Parametric Value-at-Risk analysis: Evidence from stock indices
samir mabrouk2012, The Quarterly Review of Economics and Finance
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We evaluate the performance of several volatility models in estimating one-day-ahead Value-at-Risk (VaR) of seven stock market indices using a number of distributional assumptions. Because all returns series exhibit volatility clustering and long range memory, we examine GARCH-type models including fractionary integrated models under normal, Student-t and skewed Student-t distributions. Consistent with the idea that the accuracy of VaR estimates is sensitive to the adequacy of the volatility model used, we find that AR (1)-FIAPARCH (1,d,1) model, under a skewed Student-t distribution, outperforms all the models that we have considered including widely used ones such as GARCH (1,1) or HYGARCH (1,d,1). The superior performance of the skewed Student-t FIAPARCH model holds for all stock market indices, and for both long and short trading positions. Our findings can be explained by the fact that the skewed Student-t FIAPARCH model can jointly accounts for the salient features of financial time series: fat tails, asymmetry, volatility clustering and long memory. In the same vein, because it fails to account for most of these stylized facts, the RiskMetrics model provides the least accurate VaR estimation. Our results corroborate the calls for the use of more realistic assumptions in financial modeling.
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The Quarterly Review of Economics and Finance 52 (2012) 305–321
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The Quarterly Review of Economics and Finance
journal homepage: www.elsevier.com/locate/qref
Parametric Value-at-Risk analysis: Evidence from stock indices夽
Samir Mabrouk a,∗,1 , Samir Saadi b
a
b
International Finance Group Tunisia, High School of Business, Manouba University, Manouba, Tunisia
Queen’s School of Business, Queen’s University, Kingston, Ontario, Canada
a r t i c l e
i n f o
Article history:
Received 6 January 2011
Received in revised form 14 February 2012
Accepted 13 April 2012
Available online 11 May 2012
JEL classification:
G11
G12
G15
Keywords:
Value-at-Risk
GARCH
Non-normality
Long range memory
a b s t r a c t
We evaluate the performance of several volatility models in estimating one-day-ahead Value-at-Risk
(VaR) of seven stock market indices using a number of distributional assumptions. Because all returns
series exhibit volatility clustering and long range memory, we examine GARCH-type models including
fractionary integrated models under normal, Student-t and skewed Student-t distributions. Consistent
with the idea that the accuracy of VaR estimates is sensitive to the adequacy of the volatility model used,
we find that AR (1)-FIAPARCH (1,d,1) model, under a skewed Student-t distribution, outperforms all the
models that we have considered including widely used ones such as GARCH (1,1) or HYGARCH (1,d,1). The
superior performance of the skewed Student-t FIAPARCH model holds for all stock market indices, and for
both long and short trading positions. Our findings can be explained by the fact that the skewed Student-t
FIAPARCH model can jointly accounts for the salient features of financial time series: fat tails, asymmetry,
volatility clustering and long memory. In the same vein, because it fails to account for most of these
stylized facts, the RiskMetrics model provides the least accurate VaR estimation. Our results corroborate
the calls for the use of more realistic assumptions in financial modeling.
© 2012 The Board of Trustees of the University of Illinois. Published by Elsevier B.V. All rights reserved.
“Dogmas and doctrines holding that markets worked well and
that they were self-correcting once again came to predominate.
This time, the theories were more sophisticated, but the underlying assumptions were equally irrelevant. These ideas helped
shaped the intellectual milieu which gave rise to the flawed
policies that, in turn, gave rise to the crisis.” Joseph E. Stiglitz
(2009).
1. Introduction
Risk management has become a central issue especially following the recent financial crisis when several financial firms went
bankrupt or got bailed out by their governments (e.g. Lehman
Brothers, AIG, Royal Bank of Scotland). Indeed, the substantial
losses associated with the 2007–2009 financial crisis led to calls
for a reassessment of the entire risk management system, including the tools used to measure risk. A powerful risk management
approach that gained increasing momentum since its introduction in 1994 is the Value-at-Risk (VaR). VaR provides an estimate
夽 We are grateful to H.S. Esfahani (Editor) and two anonymous referees for their
insightful comments and suggestions.
∗ Corresponding author.
E-mail addresses: [email protected] (S. Mabrouk),
[email protected] (S. Saadi).
of the possible loss of a financial portfolio at a specific risk level
over a definite holding period. In spite of the shortcomings related
to its mathematical properties and the criticisms over its potential destabilizing impact on financial activities, VaR remains the
industrial benchmark for measuring risk, and its estimation has
again attracted global attention. The aim of this paper is to evaluate the performance of VaR approach under different assumptions
of error distribution and advanced volatility modeling, and determine the best model specification that provides the most accurate
VaR estimation.
A sensible risk measure tool should take into account the stylized fact of financial time series. In fact, several empirical studies
have shown that asset returns are skewed, have fat tails and exhibit
volatility clustering as well as long memory. Since the introduction
of ARCH model, designed by Engle (1982) and then generalized
by Bollerslev (1986), the financial econometric literature developed several alternative models to capture volatility clustering in
financial time series. For example the GARCH (1,1) is known to successfully capture certain properties of asset returns such as excess
kurtosis and volatility clustering (see, for instance, Hansen & Lunde,
2005). However, because it is mainly designed to capture shortrun temporal dependencies in conditional variance, the symmetric
GARCH (p,q) models fail to capture long memory and asymmetry in financial times series (e.g. Baillie, Bollerslev, & Mikkelsen,
1996; Bollerslev & Jubinski, 1999; Breidt, Crato, & de Lima, 1998;
Davidson, 2004; Ding & Granger, 1996).
1062-9769/$ – see front matter © 2012 The Board of Trustees of the University of Illinois. Published by Elsevier B.V. All rights reserved.
http://dx.doi.org/10.1016/j.qref.2012.04.006
306
S. Mabrouk, S. Saadi / The Quarterly Review of Economics and Finance 52 (2012) 305–321
Since volatility is a key input to estimate VaR, the use of volatility models that take into account the properties of asset returns is
crucial for the accuracy of VaR estimation (e.g. Angelidis, Benos, &
Degiannakis, 2004; Bali & Theodossiou, 2007; Mabrouk & Aloui,
2010; So & Yu, 2006; Tang & Shieh, 2006; Wu & Shieh, 2007).
Accordingly, to compute VaR, we employ two popular models,
namely, FIGARCH (Baillie et al., 1996) and HYGARCH (Davidson,
2004). In fact, these models are designed to capture not only volatility clustering but also long memory in assets return volatility.
Furthermore, to account for asymmetry in return volatility, we
also use FIAPARCH model that was introduced by Tse (1998) as an
extension of FIGARCH. Finally, we measure and evaluate VaR under
normal distribution as well as two error distributions known to
better depict financial time series: Student-t and skewed Studentt. In fact, one of the shortcomings of RiskMetrics, a conventional
VaR approach proposed by J.P. Morgan, is the assumption that asset
returns are normally distributed. However, asset returns are asymmetric and have fatter tails than normal distribution.
Though the unrealistic assumptions, such as normality, are
mainly considered to allow for model tractability and to simplify
the computational side of the VaR calculation, they induce nontrivial bias in VaR estimation, and thus lead to major financial
losses. It is noteworthy that the issue of sensitivity of financial
models to pre-determined underlying assumptions has recently
attracted much attention. In fact, in his opening plenary keynote
address at the American Economic Association 2010 Meeting,
Nobel Prize Laureate Joseph E. Stiglitz (2010) asserts that the recent
financial crisis serves as a siren call for better allocation of financial economics research efforts in which models are built on more
realistic assumptions. In the same vein, Colander et al. (2009)
and Eichengreen (2008), among others, are critical of market participants for applying the results of academic research without
appropriate reservation. Academics are also blamed for not warning users of the limitations associated with its preferred models. In
particular, academics are criticized for excessive reliance on models which ignore the key elements driving outcomes in real-world
markets (e.g. Roldán, 2009; Willett, 2009).
To establish whether changes in model specification (e.g. error
distribution) influence the robustness of estimator inferences,
Leamer (1985) recommends that the researcher select alternative
assumptions and identify the corresponding interval of inferences.
Leamer’s recommendation that the researcher should exhaust all
attempts “to combat the arbitrariness associated with the choice of
prior distribution” (Leamer, 1986), takes on additional significance
when conducting data-driven empirical research. This is an important point in light of the assertion by Colander et al. (2009): “the
current approach of using pre-selected models is problematic and
we recommend a more data-driven methodology.”
In line with Leamer’s (1985) recommendation, we first model
conditional variance for seven major stock index returns using different volatility models (e.g. GARCH, HYGARCH, FIAPARCH) and
under three alternative distributions (i.e. normal, Student-t, and
skewed Student-t). We examine several returns series to make sure
that our findings are not the artefact of a specific financial market.
We select the best volatility model that fits the data based on several model selection criteria. We then assess the performance of
the selected model in estimating VaR for both in-sample and outof-sample periods and for short and long trading positions using
failure rate, the Kupiec’s (1995) likelihood ratio test and Engle
and Manganelli’s (2004) Dynamic Quantile test. We also examine
whether the choice of mean equation influences the accuracy of
VaR estimates using different orders of autocorrolation.
Our results corroborate the calls for the use of more realistic assumptions in financial modeling. In fact, we find that
skewed Student-t FIAPARCH (1,d,1) model provides more accurate
estimates of daily Value-at-Risk (VaR) returns of both long and
short trading positions than those generated using alternative
volatility models and under normal or Student-t distributions. Our
findings hold for all the financial markets. This can be explained
by the fact that the skewed Student-t FIAPARCH (1,d,1) jointly
accounts for the salient features of financial time series: fat tails,
asymmetry, volatility clustering and long memory. Furthermore,
we find that mean process specification does not play a major role
in improving VaR estimates.
The remainder of the paper is organized as follows. Section 2
provides a description of the GARCH models used in our analysis.
Section 3 introduces the VaR model and describes the evaluation
framework for VaR estimates. Section 4 presents preliminary statistics for our dataset while Section 5 provides the results of the
empirical investigation of the estimated models. Section 6 concludes the paper.
2. Modeling volatility
2.1. GARCH model
Developed by Bollerslev (1986), the GARCH model is a generalization of Engle’s (1982) ARCH model. Assuming that the returns
process is expressed as an autoregressive process of order k,
rt = ς0 +
k
i=1
ςi rt−i + εt
(1)
Conditional on information set up to time t − 1, εt is an i.i.d.
random variable with mean 0 and variance t2 , a GARCH (p,q) model
is expressed as follows:
t2 = ω +
q
i=1
˛i ε2t−i +
p
2
ˇj t−j
(2)
i=1
The lag operator allows us to specify GARCH model as:
t2 = ω + ˛(L)ε2t + ˇ(L)t2
(3)
where ˛(L) = ˛1 L + ˛2 L2 + · · · + ˛q Lq and ˇ(L) = ˇ1 L + ˇ2 L2 + · · · +
ˇp Lp
Bollerslev (1986) showed that the GARCH model is a short memory model since its autocorrelation function decays slowly with a
hyperbolic rate.
2.2. RiskMetrics model
Since its introduction in 1994 by the risk management group
at J.P. Morgan, RiskMetrics has become the standard in the field
of risk management and this despite its several limitations, such
as assuming that the error terms are normally distributed. The
RiskMetrics model is an Integrated GARCH (1,1) where ARCH and
GARCH parameters are pre-specified. RiskMetrics model can be
written as follows:
2
t2 = ω + (1 − )ε2t + t−1
(4)
where ω = 0 and fixed at 0.94 for daily data and 0.97 for weekly
data. Thus there is no estimation of volatility parameters required
in the context of RiskMetrics approach.
2.3. The Fractional Integrated GARCH model
Baillie et al. (1996) proposed a Fractional Integrated GARCH
(FIGACH) model to capture the documented evidence of long memory in financial time series. The FIGARCH model can distinguish
between short memory and infinite long memory in the conditional
S. Mabrouk, S. Saadi / The Quarterly Review of Economics and Finance 52 (2012) 305–321
variance, and this thanks to the fractionary parameter d. Formally,
the FIGARCH(p, d, q) process is specified as follows:
[ϕ(L)(1 − L)
d
]ε2t
=ω
+ [1 − ˇ(L)](ε2t
− t2 )
(5)
distribution is defined as follows:
LStud = T
ln
or
−
t2 = ω + ˇ(L)t2 + [1 − ˇ(L)]ε2t − ϕ(L)(1 − L)d ε2t
= ω[1 − L]−1 + (L)ε2t
(6)
∞
where (L) is the lag-operator, (L) =
Li and 0 ≤ d ≤ is an
i=1 i
infinite summation which, in practice, has to be truncated. According to Baillie et al. (1996), (L) should be truncated at 1000 lags.
(1 − L)d is the fractional differencing operator and it is defined as
follows:
(1 − L)d =
∞
k=0
−
(d + 1)Lk
1
= 1 − dL − d(1 − L)L2
2
(k + 1) (d − k + 1)
∞
1
ck (d)Lk
d(1 − d)(2 − d)L3 − · · · = 1 −
6
(7)
k=1
Another popular model known for capturing long memory in
conditional volatilities is the hyperbolic GARCH model (HYGARCH).
Designed by Davidson (2004), the HYGARCH model is built to test
whether the non-stationarity of the FIGARCH model holds. The
HYGARCH model extends the conditional variance of the FIGARCH
model by introducing the weights in the difference operator. The
HYGARCH model can be written as follows:
2
− ln
1
2
t=1
2
T
ln
+ ln
2.4. The hyperbolic GARCH model
+ 1
ln(t2 ) + (1 + ) ln
−
1
ln[( − 2)]
2
1+
zt2
2
t ( − 2)
(11)
where 2 < ≤ ∞ and (·) is the gamma function. In contrast to the
normal distribution, the Student-t distribution is estimated with an
additional parameter , which stands for the number of degrees of
freedom measuring the degree of fat-tails in the density.
The deviation from the normal distribution in asset returns is
mainly due to excess kurtosis and excess skewness. The Student-t
is successful in depicting the excess kurtosis in financial time series
but not excess skewness. To jointly account for the excess skewness and kurtosis, we consider the skewed Student-t distribution
proposed by Lambert and Laurent (2001). Let z∼SKST(0, 1, k, ),
the log-likelihood of the skewed Student-t distribution (SKST) L is
defined as:
LSkst = T
where c1 (d) = d, c2 (d) = (1/2)d(1 − d), etc.
307
+ 1
2
2
k + (1/k)
+(1 + ) ln 1 +
− ln
2
+ ln(s)
1+
−
−
1
ln[( − 2)]
2
T
1
ln(t2 )
2
t=1
(szt + m)2
( − 2)
k−2It
(12)
where It = 1 if zt ≥ m/s or It = −1 if zt < m/s, k is an asymmetry
Thus the HYGARCH model is a generalized FIGARCH since it
nests to GARCH when ˛ = 0 and to FIGARCH when ˛ = 1
parameter. The constants m = m(k, ) and s =
s2 (k, ) are the
mean and standard deviations of the skewed Student-t distribution:
√
(( − 1)/2) − 2
1
m(k, ) =
(13)
k−
√
k
(/2)
2.5. The fractional integrated asymmetric power ARCH model
s2 (k, v) =
t2 = ω[1 − ˇ(L)]
−1
+ {1 − [1 − ˇ(L)]
−1
(L)[1 + ˛{(1 − L)d }]}ε2t
(8)
Tse (1998) extended the FIGARCH (p,d,q) model in order to
jointly account for volatility asymmetry and long memory, and this
by adding the function (|εt | − εt )ı of the APARCH process to the
FIGARCH process. Formally, the FIAPARCH (p,d,q) can be expressed
as follows:
tı = ω[1 − ˇ(L)]
−1
+ {1 − [1 − ˇ(L)]
−1
(L)(1 − L)d }(|εt | − εt )ı (9)
k2 +
1
− 1 − m2
k2
(14)
The value of ln(k) can also represent the degree of asymmetry in
the residual distribution. We note that when ln(k) = 0, the skewed
Student-t distribution equals the general Student-t distribution,
z∼ST (0, 1, ).
3. The Value-at-Risk
where ı, and are the model parameters. The FIAPARCH process nests the FIGARCH process when ␥ = 0 and ı = 2. Thus, the
FIGARCH process is a sample case of the FIAPARCH model.
In this section we present the VaR model under a FIAPARCH
model with skewed Student-t distribution innovation. Let’s consider that
2.6. The error’s density models
rt = t + εt
Let z∼N(0, 1) be a random variable, the log-likelihood of the
normal distribution can be written as follows:
t = +
T
LNorm = −
1
[ln(2) + ln(t2 ) + zt2 ]
2
(10)
t=1
where T is the number of observations.
Several studies, however, have shown that asset returns have
fatter tails than normal distribution. To account for this property of
financial time series, we consider Student-t: Consider the random
variable z∼ST (0, 1, ), the log-likelihood function of the Student-t
m
i=1
(15)
i rt−i +
n
j εt−j
(16)
j=1
The εt = zt t is governed by a FIAPARCH (p,d,q) process and the
innovations are assumed to follow the skewed Student-t distribution if:
f (zt |k, v) =
⎧
2
⎪
⎨ k + (1/k) sg(k(szt + m)|v)
⎪
⎩
2
sg(k(szt + m)/k|v)t
k + (1/k)
,
if
zt < −m/s
zt ≥ −m/s
(17)
308
S. Mabrouk, S. Saadi / The Quarterly Review of Economics and Finance 52 (2012) 305–321
In the above equation, g(·|v) denotes the symmetrical Student-t
density and k is the asymmetry parameter. The estimated VaR for
the long and short trading positions can be expressed as follows:
˛ = P(rt < VaRt,L ) = P
˛ = P(rt > VaRt,s ) = P
r −
t
t
t
r −
t
t
t
<
>
VaRt,L − t
t
VaRt,s − t
t
(18)
(19)
Table 1
Data.
Stock index
Sample period
Observations
DAX
DOW JONES
NASDAQ
NIKKEI
CAC40
FTSE100
S&P500
01/02/1990 to 10/10/2008
01/02/1990 to 10/10/2008
01/02/1990 to 10/10/2008
01/02/1990 to 10/10/2008
01/02/1990 to 10/10/2008
01/02/1990 to 10/10/2008
01/02/1990 to 10/10/2008
4498
4734
4720
4607
4678
4727
4720
VaRt,L and VaRt,s , in Eqs. (18) and (19), are for the long and the
short trading positions, respectively. More specifically VaRt,L and
VaRt,s are as follows:
VaRt,L = t + st˛ (v, k)t
(20)
VaRt,s = t + st1−˛ (v, k)t
(21)
where st˛ (v, k) is the left quantile at the ˛% of the skewed
Student-t distribution innovation. Correspondingly, st1−˛ (v, k) is
the right quantile of the skewed Student-t distribution.1 According to Lambert and Laurent (2001) and Wu and Shieh (2007), we
can compute the one-day-ahead VaR estimated at time (t − 1) for
the long and the short trading positions. Under the hypothesis of
skewed Student-t distribution, the one-day-ahead VaR for the long
and the short trading positions are as follows:
VaR
t + st˛ (v, k)t
t,L =
t + st1−˛ (v, k)t
VaR
t,s =
1 if rt+1 < VaRt+1|t(˛)
0 if
T −N
} + 2 log
N
N
T
1−
T −N
N
T
(25)
Under the null hypothesis, LRUC has a 2 (1) as an asymptotical distribution. Thus, a preferred model for VaR prediction should
provide the property that the unconditional coverage measured by
p = E(N/T ) equals the desired coverage level p0 .
Engle and Manganelli developed the Dynamic Quantile (DQ) test
building upon a linear regression model based on the process of
centered hit function:
ı˛
t = Hitt (˛) ≡ I(yt < −VaRt (˛)|˝t−1 ) − ˛
(23)
Conditional on pre-sample values, the dynamic of the hit function is modeled as:
ı˛
t = 0 +
The accuracy of VaR estimates is sensitive to the adequacy of
the volatility model used. It is therefore important to evaluate the
performance of VaR model given the pre-selected volatility model.
We do using two approaches. The first approach consists at computing the empirical failure rate, both for the left and right tails
of the returns distribution. The prescribed probability ranges from
0.25% to 5%. The failure rate can be defined as the number of times
in which returns exceed (in absolute value) the forecasted VaR. If
the model is correctly specified, then the failure rate is equal to
the specified VaR’s level. The second approach in examining the
accuracy of VaR estimate consists at backtesting VaR using Kupiec
LR Unconditional Coverage test and the Conditional Coverage test
proposed by Engle and Manganelli (2004).
In order to test the accuracy and to evaluate the performance
of the model-based VaR estimates, Kupiec (1995) provided a likelihood ratio test (LRUC ) to examine whether the failure rate of the
model is statistically equal to the expected one (unconditional covT
erage). Consider that N =
I is the number of exceptions in the
t=1 t
sample size T, then
It+1 =
LRUC = −2 log{˛N
(1 − ˛0 )
0
(22)
3.1. Assessing the accuracy for VaR estimates
likelihood ratio statistic in the presence of the null hypothesis is
given by:
(24)
rt+1 ≥ VaRt+1|t(˛)
follows a binomial distribution, N∼B(T, ˛). If p = E(N/T ) is the
expected exception frequency (i.e. the expected ratio of violations), then the hypothesis for testing whether the failure rate of
the model is equal to the expected one is expressed as follows:
H0 : ˛ = ˛0 · ˛0 . is the prescribed VaR level. Thus, the appropriate
1
For more details, see Lambert and Laurent (2001), Giot and Laurent (2003) and
Wu and Shieh (2007).
p
i=1
(˛)
i ıt=i +
m
=1
( )
ϑi ıt=i + t
(26)
(27)
where t is an IID process. The DQ test is defined under the
hypothesis that the regressors in Eq. (27) have no explanatory
power:
T
H0 = = (0 , 1 , . . . , p , ϑ0 , ϑ1 , . . . , ϑm ) = 0
For backtesting, the DQ test statistic, in association with Wald
statics, is as follows:
DQ =
ˆ T XT
ˆ ℓ
−→ 21+p+m
˛(1 − ˛)
(28)
where X denotes he regressors matrix in Eq. (27)
4. Data and preliminary analysis
Our sample consists of daily closing prices of seven stocks
indices obtained from DataStream: Dow Jones, NASDAQ 100, and
S&P 500 from The U.S., FTSE 100 from the U.K., DAX 30 of Germany,
the CAC40 of France and the NIKKEI 225 of Japan. The sample period
and the number of observations are indicated in Table 1. The daily
returns are defined as follows:
rt = 100 ∗ log
S
t
St−1
where St denotes the daily price index.
Table 2 reports the descriptive statistics for our sample. Except
for the Nikkey stock index, all the stock index returns have a positive mean. Furthermore, all the return series are not normally
distributed as it is indicated by the 3rd and the 4th moment.
More precisely, the returns series are skewed and fat tailed. This
is also confirmed by Jarque–Bera test statistic. Fig. 1 displays the
daily returns of each index. It is clear that large price changes
tend to follow large changes, and small changes tend to follow
S. Mabrouk, S. Saadi / The Quarterly Review of Economics and Finance 52 (2012) 305–321
Daily returns-CAC40
309
Daily returns -DAX
15
8
6
10
4
2
5
0
0
-2
-4
-5
-6
-8
-10
-10
-15
-12
92
94
96
98
00
02
04
06
08
92
94
96
Daily returns - FTSE100
98
00
02
04
06
08
04
06
08
Daily returns -DOW JONES
12
8
10
6
8
6
4
4
2
2
0
0
-2
-2
-4
-4
-6
-6
-8
-10
90
92
94
96
98
00
02
04
06
08
-8
90
92
94
Daily returns -NASDAQ
96
98
00
02
Daily returns -S&P500
15
6
4
10
2
5
0
0
-2
-4
-5
-6
-10
-8
-15
-10
90
92
94
96
98
00
02
04
06
08
90
92
94
96
98
00
02
Daily returns –NIKKEI 225
15
10
5
0
-5
-10
90
92
94
96
98
00
02
04
06
08
Fig. 1. Daily returns of equity indices (CAC 40, DAX, FTSE 100, Dow Jones, NASDAQ, S&P500, Nikkei 225).
04
06
08
310
S. Mabrouk, S. Saadi / The Quarterly Review of Economics and Finance 52 (2012) 305–321
Table 2
Descriptive statistics.
Stock index
Mean
Mediane
Maximum
Minimum
SD
Skewness
Kurtosis
Jarque–Bera
DAX
DOW JONES
NASDAQ
NIKKEI
CAC40
FTSE100
S&P500
0.0262
0.0239
0.0266
−0.0317
0.0112
0.0120
0.0198
0.0741
0.0476
0.1014
−0.0232
0.0591
0.0262
0.0435
7.270
6.139
14.023
12.427
11.928
10.396
5.571
−10.374
−7.777
−10.683
−8.801
−12.595
−8.178
−9.114
1.373
1.009
1.578
1.395
1.464
1.105
1.042
−0.394
−0.387
−0.291
0.158
−0.362
−0.046
−0.432
7.673
8.139
10.165
6.324
9.924
8.129
8.630
4209
5327
10,159
2140
9448
5183
6382
Notes: S.D. is the standard deviation. For all the time series, the descriptive statistics for daily returns are expressed in percentage.
Table 3
Unit root and stationarity tests.
Index
ADF
PP
KPSS
DAX
DOW JONES
NASDAQ
NIKKEI
CAC40
FTSE100
5&P500
−46.91
−48.81
−50.13
−38.96
−50.22
−50.55
−49.29
−66.24
−68.53
−71.4
−64.96
−74.65
−73.64
−69.68
0.2586
0.3997
0.3470
0.1224
0.2100
0.2617
0.4558
existence of long memory we employ two long-range memory
tests: the log-periodogram regression (GPH) of Geweke and PorterHudak (1983) and the Gaussian semi-parametric estimate (GSP) of
Robinson and Henry (1999). Similar to several previous studies, we
use the absolute returns and the daily squared volatility returns as
proxies for daily volatility. We employ the following three bandwidth for GHP test: m = T 0.5 ; m = T 0.6 and m = T 0.7 . As for GSP test,
we use m = T /2; m = T /4 and m = T /8.
Table 4 below reports the results of the long memory tests.
Both GPH and GSP test statistics suggest the non-rejection of the
null hypothesis of short memory in returns (there is no persistence in the conditional mean), but not for the two proxies of
volatility (squared returns and absolute returns). In fact, both test
statistics indicate strong evidence of long range memory in the
conditional variance of our series. Thus, it appears that the autocorrelation function for both squared returns and absolute returns
decayed very slowly at long time lags, a property of long memory processes. Therefore, the fractionally integrated GARCH-class
models are recommended to model the dynamics of our returns
series.
Notes: MacKinnon’s 1% critical value is −3.435 for the ADF and PP tests. The KPSS
critical value is 0.739 at the 1% significance level.
small changes. This is a property of asset prices, volatility clustering (a type of heteroscedasticity) that each index seems to exhibit.
This graphical evidence is an indication of the presence of ARCH
effect in our daily returns series that should be accounted for when
estimating VaR.
To have a better understanding of the behavior of the returns
series, it is important to examine whether they are stationary.
To do so, we employ three widely used tests: the Augmented
Dickey–Fuller (1979) (ADF) unit root test, the Phillips–Perron
(1988) (PP) unit root test and Kwiatkowski, Phillips, Schmidt, and
Shin (1992) (KPSS) stationarity test. The ADF and PP unit root test
results, reported in Table 3, indicate that, for all the return series,
the null hypothesis of presence of unit root is rejected. In the same
vein, the KPSS test indicates that all the returns series are stationary. Hence, our data sample is appropriate for empirical analysis.
While our preliminary data analysis suggests using a GARCH
type model to estimate VaR, a test of the presence of long range
memory is required to refine the model selection. To examine the
5. Empirical results
5.1. Estimates GARCH-type models
In this sub-section we try to identify the volatility model that
best fits our returns data. In the next sub-section we evaluate the
accuracy of the selected model in estimating VaR of each of the
seven returns series, and this for short and long trading positions.
Before examining the choice of volatility model we first determine
Table 4
Long range memory tests.
Panel a
GPH test
m = T0.5
m = T0.6
m = T0.7
GSP test
m = T/2
m = T/4
m = T/8
Panel b
GPH Test
m = T05
m = T06
m = T0J
GSP Test
m = T/2
m = T/4
m = T/8
rt2
|rt|
DAX
DOW JONES
NASDAQ
NIKKEI
DAX
DOW JONES
NASDAQ
NIKKEI
0.62
0.52
0.45
0.48
0.44
0.43
0.49
0.47
0.38
0.25
0.35
0.29
0.59
0.47
0.41
0.35
0.35
0.37
0.45
0.38
0.36
0.26
0.27
0.24
0.303
0.394
0.501
0.271
0.360
0.480
0.282
0.366
0.447
0.276
0.372
0.455
0.260
0.315
0.424
0.232
0.280
0.389
0.237
0.288
0.355
0.198
0.282
0.322
|rt|
CAC40
FTSE 100
S&P 500
rt2
CAC40
FTSE 100
S&P 500
0.50
0.44
0.39
0.45
0.40
0.42
0.38
0.26
0.24
0.33
0.29
0.28
0.36
0.33
0.32
0.20
0.16
0.17
0.281
0.350
0.467
0.289
0.380
0.463
0.276
0.367
0.465
0.20
0.222
0.289
0.231
0.298
0.351
0.245
0.282
0.380
S. Mabrouk, S. Saadi / The Quarterly Review of Economics and Finance 52 (2012) 305–321
311
Table 5
AR (1)-RiskMetrics model estimation.
DAX
Cst(M)
AR (1)
˛1
ˇ1
ln(ℓ)
AIC
SIC
Q(20)
Q2 (20)
RBD(20)
ARCH(10)
P(60)
DOW JONES
***
0.05
(0.02)
0.03**
(0.01)
0.06
0.94
−7054.73
3.13
3.14
17.83
6.16
10.39
0.42
137.42***
***
0.04
(0.01)
0.02
(0.01)
0.06
0.94
−6152.3
2.60
2.606
21.65
12.53
18.21
0.51
128.66***
NASDAQ
NIKKEI
**
0.05
(0.02)
0.02
(0.01)
0.06
0.94
−7707.8
3.269
3.274
30.74
22.99
29.7
1.67
153.6***
0.01
(0.02)
−0.03**
(0.01)
0.06
0.94
−7764.77
3.372
3.378
21.26
17.02
25.99
0.94
77.02***
CAC40
0.03
(0.02)
−0.06***
(0.01)
0.06
0.94
−7836.76
3.352
3.357
18.74
25.65
31.41
1.78
121.89***
FTSE 100
S&P 500
***
0.03
(0.01)
−0.06***
(0.02)
0.06
0.94
−6565.55
2.779
2.785
26.57
30.03
39.35
1.78
94.89***
0.04***
(0.01)
0.0003
(0.02)
0.06
0.94
−6206.49
2.631
2.736
34.98
11.53
17.58
0.49
157.69***
Notes: ln(ℓ) is the value of the maximized log-likelihood. AIC is the Akaike Information Criterion and SIC is the Schwarz Information Criterion. Q(20) and Q2 (20) are the
Box–Pierce statistics for remaining serial correlation for respectively standardized and squared standardized residuals. RBD(20) is the residual based diagnostic for conditional
heteroscedasticity, using 20 lags. The ARCH(10) is LM-ARCH test of using 10 lags. P(60) is the Pearson goodness-of-fit statistic for 60 cells.
* Significance level at 10%.
**
Significance level at 5%.
***
Significance level at 1%.
k, the order of autocorrelation in the mean equation as indicated
in Eq. (1). Evidence from graphic representation of Akaike Information Criterion (AIC) suggests that k = 1 provides the lowest AIC
value. The choice of k = 1 is also confirmed by the Modified Qstatistics showing the residuals of the AR (1) being white noise,
and which in turn suggest that the AR (1) model accounts for all
the linearity dependence (autocorrelation) in each of the seven
index returns.2 The autocorrelation may exist due to institutional
factors such as non-synchronous trading which may induce priceadjustment.3
Based on the sample characteristics identified in Section 4,
we consider four widely used volatility models and then compare their performances based on the value of the maximized
log-likelihood function, Akaike Information Criterion (AIC) and
Schwarz Information Criterion (SIC). The models are: GARCH,
FIGARCH, FIAPARCH and HYGARCH models under normal, Studentt and skewed Student-t distributions. For the sake of comparison,
we also consider RiskMetrics which was suggested by J.P. Morgan when they introduced VaR. As indicated above, RiskMetrics
model is a pre-specified IGARCH (1,1) model where the GARCH
coefficient (ˇ1 ) is set at 0.94 with daily observations and 0.97
for weekly observations. Table 5 reports the estimation results
of AR (1)-RiskMetrics model where set ˇ1 = 0.94 since we are
dealing with daily return series. Consisting with the shortcomings of RiskMetrics approach with respect to modeling volatility
of financial time series, all the diagnostic tests indicate that the
RiskMetrics specification is not appropriate (Box–Pierce tests using
standardized residuals and squared standardized residuals, the
RBD test, the ARCH-LM test, and the Pearson goodness-of-fit).
This is in fact not surprising given the rigidity of the RiskMetrics
model and the fact that it can only be assessed under a normal
distribution. We address this limitation by considering GARCH
(1,1) under normal, as well as Student-t and skewed Student-t
distributions. Several empirical studies show that GARCH (1,1) performs particularly well in capturing some of the salient features
of returns series. For instance, Hansen and Lunde (2005) show
that GARCH (1,1) outperforms several GARCH type models and
specifications in capturing volatility clustering in finance returns
series.4
Table 6 (panels a & b) reports the estimation results of a GARCH
(1,1) process for all the returns series under the assumption that
the innovations follow either a normal distribution, Student-t or
skewed Student-t. The coefficients of the conditional variance
equation are significant at 1% level implying a strong support for
the ARCH and GARCH effects. Furthermore, the condition for the
existence of conditional variance is justified since for all the index
returns we have that ˛1 + ˇ1 < 1. Interestingly, all the model selection criteria (i.e. maximized log-likelihood function, AIC and SIC)
indicate that AR (1)-GARCH (1,1) model under a skewed Student-t
distribution provides the best fit for each of the seven index returns.
It is noteworthy, however, that the sum of the parameters estimated by the variance equation is close to one. A sum ˛1 + ˇ1 near
one is an indication of a covariance stationary model with a high
degree of persistence in the conditional variance. The sum ˛1 + ˇ1
is also an estimation of the rate at which the response function
decays on daily basis. Since the rate is high, the response function
to shocks is likely to die slowly. For instance, in the case of Dow
Jones Index, under skewed Student-t distribution, ˛1 + ˇ1 = 0.99
which means that a month after an initial shock 74% (or 0.9930 ) of
the impact remains in effect. Even six months later, 16% (or 0.99180 )
of initial shock remains persistent. The evidence of high volatility
persistence and long memory in the GARCH (1,1) model suggest
that a FIGARCH (p,d,q) model may be more adequate to describe
the data.
Table 7 provides the estimation results of FIGARCH (1,d,1)
under the three aforementioned alternative distributions. Consistent with the results from Table 4, all times series are governed by
long memory process. In fact, the fractionary integrated parameter d ranges from 0.44 to 0.65. As expected, the FIGARCH
(1,d,1) model outperform GARCH and RiskMetrics models as it
is indicated by each of the model selection criterion. Interestingly, the skewed Student-t distribution provides the most
adequate FIGARCH model for the seven time series outperforming the FIGARCH models under normal and Student-t distributions.
2
For brevity, we do not report the graphs of AIC and the Modified Q-statistics,
but they are available upon request.
3
Lo and Mackinlay (1990) argue that individual stock prices trading at different
frequencies can lead to a spurious positive autocorrelation in market-index returns.
4
It is noteworthy that Hansen and Lunde (2005) did not compare the performance
of GARCH (1,1) to those of volatility models designed to capture long range memory,
such as FIGARCH and HYGARCH. We also augment Hansen and Lunde’s work by
considering skewed Student-t distribution.
312
Table 6
AR (1)-GARCH (1,1) model estimation.
Panel a
DAX
N
Cst(M)
AR (1)
˛1
ˇ1
ln(ℓ)
AIC
SIC
Q(20)
Q2 (20)
RBD(20)
ARCH(10)
P(60)
006***
(0.01)
0.03**
(0.01)
0.08***
(0.01)
0.90***
(0.01)
–
–
–
–
−6997
3.11
3.120
18.57
5.66
5.60
0.35
137.02***
0.08***
(0.01)
0.02*
(0.01)
0.07***
(0.01)
0.92***
(0.008)
7.95***
(1.08)
−0.11***
–
−6883
3.06
3.072
19.92
6.04
5.3
0.34
100.2***
SKt
N
0.06***
(0.01)
0.01
(0.01)
0.07***
(0.08)
0.92***
(0.08)
8.47***
(1.18)
–
(0.02)
−6870
3.058
3.068
22.36
6.02
5.4
0.35
68.02**
0.04***
(0.01)
0.02*
(0.01)
0.06***
(0.01)
0.92***
(0.01)
–
–
−0.07***
–
−6113
2.585
2.591
21.42
11.96
12.41
0.55
136.7***
NASDAQ
t
SKt
0.05***
(0.01)
0.008
(0.01)
0.06***
(0.008)
0.93***
(0.009)
7.53***
(0.85)
–
–
−6018
2.545
2.553
23.18
12.08
12.52
0.53
104.9***
0.04***
(0.01)
0.002
(0.01)
0.06***
(0.008)
0.93***
(0.08)
7.87***
(0.90)
−0.16***
(0.01)
−6011
2.5452
2.5532
24.38
12.09
12.57
0.53
83.3**
N
0.06***
(0.01)
0.02**
(0.01)
0.10***
(0.02)
0.88***
(0.02)
–
–
–
–
−7656
3.247
3.254
28.47
13.51
11.81
0.76
176.6***
NIKKEI
t
SKt
0.09***
(0.01)
0.02***
(0.01)
0.07***
(0.01)
0.92***
(0.01)
7.42***
(0.82)
–
–
−7549
3.202
3.211
28.63
19.49
18.82
1.32
109***
0.06***
(0.01)
0.010
(0.01)
0.07***
(0.01)
0.92***
(0.01)
8.01***
(0.93)
−0.01
(0.02)
−7521
3.191
3.201
41.43
18.04
17.43
1.19
78.4**
N
0.015
(0.01)
0.04***
(0.01)
0.09***
(0.01)
0.88***
(0.01)
–
–
–
−7712
3.350
3.357
16.46
11.14
14.87
0.48
84***
t
SKt
0.004
(0.009)
0.03***
(0.009)
0.08***
(0.009)
0.90***
(0.009)
10.15***
(1.48)
0.001
(0.009)
0.02**
(0.009)
0.08***
(0.009)
0.90***
(0.009)
10.15***
(1.47)
–
−766
3.329
3.336
18.46
14.12
13.81
0.43
57***
(0.02)
−76,600
3.328
3.338
18.72
14.1
13.79
0.43
63***
Panel b
CAC40
Cst(M)
AR (1)
˛1
ˇ1
ln(ℓ)
AIC
SIC
Q(20)
Q2 (20)
RBD(20)
ARCH(10)
P(60)
FTSE 100
S&P 500
N
t
SKt
N
t
SKt
N
0.04***
(0.01)
−0.06***
(0.01)
0.11***
(0.02)
0.87***
(0.02)
–
–
–
–
−7765
3.322
3.329
19.70
16.71
15.66
0.86
117.84***
0.05***
(0.01)
−0.06***
(0.01)
0.08***
(0.01)
0.90***
(0.01)
7.85***
(0.95)
–
–
−7663
3.278
3.287
19.75
21.97
20.84
1.32
72.90***
0.03***
(0.01)
−0.07***
(0.01)
0.08***
(0.01)
0.90***
(0.01)
8.27***
(1.02)
–0.10***
(0.02)
−7651
3.274
3.283
21.03
21.53
20.57
1.30
51.20***
0.03***
(0.01)
−0.05***
(0.01)
0.09***
(0.01)
0.89***
(0.01)
–
–
–
–
−6526
2.763
2.770
26.92
23.40
22.61
1.07
103.04***
0.04***
(0.01)
−0.06***
(0.01)
0.08***
(0.01)
0.90***
(0.01)
9.52***
(1.27)
–
–
−6466
2.738
2.746
27.37
23.82
23.80
1.17
67.40***
0.03***
(0.01)
−0.07***
(0.01)
0.08***
(0.01)
0.90***
(0.01)
8.87***
(1.40)
−0.06**
(0.01)
−6460
2.736
2.746
27.76
23.94
23.87
1.18
51.66***
0.04***
(0.01)
0.0002
(0.01)
0.06***
(0.01)
0.93***
(0.01)
–
–
–
–
−6173
2.618
2.624
32.83
12.57
13.14
0.70
170.25***
t
0.05***
(0.01)
−0.011
(0.01)
0.05***
(0.008)
0.94***
(0.009)
7.07***
(0.77)
–
–
−6077
2.577
2.586
35.39
13.34
13.85
0.75
75.88***
SKt
0.04***
(0.01)
−0.017
(0.01)
0.05***
(0.008)
0.94***
(0.009)
7.32***
(0.81)
−0.06**
(0.01)
−6072
2.576
2.585
36.53
13.16
13.71
0.73
70.16**
Notes: ln(ℓ) is the value of the maximized log-likelihood. AIC is the Akaike (1974) Information Criterion and SIC is the Schwarz Information Criterion. Q(20) and Q2 (20) are the Box–Pierce statistics for remaining serial correlation
for respectively standardized and squared standardized residuals. RBD(20) is the residual based diagnostic for conditional heteroscedasticity, using 20 lags. The ARCH(10) is LM-ARCH test of Engle (1982), using 10 lags. P(60)
is the Pearson goodness-of-fit statistic for 60 cells. Figures between parentheses are the standard errors. N, t and SKt are respectively normal, Student-t and the skewed Student-t distribution.
*
Significance level at 10%.
**
Significance level at 5%.
***
Significance level at 1%.
S. Mabrouk, S. Saadi / The Quarterly Review of Economics and Finance 52 (2012) 305–321
DOW JONES
t
Table 7
AR (1)-FIGARCH (1,d,1) model estimation.
Panel a
DAX
DOW JONES
N
Cst(M)
AR (1)
d
ϕ1
ln(ℓ)
AIC
SIC
Q(20)
Q2 (20)
RBD(20)
ARCH(10)
P(60)
0.06***
(0.01)
0.03**
(0.01)
0.51***
(0.1)
0.13***
(0.05)
0.59***
(0.09)
–
–
–
–
−6987
3.10
3.118
18.81
4.61
4.85
0.21
154.3***
SKt
0.08***
(0.01)
0.02*
(0.007)
0.65***
(0.07)
0.1***
(0.03)
0.72***
(0.05)
8.04***
(1.01)
–
–
−6876
3.06
3.070
19.82
7.06
5.56
0.4
109.2***
N
0.06***
(0.01)
0.01*
(0.01)
0.66***
(0.07)
0.11***
(0.03)
0.74***
(0.05)
8.47***
(1.1)
−0.10***
(0.02)
−6864
3.055
3.067
22.19
7.18
5.65
0.4
75.36***
0.04***
(0.01)
0.02*
(0.01)
0.51***
(0.08)
0.22***
(0.04)
0.68***
(0.05)
–
–
–
–
−6107
2.582
2.590
21.07
12.33
13.07
0.56
137***
NASDAQ
t
SKt
0.05***
(0.01)
0.008
(0.01)
0.52***
(0.06)
0.20***
(0.03)
0.70***
(0.04)
7.52***
(0.81)
–
–
−6009
2.541
2.551
22.98
13.11
13.82
0.62
81.9***
0.04***
(0.01)
0.001
(0.01)
0.53***
(0.06)
0.20***
(0.03)
0.70***
(0.04)
7.8***
(0.86)
−0.07***
(0.01)
−6003
2.539
2.550
24.28
12.98
13.69
0.6
80.8***
NIKKEI
N
0.07***
(0.01)
0.03**
(0.01)
0.45***
(0.07)
0.28***
(0.07)
0.58***
(0.09)
–
–
–
–
−7637
3.240
3.248
31.55
8.18
8.18
0.33
143***
t
0.09***
(0.01)
0.03**
(0.01)
0.46***
(0.05)
0.27***
(0.05)
0.63***
(0.06)
7.78***
(0.82)
–
–
−7537
3.198
3.207
31.18
10.94
11.24
0.62
104***
SKt
N
0.06***
(0.01)
0.015
(0.01)
0.46***
(0.05)
0.28***
(0.05)
0.63***
(0.06)
8.43***
(0.94)
−0.15***
(0.02)
−7510
3.187
3.198
33.68
9.88
10.34
0.51
50.7***
0.017
(0.01)
0.04***
(0.01)
0.50***
(0.08)
0.18***
(0.05)
0.60***
(0.08)
–
–
–
–
−7710
3.349
3.358
17.16
13.6
13.84
0.26
794***
t
SKt
0.005
(0.01)
0.02***
(0.01)
0.61***
(0.09)
0.16***
(0.04)
0.70***
(0.06)
10.05***
(1.49)
–
–
−7659
3.328
3.337
19.10
13.41
13.06
0.35
61.6***
0.002
(0.01)
0.02***
(0.01)
0.61***
(0.09)
0.16***
(0.04)
0.7***
(0.06)
10.06***
(1.4)
−0.01
(0.02)
−7658
3.327
3.339
19.37
13.42
13.03
0.35
61.4***
Panel b
CACAO
Cst(M)
AR (1)
d
ϕ1
ˇ1
ln(ℓ)
AIC
BIC
Q(20)
Q2 (20)
RBD(20)
ARCH(10)
P(60)
FTSE 100
S&P 500
N
t
SKt
N
t
SKt
N
t
SKt
0.05***
(0.01)
−0.06***
(0.01)
0.56***
(0.08)
0.20***
(0.06)
0.64***
(0.07)
–
–
–
–
−7760
3.320
3.328
19.67
15.43
15.06
0.82
129.13***
0.05***
(0.01)
−0.06***
(0.01)
0.58***
(0.06)
0.19***
(0.03)
0.70***
(0.04)
7.81***
(0.92)
–
–
−7655
3.276
3.285
20.24
21.24
20.33
1.40
69.23***
0.03***
(0.01)
−0.07***
(0.01)
0.59***
(0.06)
0.20***
(0.04)
0.71***
(0.04)
8.20***
(0.99)
−0.09***
(0.02)
−7644
3.271
3.282
20.40
20.09
19.14
1.29
49.32***
0.03***
(0.01)
−0.05***
(0.01)
0.57***
(0.06)
0.14***
(0.05)
0.63***
(0.05)
–
–
–
–
−6518
2.760
2.768
27.98
20.74
21.03
0.95
96.64***
0.04***
(0.01)
−0.06***
(0.01)
0.55***
(0.06)
0.16***
(0.05)
0.64***
(0.05)
9.50***
(1.24)
–
–
−6460
2.736
2.745
28.67
20.57
21.25
0.95
67.58***
0.03***
(0.01)
−0.07***
(0.01)
0.55***
(0.06)
0.16***
(0.05)
0.64***
(0.05)
9.83***
(1.36)
−0.06***
(0.02)
−6455
2.734
2.745
29.02
20.63
21.20
0.95
55.26***
0.04***
(0.01)
0.001
(0.01)
0.44***
(0.06)
0.16***
(0.09)
0.57***
(0.09)
–
–
–
–
−6165
2.615
2.623
34.28
12.37
13.13
0.76
151.59***
0.05***
(0.01)
−0.012
(0.01)
0.48***
(0.05)
0.18***
(0.05)
0.64***
(0.05)
7.27***
(0.74)
–
–
−6068
2.574
2.583
36.46
15.22
16.40
1.02
83.58***
0.04***
(0.01)
−0.018
(0.01)
0.48***
(0.05)
0.17***
(0.05)
0.64***
(0.05)
7.47***
(0.77)
−0.06***
(0.01)
−6062
2.572
2.583
37.97
14.74
15.91
0.97
76.13***
313
Notes: ln(ℓ) is the value of the maximized log-likelihood. AIC is the Akaike (1974) Information Criterion and SIC is the Schwarz Information Criterion. Q(20) and Q2 (20) are the Box–Pierce statistics for remaining serial correlation
for respectively standardized and squared standardized residuals. RBD(20) is the residual based diagnostic for conditional heteroscedasticity, using 20 lags. The ARCH(10) is LM-ARCH test of Engle (1982), using 10 lags. P(60)
is the Pearson goodness-of-fit statistic for 60 cells. Figures between parentheses are the standard errors. N, t and SKt are respectively normal, Student-t and the skewed Student-t distribution.
*
Significance level at 10%.
**
Significance level at 5%.
***
Significance level at 1%.
S. Mabrouk, S. Saadi / The Quarterly Review of Economics and Finance 52 (2012) 305–321
ˇ1
t
314
Table 8
AR (1)-HYGARCH (1,d,1) model estimation.
Panel a
DAX
Cst(M)
AR (1)
d
ϕ1
ˇ1
ln(ℓ)
AIC
SIC
Q(20)
Q2 (20)
RBD(20)
ARCH(10)
P(60)
NASDAQ
t
SKt
N
t
SKt
0.06***
(0.01)
0.03**
(0.01)
0.56***
(0.11)
0.12***
(0.05)
0.61***
(0.09)
−0.02
(0.02)
–
–
–
0.08***
(0.01)
0.02*
(0.01)
0.64***
(0.07)
0.11***
(0.03)
0.72***
(0.05)
0.003
(0.01)
7.98***
(1.07)
–
0.06***
(0.01)
0.015
(0.01)
0.66***
(0.07)
0.11***
(0.03)
0.74***
(0.05)
−0.0005
(0.01)
8.48***
(1.18)
−0.10***
0.04***
(0.01)
0.02*
(0.01)
0.56***
(0.01)
0.20***
(0.05)
0.70***
(0.06)
−0.01
(0.02)
–
–
–
0.05***
(0.01)
0.008
(0.01)
0.51***
(0.07)
0.21***
(0.04)
0.69***
(0.04)
0.005
(0.05)
7.47***
(0.84)
–
0.04***
(0.01)
0.001
(0.01)
0.52***
(0.07)
0.20***
(0.04)
0.70***
(0.04)
0.0007
(0.02)
7.8***
(0.89)
−0.07***
–
−6986
3.109
3.119
18.64
4.53
4.45
0.20
155.5***
–
−6876
3.061
3.072
19.86
7.07
5.54
0.4
109.5***
(0.02)
−6864
3.056
3.068
22.19
7.18
5.65
0.41
76.94***
–
−6106
2.582
2.592
20.86
12.29
12.99
0.55
115.9***
–
−6009
2.542
2.553
23.08
13.08
13.75
0.62
86.6***
NIKKEI
N
(0.01)
−6003
2.540
2.552
24.29
12.98
13.68
0.6
78.9***
0.07***
(0.01)
0.03**
(0.01)
0.46***
(0.07)
0.27***
(0.07)
0.58***
(0.09)
−0.004
(0.02)
–
–
–
–
−7637
3.240
3.250
31.49
8.16
8.15
0.33
141***
t
0.09***
(0.01)
0.03**
(0.01)
0.44***
(0.05)
0.28***
(0.05)
0.62***
(0.07)
0.015
(0.02)
7.64***
(0.85)
–
–
−7537
3.198
3.209
31.39
10.86
10.94
0.60
106***
SKt
N
t
SKt
0.06***
(0.01)
0.015
(0.01)
0.45***
(0.05)
0.29***
(0.05)
0.63***
(0.07)
0.010
(0.02)
8.32***
(0.97)
−0.15***
0.017
(0.01)
0.03***
(0.01)
0.60***
(0.14)
0.16***
(0.06)
0.65***
(0.10)
−0.03
(0.03)
–
–
–
0.004
(0.01)
0.02**
(0.01)
0.69***
(0.14)
0.13***
(0.06)
0.74***
(0.08)
−0.02
(0.01)
10.2***
(1.52)
–
0.002
(0.01)
0.02**
(0.01)
0.69***
(0.14)
0.13***
(0.06)
0.74***
(0.08)
−0.02
(0.01)
10.2***
(1.52)
−0.01
(0.02)
−7510
3.187
3.199
33.9
9.85
10.08
0.5
47.3***
–
−7709
3.350
3.359
16.66
14.19
14.27
0.29
77.3***
–
−7658
3.328
3.339
18.63
14.01
13.48
0.37
55.5***
(0.02)
−7658
3.328
3.341
18.91
14.03
13.47
0.38
58.7***
Panel b
CAC40
Cst(M)
AR (1)
d
ϕ1
ˇ1
Log(â)
ln(ℓ)
AIC
SIC
Q(20)
Q2 (20)
RBD(20)
ARCH(10)
P(60)
FTSE 100
S&P 500
N
t
SKt
N
t
SKt
N
t
SKt
0.05***
(0.01)
−0.06***
(0.01)
0.64***
(0.09)
0.18***
(0.05)
0.67***
(0.06)
−0.03
(0.02)
–
–
–
–
−7758
3.319
3.329
19.61
15.68
14.94
0.85
128.25***
0.05***
(0.01)
−0.06***
(0.01)
0.59***
(0.06)
0.18***
(0.04)
0.70***
(0.04)
−0.004
(0.01)
7.87***
(0.96)
–
–
−7655
3.276
3.287
20.22
21.48
20.95
1.42
69.25***
0.03***
(0.01)
−0.07***
(0.01)
0.60***
(0.06)
0.19***
(0.04)
0.71***
(0.04)
−0.007
(0.01)
8.30***
(1.03)
−0.09***
(0.02)
−7644
3.272
3.284
21.36
20.45
19.70
1.32
47.17***
0.03**
(0.01)
−0.05***
(0.01)
0.59***
(0.06)
0.13***
(0.04)
0.64***
(0.05)
−0.01
(0.02)
–
–
–
–
−6517
2.760
2.770
27.98
20.78
21.01
0.96
98.88***
0.04***
(0.01)
−0.06***
(0.01)
0.58***
(0.06)
0.15***
(0.03)
0.65***
(0.15)
−0.01
(0.01)
9.67***
(1.30)
–
–
−6460
2.736
2.747
28.61
20.79
21.49
0.98
69.45***
0.03**
(0.01)
−0.07***
(0.01)
0.58**
(0.06)
0.15***
(0.03)
0.66***
(0.05)
−0.01
(0.01)
10.02***
(1.43)
−0.06***
(0.02)
−6455
2.735
2.747
28.97
20.89
21.56
0.99
50.40***
0.04***
(0.01)
0.001
(0.01)
0.42***
(0.08)
0.16***
(0.06)
0.56***
(0.11)
0.008
(0.03)
–
–
–
–
−6165
2.615
2.625
34.43
12.27
13.01
0.75
155.83***
0.05***
(0.01)
−0.012
(0.01)
0.42***
(0.06)
0.19***
(0.04)
0.61***
(0.06)
0.04
(0.03)
6.95***
(0.73)
–
–
−6066
2.574
2.585
37.49
14.09
14.80
0.94
79.18***
0.04***
(0.01)
−0.02*
(0.01)
0.43***
(0.06)
0.19***
(0.05)
0.61***
(0.06)
0.03
(0.03)
7.21***
(0.77)
−0.06***
(0.01)
−6062
2.572
2.584
38.76
13.88
14.60
0.90
64.0***
Notes: ln(ℓ) is the value of the maximized log-likelihood. AIC is the Akaike (1974) Information Criterion and SIC is the Schwarz Information Criterion. Q(20) and Q2 (20) are the Box–Pierce statistics for remaining serial correlation
for respectively standardized and squared standardized residuals. RBD(20) is the residual based diagnostic for conditional heteroscedasticity, using 20 lags. The ARCH(10) is LM-ARCH test of Engle (1982), using 10 lags. P(60)
is the Pearson goodness-of-fit statistic for 60 cells. Figures between parentheses are the standard errors. N, t and SKt are respectively normal, Student-t and the skewed Student-t distribution.
*
Significance level at 10%.
**
Significance level at 5%.
***
Significance level at 1%.
S. Mabrouk, S. Saadi / The Quarterly Review of Economics and Finance 52 (2012) 305–321
Log(â)
DOW JONES
N
Table 9
AR (1)-FIAPARCH (1,d,1) model estimation.
Panel a
Cst(M)
AR (1)
d
ϕ1
ˇ1
1
DAX
DOW JONES
NASDAQ
N
t
SKt
N
t
SKt
0.03***
(0.01)
0.03**
0.06***
(0.01)
0.02**
0.04***
(0.01)
0.02**
0.02**
(0.01)
0.02**
0.04***
(0.01)
0.02***
(0.01)
0.014
(0.01)
0.40***
0.009
(0.01)
0.41***
(0.07)
(0.07)
(0.01)
(0.01)
(0.01)
(0.01)
0.46***
(0.08)
0.19***
0.58***
(0.08)
0.14***
0.61***
(0.08)
0.14***
0.41***
(0.09)
0.30***
(0.05)
(0.03)
(0.03)
(0.05)
0.59***
(0.09)
0.42***
0.68***
(0.06)
0.70***
0.64***
(0.06)
0.31***
(0.07)
0.31***
0.29***
(0.04
0.63***
(0.06)
0.72***
0.65***
N
0.28***
(0.04
0.63***
(0.06)
0.04***
(0.01)
0.03**
NIKKEI
t
SKt
N
t
SKt
0.07***
(0.01)
0.03**
0.05***
(0.01)
0.03**
−0.02
(0.01)
−0.02
(0.01)
0.03***
−0.02
(0.01)
0.03***
(0.01)
(0.01)
0.46***
(0.09)
0.22***
0.46***
(0.09)
0.22***
(0.03)
(0.03)
0.61***
(0.07)
0.55***
0.61***
(0.07)
0.55***
(0.01)
(0.01)
(0.01)
0.38***
(0.05)
0.29***
0.36***
(0.05)
0.30***
0.37***
(0.05)
0.30***
(0.09)
(0.07)
(0.06)
0.53***
(0.01)
0.64***
0.55***
(0.09)
0.38***
0.42***
0.56***
(0.08)
0.40***
0.04***
(0.01)
0.43***
(0.08)
0.22***
(0.04
0.58***
(0.08)
0.56***
(0.07)
(0.07)
(0.15)
(0.13)
(0.13)
(0.10)
(0.08)
(0.08)
(0.11)
(0.10)
(0.10)
1.39***
(0.14)
1.43***
(0.11)
8.35***
1.43***
(0.12)
8.66***
1.58***
(0.15)
1.60***
(0.11)
8.66***
1.64***
(0.11)
9.27***
1.40***
(0.18)
–
–
1.57***
(0.13)
9.27***
1.28***
(0.16)
1.59***
(0.13)
8.73***
1.41***
(0.16)
11.9***
1.41***
(0.16)
11.9***
(1.24)
(1.36)
(1.01)
(1.06)
(1.02)
(1.15)
(1.95)
(1.95)
–
–
−6947
3.092
3.104
20.26
5.90
5.89
0.28
131.3***
ln(ℓ)
A1C
S1C
Q(20)
Q2 (20)
RBD(20)
ARCH(10)
P(60)
Panel b
Cst(M)
AR (1)
d
ϕ1
ˇ1
1
ı
ln(ℓ)
AIC
BIC
Q(20)
Q2 (20)
RBD(20)
ARCH(10)
P(60)
−0.11***
(0.02)
−6844
3.047
3.062
21.92
6.08
4.2
0.3
69.25***
–
–
−6857
3.053
3.066
20.29
6.04
4.12
0.3
85.77***
–
–
–
–
−6051
2.559
2.570
21.22
20.60
20.89
1.34
98.5***
CAC40
–
–
−5974
2.527
2.540
22.47
18.62
18.33
1.14
70.6***
−0.08***
(0.01)
−5966
2.525
2.538
22.85
10.93
18.14
1.12
47.8***
–
–
–
–
−7596
3.223
3.234
32.64
9.59
9.80
0.20
112***
–
–
−7510
3.187
3.199
31.44
8.89
5.10
0.21
96.1***
FTSE 100
−0.15***
(0.02)
−7483
3.176
3.190
32.37
8.82
9.76
0.19
50.1***
–
–
–
–
−7643
3.321
3.332
20.97
19.02
18.08
0.67
63.8***
–
–
−7605
3.305
3.318
22.01
17.43
17.13
0.56
60.2***
−0.000*
(0.02)
−7605
3.306
3.320
22.02
17.43
17.12
0.56
61.5***
S&P 500
N
t
SKt
N
t
SKt
N
t
SKt
0.008
(0.01)
−0.04***
(0.01)
0.36***
(0.06)
0.25***
(0.06)
0.51***
(0.01)
0.62***
(0.12)
1.46***
(0.11)
–
–
–
–
−7689
3.290
3.301
21.02
13.73
13.15
0.81
105.89***
0.03**
(0.01)
−0.06***
(0.01)
0.40***
(0.06)
0.25***
(0.03)
0.58***
(0.07)
0.57***
(0.10)
1.49***
(0.10)
8.89***
(1.18)
–
–
−7612
3.258
3.271
22.11
13.62
13.21
0.78
54.89***
0.01
(0.01)
−0.06***
(0.01)
0.42***
(0.06)
0.26***
(0.03)
0.59***
(0.06)
0.56***
(0.10)
1.48***
(0.10)
9.28***
(1.24)
−0.10***
(0.02)
−7600
3.253
3.267
22.41
13.62
13.22
0.77
45.50***
0.01
(0.01)
−0.06***
(0.01)
0.59***
(0.11)
0.22***
(0.05)
0.73***
(0.08)
0.44***
(0.09)
1.40***
(0.17)
–
–
–
–
−6478
2.744
2.755
26.67
18.58
18.58
0.69
86.13***
0.02**
(0.01)
−0.06***
(0.01)
0.54***
(0.07)
0.23***
(0.03)
0.70***
(0.05)
0.49***
(0.08)
1.39***
(0.14)
10.4***
(1.54)
–
–
−6422
2.721
2.733
26.96
18.78
19.33
0.75
58.31***
0.01
(0.01)
−0.06***
(0.01)
0.56***
(0.08)
0.23***
(0.03)
0.71***
(0.06)
0.50***
(0.08)
1.38***
(0.14)
10.5***
(1.62)
−0.07***
(0.02)
−6416
2.719
2.732
26.81
19.16
19.67
0.78
45.36
0.01*
(0.01)
0.02*
(0.01)
0.19***
(0.22)
−0.08
(1.17)
0.06
(1.37)
0.85***
(0.20)
1.63***
(0.53)
–
–
–
–
−6089
2.583
2.594
34.46
27.63
28.59
1.83
120.42***
0.03***
(0.01)
0.0015
(0.01)
0.27***
(0.06)
0.18***
(0.09)
0.41***
(0.13)
0.78***
(0.15)
1.57***
(0.13)
8.06***
(0.98)
–
–
−6015
2.552
2.564
36.00
17.53
18.40
1.12
48.49***
0.02**
(0.01)
−0.004
(0.01)
0.28***
(0.05)
0.18***
(0.08)
0.43***
(0.12)
0.76***
(0.14)
1.57***
(0.12)
8.32***
(1.03)
−0.07***
(0.01)
−6007
2.549
2.563
36.49
17.19
18.01
1.09
76.56***
315
Notes: ln(ℓ) is the value of the maximized log-likelihood. AIC is the Akaike (1974). Q(20) and Q2 (20) are the Box–Pierce statistics for remaining serial correlation for respectively standardized and squared standardized residuals.
RBD(20) is the residual based diagnostic for conditional heteroscedasticity, using 20lags. The ARCH(10) is LM-ARCH test of Engle (1982), using 10 lags. P(60) is the Pearson goodness-of-fit statistic for 60 cells. Figures between
parentheses are the standard errors. N, t and SKt are respectively normal, Student-t and the skewed Student-t distribution.
*
Significance level at 10%.
**
Significance level at 5%.
***
Significance level at 1%.
S. Mabrouk, S. Saadi / The Quarterly Review of Economics and Finance 52 (2012) 305–321
(0.11)
ı
316
S. Mabrouk, S. Saadi / The Quarterly Review of Economics and Finance 52 (2012) 305–321
Table 8 provides the estimation of another widely used model
known to cope with long memory in financial time series: the
HYGARCH model. It is basically a generalization of FIGARCH model.
As shown above, the HYGARCH model nests a FIGARCH model
when ˛ = 1. Thus, similar to FIGARCH (1,d,1), HYGARCH captures the long range dependence by the fractionary integrated
parameter d. Evidence of long memory is proved since the d
ranges between zero and one. This is confirmed in Table 8 where
HYGARCH’s d parameter for each index series is within 0.42 and
0.69 interval.
While FIGARCH and HYGARCH models are known to successfully deal with volatility clustering and long range memory in
financial time series, they fail to take into account the asymmetry in
returns volatility (e.g. Baillie et al., 1996, 2000; Bollerslev & Jubinski,
1999; Davidson, 2004). To capture this stylized fact, we model each
of our index returns series using FIAPARCH model. Table 9 reports
the FIAPARCH (1,d,1) model estimates results for the seven times
series under the same three distributions (normal, Student-t and
skewed Student-t). According to each of the model selection criterion, the FIAPARCH model outperforms all the alternative models
(RiskMetrics, GARCH, FIGARCH and HYGARCH). Furthermore, under
a skewed Student-t distribution, the FIAPARCH provides the best
model for all our index returns series.
As robustness checks we repeat the above analysis using others
volatility models: the EGARCH, PGARCH, GARCH-M, GJR-GARCH, VGARCH, FIEGARCH, and also bilinearity models. For each volatility
model, we run three specifications based on the three alternative
error distributions: Normal, Student-t and skewed Student-t. Our
untabulated results show that skewed Student-t AR (1)-FIAPARCH
(1,d,1) outperforms all the above volatility models. Moreover, to
see whether our results are sensitive to the choice of the mean
equation, we again repeat the same analysis using AR (k), where
k = 2, 3, 4, 5, 10, 20 and 30, and this for all the returns indices and
for all the volatility models (including RiskMetrics, GARCH, FIGARCH
and HYGARCH). Our unreported findings indicate that the skewed
Student-t AR (1)-FIAPARCH (1,d,1) yet provides better fit than the
remaining models regardless of the order of autocorrelation (k) in
the mean equation.5
We therefore use the skewed Student-t AR (1)-FIAPARCH
(1,d,1) to compute VaR estimates. We also consider the normal and Student-t distributions to see how sensitive our VaR
estimates are to the error distribution assumption. We do
so in the next section and this for short and long trading
positions.
5.2. The VaR analysis: the in-sample and out-sample VaR
estimation
In this sub-section, we estimate the in-sample one-day-ahead
VaR using AR (1)-FIAPARCH model under the three alternative
innovation’s distributions (normal, Student-t and skewed Studentt) and this for each of the seven stock index returns. We first
compute the failure rate for long and short position. The failure rate for the short trading position denotes the percentage
of positive returns larger than the VaR prediction. However, for
the long trading positions, the failure rate is the percentage of
negative returns smaller than the VaR prediction. We also compute the Kupiec’s (1995) LR tests and the Dynamic Quantile
(DQ) test. The VaR levels range (˛) from 0.05 to 0.01 for the
short trading position and from 0.95 to 0.99 for the long trading
positions.
Table 10 provides the in-sample one-day-ahead VaR results
using AR (1)-FIAPARCH model for each index series under the normal, Student-t and skewed Student-t distributions. We report the
failure rate along with the Kupiec’s (1995) LR test and the Dynamic
Quantile test and their corresponding P-values. The results clearly
indicate that VaR model based on the normal AR (1)-FIAPARCH
model fails to model large positive and negative returns. In fact, the
hypothesis of model adequacy is strongly rejected as evidenced by
the considerable difference between the prefixed level (˛) and the
failure rate. This is expected given the excess kurtosis and skewness
in our returns data, and in financial time series in general. So it is
not surprising that the FIAPARCH provides better results when we
employ Student-t. The accuracy of FIAPARCH is even higher when
we consider skewed Student-t. In fact, it outperforms both the normal FIAPARCH and Student-t FIAPARCH in estimating VaR for short
as well as long trading positions.
While in-sample performance provides a good indication about
a model forecasting accuracy, it is not a prerequisite for a good outof-sample performance (Hansen & Lunde, 2005; Pagan & Schwert,
1990). Accordingly, we provide out-of-sample evaluation of the
FIAPARCH model. Table 11 reports the out-of-sample one-dayahead VaR results using AR (1)-FIAPARCH model, and this for
short and long trading positions and under different error distribution assumptions. The out-of-sample VaR estimates are computed
based on 1000 observations (i.e. last five years). We re-estimate
the AR (1)-FIAPARCH model every 50 observations in the outof-sample period. The overall results are similar to those of the
in-sample analysis. In fact, compared to normal and Student-t,
the skewed Student-t FIAPARCH model provides the most accurate VaR estimates. However, unlike in-sample VaR results, the
out-of-sample VaR under a normal distribution outperforms the
symmetric Student-t FIAPARCH. This is because the Student-t distribution is relatively conservative. As stated above the Student-t
can capture excess kurtosis but not excess skewness.
In untabulated results, we re-estimate in-sample and out-ofsample VaR for short and long trading positions using RiskMetrics,
GARCH (1,1), FIGARCH (1,d,1) as well as several others volatility
models, in particular the EGARCH, PGARCH, GARCH-M, GJRGARCH, V-GARCH, FIEGARCH, and also bilinearity models. For
each volatility model we run three specifications based on
the three alternative error distributions: Normal, Student-t and
skewed Student-t. Our results show that skewed Student-t AR (1)FIAPARCH (1,d,1) continues to outperform all the studied volatility
models.6 We also find that RiskMetrics provides the least accurate
VaR estimates. This is likely due to the failure of RiskMetrics to
account for the main stylized facts of asset returns. We again repeat
the same analysis using AR (k), where k = 2, 3, 4, 5, 10, 20 and 30,
and this for all the returns indices and for all the volatility models.
Our findings indicate that the skewed Student-t AR (1)-FIAPARCH
(1,d,1) continues to provide the most accurate VaR estimates than
the remaining models and this regardless of the order of autocorrelation (k) in the mean equation.7 It is noteworthy that the accuracy
of VaR estimates is not sensitive to the mean process specification.
To summarize, our findings provide strong and reliable evidence that FIAPARCH model under skewed Student-t distribution
provides the most accurate VaR estimates. This can be explained
by the fact that this model takes simultaneously into account
the salient features of financial time series: fat tails, asymmetry,
volatility clustering and long memory. Our findings, hence, corroborate the calls for the use of more realistic assumptions in
6
5
All untabultated results are available upon request.
7
The results are available upon request.
The results are available upon request.
S. Mabrouk, S. Saadi / The Quarterly Review of Economics and Finance 52 (2012) 305–321
317
Table 10
In-sample VaR estimation results.
Panel a
CAC 40
Short trading position
Normal
Student-t
skSt
Long trading position
Quantile
Failure rate
Kupiec LRT
P-value
DQT
P-value
Quantile
Failure rate
Kupiec LRT
P-value
DQT
P-value
0.95
0.975
0.99
0.995
0.9975
0.95
0.975
0.99
0.995
0.9975
0.95
0.975
0.99
0.995
0.9975
0.959
0.979
0.991
0.993
0.997
0.958
0.981
0.993
0.997
0.999
0.951
0.976
0.990
0.995
0.997
9.241
4.095
0.510
1.722
0.428
7.984
8.620
6.102
4.427
6.819
0.157
0.315
0.317
0.254
0.140
0.002
0.042
0.474
0.189
0.512
0.004
0.003
0.013
0.035
0.009
0.691
0.574
0.573
0.614
0.707
16.03
13.88
35.15
109.3
113.6
13.05
19.42
57.09
60.51
5.194
3.421
6.663
33.26
93.35
29.33
0.024
0.053
0.000
0.000
0.000
0.070
0.006
0.000
0.000
0.636
0.843
0.464
0.000
0.000
0.000
0.05
0.025
0.01
0.005
0.0025
0.05
0.025
0.01
0.005
0.0025
0.05
0.025
0.01
0.005
0.0025
0.051
0.030
0.015
0.008
0.005
0.056
0.030
0.010
0.005
0.003
0.051
0.025
0.008
0.004
0.002
0.224
5.984
11.792
10.870
9.929
4.181
5.156
0.373
0.282
1.423
0.116
0.009
1.385
0.084
0.140
0.635
0.014
0.000
0.000
0.001
0.040
0.0231
0.541
0.595
0.232
0.733
0.921
0.239
0.771
0.707
5.094
10.28
22.72
19.52
27.74
7.027
10.16
4.899
6.884
3.023
2.914
3.121
4.955
8.273
0.390
0.648
0.172
0.001
0.006
0.000
0.425
0.179
0.672
0.440
0.882
0.892
0.873
0.665
0.309
0.999
Panel b
DAX
Short trading position
Normal
Student-t
skSt
Long trading position
Quantile
Failure rate
Kupiec LRT
P-value
DQT
P-value
Quantile
Failure rate
Kupiec LRT
P-value
DQT
P-value
0.95
0.975
0.99
0.995
0.9975
0.95
0.975
0.99
0.995
0.9975
0.95
0.975
0.99
0.995
0.9975
0.960
0.981
0.993
0.995
0.997
0.959
0.983
0.995
0.998
0.998
0.950
0.976
0.992
0.996
0.998
10.571
8.079
5.709
0.574
0.005
8.325
15.306
14.611
10.535
4.393
0.039
0.517
4.206
1.471
1.857
0.001
0.004
0.016
0.448
0.941
0.003
0.000
0.000
0.001
0.036
0.842
0.471
0.040
0.225
0.172
13.56
12.16
8.567
11.54
40.16
13.30
16.42
16.61
30.94
87.25
3.280
2.461
6.663
14.55
60.34
0.059
0.095
0.285
0.116
0.000
0.065
0.021
0.020
0.000
0.000
0.857
0.929
0.464
0.042
0.000
0.05
0.025
0.01
0.005
0.0025
0.05
0.025
0.01
0.005
0.0025
0.050
0.025
0.010
0.005
0.0025
0.054
0.031
0.015
0.008
0.0057
0.060
0.031
0.013
0.005
0.002
0.053
0.025
0.009
0.004
0.002
2.024
7.849
12.930
9.979
14.124
9.760
7.849
4.019
0.524
0.261
0.912
0.021
0.089
0.287
0.143
0.154
0.005
0.000
0.001
0.000
0.001
0.005
0.044
0.469
0.609
0.339
0.882
0.765
0.591
0.704
15.20
18.09
23.12
16.70
21.13
29.75
28.29
16.24
2.204
0.673
9.804
9.809
5.615
0.952
0.246
0.033
0.011
0.001
0.019
0.003
0.000
0.000
0.022
0.947
0.998
0.199
0.199
0.585
0.995
0.999
Panel c
Dow Jones
Short trading position
Normal
Student-t
skSt
Long trading position
Quantile
Failure rate
Kupiec LRT
P-value
DQT
P-value
Quantile
Failure rate
Kupiec LRT
P-value
DQT
P-value
0.95
0.975
0.99
0.995
0.9975
0.95
0.975
0.99
0.995
0.9975
0.95
0.975
0.99
0.9950
0.9975
0.956
0.979
0.988
0.995
0.997
0.954
0.980
0.994
0.998
0.999
0.947
0.976
0.990
0.996
0.998
4.088
3.432
0.905
0.019
0.374
1.773
6.508
10.446
11.980
12.579
0.773
0.355
0.038
3.671
2.322
0.043
0.063
0.341
0.889
0.540
0.182
0.010
0.001
0.000
0.000
0.379
0.550
0.844
0.055
0.127
7.305
7.668
5.660
7.632
27.13
6.007
11.96
12.06
9.388
8.269
3.961
5.154
2.848
15.60
2.262
0.397
0.362
0.579
0.366
0.000
0.538
0.101
0.098
0.225
0.309
0.784
0.641
0.898
0.029
0.943
0.05
0.025
0.01
0.005
0.0025
0.05
0.025
0.01
0.005
0.0025
0.050
0.025
0.010
0.005
0.0025
0.048
0.027
0.014
0.009
0.007
0.053
0.025
0.011
0.007
0.004
0.049
0.023
0.009
0.006
0.003
0.145
0.786
9.549
17.934
29.684
1.020
0.185
1.5116
5.562
4.671
0.012
0.618
0.414
1.567
2.773
0.702
0.375
0.002
0.000
0.000
0.312
0.667
0.218
0.0183
0.030
0.909
0.431
0.519
0.210
0.095
9.039
3.828
19.48
29.39
49.87
13.78
1.409
5.102
8.653
6.478
11.42
2.700
2.492
3.040
4.293
0.249
0.799
0.006
0.000
0.000
0.055
0.985
0.647
0.278
0.485
0.121
0.911
0.927
0.881
0.745
Panel d
FTSE 100
Short trading position
Normal
Studentt
Long trading position
Quantile
Failure rate
Kupiec LRT
P-value
DQT
P-value
Quantile
Failure rate
Kupiec LRT
P-value
DQT
P-value
0.95
0.975
0.99
0.995
0.9975
0.95
0.975
0.99
0.959
0.979
0.990
0.994
0.995
0.957
0.980
0.993
10.227
4.140
0.233
0.230
7.001
5.191
6.428
8.273
0.001
0.041
0.628
0.631
0.008
0.022
0.011
0.004
12.13
10.37
18.86
34.92
41.11
8.009
14.75
23.96
0.096
0.16
0.008
0.000
0.000
0.331
0.039
0.001
0.05
0.025
0.01
0.005
0.0025
0.05
0.025
0.01
0.050
0.031
0.013
0.009
0.005
0.053
0.030
0.011
0.059
8.110
6.013
14.04
14.302
1.068
4.633
0.675
0.808
0.004
0.014
0.000
0.000
0.301
0.031
0.411
3.126
13.07
14.02
23.99
33.03
5.630
10.46
3.905
0.873
0.070
0.050
0.001
0.000
0.583
0.163
0.790
318
S. Mabrouk, S. Saadi / The Quarterly Review of Economics and Finance 52 (2012) 305–321
Table 10 (Continued)
Panel d
FTSE 100
Short trading position
skSt
Long trading position
Quantile
Failure rate
Kupiec LRT
P-value
0.995
0.9975
0.95
0.975
0.99
0.995
0.9975
0.995
0.998
0.952
0.977
0.990
0.995
0.997
0.306
0.734
0.808
1.564
0.233
0.017
0.114
0.579
0.391
0.368
0.211
0.628
0.895
0.734
DQT
9.757
43.95
3.131
7.131
23.64
15.72
29.04
P-value
Quantile
Failure rate
0.202
0.000
0.872
0.415
0.001
0.027
0.000
0.005
0.0025
0.05
0.025
0.01
0.005
0.0025
0.005
0.003
0.050
0.024
0.009
0.003
0.002
Kupiec LRT
0.764
0.791
0.012
0.000
0.034
1.471
0.114
P-value
0.381
0.373
0.912
0.986
0.852
0.225
0.734
DQT
7.428
1.357
3.259
2.936
2.987
1.656
1.977
P-value
0.385
0.986
0.860
0.890
0.886
0.976
0.961
Panel e
NASDAQ
Short trading position
Normal
Student-t
skSt
Long trading position
Quantile
Failure rate
Kupiec LRT
P-value
DQT
P-value
Quantile
Failure rate
Kupiec LRT
P-value
DQT
P-value
0.95
0.975
0.99
0.995
0.9975
0.95
0.975
0.99
0.995
0.9975
0.95
0.975
0.99
0.995
0.9975
0.962
0.981
0.992
0.996
0.997
0.961
0.983
0.994
0.998
0.999
0.950
0.973
0.991
0.995
0.997
16.881
7.952
4.119
2.768
0.000
14.024
14.095
14.031
11.881
12.512
0.016
0.218
0.596
0.296
0.288
0.000
0.004
0.042
0.096
0.814
0.000
0.000
0.000
0.000
0.000
0.898
0.640
0.439
0.585
0.590
20.90
20.85
8.329
7.933
1.250
19.61
25.33
15.96
9.109
8.196
5.320
8.582
9.358
2.838
0.510
0.003
0.003
0.304
0.338
0.989
0.006
0.000
0.025
0.244
0.315
0.620
0.284
0.227
0.899
0.999
0.05
0.025
0.01
0.005
0.0025
0.05
0.025
0.01
0.005
0.0025
0.050
0.025
0.010
0.005
0.0025
0.054
0.031
0.015
0.012
0.009
0.059
0.032
0.014
0.009
0.005
0.050
0.023
0.011
0.006
0.003
2.117
7.274
14.054
41.385
51.664
7.850
9.781
8.166
12.89
9.719
0.005
0.562
1.571
1.161
1.351
0.145
0.006
0.000
0.000
0.000
0.005
0.001
0.004
0.000
0.001
0.941
0.453
0.210
0.281
0.245
9.500
17.64
21.50
73.23
96.40
17.54
27.02
14.16
23.31
18.40
7.648
9.492
9.132
4.097
3.517
0.218
0.013
0.003
0.000
0.000
0.014
0.000
0.048
0.001
0.010
0.364
0.219
0.243
0.768
0.833
Panel f
Nikkei 225
Short trading position
Normal
Student-t
skSt
Long trading position
Quantile
Failure rate
Kupiec LRT
P-value
DQT
P-value
Quantile
Failure rate
Kupiec LRT
P-value
DQT
P-value
0.95
0.975
0.99
0.995
0.9975
0.95
0.975
0.99
0.995
0.9975
0.95
0.975
0.99
0.9950
0.9975
0.954
0.975
0.988
0.990
0.994
0.952
0.977
0.989
0.993
0.996
0.952
0.977
0.989
0.993
0.996
1.758
0.157
1.646
12.604
11.825
0.829
1.148
0.080
1.006
1.558
0.709
0.949
0.080
1.006
1.558
0.184
0.691
0.199
0.385
0.584
0.362
0.283
0.776
0.315
0.211
0.399
0.329
0.776
0.315
0.211
4.398
8.605
13.02
33.68
29.38
5.348
11.97
15.20
6.886
4.942
4.652
11.37
15.20
6.886
4.943
0.732
0.282
0.071
0.000
0.000
0.617
0.101
0.033
0.440
0.667
0.702
0.122
0.033
0.440
0.666
0.05
0.025
0.01
0.005
0.0025
0.05
0.025
0.01
0.005
0.0025
0.050
0.025
0.010
0.005
0.0025
0.049
0.025
0.011
0.006
0.003
0.050
0.024
0.008
0.003
0.001
0.050
0.024
0.008
0.003
0.001
0.086
0.070
0.740
1.434
2.279
0.031
0.090
0.845
1.196
2.068
0.031
0.090
1.156
1.196
2.068
0.768
0.790
0.389
0.231
0.131
0.858
0.763
0.357
0.274
0.150
0.858
0.763
0.282
0.274
0.150
2.319
3.454
3.708
7.857
3.068
6.044
2.909
4.889
1.470
3.119
6.044
2.908
4.997
1.470
3.120
0.940
0.840
0.812
0.345
0.878
0.534
0.893
0.673
0.983
0.873
0.534
0.893
0.660
0.983
0.873
Panel g
S&P 500
Short trading position
Normal
Student-t
skSt
Long trading position
Quantile
Failure rate
Kupiec LRT
P-value
DQT
P-value
Quantile
Failure rate
Kupiec LRT
P-value
DQT
P-value
0.95
0.975
0.99
0.995
0.9975
0.95
0.975
0.99
0.995
0.9975
0.95
0.975
0.99
0.995
0.9975
0.958
0.977
0.990
0.995
0.997
0.953
0.980
0.994
0.997
0.999
0.947
0.974
0.991
0.997
0.999
7.554
1.295
0.031
0.299
0.003
1.319
5.851
10.325
10.066
12.521
0.440
0.034
1.940
5.720
6.958
0.005
0.255
0.86
0.584
0.953
0.250
0.015
0.001
0.001
0.000
0.507
0.852
0.163
0.016
0.008
6.848
8.668
6.774
31.68
29.32
4.843
9.895
28.09
24.21
5.388
9.631
7.823
11.67
15.46
68.81
0.444
0.277
0.452
0.000
0.000
0.679
0.194
0.000
0.001
0.612
0.210
0.348
0.111
0.030
0.000
0.05
0.025
0.01
0.005
0.0025
0.05
0.025
0.01
0.005
0.0025
0.05
0.025
0.01
0.005
0.0025
0.051
0.031
0.017
0.011
0.007
0.058
0.032
0.013
0.007
0.003
0.054
0.029
0.010
0.005
0.003
0.357
8.2082
22.373
27.143
34.304
6.456
9.751
5.435
4.814
2.810
2.098
3.300
0.164
0.777
1.347
0.550
0.004
0.000
0.000
0.000
0.011
0.001
0.019
0.028
0.093
0.147
0.069
0.684
0.377
0.245
11.37
14.66
24.20
23.95
45.37
13.14
14.61
8.605
8.679
25.05
9.166
11.35
4.176
7.140
26.32
0.123
0.040
0.001
0.001
0.000
0.068
0.041
0.282
0.276
0.000
0.240
0.124
0.759
0.414
0.000
S. Mabrouk, S. Saadi / The Quarterly Review of Economics and Finance 52 (2012) 305–321
319
Table 11
Out-of-sample VaR estimation results.
Panel a
CAC 40
Short trading position
Normal
Student-t
skSt
Long trading position
Quantile
Failure rate
Kupiec LRT
P-value
DQT
P-value
Quantile
Failure rate
Kupiec LRT
P-value
DQT
P-value
0.95
0.975
0.99
0.995
0.9975
0.95
0.975
0.99
0.995
0.9975
0.95
0.975
0.99
0.995
0.9975
0.963
0.987
0.997
0.999
0.999
0.955
0.982
0.995
0.998
0.999
0.965
0.987
0.993
0.998
0.998
3.895
7.145
6.825
4.797
1.169
0.543
2.224
3.093
2.343
1.169
5.268
7.145
1.015
2.343
0.107
0.048
0.007
0.008
0.028
0.279
0.460
0.135
0.078
0.125
0.279
0.021
0.007
0.313
0.125
0.742
7.072
10.31
5.285
3.313
1.091
7.531
6.375
2.803
1.871
1.103
9.717
10.00
2.372
1.823
0.128
0.421
0.171
0.625
0.854
0.993
0.375
0.496
0.902
0.966
0.992
0.205
0.188
0.936
0.968
0.999
0.05
0.025
0.01
0.005
0.0025
0.05
0.025
0.01
0.005
0.0025
0.05
0.025
0.01
0.005
0.0025
0.066
0.038
0.019
0.010
0.007
0.059
0.032
0.013
0.008
0.004
0.066
0.040
0.023
0.017
0.013
4.918
5.996
6.472
3.888
5.435
1.616
1.849
0.830
1.529
0.762
4.918
7.832
12.485
17.754
21.976
0.026
0.014
0.010
0.048
0.0197
0.203
0.173
0.362
0.216
0.382
0.026
0.005
0.000
0.000
0.000
18.23
21.98
18.97
20.85
33.29
12.79
14.09
11.29
16.30
3.132
15.16
29.45
33.65
52.35
88.70
0.010
0.002
0.008
0.003
0.000
0.077
0.049
0.126
0.022
0.872
0.033
0.000
0.000
0.000
0.000
Panel b
DAX
Short trading position
Normal
Student-t
skSt
Long trading position
Quantile
Failure rate
Kupiec LRT
P-value
DQT
P-value
Quantile
Failure rate
Kupiec LRT
P-value
DQT
P-value
0.95
0.975
0.99
0.995
0.9975
0.95
0.975
0.99
0.995
0.9975
0.95
0.975
0.99
0.995
0.9975
0.968
0.988
0.998
0.999
0.999
0.968
0.990
0.999
0.999
1.000
0.965
0.987
0.998
0.999
1.000
7.776
8.557
9.626
4.797
1.169
7.776
11.904
13.476
4.797
NaN
5.268
7.145
9.626
4.797
NaN
0.005
0.003
0.001
0.028
0.279
0.005
0.000
0.000
0.028
1.000
0.021
0.007
0.001
0.028
1.000
10.54
7.470
6.694
43.21
80.70
10.64
10.21
28.28
43.21
2.506
10.21
6.448
6.694
43.21
2.506
0.159
0.381
0.461
0.000
0.000
0.154
0.176
0.000
0.000
0.926
0.176
0.488
0.461
0.000
0.926
0.05
0.025
0.01
0.005
0.0025
0.05
0.025
0.01
0.005
0.0025
0.05
0.025
0.01
0.005
0.0025
0.057
0.031
0.021
0.014
0.008
0.060
0.030
0.019
0.008
0.004
0.056
0.029
0.014
0.006
0.004
0.988
1.373
9.284
10.911
7.640
1.984
0.964
6.472
1.529
0.762
0.730
0.624
1.437
0.188
0.762
0.320
0.241
0.002
0.000
0.005
0.158
0.325
0.010
0.216
0.382
0.392
0.429
0.230
0.663
0.382
8.071
12.93
41.59
60.19
58.26
8.375
12.85
38.91
25.03
6.793
7.345
11.62
23.84
3.366
6.889
0.326
0.073
0.000
0.000
0.000
0.300
0.075
0.000
0.000
0.450
0.393
0.113
0.001
0.849
0.440
Panel c
Dow Jones
Short trading position
Normal
Student-t
skSt
Long trading position
Quantile
Failure rate
Kupiec LRT
P-value
DQT
P-value
Quantile
Failure rate
Kupiec LRT
P-value
DQT
P-value
0.95
0.975
0.99
0.995
0.9975
0.95
0.975
0.99
0.995
0.9975
0.95
0.975
0.99
0.9950
0.9975
0.961
0.986
0.992
0.996
0.999
0.958
0.988
0.995
0.999
1.000
0.954
0.985
0.992
0.999
0.999
2.746
5.888
0.433
0.215
1.169
1.421
8.557
3.093
4.797
NaN
0.345
4.777
0.433
4.797
1.169
0.097
0.015
0.510
0.642
0.279
0.233
0.003
0.078
0.028
1.000
0.556
0.028
0.510
0.028
0.279
7.206
7.426
12.32
0.289
1.046
6.506
9.757
2.592
3.286
2.506
6.103
7.905
12.27
3.287
1.044
0.407
0.385
0.090
0.999
0.994
0.481
0.202
0.919
0.857
0.926
0.527
0.341
0.091
0.857
0.994
0.05
0.025
0.01
0.005
0.0025
0.05
0.025
0.01
0.005
0.0025
0.050
0.025
0.010
0.005
0.0025
0.061
0.036
0.026
0.016
0.009
0.063
0.033
0.017
0.009
0.003
0.061
0.03
0.013
0.009
0.002
2.387
4.378
17.947
15.343
10.099
3.298
2.389
4.091
2.596
0.094
2.387
0.964
0.830
2.596
0.107
0.122
0.036
0.000
0.000
0.001
0.069
0.122
0.043
0.107
0.758
0.122
0.325
0.362
0.107
0.742
17.39
24.76
53.59
36.21
20.74
24.17
24.92
24.48
5.586
2.326
22.22
17.81
2.480
5.740
0.729
0.015
0.000
0.000
0.000
0.004
0.001
0.000
0.000
0.588
0.939
0.002
0.012
0.928
0.570
0.998
Panel d
FTSE 100
Short trading position
Normal
Student-t
Long trading position
Quantile
Failure rate
Kupiec LRT
P-value
DQT
P-value
Quantile
Failure rate
Kupiec LRT
P-value
DQT
P-value
0.95
0.975
0.99
0.995
0.9975
0.95
0.975
0.99
0.995
0.968
0.988
0.998
0.999
0.999
0.968
0.990
0.999
0.999
7.776
8.557
9.626
4.797
1.169
7.776
11.904
13.476
4.797
0.005
0.003
0.001
0.028
0.279
0.005
0.000
0.000
0.028
10.54
7.470
6.694
43.21
80.70
10.64
10.21
28.28
43.21
0.159
0.381
0.461
0.000
0.000
0.154
0.176
0.000
0.000
0.05
0.025
0.01
0.005
0.0025
0.05
0.025
0.01
0.005
0.057
0.031
0.021
0.014
0.008
0.060
0.030
0.019
0.008
0.988
1.373
9.284
10.911
7.640
1.984
0.964
6.472
1.529
0.320
0.241
0.002
0.000
0.005
0.158
0.325
0.010
0.216
8.071
12.93
41.59
60.19
58.26
8.375
12.85
38.91
25.03
0.326
0.073
0.000
0.000
0.000
0.300
0.075
0.000
0.000
320
S. Mabrouk, S. Saadi / The Quarterly Review of Economics and Finance 52 (2012) 305–321
Table 11 (Continued)
Panel d
FTSE 100
Short trading position
skSt
Quantile
Failure rate
0.9975
0.95
0.975
0.99
0.995
0.9975
1.000
0.965
0.987
0.998
0.999
1.000
Long trading position
Kupiec LRT
NaN
5.268
7.145
9.626
4.797
NaN
P-value
DQT
P-value
Quantile
Failure rate
1.000
0.021
0.007
0.001
0.028
1.000
2.506
10.21
6.448
6.694
43.21
2.506
0.926
0.176
0.488
0.461
0.000
0.926
0.0025
0.05
0.025
0.01
0.005
0.0025
0.004
0.056
0.029
0.014
0.006
0.004
Kupiec LRT
0.762
0.730
0.624
1.437
0.188
0.762
P-value
DQT
P-value
0.382
0.392
0.429
0.230
0.663
0.382
6.793
7.345
11.62
23.84
3.366
6.889
0.450
0.393
0.113
0.001
0.849
0.440
Panel e
NASDAQ
Short trading position
Normal
Student-t
skSt
Long trading position
Quantile
Failure rate
Kupiec LRT
P-value
DQT
P-value
Quantile
Failure rate
Kupiec LRT
P-value
DQT
P-value
0.95
0.975
0.99
0.995
0.9975
0.95
0.975
0.99
0.995
0.9975
0.95
0.975
0.99
0.995
0.9975
0.971
0.983
0.993
0.995
0.997
0.968
0.985
0.994
0.997
0.999
0.951
0.977
0.990
0.994
0.997
10.867
2.952
1.015
0.000
0.094
7.776
4.777
1.886
0.939
1.169
0.021
0.168
0.000
0.188
0.094
0.000
0.085
0.313
1.000
0.758
0.005
0.028
0.169
0.332
0.279
0.884
0.681
1.000
0.663
0.758
11.32
3.545
1.017
0.416
1.465
9.606
4.663
1.786
1.590
3.104
7.655
2.601
0.420
0.523
1.608
0.125
0.830
0.994
0.999
0.983
0.212
0.700
0.970
0.979
0.875
0.363
0.919
0.999
0.999
0.978
0.05
0.025
0.01
0.005
0.0025
0.05
0.025
0.01
0.005
0.0025
0.050
0.025
0.010
0.005
0.0025
0.057
0.030
0.016
0.015
0.010
0.070
0.033
0.015
0.009
0.006
0.055
0.023
0.014
0.006
0.003
0.988
0.964
3.076
13.059
12.782
7.5302
2.389
2.189
2.596
3.517
0.510
0.168
1.437
0.188
0.094
0.320
0.325
0.079
0.000
0.000
0.006
0.122
0.138
0.107
0.060
0.474
0.681
0.230
0.663
0.758
2.349
13.34
28.39
72.30
60.84
10.57
13.05
28.38
12.30
5.832
3.360
14.81
28.54
0.661
0.299
0.938
0.064
0.000
0.000
0.000
0.158
0.070
0.000
0.090
0.559
0.849
0.038
0.000
0.998
0.999
Panel f
Nikkei 225
Short trading position
Normal
Student-t
skSt
Long trading position
Quantile
Failure rate
Kupiec LRT
P-value
DQT
P-value
Quantile
Failure rate
Kupiec LRT
P-value
DQT
P-value
0.95
0.975
0.99
0.995
0.9975
0.95
0.975
0.99
0.995
0.9975
0.95
0.975
0.99
0.9950
0.9975
0.955
0.980
0.993
0.996
0.998
0.948
0.980
0.995
0.997
0.999
0.949
0.981
0.995
0.998
0.999
0.543
1.099
1.015
0.215
0.107
0.0831
1.099
3.093
0.939
1.169
0.020
1.608
3.093
2.343
1.169
0.460
0.294
0.313
0.642
0.742
0.773
0.294
0.078
0.332
0.279
0.884
0.204
0.078
0.125
0.279
4.385
3.938
4.091
2.795
2.282
7.297
4.586
4.237
2.773
1.660
6.611
4.717
4.222
2.883
1.665
0.734
0.786
0.769
0.903
0.942
0.398
0.710
0.752
0.905
0.976
0.470
0.694
0.753
0.895
0.976
0.05
0.025
0.01
0.005
0.0025
0.05
0.025
0.01
0.005
0.0025
0.050
0.025
0.010
0.005
0.0025
0.043
0.022
0.014
0.006
0.003
0.048
0.023
0.009
0.003
0.001
0.048
0.024
0.010
0.003
0.001
1.080
0.384
1.437
0.188
0.094
0.085
0.168
0.104
0.939
1.169
0.085
0.041
0.000
0.939
1.169
0.298
0.535
0.230
0.663
0.758
0.770
0.681
0.746
0.332
0.279
0.770
0.838
1.000
0.332
0.279
11.56
3.960
8.344
1.318
7.076
13.63
3.050
11.26
3.955
1.619
13.63
2.750
9.692
3.994
1.614
0.115
0.784
0.303
0.987
0.421
0.058
0.880
0.127
0.784
0.977
0.058
0.907
0.206
0.780
0.978
Panel g
S&P 500
Short trading position
Normal
Student-t
skSt
Long trading position
Quantile
Failure rate
Kupiec LRT
P-value
DQT
P-value
Quantile
Failure rate
Kupiec LRT
P-value
DQT
P-value
0.95
0.975
0.99
0.995
0.9975
0.95
0.975
0.99
0.995
0.9975
0.95
0.975
0.99
0.995
0.9975
0.965
0.986
0.994
0.998
0.999
0.961
0.986
0.997
0.999
1.000
0.954
0.985
0.996
0.999
1.000
5.268
5.888
1.886
2.343
1.169
2.746
5.888
6.825
4.797
NaN
0.345
4.777
4.706
4.797
NaN
0.021
0.015
0.169
0.125
0.279
0.097
0.015
0.008
0.028
1.000
0.556
0.028
0.030
0.028
1.000
8.170
7.684
1.882
1.882
0.904
9.293
7.714
4.992
3.222
2.506
7.969
6.584
3.838
3.222
2.506
0.317
0.361
0.966
0.966
0.996
0.232
0.358
0.660
0.863
0.926
0.335
0.473
0.798
0.863
0.926
0.05
0.025
0.01
0.005
0.0025
0.05
0.025
0.01
0.005
0.0025
0.05
0.025
0.01
0.005
0.0025
0.063
0.036
0.021
0.014
0.006
0.066
0.035
0.013
0.007
0.003
0.065
0.031
0.011
0.006
0.003
3.298
4.378
9.284
10.911
3.517
4.918
3.656
0.830
0.714
0.941
4.345
1.373
0.097
0.188
0.094
0.069
0.036
0.002
0.000
0.060
0.026
0.055
0.362
0.397
0.758
0.037
0.241
0.754
0.663
0.758
26.11
44.34
27.69
65.36
7.993
32.56
40.96
7.257
2.620
1.861
31.37
21.97
0.947
1.693
1.966
0.000
0.000
0.000
0.000
0.333
0.000
0.000
0.402
0.917
0.967
0.000
0.002
0.995
0.974
0.961
S. Mabrouk, S. Saadi / The Quarterly Review of Economics and Finance 52 (2012) 305–321
financial modeling (e.g. Roldán, 2009; Stiglitz, 2010; Willett, 2009).
In the context of VaR, our results suggest that the use of realistic
assumptions can help investors and risk managers further reduce
the uncertainty associated with the maximum loss to incur.
6. Conclusion
We estimate the one-day-ahead VaR for short and long trading
position and this for seven stock indices (Dow Jones, Nasdaq100,
S&P 500, DAX30, CAC40, FTSE100 and Nikkei 225). Since volatility is a key input to estimate VaR, the use of volatility models that
take into account the properties of asset returns is crucial for the
accuracy of VaR estimation. Therefore we carry out a thorough analysis in order to choose the appropriate volatility model. Because all
returns series exhibit volatility clustering and long range memory, we examine several GARCH-type models including fractionary
integrated models. Moreover, and with the exception of RiskMetrics which can only be assed under the normal distribution, we
estimate each model under three alternative distributions: Normal, Student-t and skewed Student-t. Consistent with the idea that
the accuracy of VaR estimates is sensitive to the adequacy of the
volatility model used, we find that AR (1)-FIAPARCH (1,d,1) model,
under a skewed Student-t distribution, outperforms all the models that we have considered including widely used ones such as
GARCH (1,1) or HYGARCH (1,d,1). The superior performance of
the skewed Student-t FIAPARCH model holds for all stock market
indices, and for both long and short trading positions. Our findings
can be explained by the fact that the skewed Student-t FIAPARCH
model can jointly accounts for the salient features of financial time
series: fat tails, asymmetry, volatility clustering and long memory.
In the same vein, because it fails to account for most of these stylized facts, the RiskMetrics model which can be only assessed under
the normal distribution provides the least accurate VaR estimations. Our findings are of interest to investors and risk managers
who use forecast values to eliminate the uncertainty associated
with the maximum loss to incur. Taken together, our result corroborates the calls for the use of more realistic assumptions in
financial modeling. Our results corroborate the calls for the use
of more realistic assumptions in financial modeling.
Appendix A. Supplementary data
Supplementary data associated with this article can be found, in
the online version, at http://dx.doi.org/10.1016/j.qref.2012.04.006.
References
Angelidis, T., Benos, A., & Degiannakis, S. (2004). The use of GARCH models in VaR
estimation. Statistical Methodology, 2, 105–128.
Baillie, R. T., Bollerslev, T., & Mikkelsen, H. O. (1996). Fractionally integrated generalized autoregressive conditional heteroskedasticity. Journal of Econometrics,
73, 3–20.
Bali, T. G., & Theodossiou, P. (2007). A conditional-SGT-VaR approach with alternative GARCH models. Annals of Operations Research, 151, 241–267.
Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticty.
Journal of Econometrics, 31, 307–327.
321
Bollerslev, T., & Jubinski, D. (1999). Equity trading volume and volatility: Latent
information arrivals and common long-run dependencies. Journal of Business
and Economic Statistics, 17, 9–21.
Breidt, F. J., Crato, N., & de Lima, P. (1998). The detection and estimation of long
memory in stochastic volatility. Journal of Econometrics, 83, 325–348.
Colander, D., Föllmer, H., Haas, A., Goldberg, M. D., Juselius, K., Kirman, A., et al.
(2009). The financial crisis and the systemic failure of academic economics.
University of Copenhagen, Working Paper.
Davidson, J. (2004). Moment and memory properties of linear conditional heteroscedasticity models, and a new model. Journal of Business and Economic
Statistics, 22, 16–29.
Dickey, D., & Fuller, W. (1979). Distribution of the estimators for autoregressive time
series with unit root. Journal of American Statistic Association, 74, 427–431.
Ding, Z., & Granger, C. W. J. (1996). Modeling volatility persistence of speculative
returns: A new approach. Journal of Econometrics, 73, 185–215.
Eichengreen, B. (2008). Origins and responses to the crisis. University of California,
Berkeley, Working Paper.
Engle, R. F. (1982). Autoregressive conditional heteroskedasticity with estimates of
the variance of U.K. inflation. Econometrica, 50, 987–1008.
Engle, R. F., & Manganelli, S. (2004). CAViaR: Conditional autoregressive value
at risk by regression quantiles. Journal of Business & Economic Statistics, 22,
367–381.
Geweke, P. H. (1983). The estimation and application of long-memory time series
models. Journal of Time Series Analysis, 4, 221–238.
Giot, P., & Laurent, S. (2003). Market risk in commodity markets: A VaR approach.
Energy Economics, 25, 435–457.
Hansen, P. R., & Lunde, A. (2005). A forecast comparison of volatility models: Does
anything beat a GARCH(1,1)? Journal of Applied Econometrics, 20, 873–889.
Kupiec, P. (1995). Technique for verifying the accuracy of risk measurement models.
Journal of Derivatives, 2, 173–184.
Kwiatkowski, D., Phillips, P. C. W., Schmidt, P., & Shin, Y. (1992). Testing the null
hypothesis of stationarity against the alternative of unit root. Journal of Econometrics, 54, 159–178.
Lambert, P., & Laurent, S. (2001). Modelling financial time series using GARCH-type
models and a skewed Student-t density. Mimeo: Université de Liège.
Leamer, E. E. (1985). Sensitivity analyses would help. American Economic Review, 85
Leamer, E. E. (1986). Schwarz regression and sensitivity analysis. In D. E. Herbert, &
R. H. Myers (Eds.), Multiple regression analysis: Applications in the health sciences.
American Institute of Physics: New York.
Lo, A. W., & Mackinlay, A. C. (1990). En econometric analysis of non-synchronous
trading. Journal of Econometrics, 45, 181–211.
Mabrouk, S., & Aloui, C. (2010). One-day-ahead value-at-risk estimations with dual
long-memory models: Evidence from the Tunisian stock market. International
Journal of Financial Services Management, 4, 77–94.
Pagan, A. R., & Schwert, G. W. (1990). Alternative models for conditional stock
volatility. Journal of Econometrics, 45, 267–290.
Phillips, P. C. B., & Perron, P. (1988). Testing for a unit root in time series regression.
Biometrika, 75, 335–346.
Robinson, P. M., & Henry, M. (1999). Long and short memory conditional heteroskedasticity in estimating the memory parameter of levels. Econometric
Theory, 15, 299–336.
Roldán, J. M. (2009). Is it implementing Basel II or do we need Basel III? In BBA Annual
International Banking Conference London.
So, M. K. P., & Yu, P. L. H. (2006). Empirical analysis of GARCH models in value-at-risk
estimation. Journal of International Financial Markets, Institutions and Money, 16,
180–197.
Stiglitz, J. E. (2009). The current economic crisis and lessons for economic theory.
Eastern Economic Journal, 35, 281–296.
Stiglitz, J. E. (2010). Homoeconomicus: The impact of the economic crisis on economic theory. In American Economic Association Annual Meeting Atlanta, Georgia.
Tang, T. L., & Shieh, S. J. (2006). Long memory in stock index future markets: A
value-at-risk approach. Physica A, 366, 437–448.
Tse, Y. K. (1998). The conditional heteroscedasticity of the Yen–Dollar exchange rate.
Journal of Applied Econometrics, 13, 49–55.
Willett, T. D. (2009). How bad mental models, perverse incentives, and lazy markets interacted in generating the current financial crisis. Claremont Institute for
Economic Policy Studies, Working Paper.
Wu, P. T., & Shieh, S. J. (2007). Value-at-risk analysis for long-term interest rate
futures: Fat-tail and long memory in return innovations. Journal of Empirical
Finance, 14, 248–259.
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