This study examines long memory in Istanbul Stock Exchange (ISE) by using the structural break test in variance and ARFIMA-FIGARCH model. Our findings indicate that long memory does not exist in the equity return; however, it exits in volatility. Consequently,   ISE is found as a weak form inefficient market due to volatility as it has a predictable component

Cevik, Emrah Ismail

Korkmaz, Turhan

Özataç, Nesrin

Munich Personal RePEc Archive

MPRA
Munich Personal RePEc Archive
Testing for long memory in ISE using
Arfima-Figarch model and structural
break test
Turhan Korkmaz and Emrah Ismail Cevik and Nesrin O¨zatac¸
Mersin University, Namık Kemal University, Eastern Mediterranean
University
2009
Online at https://mpra.ub.uni-muenchen.de/71302/
MPRA Paper No. 71302, posted 16 May 2016 13:51 UTC
 1 
Testing for Long Memory in ISE Using ARFIMA-FIGARCH Model and 
Structural Break Test 
Turhan Korkmaz
a,*
, Emrah İsmail Çevika, Nesrin Özataçb 
a
 Faculty of Economics and Administrative Sciences, Zonguldak Karaelmas University, Turkey. 
b
 Faculty of Business and Economics, Eastern Mediterranean University, North Cyprus. 
Abstract 
This study examines long memory in Istanbul Stock Exchange (ISE) by using 
the structural break test in variance and ARFIMA-FIGARCH model. Our findings 
indicate that long memory does not exist in the equity return; however, it exits in 
volatility. Consequently,   ISE is found as a weak form inefficient market due to 
volatility as it has a predictable component. 
I. Introduction 
According to the efficient market hypothesis that Fama (1970) suggested all 
the information related with the financial assets affect their price and in regard to that 
information, price movements occur accordingly in the market. In an efficient 
market, the stock returns show a random walk causing it impossible to make a 
prediction from the past patterns. For this reason, many studies on the financial asset 
returns focus on if the returns have long memory features or not. If the stock returns 
have long-memory, then the previous returns will lead to make predictions on future 
returns making a weak form efficient market hypothesis to lose its validity. 
In literature, there are a huge number of studies that investigates stock market 
efficiency in developed and developing countries, using long memory models.
1
  
                                                 
*
 Corresponding author. e-mail: korktur@yahoo.com. 
Not: This paper has been prepared for publication in International Research Journal of Finance and 
Economics. The proper citation for this work would be:  
Korkmaz, T., E. İ. Çevik, ve N. Özataç “Testing for Long Memory in ISE Using ARFIMA-FIGARCH 
Model and Structural Break Test”, International Research Journal of Finance and Economics, Issue: 
26, April 2009, ss. 186-191. 
 2 
There are some studies on ISE showing the same evidence that the index return 
indicates a long memory forming a weak form inefficient market.
2
 
In this study, existence of long memory in ISE is examined using ARFIMA-
GARCH model.  Unlike other studies on ISE, presence of structural break in 
unconditional variance of index return series is examined with Iterated Cumulative 
Sums of Squares (ICSS) method. 
II. Methodology 
Multiple Structural Break Test in Unconditional Variance 
Inclan and Tiao (1994) proposed a method that based on ICSS to detect 
structural breaks in the unconditional variance of a stochastic process. In order to test 
null hypothesis of constant unconditional variance against the alternative hypothesis 
of a break in the unconditional variance, Inclan and Tiao (1994) proposed to use the 
statistic given by 
/ 2 kIT T D  (1) 
where ( / ) ( / )k k TD C C k T   and 
2
1
k
k tt
C r

  be the cumulative sum of squares of a 
series of uncorrelated random variables with mean 0 and variance 2t , t = 1, 2, ...,T . 
The value of k (k = 1, , T)  maximizes / 2 kT D  is the estimate of the break date. 
Under the variance homogeneity IT statistic behaves like a Brownian bridge 
asymptotically. At the %5 significance level, the critical value computed by Inclan 
and Tiao (1994) is C0.05 = 1.358. 
The most serious drawback of the IT test statistic is designed for independently 
and identically distributed random variables. This is very strong assumption for 
                                                                                                                                          
1
 Examples for the study are Barkoulas et al. (2000), Resende and Teixeira (2002), Tolvi (2003), 
Caporale and Gil-Alana (2004), Cajueiro and Tabak (2006), Lux and Kaizoji (2007),  McMillan and 
Thupayagale (2008), Blasco and Santamaria (1996), Vougas (2004), Gil-Alana (2006) . 
2
 Balaban (1995 and 1996), Kılıç (2004), Christodoulou-Volos and Siokis (2006), Assaf (2007), 
DiSario et al. (2008). 
 3 
financial data as most return series include conditional heteroskedasticity. For this 
purpose, Sanso et al. (2004) modified IT test statistic for conditional 
heteroskedasticity. Modified IT test statistic given by 
1/ 2
2 sup k
k
T G   (2) 
where 1/ 24ˆk k T
k
G C C
T

 
  
 
 and 
4ˆ is a consistent estimator of 4 . Non-parametric 
estimator of
4 , 
      
2
2 2 2 2 2 2
4 1
1 1 1
1 2
ˆ ˆ ˆ ˆ,
T m T
t t t
t l t l
r l m r r
T T
    
   
        (3) 
where  ,l m  is a lag window, such as the Barlett, defined as    , 1 1l m l m    , 
or the quadratic spectral. 
In the test procedure, if we are looking for only with the possible of single 
point change, then the Dk function would provide a satisfactory procedure. But when 
we are interested in finding multiple change points on an observed series the 
usefulness of the Dk function becomes questionable because of the masking effect. A 
solution is an iterative scheme based on successive application of Dk to pieces of the 
series, dividing consecutively after a possible change point is found
3
 (Inclan and 
Tiao, 1994). 
Long Memory Models 
Fractional integration first appears in literature in the studies of Granger and 
Joyeux (1980) and Hosking (1981). The model, known as Autoregressive 
Fractionally Integrated Moving Average (ARFIMA), allows for increased flexibility 
in modeling low-frequency dynamics. ARFIMA model is written as follows; 
( )(1 ) ( )d t tL L Y L    , t ~ 
2(0, )   (4) 
                                                 
3
 See Inclan and Tiao (1994) for ICSS procedure details. 
 4 
where d is fractional integration parameter as a real number, L is lag operator and εt 
is white noise residual. Polynomial structures of Equation (4) lie outside the unit 
circle, satisfying the stationarity and invariability conditions. The fractional 
differencing lag operator (1 )dL  is defined by the binomial expansion as follows. 
2 3( 1) ( 1)( 2)(1 ) 1 ...
2! 3!
d d d d d dL dL L L
  
       (5) 
ARFIMA process is nonstationary when d ≥ 0.5. For 0 < d < 0.5, ARFIMA 
process is said to exhibit long memory. The process exhibits short memory for d = 0 
and intermediate memory for d < 0.  
Baillie et al. (1996) proposed fractional integrated GARCH (FIGARCH) model 
to determine long memory in return volatility. GARCH (p, q) model in ARMA for 
squared error form given by 
2[1 ( ) ( )] [1 ( )]t tL L L v          (6) 
where 2 2t t tv    . To satisfy the covariance stationary conditions, roots 
[1 ( ) ( )]L L    and [1 ( )]L  lie outside the unit circle. FIGARCH model is derived 
from standard GARCH model with fractional difference operator, (1 )dL . 
FIGARCH model is written as follows: 
2( )(1 ) [1 ( )]      d t tL L L v  (7) 
FIGARCH ( , ,p d q ) model is transformed standard GARCH when 0d  and 
IGARCH model when 1d . 
III. Data and Empirical Results 
In this study, the data set which is comprised of daily closing prices of ISE 
with respective transaction series from 1988 to 2008, totaling 5150 observations is 
used. The closing prices of ISE 100 index is collected from The Central Bank of the 
Republic of Turkey.  The return series is adjusted by taking logarithmic difference as 
 5 
1100 ln( / )t t tr p p   . Figure 1 shows the ISE 100 index return series. The economic 
crisis in 1997 in Asia, in 1998 in Russia, in 2000 and 2001 in Turkey affect ISE 
causing its volatility to rise. After 2001 economic crisis, a stability programme has 
been applied and Turkey managed to achieve economic growth from 2003 through 
2007.  During that period ISE index return showed lower volatility. 
Figure 1: ISE 100 Index Return 
t1* t2* t3* t4*
-25
-15
-5
5
15
25
Ja
n-
88
Ja
n-
89
Ja
n-
90
Ja
n-
91
Ja
n-
92
Ja
n-
93
Ja
n-
94
Ja
n-
95
Ja
n-
96
Ja
n-
97
Ja
n-
98
Ja
n-
99
Ja
n-
00
Ja
n-
01
Ja
n-
02
Ja
n-
03
Ja
n-
04
Ja
n-
05
Ja
n-
06
Ja
n-
07
Ja
n-
08
 
The summary statistics is given in Table 1. The daily average index return is 
positive and it is 0.168. Normality test statistics indicates that distribution of return 
series is significantly different from normality and in the form of leptokurtic. 
According to the ARCH test result, conditional variance of return series exhibit 
conditional heteroscedasticity properties.
4
 
Table 1: Summary Statistics 
Series Mean Std. Dev. Skewness Kurtosis J-B Tests ARCH (5) Q (10) 
Return 0.168 2.880 -0.053 6.252 2272 [0.000] 142.9 [0.000] 90.655 [0.000] 
1) J-B indicates Jarque-Bera normality test, ARCH indicates LM conditional variance test, Q(.) indicates Box-Pierce serial 
correlation test.  
To prove the existence of structural breaks in unconditional variance, κ2 tests 
are used. We found 4 break points in variance of return series and the break point 
dates are determined as 31/12/1993, 17/06/1994, 24/10/1997 and 25/03/2003. While 
investigating the existence of the long memory as a priori, we assume that return 
series has constant unconditional variance. For this reason, investigating long 
memory with unstable unconditional variance will give deviated results. In this 
                                                 
4
 Phillips-Perron and KPSS unit root test provides that return series are stationary. 
 6 
study, in order to make unconditional variance stable the method those Noira et al. 
(2004) is applied.  
2var( )t tr   will be estimated over each of the intervals limited by the regime-
shift points of the unconditional variance as given below: 
 
2
2 2 * *
1
0
ˆ ˆ
t i i i
i
I t t t  

   
   (t = 1, , n and  i = 0, 1, ,4)   (8) 
where     
*
1
*
12 * * 2
1
ˆ 1/
it
i i i tt t
t t r 

 
   , *0 1t  , 
*
1 1498t  , 
*
2 1612t  , 
*
3 2454t  , 
*
4 3784t  , 
*
5 5150t   and  .I  is the indicator function that takes the value 1 if the argument is 
true, and 0 otherwise. To enable unconditional variance stable, return series are 
filtered as ˆ/ft t tr r   (t = 1, , n) . As a result, the return series is filtered and the 
existence of long memory is calculated according to the filtered series.
5
  
In Table 2, the GARCH model results are given. The ARMA model is 
determined with Box-Jenkins approach and Akaike information criteria. The sum of 
the GARCH parameters’ for non-filtered return series is 0.983, and for filtered return 
series is 0.928 indicating the existence of the long memory in volatility. Structural 
breaks cause volatility to increase more than it is to be expected. 
Table 2: GARCH Model Results 
 r r
f
 
Constant (mean) 0.159 [0.000] 0.060 [0.000] 
AR(1) 0.100 [0.000] 0.100 [0.000] 
Constant (variance) 0.213 [0.001] 0.077 [0.000] 
α 0.150 [0.000] 0.159 [0.000] 
β 0.832 [0.000] 0.769 [0.000] 
df 7.407 [0.000] 8.345 [0.000] 
α + β 0.983 0.928 
Q1(5) 23.987 15.743 
Q2(5) 7.448 5.221 
1) r and rf indicates non-filtered and filtered return series respectively, df indicates student t distribution parameter, Q1(.) 
indicates Box-Pierce test for standardized residuals, Q2(.) indicates Box-Pierce test for squared standardized residuals, [.] 
indicates p values.   
The presence of long memory in volatility is investigated by the FIGARCH 
model and the results are given in Table 3.  Fractional integration parameter, d, for 
                                                 
5
 The existence of the structural break is re-investigated for filtered return series and break is not 
determined. 
 7 
non-filtered return series is 0.394, for filtered series is 0.345 and it is found 
significantly different from zero.  According to these results, volatility of index 
return series has long memory and it is a predictable structure. Also fractional 
integration parameter is obtained smaller for filtered return series supports that long-
range dependence may be due to the presence of structural breaks.  
Table 3: FIGARCH Model Results 
 r r
f
 
Constant (mean) 0.162 [0.000] 0.060 [0.000] 
AR(1) 0.101 [0.000] 0.102 [0.000] 
Constant (variance) 0.389 [0.001] 0.093 [0.004] 
d  0.394 [0.000] 0.345 [0.000] 
α 0.076 [0.530] 0.009 [0.953] 
β 0.291 [0.037] 0.185 [0.326] 
df 8.114 [0.000] 8.450 [0.000] 
Q1(5) 20.848 16.647 
Q2(5) 5.080 4.751 
1) r and rf indicates non-filtered and filtered return series respectively, df indicates student t distribution parameter, Q1(.) 
indicates Box-Pierce test for standardized residuals, Q2(.) indicates Box-Pierce test for squared standardized residuals, [.] 
indicates p values.   
Finally, the existence of long memory in the mean of process is investigated by 
ARFIMA-FIGARCH model and the results can be seen on Table 4. Fractional 
integration parameter is obtained 0.035 and 0.037 respectively for non-filtered and 
filtered return series and is not significantly different from zero at %5 level.  This 
result indicates return exhibit short memory properties and cannot be estimated in the 
long run. In volatility equation, long memory properties are also significantly 
detected. 
Table 4: ARFIMA-FIGARCH Model Results 
 r r
f
 
Constant (mean) 0.164 [0.000] 0.059 [0.000] 
d 0.035 [0.069] 0.037 [0.050] 
AR(1) 0.064 [0.004] 0.063 [0.005] 
Constant (variance) 0.395 [0.001] 0.094 [0.003] 
d  0.393 [0.000] 0.346 [0.000] 
α 0.069 [0.566] 0.006 [0.969] 
β 0.284 [0.043] 0.181 [0.327] 
df 8.092 [0.000] 8.396 [0.000] 
Q1(5) 11.946 8.269 
Q2(5) 5.016 4.688 
1) r and rf indicates non-filtered and filtered return series respectively, df indicates student t distribution parameter, Q1(.) 
indicates Box-Pierce test for standardized residuals, Q2(.) indicates Box-Pierce test for squared standardized residuals, [.] 
indicates p values.   
 8 
IV. Conclusion 
In this study, the weak form market efficiency of ISE 100 index is examined by 
using the structural break test in variance and ARFIMA-FIGARCH model.  
Structural break test provides 4 break points on the return series. Related to the break 
dates, the return series is filtered and investigated whether the return and volatility 
carry long memory features or not. The results indicate that on the return, the 
existence of the long memory is not perceived but on volatility, there is long memory 
existence. According to this result, volatility has a predictable structure and it 
indicates that ISE is not a weak form efficient market. Furthermore, smaller number 
is generated from fractional integration parameter on the filtered series (for volatility 
equation) supporting the reason of spurious long memory. Finally, structural breaks 
cause volatility to increase more than it is to be expected. 
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