https://doi.org/10.31449/inf.v42i4.1862 Informatica 42 (2018) 545–553 545
  
The Heteroskedasticity Tests Implementation for Linear 
Regression Model Using MATLAB 
Lуudmyla Malyarets, Katerina Kovaleva, Irina Lebedeva, Ievgeniia Misiura and Oleksandr Dorokhov 
Simon Kuznets Kharkiv National University of Economics, Nauky Avenue, 9-A, Kharkiv, Ukraine, 61166      
E-mail: aleks.dorokhov@meta.ua 
http://www.hneu.edu.ua/ 
 
Keywords: regression model, homoskedasticity, testing for heteroskedasticity, software environment MATLAB 
Received: September 23, 2017 
The article discusses the problem of heteroskedasticity, which can arise in the process of calculating 
econometric models of large dimension and ways to overcome it. Heteroskedasticity distorts the value of 
the true standard deviation of the prediction errors. This can be accompanied by both an increase and a 
decrease in the confidence interval. We gave the principles of implementing the most common tests that 
are used to detect heteroskedasticity in constructing linear regression models, and compared their 
sensitivity. One of the achievements of this paper is that real empirical data are used to test for 
heteroskedasticity. The aim of the article is to propose a MATLAB implementation of many tests used for 
checking the heteroskedasticity in multifactor regression models. To this purpose we modified  few  open 
algorithms of the implementation of known tests on heteroskedasticity. Experimental studies for 
validation the proposed programs were carried out for various linear regression models. The models 
used for comparison are models of the Department of Higher Mathematics and Mathematical Methods 
in Economy of Simon Kuznets Kharkiv National University of Economics and econometric models which 
were published recently by leading journals. 
 Pozvetek: Avrorji prispevka se ukvarjajo s problemi ekonometričnih modelov z veliko dimenzijami, kjer 
je izračun problematičen. Razvijejo metodo v MATLABu za multifaktorske regresijske modele. 
1 Introduction 
 
In econometrics, a linear regression model is often used 
to describe different processes and phenomena. Using 
matrix notation, the linear model regression can be given 
as:  
 
 + = XB Y                           (1) 
 
where Y and  are 1  n matrices, X is ) 1 ( +  m n , 
and B is 1 ) 1 (  + m ; n is  the number of measurements 
(sample size); m is the number of independent variables 
in the regression model. 
For the i
th
 row of X (the i
th
 observation) the linear 
regression model can be written as follows:  
 
i im m i i i
x b x b x b b y  + + + + + = ...
2 2 1 1 0
          (2) 
  
where 
i
y are the values of the dependent variable, 
Y 
i
y ; i is the experiment identification number, 
n i , 1 = ; 
ij
x are the values of the independent variable 
X 
j
х ( m j , 1 = ) in the i
th
 experiment; 
0
b is the 
constant term of the equation; 
j
b are the regression 
coefficients, B 
j
b b ,
0
; 
i
 are the residuals (model 
errors).  
An error term is introduced in a regression model 
because the model does not fully represent the actual 
relationship between the variables of the model. As a 
result of this incomplete relationship, there are 
differences between the observed responses (values of 
the variable being predicted) in the given dataset and 
those predicted by a linear function of a set of 
explanatory variables. The error term is the amount at 
which the equation may differ from measurements. In 
other words that is the ‘white noise’. 
As a rule, the building a linear regression model is 
done by the method of ordinary least squares (OLS). This 
method for estimating the unknown parameters is based 
on the minimization of the sum of the squares of the 
model errors. The estimators of model parameters 
determined by OLS are known as best linear unbiased 
estimators (BLUE). The variances of the model 
parameters are determined by: 
 
jj e jj
n
i
i
j
b
z z
m n
S  =
− −
=

= 2 1
2
2
1


                (3) 
 

546 Informatica 42 (2018) 545–553 L. Malyarets et al.  
 
where 
jj
z is the diagonal element of matrix 
1
) ' (
−
= X X Z
 
which corresponds to the parameter 
j
b ; 
e

 is the 
standard error. 
The OLS application requires the realization of a 
number of conditions [1 – 3]. Only if these conditions are 
met, the estimates calculated by such a model will be 
unbiased, efficient and well-off. These conditions are 
formulated in the form of the Gauss ‒ Markov theorem.  
According to this theorem there are four principal 
assumptions which admit the using of linear regression 
models for research and prediction. One of them is the 
homoskedasticity (constant variance) of the errors in 
relation to any independent variable.  
Homoskedasticity makes the assumption that the 
errors have a constant variance: const = ) var(  and 
independent of causal variables: 0 ) , cov( = 
j
x for all j ,
m j , 1 = . The error  is a random variable distributed 
according to the normal law: ) , 0 ( ~
2

  Ν where the 
mathematical expectation of the error term is zero and 
the variance is constant. Failure to comply with this 
requirement leads to bias in the estimates obtained using 
such a regression model. Thus [4] indicate that 
estimation uncertainty may increase dramatically in the 
presence of conditional heteroskedasticity.  
The requirement of homoskedasticity also exists in 
the construction of the econometric model using the 
maximum likelihood method [5 – 7]. 
When the scatter of the errors is different, varying 
depending on the value of one or more of the 
independent variables, the error terms are 
heteroskedastic. Namely the distribution law of errors 
remains normal with a mathematical expectation equal to 
zero, but the errors of the model are a function of the 
values of the independent variables:  ~ )) ( , 0 ( X f Ν , 
where ) (X f is a function that describes the change in 
the variance of errors as a function of the values of the 
independent variables.  
A similar problem arises during the building of 
semiparametric [8 – 10] and nonparametric [11, 12] 
models. 
Heteroskedasticity makes difficult to gauge the true 
standard deviation of the forecast errors. The OLS 
estimates are no longer BLUE. Thus, if the variance of 
the errors is increasing over time, confidence intervals 
for out-of-sample predictions will tend to be 
unrealistically narrow. In particular, heteroscedasticity 
does not allow us to use equation 3 for the computation 
of 
j b
S , since it assumes a uniform dispersion of the 
errors. Under heteroskedasticity, the sample variance of 
OLS estimator is 
1 1 2
) ' ( ' ) ' ( )
ˆ
(
− −
 = X X X X X X Var
e j
b 
  (4) 
 
where Ω is the covariance matrix, the elements of which 
are defined as the variance of the model parameters. 
Under homoskedasticity,_Ω= I. Equation 4 is correct if 
there is no autocorrelation.  
For these reasons, all the conclusions obtained on the 
basis of the corresponding − t statistics and − F statistics, 
as well as interval estimates, will be unreliable. 
A unified approach to the estimation of 
heteroscedasticity is lacking. To solve this problem, a 
large number of different tests and criteria have been 
developed: the Spearman rank correlation test, the Park 
test, the Glaser test, the Goldfeld ‒ Quandt test, 
the Breusch – Pagan test, the  Leven's test, the White test, 
and so on.  
The application of all the above tests is very difficult 
for the so-called ‘manual’ account, and for a large set of 
initial data it is completely impossible. 
There are a lot of software with which you can 
identify heteroscedasticity. These are professional 
packages (SAS, BMDP), universal packages (STADIA, 
OLIMP, STATGRAPHICS, STATISTICA, SPSS) and 
specialized packages (DATASCOPE, BIOSTAT, 
MESOSAUR).  
When using economic data researcher can face two 
main problems. Firstly, all the listed software are quite 
expensive and price of the product may be an 
insurmountable barrier for the young researcher. 
Secondly, company-developer never provides the source 
code, considering that this is not necessary for an 
ordinary user. Therefore, we can not modify the built-in 
algorithms to detect and eliminate heterosquadity.  
Another drawback of the above program products is 
the outdated conceptual approaches to econometric 
methods, which are constantly being improved.  
For example, the program products SPP and 
MICROSTAT calculate the coefficient of multiple 
correlations as the square root of the coefficient of 
determination. STATGRAPHICS calculates it as the 
square root of the adjusted coefficient of determination 
[13]. While in theory the coefficients of multiple 
correlations is estimated using elements of the correlation 
matrix [2].  
Another important aspect that should be taken into 
account is the existence of different algorithms to 
identify heteroskedasticity and the specific problem of 
division by zero [14]. 
Ideal option would be to create your own software 
product that would take into account the research tasks.  
However, to write such a program, the economist 
should be an expert in algorithmic programming. But this 
happens rarely.  
In this article, we carry out a comparative analysis of 
the tests most often used to detect heteroskedasticity [1, 
2, 14] and give their source code. The use of program 
code allows you to modify the program in accordance 
with the objectives of the study. 
2 Analysis of literary data and the 
formulation of the problem 
Before starting the construction of the regression model, 
it is necessary to verify whether the conditions of the 
Gauss-Markov theorem are fulfilled.  

The Heteroskedasticity Tests Implementation for Linear... Informatica 42 (2018) 545–553 547 
One of the main methods of preliminary research on 
heteroskedasticity is a visual analysis of the graph of 
residues. On these graphs, the scattering of points can 
vary depending on the value of the independent variables 
[14, 15].  
To estimate heteroskedasticity, are used such 
quantitative tests [15 ‒ 17] as the White test, the Goldfeld 
‒ Quandt test, the Breusch ‒ Pagan test, the Park test, the 
Glazer test and also the Spearman test. Unlike other tests 
the Spearman rank correlation test is a nonparametric 
statistical test for the heteroskedasticity of random errors 
in the econometric model. The test algorithm can be 
studied in detail in [18, 19]. However, it is still not 
implemented in software products which are used to 
build multiple models [20 ‒ 27].  
In this paper we examined the software packages 
most commonly used in economic activity, which 
contain tests for heteroskedasticity [15, 28]. Indeed, these 
software products do not contain the Spearman rank 
correlation test. 
The most widely used for evaluating 
heteroscedasticity is the Park test [20, 21]. However, the 
Park test contains the assumption that the change in the 
remnants of the model is described by a functional 
dependence of a certain type. It was noted in [24, 25] that 
this can lead to unreasonable conclusions. Therefore, the 
authors propose to consider the Park test together with 
other tests.  
The software implementation of the Park test for 
multiple models also does not exist [28]. As far as we 
know software implementation of the Park test for 
multifactor models also does not exist.  
Another test that the authors of the article 
implemented in the MATLAB environment is the 
Goldfeld ‒ Quandt test. This test to check for 
heteroskedasticity of random errors is used when there is 
reason to believe that the standard deviation of errors is 
proportional to some variable.  
The test statistics has a Fisher distribution [18, 27]. 
The Goldfeld ‒ Quandt test can also be used if there is an 
assumption of intergroup heteroskedasticity, when the 
variance of errors takes, for example, only two possible 
values. In this case, for the application of the test, there is 
a need for its software implementation, since applied 
commercial software has not taken this possibility into 
account [25, 28]. 
In scientific articles on for the problem of detecting 
heteroskedasticity, the Breusch ‒ Pagan test is often 
considered [10, 29]. We also carried out research this 
problem. But it oversteps this article.  
Analysis of literature sources shows that all tests of 
heteroskedasticity detection are difficult for ‘manual’ 
application and require the development of special 
software. In turn, the software of econometric research 
does not contain built-in functions for heteroskedasticity 
testing with open source code.  
That is way the authors of this article attempted to 
implement the above tests for heteroskedasticity in the 
construction of multifactor econometric models in the 
MATLAB software environment. 
It should be noted that MATLAB does not contain 
ready-made software implementation to verify 
compliance of homoskedasticity. We chose it as a 
programming environment. For this purpose, other 
programming environments can also be used, for 
example, a the free software environment R. 
The authors have chosen MATLAB by the following 
reasons. First, MATLAB is used as a high-level 
programming language for writing scripts (Spearman.m, 
Parks.m and Gold_Quan.m). Secondly, MATLAB 
includes built-in functions for constructing regression 
models (Econometric toolbox), which gave the authors 
relief from the duty of programming the standard 
functions of regression analysis. Thirdly, the authors 
worked with data structures based on matrices. 
3 Aims and objectives of the study 
The purpose of the article is to present functions to check 
for heteroskedasticity in multifactor regression models. 
The implementation is made in MATLAB.  
To achieve this purpose, it is necessary to solve a 
number of problems. Namely:  
• writing the program code in the MATLAB 
programming environment;  
• planning and execution of computer calculations;  
• completion of programs;  
• analysis and interpretation of results;  
• comparison with the results of calculations using 
software products of leading companies.  
4 Practical implementation of the 
criteria for the detection of 
heteroskedasticity in econometric 
models in the MATLAB 
4.1 Spearman’s rank correlation test for 
multiple regression models 
The use of the Spearman’s test assumes that the variance 
of model errors will increase (or decrease) with 
increasing values of the independent variable.  
This means that the absolute values of errors 
i
 
) , 1 ( n i = and the values 
ij
x of the independent variable 
j
x ) , 1 ( m j = will correlate with each other.  
To check whether heteroskedasticity is statistically 
significant the Spearman’s test provides for the following 
stages: 
1) Estimation of the parameters of the econometric 
model by the OLS: 
 
im m i i i
x b x b x b b y + + + + = ... ˆ
2 2 1 1 0
, (5) 
 
where 
i
y ˆ is the predicted response in accordance with 
the model when the independent variables are 
) ... ; ; (
2 1 im i i
x x x
; 

548 Informatica 42 (2018) 545–553 L. Malyarets et al.  
 
2) Calculate model errors as the difference between 
the empirical and the ratchet value of the dependent 
variable: 
 
i i i
y y ˆ − =    (6) 
 
where 
i
y is the value of the dependent variable in the i
th
 
experiment;   
3) The pairs ) , (
i ij
x  are ranked in order of 
increasing values of the independent variable 
j
x ;  
4) The coefficient of rank correlation between 
i
 and 
ij
x is calculated as 
( ) 1
6 1
2
1
2
− 
 − =

=
n n
d
r
n
i
i
x 
,                (7) 
 
where 
i
d is the difference between the two ranking; 
5)  The significance of 
 x
r is tested by using − t 
statistic: 
 
2
1
2


x
x
r
n r
t
−
−
=  (8) 
 
6) In accordance with the predetermined confidence 
probability р (where р − = 1  ) the tabulated value of  
) 2 (
5 . 0 .
− = n t t
cr 
 is found. Then the calculated value is 
compared with the critical one. 
If the t-statistic value is greater than the critical 
value, we must say that heteroscedasticity is statistically 
significant. Here  is the significance level which is 
chosen to test the null hypothesis: 
0 =


x
. In the 
opposite case, the null hypothesis is non-contradictory. 
As an example of the implementation of this test, we 
can suggest the following m-file named Spearman: 
======================================== 
% initialization: 
X1 = load('data1.scv'); 
X2 = load('data2.scv'); 
X3 = load('data3.scv'); 
X4 = load('data4.scv'); 
X5 = load('data5.scv'); 
Y = load('data.scv'); 
% Formation of the source data array: 
X = [ones(n,1) X1' X2' X3' X4' X5']; 
% Construction of a linear multifactor  
% model by OLS - method: 
[b,bint,r,rint,stats] = regress(Y,X,0.05); 
y_p = b(1) + b(2).*X1 + b(3).*X2+ 
b(4).*X3+b(5).*X4+b(6).*X5 
sprintf('Model:') 
fprintf('y_p = %f + %f *X1+%f *X2+%f *X3+%f 
*X4+%f *X5',b) 
% Calculation of model remains: 
e = Y - y_p'; 
% Preparing an array for further work: 
X = [X1' X2' X3' X4' X5']; 
[n,m] = size(X);% Determining the size of the 
source data 
%========================================== 
%% Spearman rank correlation test 
% Ranking of factors: 
[Xs I] = sort(X) 
Dx = zeros(n,m); 
for j = 1:m 
    for i = 1:n 
        Dx(i,j) = i; 
    end 
end 
TMP = zeros(n,m); 
% Filling an array of factors with ranks 
% taking into account their sequence numbers: 
for j = 1:m 
    for i = 1:n 
        i1 = I(i,j); 
        TMP(i1,j) = Dx(i,j); 
    end 
end 
X = [X TMP]% Output array 
% Ranking of remains: 
[es I] = sort(e); 
es = [es ones(size(e),1)]; 
e = [e ones(size(Y),1)]; 
sprintf(' critical values t:','\n') 
t_r(:,j) = (r(:,j)*sqrt(n-1))/sqrt(1 - 
r(:,j)^2); 
end 
t_r % output array t-statistics by Spearman 
% Comparative analysis and conclusions: 
c = 0; 
for i = 1:size(e) 
    es(i,2) = i; 
end 
% Filling an array of remains with ranks 
% taking into account their sequence numbers: 
for i=1:size(e) 
    e(I,2) = es(:,2); 
end 
e% an array of remains which contains ranks 
r = zeros(1,m); 
d = zeros(n,m); 
% Calculating the difference of ranks 
for j = 1:m 
    for i = 1:n 
        d(i,j) = TMP(i,j) - e(i,2); 
    end 
end 
d % difference in rank 
% The square of the difference of ranks: 
for j = 1:m 
    d(:,j) = d(:,j).^2; 
end 
d 
Sd = zeros(1,m); 
% The sum of the difference of ranks squares 
% by the corresponding columns of ranks: 
for j = 1:m 
    Sd(:,j) = sum(d(:,j)); 
end     
Sd % output array 
% Calculating Spearman's Statistics: 
for j = 1:m 
    r(:,j) = 1 - (6*Sd(:,j))/(n*(n^2-1)); 
end 
r % Output array 
t_r = zeros(1,m); 
%% Testing of the significance of the Spearman 
coefficient: 
t_t = tinv(0.975,n-2)% tabulated value t 
for j = 1:m 
  if abs(t_r(:,j)) < abs(t_t) 
    sprintf(' Heteroskedasticity is absent ') 
  else 
    sprintf(' Heteroscedasticity is present ') 
    c = c + 1; 
  end 
end 
======================================== 

The Heteroskedasticity Tests Implementation for Linear... Informatica 42 (2018) 545–553 549 
4.2 Park's test for multiple regression 
models 
R. Park proposed a test to check for heteroskedasticity, 
which is based on some formal dependencies. Namely, it 
assumes that the heteroskedasticity may be proportional 
to some power of an independent variable 
j
x in the 
multiple models.  
Since the variance of errors ( )
i i
  
2 2
= is a 
function of the − i th value 
ij
x of the explanatory 
variable 
j
x , and for its description Park proposed the 
this dependence: i
v
ij i
x   

2 2
= .  
After computing its logarithms, we obtain the 
following relation: 
i ij i
v x + + = ln ln ln
2 2
   . Since the 
variances 
2
i
 are usually unknown, they are replaced by 
their estimates 
2
i
 . 
The Park's test provides for such effectuation stages: 
1) Estimation of the parameters of the econometric 
model by the OLS (Equation 5); 
2) Calculation of the value 
2 2
) ˆ ln( ln
i i i
y y − =  for 
each observation; 
3) Building the regression model: 
 
 
i ij i
x     + + = ln ln
2
,   (9) 
 
where 
2
ln   = . For the case of multiple regressions, 
this dependence is constructed for each explanatory 
variable; 
4) Verification of statistical significance of the 
coefficient  on the basis of − t statistics: 
 

  = t .   (10) 
 
5) In accordance with the predetermined confidence 
probability р (where р − = 1  ) the tabulated value of  
) 1 (
.
− − = m n t t
cr 
 is found. Then the calculated value is 
compared with the critical one. 
 If 1 ( − −  m n t t

, then at the level of significance 
 the coefficient  is statistically significant and there 
is a link between 
2
ln
i
 and 
i
x ln . It means that 
heteroskedasticity is present in statistical data. 
The M-file named Park's which is implementation of 
the Park test has the form: 
 
======================================= 
% initialization: 
X1 = load('data1.scv'); 
X2 = load('data2.scv'); 
X3 = load('data3.scv'); 
X4 = load('data4.scv'); 
X5 = load('data5.scv'); 
Y = load('data.scv'); 
% Formation of the source data array: 
X = [ones(n,1) X1' X2' X3' X4' X5']; 
[n, m] = size(X); 
% ==========  Park Test Algorithm ======= 
%  1 stage of the Park test 
% Construction of a linear multifactor  
% model by OLS - method: 
[b,bint,r,rint,stats] = regress(Y,X,0.05); 
y_p = b(1) + b(2).*X1 + b(3).*X2+ 
b(4).*X3+b(5).*X4+b(6).*X5 
sprintf('Model:') 
fprintf('y_p = %f + %f *X1+%f *X2+%f *X3+%f 
*X4+%f *X5') 
%   2 stage of the Park test 
ln_eps = log((Y' - y_p).^2) 
%   3 stage of the Park test 
for j=1:m 
    for i = 1:n 
        X(i,j) = log(X(i,j)); 
    end 
end 
%  4 stage of the Park test 
for i = 2:m 
 [bet, dev,stat] = glmfit(X(:,i),ln_eps); 
    t_t = tinv(0.95, n-2); 
    t_r = stat.t(2); 
% Comparative analysis and conclusions: 
        if abs(t_r) < abs(t_t) 
            sprintf(' Heteroskedasticity of %i 
factor is absent \n',i-1) 
        else 
            sprintf(' Heteroskedasticity of %i 
factor is present\n',i-1) 
        end 
end 
======================================== 
 
The Park test’s weakness is that it assumes the 
heteroskedasticity has a particular functional form. 
4.3 Goldfeld ‒ Quandt test for multiple 
regression models 
When using the Goldfeld-Quandt test for 
heteroscedasticity, it is assumed that model errors 


depend on one of the external variables 
j
x : 
2 2 2
ij
x
i
 

= 
It is also assumed that errors 
i
 are distributed 
according to the normal law, there is no autocorrelation.  
The Goldfeld-Quandt test provides for such 
effectuation stages: 
1) Estimation of the parameters of the econometric 
model by the OLS (Equation 5); 
2) Ranking of all n observations in magnitude of the 
independent variable 
j
x ; 
3) Segregation this ordered sample into three 
approximately equal parts k k n k , 2 , − , respectively;  
4) For each part of the sample that has a volume k , 
its regression equation is constructed and the sums of the 
squares of the deviations determine: 
 

=
=
k
i
i
RSS
1
2
1
  (11) 
and 

+ − =
=
n
k n i
i
RSS
1
2
3

.      (12) 
 
Than empirical meaning of the − F statistic is 
calculated: 

550 Informatica 42 (2018) 545–553 L. Malyarets et al.  
 
 
) 1 /(
) 1 /(
3
1
− −
− −
=
m k RSS
m k RSS
F               (13) 
 
5) Evidence of heteroskedasticity is based on a 
comparison of the residual sum of squares (RSS) using 
the − F statistic. The calculated value is compared with 
the critical value 
) 1 ; 1 (
.
− − − − = m k m k F F
cr 
 in 
accordance with the predetermined confidence 
probability р (where р − = 1  ).  
If ) 1 ; 1 ( − − − −  m k m k F F

, this means that at the 
level of significance  the hypothesis that there is no 
heteroskedasticity does not have grounds to reject. In the 
opposite case, the hypothesis of the absence of 
heteroskedasticity is rejected.  
For multiple regressions, we performed tests for all 
factors. The M-file named Gold_Quan which is the 
implementation of the Goldfeld ‒ Quandt test has the 
form: 
 
======================================== 
% initialization: 
X1 = load('data1.scv'); 
X2 = load('data2.scv'); 
X3 = load('data3.scv'); 
X4 = load('data4.scv'); 
X5 = load('data5.scv'); 
Y = load('data.scv'); 
% Formation of the source data array: 
X = [ones(n,1) X1' X2' X3' X4' X5']; 
[n, m] = size(X); 
%========================================= 
%% Goldfeld ‒ Quandt test: 
[Xsort Is] = sort(X); 
for i=1:size(Y) 
    Ysort(i,1) = Y(Is(i),1); 
end 
Dat = [Xsort Ysort]; 
c = fix(4*n/15); 
k = fix((n - c)/2); 
if floor(k) > 0.4 
    k = k+1; 
end 
k 
% Selective aggregate 1: 
Dat1 = Dat(1:k,:); 
[b1,dev1,stats1] = glmfit(Dat1(:,1),Dat1(:,2)); 
S1 = sum(stats1.resid.^2); 
% Selective aggregate 2: 
Dat2 = Dat(n-k+1:n,:); 
[b2,dev2,stats2] = glmfit(Dat2(:,1),Dat2(:,2)); 
S2 = sum(stats2.resid.^2); 
% Testing the hypothesis: 
if S1 > S2 
    Fp = S1/S2; 
else 
    Fp = S2/S1; 
end 
Ft = finv(0.95,k-m-1,k-m-1); 
if Fp > Ft 
     sprintf(Heteroscedasticity is present ') 
else 
    sprintf(Heteroscedasticity is absent ') 
end 
======================================== 
 
A weakness of the Goldfeld ‒ Quandt test is that the 
result is dependent on the criteria chosen for separating 
the sample measurements into their representative 
groups. 
5 Results of numerical experiments 
The problem of detecting heteroskedasticity in various 
multifactor econometric models was considered.  
For carrying out numerical simulation experiments 
we used both the models of the Department of Higher 
Mathematics, Economic and Mathematical Methods of 
KhNEU [30 ‒ 33], and econometric models which were 
published recently by leading journals [34 ‒ 36].  
To check for heteroscedasticity, we used real data. 
This is one of the advantages of this paper. However, it is 
possible to use the data obtained with the Monte Carlo 
simulation [6, 7, 37 ‒ 39]. 
Numerical experiments were performed on the 
configuration AMD Athlon 64 3200+1.5Gb Ram, 
graphic accelerator – Nvidia GeForce GTX 560 2Gb 
with using technology NVIDIA CUDA 4.2. 
Let's look at a concrete example of what happens to 
an eccentric model, if you do not take into account 
heteroskedasticity.  
As a model problem, the linear regression model was 
calculated for the cost of electronic textbooks produced 
by the Department Higher Mathematics and 
Mathematical Methods in Economy. The initial data and 
designations used in the process of correlation-regression 
analysis are shown in Figure 1, where Y is the resulting 
factor Y (cost of the electronic textbook). 
 
 
Figure 1: Initial data for model building 
 
Figure 1 shows the dependence of the cost of the 
electronic textbook (Y) on such external factors: 
 
 
     ○ - X1 (average cost of developers' wages); 
     + - X2 (publication volume); 
     × - X3 (average CD recording price); 

The Heteroskedasticity Tests Implementation for Linear... Informatica 42 (2018) 545–553 551 
     * - X4 (storage and distribution costs); 
     • - X5 (cost of the use of licensed software). 
 
The regression model was constructed using the 
built-in function Matlab-regress (y, X, alpha) with the 
code: 
 
 
 
 
======================================== 
% The program for multiple regression model 
building,  if heteroskedasticity is not taken 
into account : 
[b,bint,r,rint,stats] = regress(Y,X,0.05); 
y_p = b(1) + b(2).*X1 + b(3).*X2+ 
b(4).*X3+b(5).*X4+b(6).*X5; 
sprintf(' Heteroskedasticity is not taken into 
account:') 
fprintf('y_p = %f + %f *X1+%f *X2+%f *X3+%f 
*X4+%f *X5',b) 
======================================== 
 
The program for constructing multiple regressions, if 
you do not take into account heteroskedasticity, gives 
such a result: 
 
. 87 . 0 33 . 3 90 . 70
61 . 10 33 . 0 06 . 1864 ˆ
5 4 3
2 1
x x x
x x y
 +  +  +
+  +  + − =
 (14) 
 
The results of calculating the errors of the model 
represented by the Equation 10 are shown in Figure 2.  
 
 
Figure 2: Graphic illustration of the remnants of the 
model 
 
Analysis of the remnants of the model indicates that 
for this model the dispersion of remnants increases with 
an increasing of the value of external factors, that is, 
heteroskedasticity can not be ignored.  
Using the program procedures developed by the 
authors to identify heteroskedasticity, the following 
results were obtained:  
 
 
======================================== 
ans = Heteroskedasticity 1 is absent 
ans = Heteroskedasticity 2 is absent 
ans = Heteroskedasticity 3 is absent  
ans = Heteroskedasticity 4 is absent 
ans = Heteroskedasticity 5 is present 
======================================== 
 
The construction of the regression model, which 
takes into account the heteroskedasticity, was performed 
using the built-in function MATLAB: robustfit (X, y, 
wfun, tune,const).  
It should be emphasized that the presence or absence 
of heteroskedasticity in the initial data is determined 
automatically by using the check box.  
For this we used the code: 
 
======================================== 
%c is a parameter that takes the value 0 or 1 
%(where 0 - Heteroscedasticity is absent, 1 -
% Heteroscedasticity is present), 
%c depends on the result of the scripts’ work 
if c > 0 
X = [X1' X2' X3' X4' X5']; 
[b,stats3] = robustfit(X,Y,'fair',0.001,'on'); 
y_p = b(1) + b(2).*X1 + b(3).*X2+ 
b(4).*X3+b(5).*X4+b(6).*X5; 
sprintf('Heteroskedasticity is taken into account:') 
fprintf('y_p = %f + %f *X1+%f *X2+%f *X3+%f 
*X4+%f *X5',b) 
end 
======================================== 
 
The program for multiple regression model building, 
if heteroskedasticity is taken into account yields this 
result: 
 
. 80 . 0 18 . 4 16 . 29
33 . 10 94 . 0 85 . 27 ˆ
5 4 3
2 1
x x x
x x y
 +  +  −
−  +  + =
 (15) 
 
Thus, the above procedure allows eliminating 
heteroskedasticity. In this case, the resulting models will 
be able to adequately reflect the reality. 
Table 1 shows the results of numerical experiments 
on testing of programs which are presented in this article 
on various multifactor models.   
As can be seen from Table 1, software products 
developed by us  using MATLAB can be proposed both 
for constructing multifactor econometric models, and for 
investigating the latter for the presence of 
heteroskedasticity.  
In doing so, we used new numerical algorithms, 
developed on the basis of well-known tests of 
heteroskedasticity detection. 
Open source code allows the researcher to use this 
software to solve their own problems. 
 
0 100 200 300 400 500 600 700
0
0.5
1
1.5
2
2.5
3
3.5
4
x 10
5
Xj
e = Y - Yp
Residuals plot
X1
X2
X3
X4
X5

552 Informatica 42 (2018) 545–553 L. Malyarets et al.  
 
6 Conclusion and future work 
The article examined one of the key problems of 
regression analysis, which consists in verifying the 
fulfillment of the requirement of homoskedasticity of the 
remainders of the model. To this end we used various 
statistic tests. 
Analysis of literature sources and our own studies 
confirm the complexity of using all existing tests for 
detecting heteroskedasticity in the ‘manual account’ 
mode. Therefore, we gave our own implementation in 
MATLAB for tests used for detecting heteroskedasticity.    
This problem was successfully solved, as shown 
results of numerical experiments which are presented in 
the article. We represent all software products we have 
created with open source code, which enables each 
researcher to customize the program to their problems. 
In conclusion, we want to note that the work 
presented in this article is an on going work having the 
final purpose to create a complete and effective software   
for detecting heteroskedasticity in regression models.  
Another further development consists in developing 
a complete econometric toolbox in MATLAB.                                         
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Heteroskedasticity is 
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Heteroskedasticity is 
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Table 1: Results of testing programs on multiple models 

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554 Informatica 42 (2018) 545–553 L. Malyarets et al.