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Synthesis, structure, electrochemistry and cytotoxicity studies of Ru(II) and Pt(II)–N-heterocyclic carbene complexes of CNC-pincer ligand
Copernican Journal of Finance & Accounting
e-ISSN 2300-3065
p-ISSN 2300-1240
2015, volume 4, issue 2
Al-Hajieh H., AlNemer H., Rodgers T., & Niklewski J. (2015). Forecasting the Jordanian stock index:
modelling asymmetric volatility and distribution effects within a GARCH framework. Copernican
Journal of Finance & Accounting, 4(2), 9–26. http://dx.doi.org/10.12775/CJFA.2015.013
Heitham Al-Hajieh*
Department of Finance, King Abdulaziz University, Saudi Arabia
Hashem AlNemer**
Department of Finance and Insurance, University of Jeddah, Saudi Arabia
Timothy Rodgers***
School of Economics, Finance and Accounting, Coventry University, UK
Jacek Niklewski****
School of Economics, Finance and Accounting, Coventry University, UK
forecasting the jordanian stock index:
modelling asymmetric volatility
and distribution effects within a garch framework
Keywords: GARCH, asymmetry, distributions.
J E L Classification: C01, C58, G15.
Date of submission: May 16, 2015; date of acceptance: October 26, 2015.
Contact information: Haawadh@kau.edu.sa, Department of Finance, King Abdulaziz
University, Abdullah Sulayman, Jeddah 21589, Saudi Arabia, phone: +966 2 695 2000.
**
Contact information: Halnemer@kau.edu.sa, Department of Finance and Insurance, University of Jeddah, Saudi Arabia.
***
Contact information: T.Rodgers@coventry.ac.uk, School of Economics, Finance
and Accounting, Coventry University, UK.
****
Contact information: J.Niklewski@coventry.ac.uk, School of Economics, Finance
and Accounting, Coventry University, UK.
*
10
H. Al-Hajieh, H. AlNemer, T. Rodgers, J. Niklewski
Abstract: The modelling of market returns can be especially problematical in emerging
and frontier financial markets given the propensity of their returns to exhibit significant non-normality and volatility asymmetries. This paper attempts to identify which
representations within the GARCH family of models can most efficiently deal with these issues. A number of different distributions (normal, Student t, GED and skewed Student) and different volatility of returns asymmetry representations (EGARCH and GJR-GARCH) are examined. Our data set consists of daily Jordanian stock market returns
over the period January 2000 – November 2014. Using both the Superior Predicative
Ability (SPA) and Model Confidence Set (MCS) testing frameworks it is found that using
GJR-GARCH with a skewed Student distribution most accurately and efficiently forecasts Jordanian market movements. Our findings are consistent with similar research undertaken in respect to developed markets.
Introduction
The global financial crisis of 2007-09 and subsequent shocks in the Euro-area
and beyond has led researchers to examine again the ways in which they model stock market returns. It has become increasingly apparent that the ‘standard assumptions’ made in respect to the ways in which statistical series are
distributed are not applicable in financial markets. As the global economy becomes more integrated it is also becoming increasingly important to understand how emerging and frontier markets react in periods of high volatility. In
this paper we explore which elements of the GARCH family of models can be
used to efficiently and effectively model markets in Jordan from January 2000
to November 2014.
The paper begins with a brief review of the literature in the second Section. This is followed in the subsequent section by a description of the data
and methodology. The most efficient model is then identified using the Superior Predictive Ability (SPA) and Model Confidence Set (MCS) prediction frameworks before, finally, some brief conclusions are drawn.
Literature: volatility modelling in the GARCH framework
The GARCH model was first introduced by Bollerslev (1986). Much of the subsequent research in this area has focused on developing the model to better reflect the data found in real-world settings, such as, for example, financial markets. The finance-related literature focuses principally on modelling (i) the
structure of the volatility (ii) the nature of the distribution of the returns.
Forecasting the Jordanian stock index…
11
Much of the work on modelling the structure of volatility relates to the asymmetries found in stock market returns. For example, Engle and Ng (1993) found
evidence supporting the Quadratic-GARCH model. Others, such as Brailsford
and Faff (1996), found evidence to support GJR-GARCH and Heynen and Kat
(1994) argued that EGARCH has a superior predictive ability. Although the literature does not show one individual asymmetry specification as being clearly
superior to others, Awartani and Corradi (2005) argue that they generally outperform non-asymmetric specifications in financial market prediction. However, it can be noted that the evidence is not unequivocal; McMillan, Speight,
and Apgwilym (2000) found GARCH, moving average and exponential smoothing models to provide marginally superior daily volatility forecasts. Their work
also strongly suggested that EGARCH does not necessarily outperform simple
GARCH model in forecasting market volatility.
For the purposes of our paper methodologies with the greatest out-of-sample forecasting accuracy are the most desirable. Balaban (2004) tested a series of both symmetric and asymmetric models (included ARCH, GARCH, GJRGARCH and EGARCH). Their results suggest that all models are biased and
generally over-predict volatility. Model performance in these latter respects
was best for GARCH and worst for GJR-GARCH. However, they also noted that if
avoidance of under-prediction was the key decision criteria, ARCH was the preferred model.
A further issue that is important to consider is that the nature of volatility
in emerging and frontier markets differs considerably from that found in developed markets (Andrikopoulos, Niklewski and Rodgers forthcoming). Given
that the focus of this paper is Jordan, it is important to consider how different
GARCH specifications perform in these market-types. Gokcan (2000) examined
seven emerging market (Argentina, Brazil, Colombia, Malaysia, Mexico, Philippines, Taiwan) and found that GARCH(1,1) outperformed EGARCH everywhere
with the exception of Brazil.
The most compelling conclusion we draw from the literature in respect to
modelling volatility structure is that it is difficult to identify one single model
that is clearly superior to others. This indicates to us that it may be necessary
to test a number of volatility specifications.
A standard feature of most financial markets is that their returns are nonnormal with distributions exhibiting ‘fat-tailed’ characteristics (Mittnik,
Paolella, and Rachev 2000). This appears to be particularly an issue in emerging markets. Brooks (2007) studied a set of such markets (including MENA re-
12
H. Al-Hajieh, H. AlNemer, T. Rodgers, J. Niklewski
gion countries) using the Asymmetric Power ARCH model. He found that unlike
developed markets, where non-normal conditional error distributions appear
to fit the data well, there were a set of emerging markets where estimation
problems arise using a conditional t distribution. It was also found that the degree of volatility asymmetry appears to vary across markets, with the Middle
Eastern and African markets having very different volatility asymmetry characteristics to Latin American markets.
Brooks (2007) found that a fat-tailed t-distribution was needed to model
the distribution of returns in most MENA markets. However, there were differences. For example, Turkey, Egypt and Morocco display much larger kurtosis
and exhibit fatter tails than Jordan. Likelihood ratio tests were found to clearly
favour the APARCH with t-distribution rather than a normal distribution. It is
possible that such differences may reflect the Islamic nature of these markets.
For example, Al-Hajieh, Redhead, and Rodgers (2011) found that the month of
Ramadan (Islamic holy month) shows high level of volatility and the overall
impact of Ramadan on returns is statistically significant in most Middle East
countries.
We conclude the literature review by identifying that we are aware of no
studies of volatility forecasting in emerging markets that have examined the
combined issues of the distribution of returns and the GARCH model specification. We have also identified from the literature that (i) no single GARCH model specifications clearly outperforms other forms in all circumstances and (ii)
evidence to suggest that the distribution of returns in emerging markets (like
Jordan) can follow a number of possible forms and that these can interact with
volatility models in different ways. In this paper we therefore test a number of
possible distribution-types (Normal, Student-t, GED, and Skewed Student) and
GARCH model types (GARCH, GJR-GARCH and EGARCH) in order to identify the
most efficient volatility forecasting model for Jordan.
Data description
The empirical investigation is undertaken in respect to daily closing price data
for the Jordanian Amman Stock Exchange Index (ASE) covering the period 2nd
January 2000 to 27th November 2014. The data source is the Thomson-Reuters
Eikon database and the dataset comprises of a total of 3655 trading days. Daily
returns computed as the log-difference of the daily closing prices:
Amman Stock Exchange Index (ASE) covering the period 2nd January 2000 to
mber 2014. The data source is the Thomson-Reuters Eikon database and the
mprises of a total of 3655 trading days. Daily returns computed as the logForecasting the Jordanian stock index…
of the daily closing prices:
Rt ln Pt ln Pt 1
13
[1]
[1]
ASE Index closing prices are presented in Figure 1 and the daily returns in
dex closing prices are presented in Figure 1 and the daily returns in Figure 2. A
Figure 2. A number of volatility clusters can be observed in the returns data; for
volatility clusters can be observed in the returns data; for example, a cluster
example, a cluster corresponding to the 2007-09 global financial crisis.
ing to the 2007-09 global financial crisis.
Figure 1. Daily Jordanian Price Index January 2000–November 2014
Figure 1. Daily Jordanian Price Index January 2000–November 2014
S o u r c e : created by the authors using OxMetricsTM 7 software and data from Thomson Reuters
EikonTM.
A preliminary statistical analysis of the daily returns is presented in Table 1. It can be noted that average daily returns are small relative to the standard deviation. The series also displays negative skewness and strong positive kurtosis; these are indicative of a heavy tailed non-Gaussian distribution.
Table 1. Descriptive Statistics of Daily Returns January 2000–November 2014
Mean
Std.
Dev
Min
Max
Skewness
Excess
Kurtosis
J-B
Test
ARCH
Test1
L-B
Test1
ADF
Test
0.025
0.946
-6.428
6.198
-0.363
6.167
5872.2**
107.9**
2571**
-32.8**
** Significant at 1%.
Both the Ljung-Box and the ARCH tests use 10 lags.
1
S o u r c e : estimated by the authors using OxMetricsTM 7.
14
H. Al-Hajieh, H. AlNemer, T. Rodgers, J. Niklewski
Figure 2. Jordanian Daily Percentage Returns January 2000–November 2014
S o u r c e : created by the authors using OxMetricsTM 7.
This is confirmed by the Jarque-Bera test which rejects unconditional normality and further confirmatory evidence is provided in the related histogram (Figure 3).
Figure 3. Histogram of Jordanian Daily Returns January 2000–November 2014
S o u r c e : created by the authors using OxMetricsTM 7.
Forecasting the Jordanian stock index…
15
The Ljung-Box Q-test and the ARCH tests suggest autocorrelation and hetroskedasticity within the data and the ADF (Augmented Dickey-Fuller) unit root test
rejects the null hypothesis of data non-stationary.
The research methodology
A total of 12 GARCH-model-specification/distribution pairs are tested with
the results being presented in Tables 3–7. The models tested are: GARCHbased specifications (GARCH-N, GARCH-T, GARCH-GED and GARCH-ST)1;
and two sets of asymmetry-type specifications: (i) EGARCH (EGARCH-N,
EGARCH-T, EGARCH-GED and EGARCH-ST) and (ii) GJR-GARCH (GJR-GARCH-N,
GJR- GARCH- T, GJR- GARCH-GED and GJR- GARCH-ST).
The alternative models are subsequently evaluated by: (i) an evaluation of
model parameters and (ii) an evaluation of model forecasting performance. For
robustness, the latter undertakes a series of tests using both the Superior Predictive Ability (SPA) test Hansen (2005) and the Model Confidence Set (MCS)
test (Hansen, Lunde, and Nason 2011). Both are available in the OxMetricsTM 72
software package used in this paper. The forecast-based tests use a ‘loss-function’ to identify the most efficient model. The loss function can be estimated using Mean Squared Error (MSE) and Mean Absolute Deviation (MAD) statistics.
SPA identifies the ‘best’ model in terms of predictive ability and MCS identifies
the ‘best’ model set.
The model specifications and the distributions tested are identified below.
They consist of three different conditional volatility specifications and four different statistical distributions.
(i) GARCH
The GARCH model, as introduced by Bollerslev (1986), is a generalisation of
the ARCH specification of Engle (1982). The model specifies that the conditional variance is a function of the lagged squared residuals as well as of its past
conditional variances. Although the equation may be specified with a number
1
N stands for the Normal distribution, T the Student t distribution, GED the Generalised Error Distribution and ST the Standardised Skewed Student distribution.
2
The MULCOM 3.0 package running SPA and MCS was developed by Hansen and
Lunde (2014).
ons.
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Where
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here the log-likelihood function of the distribution
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here the log-likelihood
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the log-likelihood
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of the distribution
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Where Where
the log-likelihood
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of the distribution
is:
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thez tlog-likelihood
is: 2
1
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[ 72]
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log
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)
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LGED log 0 .5
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(zero mean
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[[7]
7]
the log-likelihood
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Where Where
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18
H. Al-Hajieh, H. AlNemer, T. Rodgers, J. Niklewski
Results:
model
evalutation
(vii)
Standardized
(zero mean and unit variance) skewed-Student distribution
Results:
model
evalutation
We evaluate
the alternative
models
an assessment
of parameters
the parameters
associated
We evaluate
the alternative
models
by (i)by
an(i)
assessment
of the
associated
Where
themodel
log-likelihood
function
of the
distribution
is: performance.
withmodel
each
an evaluation
of each
model
forecasting
with each
set (ii)set
an(ii)
evaluation
of each
model
set’sset’s
forecasting
performance.
1
2
2
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s )
T log
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log
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T log
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1
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1
2
T
( sz m )
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2
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2
(
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t 1
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t
2
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t
(i)
Parameter Based Evaluation
Parameter Based Evaluation
Tables 2, 3 and 4, present
the parameter
and associated significance tests for the
Results:
modelvalues
evalutation
Tables 2, 3 and 4, present the parameter values and associated significance tests for the
GARCH, EGARCH and GJR-GARCH specified model sets. For the first and third model
We evaluate
the alternative
models
by (i)model
an assessment
the parameters
asGARCH,
andinGJR-GARCH
specified
sets. parameters
For the of
first
third and
model
setsEGARCH
the constants
the mean equations
and the variance
areand
positive
statis-
(i)
sociated
with in
each
model
set (ii)and
an the
evaluationparameters
of each model
set’sand
forecasting
sets
thetically
constants
the mean
statissignificant
for allequations
distributions.
Forvariance
the EGARCH modelare
setpositive
the constants
for the
performance.
tically mean
significant
for are
all not
distributions.
the EGARCH model set the constants for the
equations
statisticallyFor
significant.
mean equations
are not
statistically
significant.
The alpha
coefficient
for all
models and distributions is statistically significant at the
(i) Parameter Based Evaluation
The99%
alpha
coefficient
for allThis
models
andthe
distributions
statistically
significant
the
level
of confidence.
implies
existence ofisthe
ARCH process
in the at
residuals
99% Tables
level
confidence.
This
implies
existence
the ARCH
process
in the
residuals
term.ofThe
time-varying
volatilityofvalues
clustering;
this
indicates
that
periods of
2,
3returns
and 4,exhibit
present
thetheparameter
and
associated
significance
term.
time-varying
clustering;
this indicates
that periods
volatility
areexhibit
followed
by periods
ofvolatility
relative
calm.
testsThe
for returns
the GARCH,
EGARCH
and
GJR-GARCH
specified
model sets.
For theoffirst
The
beta coefficients
of the
threein
models
in all equations
distribution are
and third
sets
constants
the
mean
andalso
thestatistically
variance signifiparamvolatility
aremodel
followed
bythe
periods
of relative
calm.
eters
are
significant
for all
thesignifiEGARCH
atcoefficients
the 99%and
level
of three
confidence.
This
indicates
thatdistributions.
the
is For
dependent
on its
Thecant
betapositive
ofstatistically
the
models
in all
distribution
arevariance
also statistically
model
set99%
the
constants
for theofmean
equations
are
statistically
significant.
average.
a subset
the models
the that
sum
of not
alpha
andisbeta
is close
to unity,
cant
atmoving
the
level ofIn confidence.
This
indicates
the
variance
dependent
on
its
The
alpha
coefficient
for
all
models
and
distributions
is
statistically
signifiimplies
these cases
volatility
quiteand
persistent
suggests
that a
movingwhich
average.
In ainsubset
of thethat
models
the shocks
sum ofare
alpha
beta is and
close
to unity,
cant at
thepositive
99% level
of confidence.
This
implies
the existence
of the
ARCH
prolarge
(or
negative)
return
will
lead
future
forecasts
of
the
variance
to
be
high
which implies in these cases that volatility shocks are quite persistent and suggests that a for
cess in
the
residuals
term.
The
returns
exhibit
time-varying
volatility
clustering;
an extended
period. return will lead future forecasts of the variance to be high for
large positive
(or negative)
this indicates
that
volatility
are
followed
byterm
periods
ofinrelative
Theperiod.
GARCH periods
coefficientof(beta)
is larger
than
the ARCH
(alpha)
all three calm.
model
an extended
The
beta
coefficients
of
the
three
models
in
all
distribution
are
also statistisets. This is a further indication that the conditional variance will exhibit long persistence
The GARCH coefficient (beta) is larger than the ARCH term (alpha) in all three model
cally
significant at the 99% level of confidence. This indicates that the variance
of volatility.
sets.
This is a further
indication
the conditional
variance
willmodels
exhibit long
is dependent
on its
movingthat
average.
In a subset
of the
the persistence
sum of alpha
of
volatility.
and beta is close to unity, which implies in these cases that volatility shocks are
quite persistent and suggests that a large positive (or negative) return will lead
future forecasts of the variance to be high for an extended period.
The GARCH coefficient (beta) is larger than the ARCH term (alpha) in all
three model sets. This is a further indication that the conditional variance will
exhibit long persistence of volatility.
19
Forecasting the Jordanian stock index…
The GJR models show no evidence of asymmetry effects being statistically
significant. EGARCH models however, indicate significance in the magnitude
effect but not in the sign effect.
Const.
(M)α
Distribution
Normal
Student
GED
Skewed
Student
α
Table 2. The GARCH model set
Const.
(V)β
ARCH
(Alpha)
GARCH
(Beta)
Student
(DF)µ
GED
(DF)µ
Asymm.
Tail
Coefficient
0.02
0.01
0.11
0.89
NA
NA
NA
NA
p-value
0.05
0.02
0
0
NA
NA
NA
NA
Coefficient
0.02
0.01
0.14
0.86
6.23
NA
NA
NA
p-value
0.02
0.02
0
0
0
NA
NA
NA
Coefficient
0.02
0.01
0.12
0.88
NA
1.36
NA
NA
p-value
0.02
0.02
0
0
NA
0
NA
NA
Coefficient
0.02
0.01
0.14
0.86
NA
NA
-0.01
6.22
p-value
0.05
0.02
0
0
NA
NA
0.69
0
Mean equation, β Variance equation, µ Degrees of freedom.
S o u r c e : estimated by the authors using OxMetricsTM 7.
Table 3. The EGARCH model set
Const. Const. ARCH GARCH
(M)α
(V)β (Alpha) (Beta)
Student
(DF)µ
GED
(DF)µ
Coefficient
0.01
0.12
-0.53
0.99
NA
NA
NA
NA
0
0.39
p-value
0.16
0.74
0
0
NA
NA
NA
NA
0.8
0
Coefficient
0.02
-0.59
-0.52
0.99
6.47
NA
NA
NA
-0.01
0.43
p-value
0.17
0.02
0
0
0
NA
NA
NA
0.55
0
Coefficient
0.02
-0.68
-0.53
0.99
NA
1.38
NA
NA
0
0.41
p-value
0.11
0
0
0
NA
0
NA
NA
0.93
0
Coefficient
0.01
-1.11
-0.53
0.99
NA
NA
-0.02
6.44
-0.01
0.43
p-value
0.15
0.15
0
0
NA
NA
0.48
0
0.65
0
Distribution
Normal
Student
GED
Skewed
Student
α
Mean equation, β Variance equation, µ Degrees of freedom.
S o u r c e : estimated by the authors using OxMetricsTM 7.
Asymm. Tail
EGARCH EGARCH
(Theta1) (Theta2)
20
H. Al-Hajieh, H. AlNemer, T. Rodgers, J. Niklewski
Table 4. The GJR-GARCH model set
Const.
(M)α
Const.
(V)β
ARCH
(Alpha)
GARCH Student
(Beta)
(DF)µ
GED
(DF)µ
Asymm.
Tail
GJR
(Gamma)
Coefficient
0.02
0.01
0.11
0.89
NA
NA
NA
NA
-0.01
p-value
0.03
0.03
0
0
NA
NA
NA
NA
0.53
Coefficient
0.02
0.01
0.14
0.86
6.22
NA
NA
NA
0.01
p-value
0.03
0.02
0
0
0
NA
NA
NA
0.6
Coefficient
0.02
0.01
0.12
0.88
NA
1.36
NA
NA
0
p-value
0.02
0.03
0
0
NA
0
NA
NA
0.92
Coefficient
0.02
0.01
0.14
0.86
NA
NA
-0.01
6.21
0.01
p-value
0.05
0.03
0
0
NA
NA
0.75
0
0.62
Distribution
Normal
Student
GED
Skewed
Student
α
Mean equation, β Variance equation, µ Degrees of freedom.
S o u r c e : estimated by the authors using OxMetricsTM 7.
We turn now to the issue of identifying the most efficient model(s) from
the groups that have been tested. The diagnostic tests of the standardized residuals (Table 5) give us little help in this respect. Both ARCH(10) and Q2(10)
statistics indicate that hetroskedasticity has not been fully accounted for by
the models which means the estimated volatility equations have to be treated
with some degree of caution. Furthermore, Log likelihood and Akaike information criteria based tests all have approximately similar results; this suggests
they provide minimal help in distinguishing between model sets on the basis of
model fit. We can conclude from this that alternative forecasting-based testing
procedures will be required. It can be noted as a caveat to the above conclusion
however, that for all model sets, Akaike indicates that the normal distribution
produces the worst performance. From this it can possibly be concluded that
this distribution can probably be discounted at the outset.
21
Forecasting the Jordanian stock index…
Model
GARCH
EGARCH
GJR-GARCH
α
Table 5. Diagnostic Tests Of The Standardised Residuals
Distribution
Log Likelihood
Q 2(10)α
ARCH(10)α
Akaike
Normal
-4044.15
37.14
(0)
3.49
(0)
2.22
Student
-3963.3
31.31
(0)
2.92
(0)
2.18
GED
-3966.62
33.86
(0)
3.15
(0)
2.17
Skewed Student
-3963.22
31.29
(0)
2.92
(0)
2.17
Normal
-3980.23
21.62
(0)
2.14
(0.01)
2.18
Student
-3893.84
21.01
(0)
2.11
(0.02)
2.13
GED
-3900.81
21.28
(0)
2.12
(0.01)
2.13
Skewed Student
-3890.96
21.04
(0)
2.11
(0.02)
2.13
Normal
-4043.82
38.11
(0)
3.57
(0)
2.22
Student
-3963.12
30.56
(0)
2.86
(0)
2.17
GED
-3966.62
34.01
(0)
3.17
(0)
2.17
Skewed Student
-3963.07
30.59
(0)
2.86
(0)
2.17
The p-values are shown in brackets.
S o u r c e : estimated by the authors using OxMetricsTM 7.
(ii) Forecasting Based Evaluation
The Superior Predictive Ability test can be used for comparing the performances of two or more forecasting models. Forecasts are evaluated using
a pre-specified loss function with the ‘best’ forecast model being the one producing the smallest expected loss. An important issue that researchers face is
identifying what the loss function is estimated against. For the purposes of this
paper losses are estimated relative to the observed returns.
The two potential loss functions used by SPA are the Mean Squared Error
(MSE) and Mean Absolute Deviation (MAD). The losses estimated from the cho-
22
H. Al-Hajieh, H. AlNemer, T. Rodgers, J. Niklewski
sen function are then compared against a benchmark model. Identifying an appropriate benchmark to use is another important issue in respect to this methodology. In this paper we benchmark against a random walk.
The result presented in Table 6 below identify that GJR-GARCH with Skewed
distribution produces the smallest loss and is therefore the best fitting model. These findings are consistent with those of Marcucci (2005) and Awartani and Corradi (2005). The latter found GARCH-N to be outperformed by both
EGARCH and GJR-GARCH models across different forecast horizons.
Performance
Table 6. Superior Predictive Ability Tests
Model
Sample Loss
(MSE*103)
t-statistic
p-value
Most Significant
GJR-GARCH-Skewed Student
0.00039
16.75819
0
Best
GJR-GARCH-Skewed Student
0.00039
16.75819
0
Model_25%
GJR-GARCH-Student-t
0.00043
16.75729
0
Median
GARCH-GED
0.00049
16.75627
0
Model_75%
EGARCH-Normal
0.02761
16.63402
0
Worst
EGARCH-Skewed Student
0.03208
16.59516
0
S o u r c e : estimated by the authors using OxMetricsTM 7 and MULCOM 3.0 package (Hansen, and
Lunde 2014).
Forecasts can also be tested using the model confidence set (MCS) procedure. This uses the same loss function as SPA but requires no benchmark. It
identifies efficient model sets at different levels of confidence. Hansen and Lunde state “the set,, that consists of the ‘best’ model(s) from a collection of models,; where ‘best’ is defined in terms of a criterion that is user-specified. The
MCS procedure yields a model confidence set,, that is a set of models constructed to contain the best models with a given level of confidence. The models in
are evaluated using sample information about the relative performances of the
models in ”. (Hansen and Lunde 2014, 16). The result of MCS presented in Table
7 identifies that the 90% confidence model set consists of a single model; namely, GJR-GARCH with Skewed Student distribution.
23
Forecasting the Jordanian stock index…
Table 7. Model Confidence Set Tests
Model
*
MSE*103
p-value
GARCH-Normal
0.00041
0
GARCH-Student-t
0.00047
0
GARCH-GED
0.00049
0
GARCH-Skewed Student
0.00041
0
EGARCH-Normal
0.02761
0
EGARCH-Student-t
0.03117
0
EGARCH-GED
0.02926
0
EGARCH-Skewed Student
0.03208
0
GJR-GARCH-Normal
0.00047
0
GJR-GARCH-Student-t
0.00043
0
GJR-GARCH-GED
0.0005
0
GJR-GARCH-Skewed Student
0.00039
1.0000*
90% model confidence set.
S o u r c e : estimated by the authors using OxMetricsTM 7 and MULCOM 3.0 package (Hansen, and
Lunde 2014).
Discussion of findings and conclusions
Given the large differences between developed and emerging markets, it is possibly a little surprising that the result of our study, made in respect to Jordan,
are consistent with previous studies made of developed markets. For example,
Engle and Ng (1993), examining Japanese stock return also found strong support for the GJR-GARCH model. Similarly, Bentes, Menezes, and Ferreira (2013)
examining NIKKEI 225, S&P 500 and STOXX 50 from 1987–2013 found all stock
index returns tested exhibited asymmetry.
Further similarities can be identified between our results and other studies. Liu & Hung (2010) investigated the performance of one-step-ahead forecasting using asymmetric GARCH models with different distribution assumptions. Their work, in respect to United States data, concluded that GJR-GARCH
generated volatility forecasts were more accurate that those produced by their
EGARCH counterparts. Furthermore, their results indicated that modelling the
asymmetric component was much more important than specifying the correct
24
H. Al-Hajieh, H. AlNemer, T. Rodgers, J. Niklewski
error distribution when it came to improving volatility forecasting. This was
especially the case in the presence of fat-tails, leptokurtosis, skewness and the
leverage effect.
A number of other studies have examined volatility in MENA countries like
Jordan. Assaf (2015), for example, examined the forecasting performance of
the Value-at-Risk (VaR) models in Egypt, Jordan, Morocco, and Turkey. Their
results suggested that returns had a significantly fatter tails than the normal
distribution and that Student APARCH model produced more accurate results
than those generated using Normal APARCH models.
The considerable variety of results found in these different studies suggests
to us that it is difficult to conclude that there is a ‘one size fits all’ model that can
be used to model asymmetry affects in stock market returns.
We believe that our study contributes significantly to the literature by examining the relative forecasting performances of different distribution-type (Normal, Student-t, GED, and Skewed Student) and asymmetry-type (GJR- GARCH
and EGARCH) GARCH models. Both our Superior Predictive Ability and Model
Confidence Set results identify that GJR-GARCH with Skewed Student distribution is the best fitting model for Jordan. The finding in our research chimes
with the findings of similar studies undertaken in different market contexts;
such as Liu & Hung (2010) work in respect to the United States.
References
Al-Hajieh, H., Redhead, K., & Rodgers T. (2011). Investor sentiment and calendar anomaly effects: A case study of the impact of Ramadan on Islamic Middle Eastern markets. Research in International Business and Finance, 25 (3), 345-356. http://dx.doi.
org/10.1016/j.ribaf.2011.03.004.
Andrikopoulos, P., Niklewski, J., & Rodgers T. (forthcoming). The portfolio diversification benefits of frontier markets: an investigation into regional effects. Handbook
of Frontier Markets. Ed. by Andrikopoulos P., Gregoriou G., and Kallinterakis V. Elsevier.
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