We will here study the stock market indexes, in the context of a random walk test with ARCH (q) disturbances. This model based on these theoretical predictions has been valuated from the Tunis Stock market data. The coherence of the parameters signs and the statistical relevance of the estimations are validating the choice of the conditionally heteroskedastic random walk mode

Ben Naceur, Hassen

Munich Personal RePEc Archive

MPRA
Munich Personal RePEc Archive
Stock Market Indexes: A random walk
test with ARCH (q) disturbances
Hassen Ben Naceur
Universite´ Tunis El Manar Faculte´ des Sciences Economiques et de
Gestion de Tunis, Tunisia
September 2014
Online at https://mpra.ub.uni-muenchen.de/78978/
MPRA Paper No. 78978, posted 7 May 2017 06:55 UTC
International Journal of Innovation and Scientific Research 
ISSN 2351-8014 Vol. 8 No. 2 Sep. 2014, pp. 305-316 
© 2014 Innovative Space of Scientific Research Journals 
http://www.ijisr.issr-journals.org/ 
 
Corresponding Author: Hassen Ben Naceur 305 
 
 
Stock Market Indexes: A random walk test with ARCH (q) disturbances 
Hassen Ben Naceur 
Université Tunis El Manar 
Faculté des Sciences Economiques et de Gestion de Tunis, Tunisia 
 
 
 
Copyright © 2014 ISSR Journals. This is an open access article distributed under the Creative Commons Attribution License, 
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 
 
 
ABSTRACT: We will here study the stock market indexes, in the context of a random walk test with ARCH (q) disturbances. 
This model based on these theoretical predictions has been valuated from the Tunis Stock market data. The coherence of the 
parameters signs and the statistical relevance of the estimations are validating the choice of the conditionally 
heteroskedastic random walk model. 
KEYWORDS: white noise, index, random walk, ARCH (or GARCH) model. 
1 INTRODUCTION 
We will, here, study the series of the stock market quotations. The choice of the topic is dictated by considerations 
originating, a priori, in the economic theory and the availability of the series of observations. 
2 THE CONDITIONALLY HETEROSKEDASTIC RANDOM WALK MODEL 
We will, here, achieve a random walk test. We will assume that the data-generating process is a random walk process, 
that is to say: 
ttt eyy += −1      
The disturbances  te  are checking the following   hypotheses: 
[ ] 0=teE  
[ ]


 =
=
−
           otherwise      0
0   if   2 s
eeE stt
σ
 
The first interpretation of the random walk hypothesis is dictated by the conditional expectation, that is to say: 
1
1
1
1
                   
−
−
−
−
=



 +
=




t
t
tt
t
t
y
y
eyEy
yE
 
Then, the best prediction of the stock market quotations corresponds to its past value. 
The choice of a random walk process means that in average, the change of quotation is zero. 
This second interpretation is described in terms of the conditional expectation of the increases of the series: 
Stock Market Indexes: A random walk test with ARCH (q) disturbances 
 
 
ISSN : 2351-8014 Vol. 8 No. 2, Sep. 2014 306 
 
 
      0                              
11
1
=




=



 −
−−
−
t
t
t
tt
y
eEy
yyE
 
We will achieve, as a first step, the conditionally heteroskedastic random walk   test: 
 




+Σ+=
+=
−
=
−
    
2
10
2
1
titi
q
it
ttt
ecce
eyy
ω
 
:                            
  as, Then, 1
*
t
ttt
e
yyy
=
−=
−
 
We will, here, study the data-generating process as a conditionally heteroskedastic white noise model: 
  
     
2
10
2
*




+Σ+=
=
−
=
titi
q
it
tt
ecce
ey
ω
 
We will, here, achieve the white noise test in the series studied: 
        
1
*
t
ttt
e
yyy
=
−=
−
 
[ ] [ ]
0                       
   Then, *
=
= tt eEyE  
[ ] [ ]
( )s
eeEyyE sttstt
γ=
=
−−
          
**
 
 
( ) [ ]
( )


 =
=
=
−
.      otherwise         0
0   if    0
          
**
s
yyEs stt
γ
γ
 
Then: 
            
( )
( )


 =
=
=
.       otherwise     0
0   if     1
          
0
s
s
s γ
γρ
 
*
ty  Is a white noise if 0=sρ , ,...2,1=s  
Then, we will achieve a white noise test in the series 
*
ty  : 
                 
    0:
0:0
≠
=
i
i
a
i
i
H
H
ρ
ρ
 
,...2,1=i  
Hassen Ben Naceur 
 
 
ISSN : 2351-8014 Vol. 8 No. 2, Sep. 2014 307 
 
 
Under 0:0 =i
iH ρ , the data-generating process is a white noise. 
This propriety of the process studied enables as to get interested to the future values of the stock market indexes series: 
t
t
t yy
yE =



 +1  
As the matter of fact, the prediction to the time t  of the process studied corresponds to the current value. 
The statistics associated to this test are then: 
( )
[ ]ii
i
ii
T
t
i
ρρ
ρσ
ρρ
ρ
−=
−
=
ˆ
     
ˆˆ
ˆ
2
1
ˆ
 
Under 0:0 =i
iH ρ  ,  ( )1,0ˆ21 NT i ≈ρ  
( )
( )0ˆ
ˆ
ˆ
γ
γρ ii =  
( ) itt
t
ee
iT
i
−
Σ
−
= ˆˆ
1γˆ . 
We will reject  0:0 =i
iH ρ    if   
2
ˆ αρ tt i > . 
If 05,0=α  ,   96,1
2
=αt . 
The hypothesis 0:0 =i
iH ρ  validates the white noise hypothesis. 
We will, here, achieve a second white noise test: 
     
0:
0
10
c
a
s
HH
H
=
=== ρρ K
 
The alternative hypothesis indicates the complementary of 0H . 
The statistic associated to this test is then: 
2
1
ˆi
s
i
TQ ρ
=
Σ= . 
Under  0: 10 === sH ρρ K  , ( )sQ 2αχ≈ . 
We will determine the value of the Ljung –Box statistic, 
*Q  : 
( ) 2
1
*
ˆ
12 i
s
i iT
TTQ ρ
−
Σ+=
=
      . 
*Q is following a Khi –square distribution with s degrees of freedom. 
 
 
 
Stock Market Indexes: A random walk test with ARCH (q) disturbances 
 
 
ISSN : 2351-8014 Vol. 8 No. 2, Sep. 2014 308 
 
 
3 PROPRIETIES OF THE SERIES STUDIED-TESTS 
3.1 PROPRIETIES OF THE SERIES STUDIED 
The series of the stock market quotations is subject to an instantaneous variability.  
We will study the series available, tp , over the period, [1997, 2008].  
The data constitute the daily frequencies.  
i. Inference  statistics 
We will, at a first step, determine the moments of order one and two, the minimum and the   maximum, the of skewness 
( )s  and kurtosis ( )k  parameters …: 
 
tp  ty  
m  1146,25 0,0005 
Median 1031,85 0,0002 
Minimum 449,64 -0,05 
Maximum 2346,11 0,04 
σ  473,10 0,0077 
s  0,64 0,08 
k  2,64 6,24 
N  2742 2742 
 
We will study, as a second step, in terms of rate increase, the stock market series: 






=
−1
*
t
t
t p
p
Logp . 
We will compare the skewness ( )s  and the kurtosis ( )k  coefficients to the parameters values   associated to a 
gaussian process: 0=s  and 3=k . 
The values of these coefficients, corresponding to 08,0=s  and 24,6=k , do not belong to the reference values of a 
gaussian process. 
As a matter of fact, the series of the stock market quotations is not following a normal distribution. 
The value of kurtosis, which is quite high, expresses the leptokurtic aspect of the series studied. 
The coefficient s  does not belong to the proximity of zero. This hypothesis is significant of non-linearity. 
 ii. The graphic study: 
The series of the stock market quotations, tp  is not, a priori, stationary (Graph a). 
In the other hand, the series 
*
tp  is stationary (Graph b).  
The graphic representation of this series is expressing the regroupings of volatility: 
The values of this series, with unpredictable signs are followed by values with the same signs. 
The variability of the series persists in time. These predictions are validating, a priori, of an ARCH model hypothesis, 
representative of a data-generating process. 
 
Hassen Ben Naceur 
 
 
ISSN : 2351-8014 Vol. 8 No. 2, Sep. 2014 309 
 
 
 
 
Graph a                       Graph b 
3.2 TESTS  
We will, here, achieve a whole range of tests: a stationarity test, a normality test, 
i. Stationarity test  
 ADF  pp −  
tp  ( )1I  ( )1I  
*
tp  ( )0I  ( )0I  
 
The series of the stock market quotations tp  is not stationary. 
The series  
*
tp  is stationary. Then, the series  tp  is ( )1I . 
ii. Normality test  
Test 1 :  Jarque –Béra test: 
We will, here, achieve the test: 



=−
=
03
0
:0 k
s
H  
c
a HH 0= . 
The statistics associated to this test are then: 
( ) 



−+=− 22 3
4
1
6
ksTBJ . 
Under ( )      .     2 :           03
0
: 20



≈=−
=
αχJBk
s
H  
s  and  k  indicate the skewness and the kurtosis coefficients. 
 Statistics of the  test p  
*
tp  01,1197  000000,0  
 
 
-.06
-.04
-.02
.00
.02
.04
.06
500 1000 1500 2000 2500
Y
400
800
1,200
1,600
2,000
2,400
500 1000 1500 2000 2500
P
Stock Market Indexes: A random walk test with ARCH (q) disturbances 
 
 
ISSN : 2351-8014 Vol. 8 No. 2, Sep. 2014 310 
 
 
As 05,0≤p , we will, here, reject:  



=
−=
0   
3
:0
ks
H  
The series  ty  is not following a normal distribution. 
Test 2 :  Granger –Newbold test 
We will, afterwards, achieve the normality test of Granger –Newbold. 
We will determine the autocorrelation coefficients of order j, :jρ  
j  ( )tj yρ  ( )2tj yρ  ( )( )2
2
tj
tj
y
y
ρ
ρ
 
24
23
4
3
2
1
M
 
031,0
015,0
024,0
041,0
111,0
313,0
−
−
−
M
 
034,0
024,0
199,0
215,0
340,0
494,0
M
 
0282,0
01,0
0028,0
0080,0
04,0
2,0
M
 
 
The ratio between  and ( )2tj yρ   does not belong to the proximity of one the series studied is not following a 
normal distribution. It is constitutes the general characteristic of the financial series. 
It is agreed to underline that: 
*
tt py =  
4 ESTIMATION 
In the choice of model, we realize a whole class of tests. 
4.1 TESTS ON THE MODEL  
tt eby += 0  
i. White noise test 
We will, here, achieve, on the series
2
ˆ te , the test: 
.               
)0:(or                       0:
0
100
c
a
si
HH
HH
=
==== ρρρ K
 
We will reject, 0: 10 =ρH  (and 0: 210 == ρρH ). Then, 2ˆte  is, a priori, following a second order   autoregressive 
process: The disturbances te  are following an ARCH (2) (or GARCH (1, 1)) model. 
 
 
ii. Test  
( )tj y2ρ
Hassen Ben Naceur 
 
 
ISSN : 2351-8014 Vol. 8 No. 2, Sep. 2014 311 
 
 
0....: 10 === qccH  
            
cHHa 0=  
     We will achieve, in the context of this hypothesis, the regression of de 
2
ˆte  on the values  
2
ˆ ite −  : 
titi
q
it
ecce ω+Σ+=
−
=
2
10
2
ˆˆ . 
 
( )qARCH  0c  1c  2c  F  2NR  
( )1ARCH  
( )30,12
1005,3 5−×
 ( )25,29
48,0
 
 56,255  34,652  
 
( )38,10
1062,2 5−×
 ( )15,22
42,0
 ( )44,7
14,0
 
71,463  40,693  
 
The numbers in brackets are the  t  of Student. 
 
 2R  LogL  AIC  SIC  HQC  DW  
( )1ARCH  24,0  52,20872  24,15−  22,15−  24,15−  14,2  
 25,0  86,20891  25,15−  25,15−  25,15−  00,2  
 
The values of  LogL  and  2R  and the information criteria validate the hypothesis of an ( )2ARCH model. 
We will, then, achieve the test: 
0:
0:
2
20
≠
=
cH
cH
a
 
We will, here, determine  iSCR  : 
5
0 1088,3
−×=SCR  
51080,3 −×=aSCR     . 
The statistics associated to this test is then: 
( )
( )3
1
0
−
−
=
N
SCR
SCRSCR
F
a
a
   . 
This statistics is following a Fisher distribution with 1and 3−N  degrees of freedom. 
( ) ( )
66,57    
3
1080,3
1080,388,3
5
5
=
−
×
×−
=
−
NF
 
( )3,1 −> NFF α  : We will reject 0: 20 =cH . 
( )2ARCH
( )2ARCH
Stock Market Indexes: A random walk test with ARCH (q) disturbances 
 
 
ISSN : 2351-8014 Vol. 8 No. 2, Sep. 2014 312 
 
 
The test validates an ( )2ARCH   process. 
At the first time, the choice of model validates the following estimations: 
 
0
ˆb  
0
ˆbt  
p  
000534,0  62,3  0003,0  
 
163294,0=SCR . 
( ) 30,9443,, 20 =σbyLogL  
37,1=DW  
8896,6−=AIC  
8875,6−=SIC  
8888,6−=HQC  
Nous réalisons le test : 
0:
0:
0
00
≠
=
bH
bH
a
 
Under 0: 00 =bH ,    ( )      62,3ˆˆ
ˆ
0
0
ˆ
0
==
b
b
tb σ
 
as 05,0<p , we reject 0: 00 =bH . 
 Dans un second temps, nous réalisons le test d’un processus ( )qARCH  : 
 
      
  
ˆˆ
2
110
2
0



++=
+=
− ttt
tt
ecce
eby
ω
 
  p  
0
ˆb  
( )62,1
000184,0
 
1034,0  
0c  
 
 
1c  
 
( )14,36
105,2 5−×
 
 
( )90,15
62,0
 
0000,0  
 
 
0000,0  
 
The parameter 0b is not significantly different from zero. 
 
 
4.2 RANDOM WALK WITH DISTURBANCES GARCH(P, Q)-TESTS 
Hassen Ben Naceur 
 
 
ISSN : 2351-8014 Vol. 8 No. 2, Sep. 2014 313 
 
 
We will study the statistical pertinence of estimations of the two classes of models: 
i- Random walk model with disturbances GARCH (p, q), 
ii- Random walk model with disturbances TGARCH (p, q). 
The estimations of these models are the following: 
ARCH (1) model: 



++=
+=
−
−
    
ˆˆ
2
110
2
1
ttt
ttt
ecce
eyy
ω
 
( ) ( )60,15             52,10     
ˆ44,0108,4ˆ 2 1
52
−
− +×= tt ee
 
 
ARCH (2) model: 
- 
    
       
ˆˆˆ
2
22
2
110
2
1



+++=
+=
−−
−
tttt
ttt
ececce
eyy
ω
 
( ) ( ) ( )10,84     45,14             38,85     
       
ˆ20,0ˆ41,0108,3ˆ 2 2
2
1
52
−−
− ++×= ttt eee
 
 
ARCH (6) model: 
-        
        
ˆˆ
2
6
10
2
1




+Σ+=
+=
−
=
−
titiit
ttt
ecce
eyy
ω
 
 
 icˆ  ict ˆ  
0c  
5108,1 −×  20,84 
1c  0,41 14,52 
2c  0,20 11,11 
3c  0,08 8,10 
4c  0,14 8,12 
5c  0,10 6,48 
6c  0,04 4,84 
 
GARCH (p, q) model: 
- 



+++=
+=
−−
−
tttt
ttt
haecce
eyy
ω11
2
110
2
1
ˆˆ
 
 
 0c  1c  1a  
Stock Market Indexes: A random walk test with ARCH (q) disturbances 
 
 
ISSN : 2351-8014 Vol. 8 No. 2, Sep. 2014 314 
 
 
( )1,1GARCH  
( )72,11
104,3 6−×
 ( )72,19
28,0
 ( )82,61
72,0
 
TGARCH (p, q) model: 
 
 0c  
+
1c  
−
1c  2c  
( )1,1TARCH  
( )32,51
108,4 5−×
 ( )76,6
26,0
 ( )42,5
35,0
 
 
( )1,1TGARCH  
( )30,11
104,3 6−×
 ( )45,10
25,0
 ( )12,2
06,0
 ( )56,57
72,0
 
 
The numbers in brackets are the  t  of Student. 
The criterion of the maximum of likelihood and information leads us to the choice of a random walk model with 
disturbances ARCH (or TARCH). These criteria are determining bay the following values: 
 
 ( )2,, σcyLogL  AIC  SIC  HQC  
( )1ARCH  60,9264  7602,6−  7538,6−  7580,6−  
 62,9305  7895,6−  7808,6−  7864,6−  
( )6ARCH  97,9417  8715,6−  8628,6−  8684,6−  
( )1TARCH  90,9273  7664,6−  7577,6−  7632,6−  
( )1,1TGARCH  6,9419  8717,6−  8608,6−  8678,6−  
( )1,1GARCH  97,9417  8715,6−  8628,6−  8684,6−  
ARCH-M 9930,98 -7,2425 -7,2318 -7,2387 
 
 
The series of stock market quotations is following a random walk process. 
It is a random walk model with disturbances ARCH (or TARCH). 
 
 
 
 
 
 
 
 
 
 
 
 ( )2ARCH  ( )6ARCH  ( )1,1GARCH  ( )1,1TARCH  ( )1,1TGARCH  
( )2ARCH
Hassen Ben Naceur 
 
 
ISSN : 2351-8014 Vol. 8 No. 2, Sep. 2014 315 
 
 
0c  
( )38,85 
108,3 5−×
 ( )20,84 
108,1 5−×
 ( )11,72 
104,3 6−×
 ( )51,32 
108,4 5−×
 ( )11,30 
104,3 6−×
 
1c  ( )14,45 
41,0
 ( )14,52 
41,0
 ( )19,72 
28,0
 
  
2c  ( )10,85 
20,0
 ( )11,11 
20,0
 
   
3c   ( )8,10 
08,0
 
   
4c   ( )8,12 
14,0
 
   
5c   ( )6,48 
10,0
 
   
6c   ( )4,84 
04,0
 
   
+
1c  
   
( )6,76 
27,0
 ( )10,45 
25,0
 
−
1c  
   
( )5,42 
35,0
 ( )2,12 
06,0
 
1a    ( )61,82 
72,0
 
 
( )57,56 
72,0
 
 
The value of the likelihood function and the information criteria validate the random walk hypothesis with disturbances
( )1,1TGARCH . 
5 CONCLUSION  
The proprieties of the series studied are conditioning the inference methods: 
- The series of stock market quotations, as any financial series, is subject to an instantaneous variability. The regrouping 
of volatility is described by the values of the series with unpredictable signs, which are followed by values with the 
same signs. 
- The skewness ( )s  and the kurtosis ( )k  coefficients do not validate the   hypothesis of a Gaussian process. 
- In the tests on the residues, we will accept the hypothesis of an autoregressive conditionally heteroskedastic process. 
- The choice of the ARCH model is dictated by the proprieties of the series studied.  
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Stock Market Indexes: A random walk test with ARCH (q) disturbances 
 
 
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Stock Market Indexes: A random walk test with ARCH (q) disturbances

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Stock Market Indexes: A random walk test with ARCH (q) disturbances

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