Autocorrelated Error. CAUCHY: Jurnal Matematika Murni dan
Aplikasi, 5(3); 95–101

Feng, Y., Yang ; Chen and X. He (2015). Bayesian quantile regres-
sion with approximate likelihood. Bernoulli, 21(2); 832–850

Muharisa, C., F. Yanuar, and D. Devianto (2018). Simulation Study
The Using of Bayesian Quantile Regression in Nonnormal
Error. CAUCHY: Jurnal Matematika Murni dan Aplikasi, 5(3);
121–126

Oh, M., J. Choi, and E. Sug (2016). Bayesian variable selection
in quantile regression using the Savage–Dickey density ratio.
Journal of the Korean Statistical Society, 45(3); 466–476

Yanuar, F., K. Ibrahim, and A. A. Jemain (2013). Bayesian struc-
tural equation modeling for the health index. Journal of Ap-
plied Statistics, 40(6); 1254–1269

Yanuar, F., H. Yozza, F. Firdawati, I. Rahmi, and A. Zetra (2019a).
Applying bootstrap quantile regression for the construction of
a low birth weight model. Makara Journal of Health Research,
23(2); 5

Yanuar, F., H. Yozza, and R. V. Rescha (2019b). Comparison of
Two Priors in Bayesian Estimation for Parameter of Weibull
Distribution. Science and Technology Indonesia, 4(3); 82–87

Yanuar, F., H. Yozza, and A. Zetra (2019c). Bayesian Quantile
Regression Methods in Handling Non-normal and Heteroge-

© 2020 The Authors. Page 73 of 74



Desviona et. al. Science and Technology Indonesia, 5 (2020) 70-74

neous Error Term. Asian Journal of Scienti�c Research, 12(3);
346–351

Yanuar, F., A. Zetra, C. Muharisa, D. Devianto, A. R. Putri, and
Y. Asdi (2019d). Bayesian Quantile Regression Method to

Construct the Low Birth Weight Model. In Journal of Physics:
Conference Series, volume 1245. IOP Publishing, page 012044

© 2020 The Authors. Page 74 of 74


	INTRODUCTION
	EXPERIMENTAL SECTION
	RESULTS AND DISCUSSION
	CONCLUSIONS