Empirical distributions of daily equity index returns: A comparison


Expert Systems With Applications 54 (2016) 170–192

Contents lists available at ScienceDirect

Expert Systems With Applications

journal homepage: www.elsevier.com/locate/eswa

Empirical distributions of daily equity index returns: A comparison

Canan G. Corlu a, Melike Meterelliyoz b,∗, Murat Tiniç c

a Metropolitan College, Boston University, Boston, MA 02215 USA
b Department of Business Administration, TOBB University of Economics and Technology, Ankara 06560 Turkey
c Department of Management, Bilkent University, Ankara 06800 Turkey

a r t i c l e i n f o

Keywords:

Index returns

Generalized lambda

Johnson translation system

Skewed-t

Normal inverse Gaussian

g-and-h

a b s t r a c t

The normality assumption concerning the distribution of equity returns has long been challenged both

empirically and theoretically. Alternative distributions have been proposed to better capture the char-

acteristics of equity return data. This paper investigates the ability of five alternative distributions to

represent the behavior of daily equity index returns over the period 1979–2014: the skewed Student-t

distribution, the generalized lambda distribution, the Johnson system of distributions, the normal inverse

Gaussian distribution, and the g-and-h distribution. We find that the generalized lambda distribution is a

prominent alternative for modeling the behavior of daily equity index returns.

© 2016 Elsevier Ltd. All rights reserved.

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1. Introduction

The assumption that stock price changes follow a stable distri-

bution forms the basis for major asset pricing and option pricing

models. Early models by Bachelier (1900) take normality as a

fundamental assumption for modeling stock price movements. In

line with this assumption, Osborne (1959) shows that logarithms

of the changes in the stock prices are mutually independent with

a common probability distribution (i.e., they conform to a random

walk). He then suggests that stock price changes must follow a

normal distribution. However, these findings have been challenged

both theoretically and empirically.1

An early work by Mandelbrot (1967) proposes that stock price

returns belong to the family of stable Paretian distributions be-

cause they have fatter tails. Fama (1963; 1965) provides empirical

evidence that supports this claim and demonstrates that stock

price changes indeed have fatter tails and have higher peaks than

the normal distribution. More recently, Rachev, Stoyanov, Biglova,

and Fabozzi (2005) compared the stable Paretian distribution

to the normal distribution using 382 US stock returns over the

period 1992–2003. The authors investigated the daily returns

using two probability models: the homoskedastic independent and

identically distributed model and the conditional heteroskedastic

ARMA-GARCH model. Normality was rejected for both models.

However, Officer (1972) found that normality holds for monthly
∗ Corresponding author. Tel.: +90 312 2924214.
E-mail addresses: canan@bu.edu (C.G. Corlu), mkuyzu@etu.edu.tr (M. Meterel-

liyoz), tinic@bilkent.edu.tr (M. Tiniç).
1 A similar line of argument holds for exchange rate returns.

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http://dx.doi.org/10.1016/j.eswa.2015.12.048

0957-4174/© 2016 Elsevier Ltd. All rights reserved.
eturns and that the standard deviation of the returns is inconsis-

ent with the stable hypothesis. To support this argument, Praetz

1972) then suggested the Student-t distribution as an alternative

o the stable Paretian because the stable Paretian distribution has

n infinite variance property and the density function of the stable

aretian is unknown. Over an eight-year period, Praetz (1972)

xamined weekly data from Sydney Stock Exchange and showed

hat the Student-t distribution can be used as an alternative to

xplain the stock price behavior.

The Student-t distribution was also compared with the nor-

al distribution and the Cauchy distribution by Blattberg and

onedes (1974). Contrary to Praetz (1972), they used both daily

nd weekly returns of stocks of the Dow Jones Industrial (DJI), and

hey used the maximum likelihood estimation method for esti-

ating the parameters of the distributions. Blattberg and Gonedes

1974) showed that the Student-t distribution performs better than

he normal distribution on daily returns. However, normality is not

ejected for monthly return data. Hagerman (1978) tested the nor-

ality hypothesis on both individual stocks of the American and

ew York Stock Exchanges on portfolios that contain these stocks,

nd found that they do not behave in line with the normal dis-

ribution. Hagerman (1978) proposed that the mixture of normal

istributions and the Student-t distribution can be an alternative

o representing the characteristics of stock return data. However,

he performance of these two distributions against each other was

ot investigated in Hagerman’s work.

Kon (1984) compared the discrete mixture of normal distribu-

ions and the Student-t distribution over a period of almost 19

ears, examining daily returns of 30 stocks from DJI and Standard

Poor’s (S&P) value- and equal-weighted stock market indexes. A

http://dx.doi.org/10.1016/j.eswa.2015.12.048
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mailto:canan@bu.edu
mailto:mkuyzu@etu.edu.tr
mailto:tinic@bilkent.edu.tr
http://dx.doi.org/10.1016/j.eswa.2015.12.048


C.G. Corlu et al. / Expert Systems With Applications 54 (2016) 170–192 171

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iscrete mixture of normal distributions is shown to have greater

alidity than the Student-t distribution in modeling the data. Simi-

ar to Blattberg and Gonedes (1974) and Akgiray and Booth (1987);

fficer (1972) stated that the monthly returns of stock prices can

e assumed to be normally distributed. However, for the daily data

hey found that the mixed diffusion process and the mixture of

ormal distributions perform better than the stable distributions.

Bookstaber and McDonald (1987) proposed the generalized beta

GB2) distribution to explain the behavior of stock returns. This

as chosen because the GB2 is a flexible distribution and acknowl-

dges various distributions as special cases. They found that the

B2 distribution is significantly better than the lognormal dis-

ribution, especially in relation to short time intervals. Badrinath

nd Chatterjee (1988) examined the Center for Research in Secu-

ity Prices (CRSP) value-weighted market index returns between

962 and 1985 and concluded that returns of stock prices follow

skewed g-and-h distribution.2 Similarly, Mills (1995) found that

he g-and-h distribution accurately fits a dataset that consists of

hree London Stock Exchange indices: FTSE 100, Mid 250, and FTSE

50.

A more general comparison of distributions with finite vari-

nces over equity stocks was conducted by Gray and French (1990).

hey compared the scaled-t distribution, the logistic distribution,

he exponential power distribution, and the normal distribution

ver the log-returns of daily S&P 500 Composite index values for

he period 1979–1985. Among four alternatives, the exponential

ower distribution was found to be the best fit. Lau, Lau, and Win-

ender (1990) showed that series of returns of stock prices that are

aken from the CRSP yield higher kurtosis and skewness than the

ormal distribution. They proposed the lognormal, beta, Weibull,

earson Types IV and VI, and Johnson system of distributions as al-

ernatives. A general comparison of the normal distribution to the

caled-t distribution and to the mixture of two normal distribu-

ions was conducted by Aparicio and Estrada (2001) using the daily

eturns of 13 different European stock markets. It was found that

he scaled-t distribution is a significantly better fit for the data, and

he partial mixture of two normal distributions also performs well.

ormality is rejected in all cases.

Linden (2001) introduced the Laplace mixture distribution,

hich is derived by conditioning the standard deviation of the nor-

al distribution as an exponentially distributed random variable.

inden (2001) used this distribution to represent the daily, weekly,

nd monthly returns of the 20 most traded shares and the index

f the Helsinki Stock Market. The normality assumption is not al-

ays rejected for the weekly and monthly returns. However, for

he daily returns, an asymmetric Laplace distribution is found to

e a better candidate than the normal distribution.

Harris and Küçüközmen (2001a) and Harris and Küçüközmen

2001b), respectively, examined the skewed generalized-t distribu-

ion (SGT) and the exponential generalized beta distribution (EGB)

sing daily UK, US, and Turkish equity returns. Consequently, they

ound that the SGT outperforms the EGB. In both studies, the au-

hors rejected the hypothesis that the daily returns are distributed

ith the Student-t, power of exponential, or logistic distribution.

n addition, for the daily Turkish returns, the Laplace distribution

as also rejected. For the UK returns, the skewed-t distribution

as preferred, whereas for the US returns, the generalized-t distri-

ution was preferred. More recently, Behr and Pötter (2009) com-

ared the generalized hyperbolic distribution, the generalized logF

istribution, and the finite mixture of Gaussians on monthly S&P

00 index returns over the years 1871–2005 and daily returns over

he years 2001–2005. For the monthly returns, the two-component
2 Badrinath and Chatterjee (1988) also provide an excellent review of the litera-

ure.

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aussian mixture distribution described the empirical distribution

f the returns better than alternative distributions. Although the

eneralized hyperbolic distribution is the poorest performer for

onthly returns, it performs best for daily data. However, as the

aily data examined by Behr and Pötter (2009) is almost symmet-

ic, the Laplace distribution, which does not have a parameter to

apture the asymmetries, fits as well as the generalized hyperbolic

istribution.

Finally, as an alternative to the stable distribution and the

tudent-t distribution, Chalabi, Scott, and Würtz (2010) use the

eneralized lambda distribution (GLD) for modeling equity returns.

tarting with Eberlein and Keller (1995), the normal inverse Gaus-

ian (NIG) distribution is used to model financial returns and

articularly for modeling 30 stocks at the German Stock Index.

rause, Zentrum, and Modellbildung (1997) show the applicability

f the NIG distribution in modeling German stock and US Stock In-

ex data. Bølviken and Benth (2000) used the NIG distribution to

odel 8 Norwegian stocks.

In Table 1, we summarize the papers that performed compar-

son studies to investigate the behavior of stock returns. We find

hat the outcomes differ and are often conflicting. Based upon this,

ur goal in this study is to fill this gap in the literature by ad-

ressing which distribution is best for modeling daily equity index

eturn data. To this end, we consider the following flexible distri-

utions that are commonly used in finance: the skewed Student-

distribution, the GLD, the NIG distribution, the Johnson system

f distributions, and the g-and-h distribution. We conduct a com-

rehensive numerical analysis to compare the overall suitability

f these five distributions on the equity index returns of twenty

ifferent countries over the period 1979–2014, which is divided

nto twelve three-year sub-periods. We also include the normal

istribution in our experimental design. The overall suitability is

nitially compared using the Kolmogorov–Smirnov (KS) test statis-

ic (Chakravarti & Laha, 1967) and the Anderson–Darling (AD) test

tatistic (Anderson & Darling, 1954). Furthermore, we conduct p-

alue tests in order to assess the significance of these KS and

D statistics. In addition, the explanatory power of the models is

ested using in-sample Value-at-Risk (VaR) failure rates. Consistent

ith other studies in the previous research, we find that normal-

ty is rejected in all sub-periods for all markets. Our p-value tests

nd the in-sample VaR test suggest that GLD performs best for all

arkets over all time periods.

The remainder of the paper is organized as follows. Section 2

resents the data. Section 3 presents the distributions along with

he fitting methods that are used to estimate the parameters of

he distributions. Section 4 discusses our numerical study and

ection 5 presents key conclusions.

. Description of the data

We create a diversified sample from ten developed and ten

merging market indexes. The selected developed stock market in-

exes are: S&P/ASX 200 Index (Australia), S&P/Toronto Stock Ex-

hange Index (Canada), CAC 40 (France), DAX (Germany), NIKKEI

25 (Japan), the Straits Times Index (Singapore), IBEX 35 (Spain),

MI (Switzerland), FTSE 100 (UK), and S&P 500 (US), while the

merging stock market indexes are the Ibovespa Index (Brazil),

PSA Index (Chile), SHSZ 300 (China), BSE 500 (India), KOSPI Index

Korea), FBMKLCI Index (Malaysia), the Mexican IPC Index (Mex-

co), MICEX Index (Russia), JALSH Index (South Africa), and BIST

00 (Turkey). The daily closing index levels from January 1979 to

ugust 2014 are collected using the Bloomberg Terminal.

Bloomberg provides index levels for the S&P/ASX 200, CAC

0, DAX and IBEX 35 prior to their establishment date. This can

appen due to two reasons. First, the index levels can be adjusted

ith respect to their ancestor indices. For instance, the DAX



172 C.G. Corlu et al. / Expert Systems With Applications 54 (2016) 170–192

Table 1

Summary of articles that performed comparison studies to investigate the behavior of stock returns. The first column of this

table lists the articles that compare several distributions for modeling stock returns. The second column of the table presents

the distributions considered in the corresponding study.

Article Compared distributions

Fama (1963), Fama (1965) and Rachev et al. (2005) Stable Paretian∗ , normal
Praetz (1972) Scaled-t∗ , compound process , normal
Hsu, Miller, and Wichern (1974) Normal∗ , stable Paretian
Blattberg and Gonedes (1974) Student-t∗ , Cauchy, normal
Kon (1984) Discrete mixture of normals∗ , Student-t
Eberlein and Keller (1995) Normal inverse Gaussian∗ , normal
Akgiray and Booth (1987) Mixed diffusion process∗ , discrete mixture of normals
Bookstaber and McDonald (1987) Generalized beta∗ , lognormal
Gray and French (1990) Exponential power∗ , logistic, Scaled-t, normal
Aparicio and Estrada (2001) Scaled-t∗ , mixture of normal, normal
Linden (2001) Asymmetric Laplace∗ , normal
Harris and Küçüközmen (2001a), (2001b) SGT∗ , EGB, exponential power, logistic, Student-t, Laplace
Behr and Pötter (2009) Finite mixture of Gaussians∗ , generalized hyperbolic∗∗ , generalized logF

∗ represents the outperforming distribution.
∗∗ In Behr and Pötter (2009), a mixture of two Gaussians describes the monthly data better than the others, whereas a

generalized hyperbolic distribution is the best fit for the daily data.

Table 2

Markets considered in each sub-period. This table presents the emerging and developed markets consid-

ered in each sub-period. For instance, in sub-period 1979–1981, data are available only for the developed

markets Canada, Germany, Japan, and the US and the emerging markets Korea and Malaysia. Beginning

from the sub-period 2000–2002, we have access to data for all ten developed and ten emerging markets.

Periods Developed markets Emerging markets

1979–1981 Canada, Germany, Japan, US Korea, Malaysia

1982–1984 Canada, Germany, Japan, UK, US Korea, Malaysia

1985–1987 Canada, France, Germany, Japan, Spain, UK, US Korea, Malaysia

1988–1990 Canada, France, Germany, Japan, Chile, Korea, Malaysia, Turkey

Spain, Switzerland, UK, US

1991–1993 Australia, Canada, France, Germany Brazil, Chile, Korea, Malaysia, Turkey

Japan, Spain, Switzerland, UK, US

1994–1996 Australia, Canada, France, Germany Brazil, Chile, Korea, Malaysia

Japan, Spain, Switzerland, UK, US Mexico, South Africa, Turkey

1997–1999 Australia, Canada, France, Germany, Japan Brazil, Chile, India, Korea, Malaysia

Singapore, Spain, Switzerland, UK, US Mexico, Russia, South Africa, Turkey

2000–2014 All markets All markets

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follows its ancestor indices Börsenzeitungs Index and Hardy In-

dex.3 Second, the index levels can be recalculated to an earlier

date. For instance, the CAC 40 is established in December 31, 1987;

however, CAC 40 index levels are recalculated on a daily basis to

September 7, 1987 (Shilling, 1996). Similarly, the S&P/ASX 200 is

established in April 3, 2000; however, S&P/ASX 200 index levels

are recalculated to May, 1992.4 IBEX 35 is launched in January 14,

1992 and IBEX 35 index levels are recalculated to January 5, 1987

(Fernandez, Aguirreamalloa, & Avendaño, 2011).

The daily logarithmic returns are calculated as Xt =
log(St /St−1 ), where St is the closing index level at time t. We
divide the data for each market into twelve three-year nonover-

lapping sub-periods. Our sample period contains both regional and

global financial crises such as the 1997 Asian financial crisis, the

Global Financial Crisis (2006–2008), and the Euro-zone Sovereign

Debt Crisis (2009–2011). Unfortunately, return data is not available

for every market during all of these sub-periods. In Table 2, we

list data that are available for the developed and the emerging

markets in each sub-period. For example, in sub-period 1979–

1981, data are available only for the developed markets Canada,

Germany, Japan, and the US and the emerging markets Korea and

Malaysia. Beginning from the sub-period 2000–2002, we have

access to data for all ten developed and ten emerging markets.

Fig. 1 presents the descriptive statistics for the developed mar-

kets and the emerging markets over different sub-periods. The
3 Based on communications with Deutsche Börse Group.
4 Based on communications with Standard & Poor’s Dow Jones Indices. a
umber in each circle represents the sub-period number, where 1

s for the sub-period 1979–1981 and 13 is for the entire sample.

he skewness equals to s = (m3/m2 )3/2 and the kurtosis is given
y k = m4/m22, where mi is the estimate of the ith moment around
he mean. The skewness values of the returns in the figure are

lustered either on the right or left hand-side of the vertical line

rawn at zero, indicating that all returns of all indices demonstrate

skewed (either left or right) behavior. Similarly, the kurtosis of

eturns are all clustered above the line drawn at three (some be-

ng very far from three) meaning that the return distributions are

ore peaked than the normal distribution.

For the developed markets in Fig. 1, the fourth sub-period (i.e.,

988–1990) and the thirteenth sub-period (i.e., the entire sample)

xhibit skewness and kurtosis values that are significantly far from

he skewness and kurtosis values of the normal distribution. For

he emerging markets, there are more sub-periods that exhibit

igh skewness and kurtosis values, such as fourth, seventh, eighth,

inth, twelfth, and thirteenth sub-periods (i.e., 1988–1990, 1997–

999, 2000–2002, 2003–2005, 2012–2014, entire sample). These

wo figures alone indeed suggest that the normal distribution is

ot a good model for describing the distribution of daily equity in-

ex returns. Additionally, we calculate the Bera-Jarque (BJ) statistic

or all of the countries over all sub-samples in order to test for any

eparture from normality. The results indicate that the normality

s rejected in 96% of the instances, at the 1% significance level.5
5 To save space, we do not present the BJ test statistics here. However, they are

vailable upon request.



C.G. Corlu et al. / Expert Systems With Applications 54 (2016) 170–192 173

Fig. 1. Skewness and kurtosis values of each market over all sub-samples and the entire sample. The top figure presents the skewness (on the horizontal axis) and the

kurtosis (on the vertical axis) values of each developed market over all sub-samples and the entire sample. The bottom figure does the same for emerging markets. The

number in each circle represents the sub-period number with 1 presenting the sub-period 1979–1981 and 13 presenting the entire sample. The skewness equals to s =
(m3 /m2 )

3/2 and the kurtosis is given by k = m4 /m22 , where mi is the estimate of the ith moment around the mean.

3

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. Flexible distributions

We describe the generalized lambda distribution (GLD) in

ection 3.1, the Johnson translation system in Section 3.2, the

kewed Student-t distribution in Section 3.3, the normal inverse

aussian (NIG) distribution in Section 3.4, and the g-and-h distri-

ution in Section 3.5.

.1. The Generalized Lambda distribution

The GLD (Filliben, 1975; Joiner & Rosenblatt, 1971; Ramberg &

chmeiser, 1974), which is an extension of Tukey’s lambda distri-

ution (Hastings, Mosteller, Tukey, & Winsor, 1947), is defined by
he following inverse cumulative distribution function:

−1(u; λ1, λ2, λ3, λ4 ) = λ1 +
uλ3 − (1 − u)λ4

λ2
, (3.1)

here 0 ≤ u ≤ 1; λ1 is the location parameter, λ2 is the scale
arameter, and λ3 and λ4 are related to skewness and kurtosis, re-
pectively. This representation is denoted as Ramberg–Schmeiser

eneralized Lambda Distribution (RS GLD) in reference to the

arametrization of Ramberg and Schmeiser (1974). However, the

robability density function (pdf) associated with equation (3.1)

oes not provide a valid pdf for all combinations of λ3 and λ4
Fournier et al., 2007).

In order to avoid this problem, Freimer, Kollia, Mudholkar,

nd Lin (1988) proposed a different parametrization of the GLD,



174 C.G. Corlu et al. / Expert Systems With Applications 54 (2016) 170–192

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denoted as Freimer–Mudholkar–Kollia–Lin Generalized Lambda

Distribution (FMKL GLD). This is given by:

F −1(u; λ1, λ2, λ3, λ4 ) = λ1 +
1

λ2

(
uλ3 − 1

λ3
− (1 − u)

λ4 − 1
λ4

)
.

Both the FMKL and RS representations are used in practice as

they offer a wide variety of shapes. However, the FMKL representa-

tion is preferable as a result of its ease of use. In addition, it is well

defined over all parameter values with the only restriction being

that λ2 must be positive. Also, the condition min(λ3, λ4 ) > −1/k
must hold to have a finite kth moment. In this study, we use the

FMKL representation. The FMKL GLD curves are classified into five

categories: Class-I family (λ3 < 1, λ4 < 1) represents unimodal
densities with continuous tails, Class-II family (λ3 > 1, λ4 < 1)
represents monotone pdfs similar to the exponential distribution,

Class-III family ( 1 < λ3 < 2, 1 < λ4 < 2) represents U-shaped
densities with truncated tails, Class-IV family (λ3 > 2, 1 < λ4 <
2) represents S-shaped densities, and finally Class-V family (λ3 >
2, λ4 > 2) represents unimodal densities with truncated tails. We
find that the daily equity return data belongs to the Class-I family.

Prior studies propose different fitting techniques for estimat-

ing the parameters of the GLD.6 In this study, we use the maxi-

mum likelihood estimation method in the GLDEX package of R (Su,

2007).

3.2. The Johnson translation system

A random variable X from the Johnson translation system is

represented by (Johnson, 1949):

X = ξ + λr−1
(

Z − γ
δ

)
,

where Z is a standard normal random variable, γ and δ are shape
parameters, ξ is a location parameter, λ is a scale parameter, and
r( · ) is one of the following transformations:

r(X ) =

⎧⎪⎨
⎪⎩

x for the SN (normal) family
log(x) for the SL (lognormal) family
log(x/(1 − x)) for the SB (bounded) family
log(x +

√
x2 + 1) for the SU (unbounded) family.

The range of the random variable X is defined by the family of in-

terest: X > ξ and λ = 1 for the SL family; ξ < X < ξ + λ for the SB
family; and −∞ < X < ∞ for the SN and SU families. For each fea-
sible combination of the skewness and the kurtosis values there is

a unique family that depends on the choice of r. In this study, we

only consider the SU family of the Johnson translation system be-

cause the equity index returns have skewness and kurtosis values

that conform with the SU family.

There are alternative methodologies proposed in previous re-

search to estimate the parameters of the SU.
7 We use the method-

ology proposed in Tuenter (2001), which resembles the method of

moments, where the sample skewness and kurtosis are equalized

with the theoretical skewness and kurtosis.

3.3. The skewed Student-t distribution

A number of skewed Student-t distributions have been pro-

posed in previous research. In this study, due to its simplicity, we

follow the parametrization used in Azzalini and Capitanio (2003).
6 A detailed review can be found in Corlu and Meterelliyoz (2015).
7 A detailed review can be found in Corlu and Biller (2015).

w

k

r

random variable X from the skewed Student-t distribution (here-

fter, skewed-t distribution) has a density of the form:

f (x; δ, ν, μ, β ) = 1
δ

tν

(
x − μ

δ

)
2Tν+1

(
β
(

x − μ
δ

)√
ν + 1(

x−μ
δ

)2 + ν
)

,

here μ, δ, and β represent the location, scale, and skewness pa-
ameters, respectively. tν is the density of the standard Student-t

istribution with ν degrees of freedom and Tν+1 is the distribution
unction of the standard Student-t distribution with ν + 1 degrees
f freedom.

Estimates of the parameters of the skewed-t distribution are ob-

ained using the maximum likelihood estimation method proposed

y Azzalini and Capitanio (2003). The maximization of the likeli-

ood function is conducted by the Nelder–Mead algorithm.

.4. The Normal Inverse Gaussian distribution

The NIG distribution is obtained from a more general distribu-

ion called the generalized hyperbolic distribution (GHD), whose

ensity is given by:

f (x; λ, α, β, μ, δ)

=
(
δ
√

α2 − β2
)λ(

δα
)1/2−λ

√
2πδKλ

(
δ
√

α2 − β2
)

(
1 + (x − μ)

2

δ2

)λ/2−1/4

× exp(β(x − μ))Kλ−1/2
(

αδ

√
1 + (x − μ)

2

δ2

)
,

here Kλ is the modified third-order Bessel function; μ and δ rep-
esent location and scale parameters, respectively; λ is the class-
efining parameter; α is a parameter related to tail heaviness; and

is the asymmetry parameter. The density is defined under the

ollowing parameter restrictions:

δ ≥ 0 and |β| < α if λ > 0
> 0 and |β| < α if λ = 0

δ > 0 and |β| ≤ α if λ < 0
faff, McNeil, and Ulmann (2013) indicate that the GHD can rep-

esent skewed distributions as well as heavy tails. The variants of

he GHD can be obtained by changing the value of the parame-

er λ; this is why λ is called the class-defining parameter. The NIG
istribution is obtained from the GHD by setting λ = −1/2, and its
ensity is given by:

f (x; α, β, μ, δ) =
αδK1

(
α
√

δ2 + (x − μ)2
)

π
√

δ2 + (x − μ)2
eδ

√
α2 −β2 +β(x−μ)

,

ith |β| ≤ α and δ > 0 (Prause, 1999).
We use the ghyp package of R to estimate the parameters of

he NIG.

.5. The g-and-h distribution

The g-and-h distribution is a functional transformation of the

tandard normal distribution. A random variable from the g-and-h

istribution is obtained by transforming the standard normal ran-

om variable Z to the following form (Tukey, 1977):

g,h(Z) =
(
egZ − 1

) exp (hZ2/2)
g

,

here g, h ∈ R. The g and h parameters account for skewness and
urtosis, respectively. When a location parameter A and a scale pa-

ameter B are incorporated, a random variable from the g-and-h



C.G. Corlu et al. / Expert Systems With Applications 54 (2016) 170–192 175

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1

4

E
n

ti
re

 S
a

m
p

le

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

Canada

Skewed−t

GLD

NIG
Johnson Su
g−and−h

Normal
1

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1

1
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0
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2
0

0
9

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0
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1

2
0

1
2

−
2

0
1

4

E
n

ti
re

 S
a

m
p

le

0.00

0.02

0.04

0.06

0.08

0.10

Germany

Skewed−t

GLD

NIG
Johnson Su
g−and−h

Normal

Fig. 2. Kolmogorov–Smirnov (KS) statistics of all six probability distributions for Canada and Germany. The top figure shows the KS statistics of the skewed-t distribution,

generalized lambda distribution (GLD), normal inverse Gaussian (NIG) distribution, Johnson SU family, g-and-h distribution, and normal distribution for Canada over all sub-

periods starting with 1979–1981. The last tick mark on the horizontal axis presents the KS statistic for the entire sample over the period 1979–2014. The bottom plot does

the same for Germany.

d

X

f

l

(

c

g

g

w

b

i

o

n

d

X

X

S

t

istribution takes the following form:

g,h(Z) = A + B
(
egZ − 1

) exp (hZ2/2)
g

. (3.2)

Among the alternative fitting methodologies that are proposed

or estimating the parameters of the g-and-h distribution, we fol-

ow the procedure used in Mills (1995) and Dutta and Babbel

2002) as a result of its tractability. Specifically, the estimation pro-

edure starts with identifying the pth percentile of the parameter

as follows:

p = −
(

1

Zp

)
ln

(
X1−p − X0.5
X0.5 − Xp

)
, (3.3)

here Xp and Zp are the pth percentile of the empirical distri-

ution and the standard normal distribution, respectively. It is

mportant to note that by using different values of p, one can
btain multiple estimates of g. Following (Mills, 1995), we use

ine percentiles and we choose percentiles using letter values; i.e.,

p = 1/2, 1/4, 1/16, . . .. The estimate for g is calculated as the me-
ian of gp values.

Using equation (3.2), it is trivial to derive

p = A + B
(
egZp − 1

) exp (hZ2p/2)
g

and (3.4)

1−p = A + B
(
egZ(1−p) − 1

) exp (hZ2
(1−p)/2

)
g

. (3.5)

ince X0.5 = A, the location parameter A is given by the median of
he data set.



176 C.G. Corlu et al. / Expert Systems With Applications 54 (2016) 170–192

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4

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9

6

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9
7

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2

0
1

4

E
n

ti
re

 S
a

m
p

le

0.00

0.02

0.04

0.06

0.08

Mexico

Skewed−t

GLD

NIG
Johnson Su
g−and−h

Normal

1
9

8
8

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9
9

0

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9
1

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3

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0
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5

2
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9

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0
1

1

2
0

1
2

−
2

0
1

4

E
n

ti
re

 S
a

m
p

le

0.00

0.01

0.02

0.03

0.04

0.05

0.06

Turkey

Skewed−t

GLD

NIG
Johnson Su
g−and−h

Normal

Fig. 3. Kolmogorov–Smirnov (KS) statistics of all six probability distributions for Mexico and Turkey. The top figure shows the KS statistics of the skewed-t distribution,

generalized lambda distribution (GLD), normal inverse Gaussian (NIG) distribution, Johnson SU family, g-and-h distribution, and normal distribution for Mexico over all sub-

periods starting with 1994–1996. The last tick mark on the horizontal axis presents the KS statistic for the entire sample over the period 1994–2014. The bottom plot does

the same for Turkey over all sub-periods starting with 1988–1990.

t

4

o

m

K

t

S

4

c

Subtracting (3.5) from (3.4) and letting Zp = −Z1−p provides the
following result:

ln
g
(
Xp − X1−p

)
egZp − e−gZp = ln(B) + h(Z

2
p/2). (3.6)

If the data is positively skewed, the left-hand side of (3.6)

can be replaced with the upper half-spread (UHS), as defined in

Hoaglin (2006):

UHS = g(X1−p − X0.5 )(
e−gZp − 1

) (3.7)
Conversely, if the data is negatively skewed, then a lower half-

spread (LHS) can be used on the left-hand side of (3.6):

LHS = g(X0.5 − Xp)(
1 − egZp

) (3.8)
The estimates of h and B are obtained from the coefficient and

he intercept of the linear regression of In(UHS) and In(LHS) on

(Z2p/2).

. Performance and risk estimation

The goal of this section is to evaluate the performances

f the density functions described in Section 3 in order to

odel the daily equity index returns. To this end, we use the

olmogorov–Smirnov (KS) and Anderson–Darling (AD) test statis-

ics in Section 4.1 and we present the Value-at-Risk (VaR) levels in

ection 4.2.

.1. Goodness-of-fit

In this section, we first use the KS and AD test statistics to

ompare the goodness-of-fit of the distributions of interest. Both



C.G. Corlu et al. / Expert Systems With Applications 54 (2016) 170–192 177

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n

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a

m
p

le

0.0

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0.8

1.0

1.2

Canada

Skewed−t

GLD

NIG
Johnson Su
g−and−h

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−
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0
1

4

E
n

ti
re

 S
a

m
p

le

0.0

0.5

1.0

1.5

2.0

Germany

Skewed−t

GLD

NIG
Johnson Su
g−and−h

Fig. 4. Anderson–Darling (AD) statistics of five probability distributions for Canada and Germany. The top figure shows the AD statistics of the skewed-t distribution, gen-

eralized lambda distribution (GLD), normal inverse Gaussian (NIG) distribution, Johnson SU family, and g-and-h distribution for Canada over all sub-periods starting with

1979–1981. The last tick mark on the horizontal axis presents the AD statistic for the entire sample over the period 1979–2014. The bottom plot does the same for Germany.

o

t

l

c

i

t

w

c

s

m

s

i

t

t

e

h

a

s

t

b

A

w

d

f

e

a

g

t

b

t

f these test statistics summarize the difference between the fit-

ed cumulative distribution function F̂ and the empirical cumu-

ative distribution function Fe. In particular, the KS test statistic

orresponds to the largest distance between Fe(x) and F̂ (x), that

s, supx {|Fe (x) − F̂ (x)|}, while the AD test statistic corresponds to
he weighted average of the squared differences, (Fe (x) − F̂ (x))2,
here the weights are chosen in such a way that the discrepan-

ies in the tails are emphasized. The smaller the KS and AD test

tatistics, the better the fit.

Fig. 2 presents the computed KS test statistics for the developed

arkets of Canada and Germany under the six distributional as-

umptions. Fig. 3 does the same for the emerging markets of Mex-

co and Turkey. Fig. 4 presents the computed AD test statistics for

he developed markets of Canada and Germany using all distribu-

ions except the normal distribution. Fig. 5 does the same for the

merging markets of Mexico and Turkey. The markets on the plots
ave been selected as they are the most representative of the over-

ll results. We excluded the normal distribution from Figs. 4 and 5

ince the AD test statistic values for the normal distribution are

oo large compared to the respective results for all other distri-

utions under consideration, and when the normal distribution’s

D statistics are included, the scale of y-axis becomes very large,

hich prevents to reveal the differences of AD statistics for other

istributions. KS and AD plots for the remaining markets can be

ound in the Appendix.

We find that for every market in each sub-sample, and in the

ntire sample, the KS statistic in the normal case is almost always

bove the KS statistics in all other five distributions. A similar ar-

ument also holds for the AD statistics. Furthermore, we observe

hat, in general, all five distributions excluding the g-and-h distri-

ution perform very similarly to each other; the g-and-h distribu-

ion marginally underperforms in many sub-samples.



178 C.G. Corlu et al. / Expert Systems With Applications 54 (2016) 170–192

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E
n

ti
re

 S
a

m
p

le

0.0

0.5

1.0

1.5

2.0

2.5

Mexico

Skewed−t

GLD

NIG
Johnson Su
g−and−h

1
9

8
8

−
1

9
9

0

1
9

9
1

−
1

9
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3

1
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4

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6

1
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7

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2
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0
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0
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5

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2
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0
9

−
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0
1

1

2
0

1
2

−
2

0
1

4

E
n

ti
re

 S
a

m
p

le

0

1

2

3

4

Turkey

Skewed−t

GLD

NIG
Johnson Su
g−and−h

Fig. 5. Anderson–Darling (AD) statistics of five probability distributions for Mexico and Turkey. The top figure shows the AD statistics of the skewed-t distribution, generalized

lambda distribution (GLD), normal inverse Gaussian (NIG) distribution, Johnson SU family, and g-and-h distribution for Mexico over all sub-periods starting with 1994–1996.

The last tick mark on the horizontal axis presents the AD statistic for the entire sample over the period 1994–2014. The bottom plot does the same for Turkey over all

sub-periods starting with 1988–1990.

d

d

m

w

fi

t

t

t

α
t

t

i

8 We must express our gratitude to an anonymous referee for his/her observation.
Next, we compare the distributions of interest by plotting the

empirical histogram of the daily returns together with the esti-

mated distribution functions. Fig. 6 presents the empirical his-

togram of the log-returns of the Ibovespa Index (Brazil) over the

sub-period 1997–1999 together with six estimated distribution

functions for the same index. One notable observation is that the

normal distribution significantly underperforms other distributions

in modeling the Ibovespa Index. In particular, the normal distribu-

tion cannot capture the peakedness and the tails of the data. Other

distributions including the generalized lambda distribution (GLD),

the skewed-t distribution, the normal inverse Gaussian (NIG) dis-

tribution, and the g-and-h distribution perform very similarly in

modeling the data. However, the fit of the Johnson translation sys-

tem is slightly more peaked than the histogram.

As the computed KS and AD statistic values are very close to

each other in all distributions, it is difficult to identify whether the
ifferences in these statistics are significant.8 Furthermore, Fig. 6

oes not provide an answer to the question of which distribution

odels the daily equity return data best. To address this problem,

e conduct a power test, where we calculate the p-value of each

t. The null hypothesis is that the observed data originates from

he hypothesized distribution and the alternative hypothesis is that

he observed data does not belong to the hypothesized distribu-

ion. The null hypothesis is rejected if the p-value is lower than

= 5%. It is important to recognize here that for many distribu-
ions under consideration, the critical values of both the KS statis-

ic and AD statistic do not exist. The typical approach in this case

s to compute the p-values and the critical values by means of a



C.G. Corlu et al. / Expert Systems With Applications 54 (2016) 170–192 179

Log−Returns

F
re

q
u

e
n

c
y

−0.1 0.0 0.1 0.2 0.3

5
1

0
1

5
2

0
2

5
Brazil (1997−1999)

GLD

Normal

Skewed−t
Johnson Su
NIG
g−and−h

Fig. 6. Histogram and distribution fits for a selected market. This figure plots the

histogram of the Brazil log-returns together with the estimated distribution func-

tions of the generalized lambda distribution (GLD), normal distribution, skewed-t

distribution, Johnson SU family, normal inverse Gaussian (NIG) distribution, and g-

and-h distribution over the sub-period 1997–1999. The bins of the histogram are

calculated with the Scott method in R.

M

2

A

d

T

(

c

s

a

r

i

T

G

t

i

i

m

e

o

r

r

t

c

t

a

d

fi

G

(

m

b

N

s

s

6

o

f

t

a

2

a

4

v

p

i

i

{

r

t

s

d

t

p

i

2

w

(

v

6

t

m

m

t

t

l

a

l

T

u

n

A

t

t

O

m

onte Carlo simulation for each hypothesized distribution (Ross,

004). We follow the following steps to calculate the p-value:

Step 1. Use the data to estimate the parameters of the hypoth-

esized distribution and compute the value of the test statis-

tics denoted by G (such as KS and/or AD statistic).

Step 2. Generate a sample of size n that is equal to the number

of observed data from the fitted distribution.

Step 3. Fit the hypothesized distribution to the data generated

in Step 2 and estimate the sample goodness-of-fit statistics.

Step 4: Repeat Step 1 through Step 3 1000 times and calculate

the p-value as the proportion of times the sample statistics

values exceed the observed value G of the original sample.

Step 5: Reject the null hypothesis if p-value is smaller than

0.05.

Table 3 tabulates the number of rejections using both KS and

D test statistics in each sub-period for each distribution in the

eveloped markets. Table 4 does the same for emerging markets.9

he last row of both tables tabulates the percentage of sub-periods

out of 12 sub-periods) in which a particular distribution is ac-

epted by the p-value test using both KS and AD statistics. The re-

ults reported in both tables suggest that the GLD is a prominent

lternative for fitting the daily equity index return data. We fail to

eject the GLD according to both KS and AD test statistics for all

ndex returns in all sub-samples, as well as in the entire sample.

herefore, according to the p-value test, we can conclude that the

LD is a powerful distribution in modeling both the center and the

ails of the daily index return data.

We conclude this section by comparing the models considered

n this paper according to their stability.10 Stability is particularly

mportant for applications of portfolio analysis and risk manage-

ent (Rachev & Mittnik, 2000). In particular, stable distributed
9 To save space, we do not report the results for each specific market here. How-

ver, these results are available upon request.
10 We must express our gratitude to an anonymous referee who brought this to

ur attention.

j

m

t

S

t

eturns possess the property that linear combinations of return se-

ies, such as portfolios, follow a stable distribution. We investigate

he relative stability of the models of interest by comparing their

apability to represent the variety of shapes taken by return dis-

ributions over twelve sub-periods. We find that in both emerging

nd developed markets, GLD can adequately describe the empirical

istributions in all sub-periods. Other distributions achieve a good

t in smaller share of periods. Specifically, in developed markets,

LD is followed by Johnson SU family (92%), g-and-h distribution

67%), skewed-t distribution (42%), NIG distribution (41%), and nor-

al distribution (0%) under the KS statistic and by g-and-h distri-

ution (67%), Johnson SU family (42%), skewed-t distribution (25%),

IG distribution (17%), and normal distribution (0%) under the AD

tatistic (the last row of Table 3). In emerging markets, both John-

on SU family and g-and-h distribution follow GLD with a share of

7%, and skewed-t distribution and NIG distribution with a share

f 25% under the KS statistic. When the AD statistic is used to per-

orm the p-value test, the GLD is accepted in all of the sub-periods,

he Johnson SU family and g-and-h are accepted in 58% of periods,

nd the skewed-t distribution and NIG distribution are accepted in

5% and 17% of periods, respectively. The normal distribution is not

ccepted in any of the sub-periods (the last row of Table 4).

.2. Risk estimation

We examine the behavior of the distributions at the extreme

alues of each market index return using the VaR measure. The

urpose of this investigation is to observe the risk that an investor

s facing when she has a long or short position on the market

ndexes from our sample. The risk levels are determined as α ∈
0.005, 0.01, 0.05, 0.95, 0.99, 0.995}, in which the first three levels

epresent the lower extreme of returns (long position) and the last

hree represent the upper extreme of returns (short position). In-

ample VaR(α) values are calculated to observe the behavior of the
istributions at the tails. We then apply the Kupiec likelihood ratio

est given in Kupiec (1995), which tests whether the expected pro-

ortion of violations is equal to α. The likelihood ratio test statistic
s given by:

log((τ (α)/n)τ (α)(1 − τ (α)/n)n−τ (α) ) − 2 log(ατ (α)(1 − α)n−τ (α) ),
here τ (α) is the number of times the observed returns are above

short positions) or below (long positions) the theoretical VaR

alue and n is the sample size.

The results of the Kupiec test are represented in Tables 5 and

. For example, Table 5 lists the number of times each distribu-

ion is rejected at each significance level for all of the developed

arkets in each sub-period. Table 6 does the same for emerging

arkets. Focusing on the sub-period 1979–1981 in Table 5, we see

hat the GLD, Johnson SU family, skewed-t distribution, NIG dis-

ribution, and g-and-h are not rejected in any of the significance

evels. However, the normal distribution is rejected by 3 markets

t the significance level of 0.005, by 2 markets at the significance

evel of 0.01, and by 1 market at the significance level of 0.95.

he total number of rejections is tabulated at the far right col-

mn of the table. For the normal distribution, for instance, the total

umber of rejections is equal to 6 over the sub-period 1979–1981.

dding up the number of rejections over the sub-periods, we see

hat the GLD is the least rejected model with only 7 rejections in

he developed markets and 8 rejections in the emerging markets.

n the other hand, the normal distribution is the most rejected

odel with 207 rejections in the developed markets and 213 re-

ections in the emerging markets. In both developed and emerging

arkets, the g-and-h distribution, NIG distribution, and skewed-

distribution perform similarly to each other, while the Johnson

U family underperforms other distributions with 65 rejections in

he developed markets and 57 rejections in the emerging markets.



180 C.G. Corlu et al. / Expert Systems With Applications 54 (2016) 170–192

Table 3

Number of times that each distribution is rejected according to Kolmogorov–Smirnov (KS) and Anderson–

Darling (AD) test statistics in developed markets. The first column of this table (excluding the last row) lists

the periods under consideration; the second column (excluding the last row) presents the number of devel-

oped markets considered in each period; the remaining columns (excluding the last row) presents the number

of times a distribution is rejected by the p-value test according to the KS and AD test statistics. For instance, in

the sub-period 1979–1981, the generalized lambda distribution (GLD), Johnson SU family, and g-and-h distribu-

tion are rejected neither according to the KS statistic nor according to the AD statistic; the skewed-t distribution

is rejected once according to both KS and AD test statistics; the normal inverse Gaussian (NIG) distribution is

not rejected according to the KS statistic but rejected once according to the AD statistic; and the normal distri-

bution is rejected 3 times according to both KS and AD statistics. The very last row of this table presents the

percentage of periods in which a particular distribution is accepted by the p-value test. For instance, the GLD is

never rejected in any of the sub-periods; therefore, its percentage of acceptance is 100% according to both KS

and AD statistics. The Johnson SU family is rejected in the sub-period 2003–2005 according to the KS statistic

(rejected only once in 12 sub-periods; thus, the percentage of acceptance is 11/12 × 100 ≈ 92%) and is not
rejected in the sub-periods 1979–1981, 1988–1990, 1994–1996, 2009–2011, and 2012–2014 according to the AD

statistic (accepted in 5 sub-periods out of 12 sub-periods; thus, the percentage of acceptance is 5/12 × 100 ≈
42%).

Period # Markets GLD Johnson SU Skewed-t NIG g-and-h Normal

KS AD KS AD KS AD KS AD KS AD KS AD

1979–1981 4 0 0 0 0 1 1 0 1 0 0 3 3

1982–1984 5 0 0 0 1 0 0 0 0 0 0 4 4

1985–1987 7 0 0 0 1 1 2 3 4 0 0 7 7

1988–1990 8 0 0 0 0 0 2 1 4 0 0 8 8

1991–1993 9 0 0 0 1 2 3 1 3 1 0 8 8

1994–1996 9 0 0 0 0 1 1 0 1 0 0 8 8

1997–1999 10 0 0 0 1 0 0 2 0 0 0 8 9

2000–2002 10 0 0 0 1 0 0 0 1 0 0 9 10

2003–2005 10 0 0 2 7 2 3 2 2 1 1 10 10

2006–2008 10 0 0 0 3 0 2 3 4 2 1 10 10

2009–2011 10 0 0 0 0 1 2 0 1 2 2 9 10

2012–2014 10 0 0 0 0 1 3 1 1 0 0 10 10

Entire Sample 10 0 0 1 2 2 7 2 7 1 1 10 10

% of acceptance 100 100 92 42 42 25 41 17 67 67 0 0

Table 4

Number of times that each distribution is rejected according to Kolmogorov–Smirnov (KS) and Anderson–

Darling (AD) test statistics in emerging markets. The first column of this table (excluding the last row) lists

the periods under consideration; the second column (excluding the last row) presents the number of emerging

markets considered in each period; the remaining columns (excluding the last row) presents the number of

times a distribution is rejected by the p-value test according to the KS and AD statistics. For instance, in the

sub-period 1979–1981, the generalized lambda distribution (GLD), Johnson SU family, normal inverse Gaussian

(NIG) distribution, and g-and-h distribution are rejected neither according to the KS statistic nor according to

the AD statistic; the skewed-t distribution is rejected once according to the AD statistic and normal distribu-

tion is rejected 2 times according to both KS and AD statistics. The very last row of this table presents the

percentage of periods in which a particular distribution is accepted by the p-value test. For instance, the GLD is

never rejected in any of the sub-periods; therefore, its percentage of acceptance is 100% according to both KS

and AD statistics. The Johnson SU family is rejected in the sub-periods 1982–1984, 1985–1987, 1988–1990, and

2009–2011 according to the KS statistic (rejected 4 times in 12 sub-periods; thus, the percentage of acceptance

is 8/12 × 100 ≈ 67%) and is rejected in the sub-periods 1985-1987, 1991-1993, 1997-1999, 2003-2005, and
2009–2011 according to the AD statistic (rejected in 5 sub-periods out of 12 sub-periods; thus, the percentage

of acceptance is 7/12 × 100 ≈ 58%).

Period # Markets GLD Johnson SU Skewed-t NIG g-and-h Normal

KS AD KS AD KS AD KS AD KS AD KS AD

1979–1981 2 0 0 0 0 0 1 0 0 0 0 2 2

1982–1984 2 0 0 1 0 0 0 1 1 0 0 2 2

1985–1987 2 0 0 1 2 2 2 2 2 0 0 2 2

1988–1990 4 0 0 1 0 2 2 2 1 0 1 4 4

1991–1993 5 0 0 0 1 0 0 0 0 1 1 4 4

1994–1996 7 0 0 0 0 2 1 1 1 0 0 6 6

1997–1999 9 0 0 0 2 3 2 2 2 1 1 8 9

2000–2002 9 0 0 0 0 1 0 0 2 0 0 8 8

2003–2005 10 0 0 0 2 2 2 1 2 1 1 7 8

2006–2008 10 0 0 0 0 2 2 1 2 2 2 10 10

2009–2011 10 0 0 1 1 1 1 1 1 0 0 10 10

2012–2014 10 0 0 0 0 1 1 2 2 0 0 9 9

Entire Sample 10 0 0 1 2 4 5 4 5 4 4 10 10

% of acceptance 100 100 67 58 25 25 25 17 67 58 0 0



C.G. Corlu et al. / Expert Systems With Applications 54 (2016) 170–192 181

Table 5

Value-at-Risk (VaR) failure rate results for the developed markets. This table presents the number of times

each distribution including the generalized lambda distribution (GLD), Johnson SU family, skewed-t distribu-

tion, normal inverse Gaussian (NIG) distribution, g-and-h distribution, and normal distribution is rejected at

various significance levels for all developed markets in each sub-period according to the Kupiec likelihood

ratio test.

Period # Market Method Significance levels Total # Rejections

0.005 0.01 0.05 0.95 0.99 0.995

1979–1981 4 GLD 0 0 0 0 0 0 0

Johnson SU 0 0 0 0 0 0 0

Skewed-t 0 0 0 0 0 0 0

NIG 0 0 0 0 0 0 0

g-and-h 0 0 0 0 0 0 0

Normal 3 2 0 1 0 0 6

1982–1984 5 GLD 0 0 0 0 0 0 0

Johnson SU 1 1 0 0 1 2 5

Skewed-t 0 0 0 0 0 0 0

NIG 0 0 0 0 0 0 0

g-and-h 0 0 0 0 0 0 0

Normal 1 0 0 0 2 3 6

1985–1987 7 GLD 0 0 0 0 0 0 0

Johnson SU 1 0 0 0 0 0 1

Skewed-t 0 0 0 0 0 0 0

NIG 0 0 1 0 0 0 1

g-and-h 1 1 0 0 0 0 2

Normal 1 1 3 4 0 0 9

1988–1990 8 GLD 0 0 0 1 0 0 1

Johnson SU 1 2 0 0 1 1 5

Skewed-t 0 0 0 0 0 0 0

NIG 0 0 0 0 0 0 0

g-and-h 0 0 0 0 0 0 0

Normal 4 4 2 5 0 1 16

1991–1993 9 GLD 0 0 0 0 0 0 0

Johnson SU 2 2 0 0 1 1 6

Skewed-t 0 0 0 0 0 0 0

NIG 0 0 0 1 0 0 1

g-and-h 0 0 0 0 0 0 0

Normal 1 0 2 1 0 4 8

1994–1996 9 GLD 0 0 0 0 0 0 0

Johnson SU 1 0 0 0 1 0 2

Skewed-t 0 0 0 0 0 0 0

NIG 0 0 0 0 0 0 0

g-and-h 0 0 0 0 0 0 0

Normal 5 3 0 0 0 1 9

1997–1999 10 GLD 0 0 0 0 0 0 0

Johnson SU 2 0 0 0 1 1 4

Skewed-t 0 0 0 0 0 0 0

NIG 0 0 0 0 0 0 0

g-and-h 0 0 0 0 0 0 0

Normal 5 4 0 3 0 0 12

2000–2002 10 GLD 0 0 0 0 0 0 0

Johnson SU 1 0 0 0 0 1 2

Skewed-t 0 0 0 0 0 1 1

NIG 0 0 0 0 0 0 0

g-and-h 0 0 0 0 0 0 0

Normal 2 2 0 0 2 4 10

2003–2005 10 GLD 0 0 0 0 0 0 0

Johnson SU 1 1 0 0 1 3 6

Skewed-t 0 0 0 0 0 0 0

NIG 0 0 0 0 0 0 0

g-and-h 0 0 0 0 0 0 0

Normal 9 7 0 2 0 2 20

2006–2008 10 GLD 1 0 0 0 1 0 2

Johnson SU 0 0 0 0 2 3 5

Skewed-t 0 0 0 0 2 0 2

NIG 0 0 0 0 1 0 1

g-and-h 0 0 0 0 1 0 1

Normal 10 10 1 8 2 6 37

2009–2011 10 GLD 0 0 0 0 0 0 0

Johnson SU 2 1 0 0 0 2 5

Skewed-t 0 0 0 0 0 0 0

NIG 0 0 0 0 0 0 0

g-and-h 1 0 0 0 0 0 1

Normal 8 7 0 0 0 2 17

2012–2014 10 GLD 1 0 0 0 0 0 1

Johnson SU 4 3 0 0 3 2 12

Skewed-t 1 0 0 0 0 0 1

NIG 0 0 0 0 0 0 0

g-and-h 0 0 0 0 0 0 0

Normal 7 7 0 0 1 1 16

Entire Sample 10 GLD 0 2 1 0 0 0 3

Johnson SU 3 2 2 2 2 2 13

Skewed-t 1 2 1 0 1 0 5

NIG 1 2 2 0 1 0 6

g-and-h 1 0 0 0 0 0 1

Normal 10 10 9 5 7 10 51



182 C.G. Corlu et al. / Expert Systems With Applications 54 (2016) 170–192

Table 6

Value-at-Risk (VaR) failure rate results for the emerging markets. This table presents the number of times each

distribution including the generalized lambda distribution (GLD), Johnson SU family, skewed-t distribution, nor-

mal inverse Gaussian (NIG) distribution, g-and-h distribution, and normal distribution is rejected at various

significance levels for all emerging markets in each sub-period according to the Kupiec likelihood ratio test.

Period # Market Method Significance levels Total # Rejections

0.005 0.01 0.05 0.95 0.99 0.995

1979–1981 2 GLD 0 0 0 0 0 0 0

Johnson SU 0 0 0 0 0 1 1

Skewed-t 0 0 0 0 0 0 0

NIG 0 0 0 0 0 0 0

g-and-h 0 0 0 0 0 0 0

Normal 1 1 0 1 0 1 4

1982–1984 2 GLD 0 0 0 0 0 0 0

Johnson SU 0 0 0 0 0 0 0

Skewed-t 0 0 0 0 0 0 0

NIG 0 0 0 0 0 0 0

g-and-h 0 0 0 0 0 0 0

Normal 1 0 1 0 2 2 6

1985–1987 2 GLD 0 0 0 0 0 0 0

Johnson SU 1 0 0 0 0 0 1

Skewed-t 0 0 0 0 0 0 0

NIG 0 0 0 0 0 0 0

g-and-h 0 0 0 0 0 0 0

Normal 2 1 1 1 0 0 5

1988–1990 4 GLD 0 0 0 0 0 1 1

Johnson SU 2 1 0 0 1 2 6

Skewed-t 0 0 0 0 1 2 3

NIG 0 0 0 0 1 2 3

g-and-h 0 0 0 0 0 0 0

Normal 3 2 1 1 2 2 11

1991–1993 5 GLD 0 0 0 0 0 2 2

Johnson SU 0 0 0 0 1 2 3

Skewed-t 0 0 0 0 0 2 2

NIG 0 0 0 0 0 2 2

g-and-h 0 0 0 0 0 1 1

Normal 1 1 1 0 3 4 10

1994–1996 7 GLD 0 0 0 0 0 0 0

Johnson SU 2 2 0 0 3 2 9

Skewed-t 0 0 0 0 0 1 1

NIG 0 0 0 0 0 1 1

g-and-h 0 0 0 0 0 0 0

Normal 3 1 0 0 4 4 12

1997–1999 9 GLD 0 0 0 0 0 1 1

Johnson SU 1 1 0 0 3 3 8

Skewed-t 0 0 0 0 0 1 1

NIG 0 0 0 0 0 1 1

g-and-h 0 0 0 0 2 0 2

Normal 5 3 2 3 3 5 21

2000–2002 9 GLD 0 0 0 0 0 0 0

Johnson SU 2 2 0 0 0 0 4

Skewed-t 0 0 0 0 0 0 0

NIG 0 0 0 0 0 0 0

g-and-h 0 0 0 1 0 0 1

Normal 4 3 2 2 1 5 17

2003–2005 10 GLD 0 0 0 0 0 0 0

Johnson SU 0 1 0 0 1 2 4

Skewed-t 0 0 0 0 0 0 0

NIG 0 0 0 0 0 0 0

g-and-h 0 0 0 0 0 0 0

Normal 4 4 1 4 1 2 16

2006–2008 10 GLD 1 1 0 0 0 0 2

Johnson SU 2 2 0 0 1 1 6

Skewed-t 1 1 0 0 1 1 4

NIG 0 0 0 0 0 0 0

g-and-h 0 0 0 0 1 1 2

Normal 10 10 1 6 2 5 34

2009–2011 10 GLD 1 0 0 0 0 0 1

Johnson SU 2 2 0 0 0 0 4

Skewed-t 1 0 0 0 0 0 1

NIG 1 0 0 0 0 0 1

g-and-h 1 0 0 0 0 0 1

Normal 6 7 0 0 3 3 19

2012–2014 10 GLD 0 1 0 0 0 0 1

Johnson SU 1 2 0 0 2 1 6

Skewed-t 0 1 0 0 0 0 1

NIG 0 1 0 0 0 0 1

g-and-h 0 0 0 0 0 0 0

Normal 3 2 0 0 1 3 9

(continued on next page)



C.G. Corlu et al. / Expert Systems With Applications 54 (2016) 170–192 183

Table 6 (continued)

Period # Market Method Significance levels Total # Rejections

0.005 0.01 0.05 0.95 0.99 0.995

Entire Sample 10 GLD 0 0 0 0 0 0 0

Johnson SU 1 1 0 0 1 2 5

Skewed-t 0 0 0 1 0 2 3

NIG 0 0 0 1 0 0 1

g-and-h 0 1 1 1 3 1 7

Normal 10 10 7 4 8 10 49

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hus, we conclude that the GLD also outperforms other distribu-

ions in terms of VaR performance.

. Conclusion

An analysis of the empirical distribution of equity returns is

mportant to both academic researchers and financial experts and

as many implications in the calculation of risk measures such as

alue-at-Risk (VaR) and the pricing of equity options. Despite its

opularity in finance applications, it is now well known that the

ormal distribution fails to capture certain characteristics of equity

eturn data. This has motivated researchers to investigate flexible

istributions with the ability to better represent stock return data.

evertheless, on a practical level, which distribution best suited for

odeling equity stock return data has remained an open question.

This paper contributes to the literature in the following two

ays: First, we investigate the relative performance of five widely

sed flexible distributions in finance, specifically, the generalized

ambda distribution (GLD), Johnson system of distributions, skewed

tudent-t distribution, normal inverse Gaussian (NIG) distribution,

nd g-and-h distribution, for modeling the distribution of daily

quity index returns of ten developed and ten emerging markets

ver the years 1979–2014. We also use the normal distribution as

benchmark in our analysis. Second, we evaluate the implication

f our results for the implementation of the well-known risk

easure, VaR. Our analyses support the empirical evidence in the

revious research that the behavior of equity returns are far from

eing normally distributed. We further show that the marginal

istribution of daily equity index returns can be well described by

he GLD. The relative stability of the GLD also makes it more favor-

ble among other distributions. From a practical expert intelligent

ystem perspective, the GLD has the following advantages: (i) The

epresentation of the GLD as an inverse cumulative distribution

unction makes it easy to quickly generate random variates from

he GLD in a Monte Carlo simulation, which is widely used in

isk management and the pricing of derivative securities. (ii)

he percentile representation of the GLD makes it convenient to

stimate the risk measures, such as VaR and expected shortfall.

In fitting the distributions to the observed data, we must

hoose among several fitting methods from the previous research.

n making this choice, we consider the availability of off-the-shelf

lgorithms that can be easily used by expert and intelligent sys-

ems. If such an algorithm is not available, then we implement the

asiest method. One potential limitation of our work is that the

mplemented fitting methods may not be the “best” methods. This

imitation could be overcome by performing an intensive study to

xamine the relative performance of fitting methods for each dis-

ribution considered in this paper. One such study that we are

ware of concerns the estimation of the parameters of the GLD

Corlu & Meterelliyoz, 2015).

Our focus in this paper is on the unconditional distribution of

quity returns. As stock returns typically exhibit temporal depen-

ence, considering conditional homoskedastic models such ARMA

nd conditional heteroskedastic models such as ARCH, GARCH, and

RMA-GARCH models, where the residuals in these models follow
exible distributions considered in this paper, could be of inter-

st from an expert and intelligent systems perspective. Another

uture research avenue is related to the stability of the probabil-

ty distributions. As mentioned in the paper, stability is impor-

ant, especially for portfolio analysis and risk management. An

nteresting problem to consider is how the performance of the

LD would compare with some of the popular stable distributions

sed in finance, such as stable Paretian laws. In addition, studying

he performance of the considered distributions using weekly and

onthly equity return data may produce different insights, just as

eekly and monthly data may exhibit different characteristics than

he daily data. Finally, another research idea is to extend our anal-

sis on the behavior of distributions at the extreme values of each

arket index return using risk measures other than VaR. Despite

eing a widely used risk measure, VaR has been criticized for not

roperly presenting the full picture of the risks a company faces.

n particular, a well-known shortcoming of VaR is that it is not

coherent risk measure (Artzner, Delbaen, Eber, & Heath, 1999).

n alternative coherent risk measure is the expected shortfall, also

nown as conditional VaR or tail loss. The extension of our analy-

is using the expected shortfall as the risk measure may yield ad-

itional insights.

ppendix A.1. Plots obtained using the Kolmogorov–Smirnov

est statistic

This Appendix presents the computed Kolmogorov–Smirnov

KS) test statistic for developed and emerging markets consid-

red in the paper under several distributions. Specifically, Fig. 1

resents the computed KS statistics for the developed markets of

ustralia, France, Japan, and Singapore; Fig. 2 presents the com-

uted KS statistics for the developed markets of Spain, Switzer-

and, UK, and US; Fig. 3 presents the computed KS statistics for

he emerging markets of Brazil, Chile, China, and India; and Fig. 4

resents the computed KS statistics for the emerging markets of

orea, Malaysia, Russia, and South Africa. In each figure, the KS

tatistic is compared with respect to the following six distribu-

ions: the skewed-t distribution, the generalized lambda distribu-

ion (GLD), the normal inverse Gaussian (NIG) distribution, Johnson

U family, g-and-h distribution, and normal distribution.

ppendix A.2. Plots obtained using the Anderson–Darling test

tatistic

This Appendix presents the computed Anderson–Darling (AD)

est statistic for developed and emerging markets considered in

he paper under several distributions. Specifically, Fig. 5 presents

he computed AD statistics for the developed markets of Aus-

ralia, France, Japan, and Singapore; Fig. 6 presents the computed

D statistics for the developed markets of Spain, Switzerland, UK,

nd US; Fig. 7 presents the computed AD statistics for the emerg-

ng markets of Brazil, Chile, China, and India; and Fig. 8 presents

he computed AD statistics for the emerging markets of Korea,

alaysia, Russia, and South Africa. In each figure, the AD statistic

s compared with respect to the following five distributions: the



184 C.G. Corlu et al. / Expert Systems With Applications 54 (2016) 170–192

Fig. 1. KS statistics for the developed markets of Australia, France, Japan, and Singapore.



C.G. Corlu et al. / Expert Systems With Applications 54 (2016) 170–192 185

Fig. 2. KS statistics for the developed markets of Spain, Switzerland, UK, and US.



186 C.G. Corlu et al. / Expert Systems With Applications 54 (2016) 170–192

Fig. 3. KS statistics for the emerging markets of Brazil, Chile, China, and India.



C.G. Corlu et al. / Expert Systems With Applications 54 (2016) 170–192 187

Fig. 4. KS statistics for the emerging markets of Korea, Malaysia, Russia, and South Africa.



188 C.G. Corlu et al. / Expert Systems With Applications 54 (2016) 170–192

Fig. 5. AD statistics for the developed markets of Australia, France, Japan, and Singapore.



C.G. Corlu et al. / Expert Systems With Applications 54 (2016) 170–192 189

Fig. 6. AD statistics for the developed markets of Spain, Switzerland, UK, and US.



190 C.G. Corlu et al. / Expert Systems With Applications 54 (2016) 170–192

Fig. 7. AD statistics for the emerging markets of Brazil, Chile, China, and India.



C.G. Corlu et al. / Expert Systems With Applications 54 (2016) 170–192 191

Fig. 8. AD statistics for the emerging markets of Korea, Malaysia, Russia, and South Africa.



192 C.G. Corlu et al. / Expert Systems With Applications 54 (2016) 170–192

H

H

H

H

H

H

J

J

K

K

L

L

M

M

O

O

P

P

P

P

R

R

R

R
S

S

T

T

skewed-t distribution, the generalized lambda distribution (GLD),

the normal inverse Gaussian (NIG) distribution, Johnson SU family,

and g-and-h distribution.

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	Empirical distributions of daily equity index returns: A comparison
	1 Introduction
	2 Description of the data
	3 Flexible distributions
	3.1 The Generalized Lambda distribution
	3.2 The Johnson translation system
	3.3 The skewed Student-t distribution
	3.4 The Normal Inverse Gaussian distribution
	3.5 The g-and-h distribution

	4 Performance and risk estimation
	4.1 Goodness-of-fit
	4.2 Risk estimation

	5 Conclusion
	Appendix A.1 Plots obtained using the Kolmogorov-Smirnov test statistic
	Appendix A.2 Plots obtained using the Anderson-Darling test statistic
	 References