Empirical Stylized Facts Modelling and Forecast Evaluations for Energy and Stock Markets

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Empirical Stylized Facts Modelling andForecast Evaluations for Energy andStock MarketsKim Leng Tana, Wen Cheong Chinb & Siow Hooi Tana

a Faculty of Management Multimedia University 63100 Cyberjaya,Selangor Malaysia. E-mail:b Faculty of Computing and Informatics Multimedia University 63100Cyberjaya, Selangor MalaysiaPublished online: 13 Nov 2014.

To cite this article: Kim Leng Tan, Wen Cheong Chin & Siow Hooi Tan (2014) Empirical StylizedFacts Modelling and Forecast Evaluations for Energy and Stock Markets, Journal of Statistics andManagement Systems, 17:4, 311-347, DOI: 10.1080/09720510.2014.914296

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*E-mail: [email protected]

Empirical Stylized Facts Modelling and Forecast Evaluations for Energy and Stock Markets

Kim Leng Tan 1,*

Wen Cheong Chin 2

Siow Hooi Tan 1

1 Faculty of ManagementMultimedia University63100 Cyberjaya, SelangorMalaysia2 Faculty of Computing and InformaticsMultimedia University63100 Cyberjaya, SelangorMalaysia

AbstractThis paper applies autoregressive heteroscedasticity (ARCH) family models for the

purpose of comparing stylized facts such as volatility clustering, leverage eff ect, long mem-

ory volatility and risk-return tradeoff for energy and stock markets. Empirical results have

found that the presences of volatility clustering in both markets and the impact of volatility

shocks to the conditional volatility display hyperbolic rather than exponential rate of decay.

Meanwhile, only stock markets denote the leverage eff ect, which implies that ‘bad’ news

has a greater impact on volatility than ‘good’ news at the same magnitude. Additionally,

empirical results also highlighted that Kerosene and Brent crude oil are the only energy

commodities exhibit risk-return tradeoff . For forecast evaluations, the FIAPARCH model in-

dicates superior out of sample forecasts over short and long time horizon for stock markets.

Nevertheless, FIAPARCH model suits better over long term as compared to short term for

energy markets. Finally, the Superior Predictive Ability (SPA) tests suggested that overall

asymmetric long memory GARCH models display higher forecasting accuracy than the stan-

dard GARCH models.

Keywords: Stylized facts, Energy markets, Stock markets, Comparison, ARCH.

Journal of Statistics & Management SystemsVol. 17 (2014), No. 4, pp. 311–347

©

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312 K. L. TAN, W. C. CHIN AND S. H. TAN

1. Introduction

In recent years, interests of researchers regarding stylized facts have

been constantly growing especially in energy markets and stock markets.

Empirical stylized facts refer to observations that are consistent and have

been made in many contexts (e.g., across a wide range of instruments,

markets and time periods). They are accepted as truths, to which theories

must fi t (Cont 2001; 2008, Mukherjee et al. 2011). The common empirical

stylized facts include fat-tail asset returns, volatility clustering, long mem-

ory volatility and leverage eff ect in global fi nancial markets. Theoretically,

stylized facts are obtained via applying a common denominator among

the properties observed in studies of diff erent markets and instruments.

However, stylized facts exhibit ineff iciency especially in distinguishing

among the various types of parametric models. In this study, we are focus-

ing on some common stylized facts such as volatility clustering, leverage

eff ects, long memory volatility and risk-return tradeoff for both the energy

and stock markets.

One of the empirical stylized facts is clustering volatility. Persistence

of volatility is a well established stylized fact and remains the salient

feature of many models (Chen and Ghysels, 2010). Volatility clustering

simply refers to the property that there are periods of high and low (con-

ditional or unconditional) variance. As noted by Mandelbrot (1963), “large

changes tend to be followed by large changes, of either sign, and small

changes tend to be followed by small changes”. Therefore, it is crucial in

predicting the price volatility while deeply understanding and managing

the risk associated with price volatility in order to allow better targeted

policy (Regnier, 2007). Commonly, volatility clustering and leptokurtosis

are observed in the fi nancial time series and magnitude for the response

of future volatility diff ers depending on whether past news are positive or

negative (Li, Hamill and Opong 2010; Black 1976 and Pagan 1996).

The relation between price movement and volatility is also an inter-

esting stylized facts that had been widely studied in fi nancial research.

Empirical research found that information arrival is positively correlated

with volatility according to Aguilar and Ringgenberg (2011). Information

may aff ect prices which are unknowable in the present and appears ran-

domly in the future (Lee and Lee, 2009; Kaufmann and Laskowski, 2005).

Hence, with only historical data for prices and returns, future prices are

best predicted using current prices. While for the return (or changes in

price), it tends to be zero and thus unpredictable. This is the essence of

weak-form Eff icient Market Hypothesis (EMH) which well related to ran-

dom walk.

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EMPIRICAL STYLIZED FACTS MODELLING 313

Negative asymmetric volatility or leverage eff ect is another stylized

facts commonly found in market returns in response to fl owing of new

information. In general, it is believed that positive or negative shocks tend

to aff ect the volatility of market returns. Specifi cally, volatility tends to

respond more to negative shocks or ‘bad’ news rather than the positive

shocks or ‘good’ news under the same magnitude. According to Paytakhti

and Shamsavari (2011), asymmetric is tentatively important in fi nancial

decision making especially in selecting portfolio, managing risk, hedging

and pricing the fi nancial derivatives. In term of volatility modeling, the

existence of leverage eff ects suggest that investor’s response to shocks is

not symmetric, which is contrary with the assumptions behind autore-

gressive conditional heteroscedasticity (ARCH) model and the gener-

alized autoregressive conditional heteroscedasticity (GARCH) model

(Ismail 2011). Hence, asymmetric ARCH models such as TGARCH model

and GJR GARCH model served this purpose better than the symmetric

GARCH model.

Long memory volatility is one of the interesting stylized facts in

fi nance. In an informationally ineff iciency market, long memory has

become one of the important considerations in the modeling specifi ca-

tion. It describes the correlation structure of a series at long lags whereby

such series normally characterized by distinct but non-periodic cyclical

patterns. According to Butler, Gerken and Okada (2011), long memory is

characterized by slow hyperbolic decay in the autocovariances of squared

returns or transformations of squared returns in the conditional volatility

of a univariate fi nancial time series.

Another well-known stylized fact is known as risk-return tradeoff

or risk premium which represents a factor including in the estimation of

shareholders equity, cost of debt (by adding the credit risk spread) used

in corporate fi nance and in the valuation of the fi nancial assets. Risk pre-

mium in fi nance is defi ned as the returns over and above risk free rate of

return that an investor expects in exchange for each additional unit of risk.

Markowitz portfolio theory indicates that rational investors only accept

additional risk if they expect a greater return and hence refers this greater

return as risk-return tradeoff . Literally, it is acceptable in general that ex-

pected returns of the market is positively and proportionally related to

the conditional volatility. In other words, risk-averse investors normally

require a higher expected return (higher risk premium) as the compensa-

tion of higher expected risks.

The above discussed stylized facts are the most common identifi ed

stylized facts. However, there are other stylized facts such as heavy tails,

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intermittency, aggregational Gaussianity and asymmetry in time scales

that are not discussed in this study. Empirical evidence showed that

there has an existence of fundamentally diff erent ubiquitous properties

called “stylized facts”, such as fat-tailed distribution, volatility clustering,

and scaling / multiscaling in fi nancial time series (He and Zheng, 2008).

Specifi cally the recent research on the similarities and diff erences between

the energy and stock market are such as Dellate and Lopez (2012) who

modeled the dependence between commodity and stock markets from

January 1990 to February 2012. They found that the dependence was time-

varying, symmetric and occurred most of the time. Besides that, Creti, Marc

and Valerie. (2012) studied the volatility together with the link between

stock and commodity markets and shown that correlations between com-

modity and stock markets was time-varying and highly volatile. However,

Gorton and Rouwenhorst (2005), Chong and Miff re (2010) and also Hong

and Yogo (2009) reached similar conclusions over the more recent studies.

These studies concluded that equity and commodity future contracts had

the same average returns along with a negative correlation for bonds and

equities and thus presented less volatile returns. On the other hand, study

of Chong and Miff re (2010) showed that there was weakness correlation

between commodity and conventional stock and bond returns which did

not happened in the study of Silvennoinen and Thorp (2010). Nevertheless,

Gorton and Rouwenhorst (2005) indicated also risk premium on commod-

ity futures was essentially same as equities. However, there is still lack of

consensus in the empirical papers investigating and comparing the simi-

larities and diff erences of stylized facts between energy and stock markets.

This paper attempts to model and compare some common stylized

facts between energy and stock markets using the ARCH family models.

After that, a series of forecast evaluation measurements are conducted

to select the most appropriate models which suit both the fi nancial mar-

kets. The rest of the paper is organized as follows. Part 2 literally reviews

the previous related studies and Part 3 explains the data involved and

methods used in this study. Meanwhile, Part 4 presents and discusses the

empirical results and at last but not least, Part 5 of this paper provides

conclusions and suggestions for future expansion.

2. Methodology

2.1. Data Source

The study period starts from January 1999 to December 2007 whereby

the data for last year, (i.e., January 2007 to December 2007) is reserved for

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out-of-sample forecasting evaluation. Dataset for this study consists of the

following fi nancial markets.

Energy Commodities Stock Indices

1. Brent Crude Oil 1. FTSE 100 (UK)

2. West Texas 2. FTSE Bursa

Intermediate (WTI) Crude Oil Malaysia KLCI (Malaysia)

3. U.S. Gulf Coast Kerosene 3. Nasdaq 100 (US)

The dataset for energy market commodities are obtained from the

United States Energy Information Administration (EIA) whereas the

stock markets dataset are obtained from DataStream database. This study

focuses on oil markets as oil plays a crucial role in the world economy

and has been the driver of industrialization and modernization. Due

to that, price volatility of this commodity will signifi cantly aff ect the

stock markets. Hence, the emerging stock markets such as FTSE 100,

KLCI and Nasdaq 100 are taking into consideration for the purpose of

comparison.

2.2. Model Framework

The return series, tr that is mainly used for conditional mean is de-

fi ned as the percentage continuously compounded inter-day in terms of

close-to-close index on consecutive trading days:

In Inr P P100 , ,t close t close1= - -t ^ h (1)

where Pt,close is the daily closing index and is the return for commodities or

stocks indices for 1,2, .,…t T= .

The Generalized ARCH model is one of the most commonly used

time series models employed in the recent fi nance literature for studying

the volatility. It is an extended model by Bollerslev (1986) from ARCH

model introduced by Engle (1986) as the fi rst model providing systematic

framework for volatility modeling. The standard GARCH (p,q) model can

be expressed as

r n f= +t tt (2)

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where nt denotes the conditional mean and zf v=t t t with ( , )z NID 0 1+t

or ( , )z GED 0 1+t for our study. Meanwhile, conditional variance, 2vt is

given by

2 2 2v ~ af bv= + +t t t1 1- - , (3)

with the restrictions of , ,0 0 02 2 2~ a b and .11a b+

The standard GARCH model only focuses on short-term volatility

specifi cations and forecasting. In order to include the long memory styl-

ized fact in the modeling, fractionally integrated generalized ARCH (FI-

GARCH) model has been introduced by Baillie, et al. (1996). The standard

FIGARCH (1,d,1) model is given by

( ) ( ) ( )L L L1 1 1 1 d2 2 1 2v ~ bv b { f= + + - - - --t t T1- 6 @ (4)

where , ,0 1 12 1 1~ { b and .d0 1# # L represents the lag operation

and d is the parameter for fractional integration which allows autocorre-

lations to decay at a slow hyperbolic rate, and therefore characterize the

long memory property in volatility. When 0,d 2 the main characteristic

of this model is non-stationary. The non-integer value of d can be written

in a binomial expansion as follow:

( 1) ( 1)

( )L

d kkd

L11d

k 0x x

x-

+ - +=

+3

=

k^ h /

( ) ( ) ( )dL d d L d d d L121 1 1 1 2

6f= - - - - - -2 3

k ( )c d Lk

=3

=1

k/ (5)

Fractionally Integrated Asymmetric Power ARCH (FIAPARCH)

model was developed by Tse (1998) in order to allow news impact and

long memory volatility. Specifi cations for FIAPARCH (1,d,1) model can be

represented by

( ) ( )

( )L

k d kd

L11 1

1d

k 0

vx x

x- =

+ - ++3

d

=t

k^ h / (6)

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with the restriction of , , , ,0 1 0 1 1 12 1 2 1 1 1~ { d b c- and

.d0 1# # When 0c = and 2c = , the FIAPARCH process will be reduced

to FIGARCH. Another advantage of this specifi cation is the volatility can

be expressed by power factor of standard deviation. Hence, the volatility

representation is not restricted to variance and standard deviation.

2.3. Forecasting Evaluations

The performance of forecasts are normally based on the out-of-sam-

ple forecast evaluations whether the volatility model is able to control the

over fi tting or parameterization problems. Various forecasting criteria or

loss functions can be considered to assess the predictive accuracy of vola-

tility models. For this purpose, daily actual volatility (variance) is assessed

using the daily square returns, r2t and indicated as whereas the estimated

volatility forecast is denoted by 2vtt . Six diff erent accurate loss functions

are used to evaluate the forecast performance of the respected individual

time series :

MSE nt

n1 2 2

1

2v v= --

=t t

t^ h/ (7)

MSE nt

n1 2 2

1

2v v= --

=t t

t^ h/ (8)

/HMSE n 1t

n1 2 2 2

1

v v= --

=t t

t^ h/ (9)

/HMSE n 1t

n1 2 2 2

1

v v= --

=t t

t^ h/ (10)

In( ) /QLIKE nt

n1 2 2 2

1

v v v= -

=t t tt t^ h/ (11)

In /R LOG nt

n2 1 2 2

1

v v= -

=t t

2t6 @/ (12)

where

n – Number of forecasting data;

MSE – Mean Square Error;

MAE – Mean Absolute Error;

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HMSE – Hetoroscedasticity Mean Square Error;

HMAE – Hetoroscedasticity Mean Absolute Error;

QLIKE – Quasi Likelihood; and

R2LOG – Return Square Logarithm

The Superior Predictive Ability (SPA) test can be implemented to

compare the performances of two or more forecasting models. This test

is extended from White (2000) by Hansen (2005). Besides that, SPA test

can also be applied to economic theory that places restrictions on the pre-

dictability of certain variables, such as eff icient market hypothesis (EMH)

(Sullivan, Timmermann, and White, 1999; Hansen, 2005). According to

Wei, Wang and Huang (2010), this test has been proven to be more ro-

bust and have good power properties than other approaches. Hence, it has

been applied by Koopman, Jungbacker and Hol (2005) and Hansen and

Lunde (2005) in their research.

In SPA test, a pre-specifi ed loss function is used to evaluate the fore-

casts, and model that produces the smallest expected loss is the ‘best’ fore-

cast model. The loss function relative to the benchmark model at time t,

can be expressed as:

, , 1, .., 1, ,, … ……d L Y Y Y Y m t nL k= =- =, ,k t t t k t,t0t t^ ^h h (13)

where ( , )L Y Y ,t t0t is the value of the loss function at time t for a benchmark

model and k( , )L Y Y ,t tt is the value of loss function at time t for another com-

petitive model, k. The issue here is whether any of the models, 1, ..,…k m= are outperform as compared to benchmark model. In order to analyze this

issue, testable hypothesis that the benchmark model is the best forecasting

model can be formulated as

: 0 Or : [ ] 0 for all 1, ..,…maxH H E d k m0 # #n =,k t0k (14)

One way to test this hypothesis according to Hansen (2005) is to con-

sider the test statistic of

k

T n d/k

1 2

~=n

k

maxSPA

t (15)

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where kd n d1

t

n

1

==

,k t/ and kvar n d/2 1 2~ =kt t ^ h is a consistent estimator of the

asymptotic variance klim v r n da /2 1 2~ =k n "3 ^ h. Under the regularity condi-

tion, it holds that

k k

n T d/ k p1 2

~ ~n

=-n

k kmax max kSPA

t (16)

which is greater than zero if and only if 0>nk for some k. Therefore, we

can test H0 using the test statistics TnSPA .

3. Results and Discussions

3.1. Descriptive Statistics

Table 1 represents the descriptive statistics for daily returns of en-

ergy markets and stock markets. Sample mean of the return series for

both markets are positive with the values close to zero. As a comparison,

sample mean of stock markets is relatively smaller than energy markets.

Meanwhile, standard deviation of energy markets is slightly higher as

compared to stock markets which implied that energy markets are more

volatile than stock markets for the selected studies period. Overall, energy

markets indicate higher kurtosis and heavier tails as compared to stock

markets. Finally, Jarque-Bera statistics indicate that the null hypothesis of

normality is rejected at one percent signifi cance level.

3.2. Estimation Results

Table 2 to Table 5 display the estimation results for both the markets

using normal and generalized error (GED) distributions of GARCH, GJR

GARCH, FIGARCH and FIAPARCH models. The fi rst-order autocorrela-

tion AR (1) indicates a mixture of positive and negative parameters in both

energy and stock markets. For energy markets, all the fi rst lag of returns is

insignifi cant at 5 percent level under normal and GED distribution. On the

other hand, stock markets denote signifi cance of joint estimated AR (1) co-

eff icient at 1 percent or 5 percent signifi cant level under both normal and

GED distributions. However, Nasdaq 100 displays insignifi cant coeff icient

for FIAPARCH model under normal and GED distributions.

For tail distribution analysis, GED distributions of GARCH, GJR

GARCH, FIGARCH and FIAGRACH models exhibit heavy-tails which

could be highlighted by degrees of freedom (v). All energy commodities

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Tabl

e 1

Des

crip

tive

Stat

istic

s fo

r Dai

ly R

etur

ns

E

ner

gy

Mark

ets

Sto

ck M

ark

ets

B

ren

t K

ero

sen

e W

TI

FT

SE

100

KL

CI

Nasd

aq

100

Mea

n

0.0

96200

0.0

95875

0.0

92913

0.0

038828

0.0

40701

0.0

056332

Std

. D

ev.

2.3

422

2.8

029

2.4

134

1.1

378

1.0

464

2.2

071

Min

imu

m

-19.8

91

-16.7

16

-17.0

92

-5.8

853

-6.3

422

-10.2

39

Max

imu

m

12.8

53

15.6

75

12.4

43

5.9

026

5.8

505

17.2

03

Sk

ewn

ess

-0.4

5577

0.2

8057

-0.5

5487

-0.1

9896

-0.1

6044

0.2

5792

Ku

rto

sis

3.7

973

2.9

225

3.6

746

2.8

917

5.8

830

3.6

434

Jarq

ue-

Ber

a

1286.7

**

719.5

5**

1212.1

**

717.1

3**

2847.8

**

1131.8

**

No

. o

f O

bs.

2271

2201

2225

2273

2217

2257

Not

es: **

in

dic

ate

s 1 p

erce

nt

lev

el o

f si

gn

ifi c

an

ce

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Tabl

e 2

GA

RC

H M

odel

Est

imat

ion

Res

ults

E

ner

gy

Mark

et

Sto

ck M

ark

et

B

ren

t K

ero

sen

e W

TI

FT

SE

100

KL

CI

Nasd

aq

100

N

orm

al

GE

D

No

rmal

GE

D

No

rmal

GE

D

No

rmal

GE

D

No

rmal

GE

D

No

rmal

GE

D

Con

ditio

nal M

ean

0a

0.1

43090**

0.1

57778**

0.1

29459*

0.1

55862**

0.1

13020*

0.1

60110**

0.0

33547*

0.0

39307*

0.0

44103*

0.0

35682*

0.0

48219

0.0

50484

(0

.049683)

(0.0

53553)

(0.0

56291)

(0.0

55483)

(0.0

57115)

(0.0

54186)

(0.0

16909)

(0.0

17007)

(0.0

19211)

(0.0

17447)

(0.0

30600)

(0.0

30781)

a1

0.0

14980

0.0

06840

-0.0

20856

-0.0

31713

-0.0

28060

-0.0

40036

-0.0

61331**

-0

.056296*

0.1

81204**

0.1

32625**

-0

.050284*

-0.0

47449*

(0

.023001)

(0.0

20903)

(0.0

23734)

(0.0

22819)

(0.0

24986)

(0.0

21737)

(0.0

23278)

(0.0

22932)

(0.0

26697)

(0.0

24913)

(0.0

22027)

(0.0

21569)

AR

CH

-M

-0.2

26986

-0.2

99995

-0.2

78629*

-0.2

68262*

-0.0

46868

-0.0

35800

0.0

18441

0.0

11713

0.0

61661

0.0

05198

0.0

17087

0.0

14019

(0

.19046)

(0.2

0324)

(0.1

3014)

(0.1

2640)

(0.1

9410)

(0.2

2140)

(0.0

57835)

(0.0

59653)

(0.0

69743)

(0.0

33-3

50)

(0.0

50660)

(0.0

49984)

Con

ditio

nal V

aria

nce

0a

0.3

72137**

0.3

52675**

0.3

29556**

0.3

31335**

0.4

87739

0.2

37341

0.0

10804*

0.0

10463*

0.0

03949

0.0

04928

0.0

03583

0.0

03415

(0

.10809)

(0.1

1661)

(0.1

1143)

(0.1

1836)

(0.5

2757)

(0.2

6410)

(0.0

04230)

(0.0

040618)

(0.0

037115)

(0.0

043584)

(0.0

034632)

(0.0

033

955)

a1

0.0

79755**

0.0

69595**

.0

82882**

0.0

79417*

0.0

75455

0.0

45836

0.0

98261**

0.0

96296**

0.0

69431*

0.0

76979*

0.0

45451**

0.0

44925**

(0

.022

905)

(0.0

19690)

0(0

.015730)

(0.0

15874)

(0.0

46874)

(0.0

31381)

(0.0

17496)

(0.0

16707)

(0.0

33577)

(0.0

37748)

(0.0

097013)

(0.0

094923)

b1

0.8

56017**

0.8

67933**

0.8

76034**

0.8

78646**

0.8

44132**

0.9

13532**

0.8

94016**

0.8

96169**

0.9

29528**

0

.920842**

0.9

53664**

0.9

54223**

(0

.02691)

(0.0

32269)

(0.0

24829)

(0.0

264667)

(0.1

2828)

(0.0

73723)

(0.0

17846)

(0.0

17134)

(0.0

33750)

(0.0

38170)

(0.0

093398)

(0.0

031354)

ab

+1

1

0.9

3577

0.9

3753

0.9

5892

0.9

5806

0.9

1959

0.9

5937

0.9

9228

0.9

9247

0.9

9896

0.9

9782

0.9

9911

0.9

9915

v

1.5

16652**

1.6

15527**

1.4

14800**

1.7

61313**

1.2

92311**

1.8

60363**

(0.0

80806)

(0

.082417)

(0

.075042)

(0

.093432)

(0

.058245)

(0

.10218)

(Con

tinue

d )

Dow

nloa

ded

by [

Kim

Len

g T

an]

at 1

9:32

20

Mar

ch 2

015

Mod

el S

elec

tion

Lo

g (

L)

-4589.5

19

-4565.9

53

-4718.6

26

-4707.1

64

-4534.2

15

-4495.5

80

-2771.2

84

-2767.4

85

-2491.4

68

-2432.9

03

-4025.1

02

-4024.0

12

AIC

4.

537796

4.5

15509

4.8

44744

4.8

34014

4.5

94347

4.5

56256

2.7

48796

2.7

46025

2.5

35773

2.4

77302

4.0

18048

4.0

17958

SIC

4.5

51656

4.5

32142

4.8

59040

4.8

51170

4.6

08489

4.5

73226

2.7

62684

2.7

62691

2.5

49956

2.4

94321

4.0

32016

4.0

34720

Dia

gn

ost

ic

Q(1

0)

6.4

0047

6.7

1329

4.3

3626

4.8

4011]

6.3

3255

6.9

5763.

58268

3.3

0949

14.7

272

24.0

471**

9.4

5223

9.3

8792

[0

.6992639]

[0.6

669418]

[0.8

879207]

[0.8

480176

[0.7

062293

[0.6

415311]

[0.9

366752]

[0.9

507513]

[0.0

987082]

[0.0

042274]

[0.3

9661

87]

[0.4

022630]

Q2(

10)

15.4

262

17.0

140*

3.3

8038

3.4

5297

5.6

5645

10.2

139

15.1

265

15.3

595

23.3

488**

18.9

363*

16.4

769*

16.5

579*

[0

.0513684]

[0.0

299634]

[0.9

082723]

[0.9

028089]

[0.6

856514]

[0.2

503310]

[0.0

567325]

[0.0

525211]

[0.0

029442]

[0.0

152042]

[0.0

36

0414]

[0.0

50568]

AR

CH

(10)

1.4

791

1.6

339

0.3

3841

0.3

4398

0.5

2777

0.9

5327

1.4

930

1.4

730

2.2

326*

1.8

176

1.6

222

1.6

330

[0

.1408]

[0.0

912]

[0.9

708]

[0.9

302]

[0.8

716]

[0.4

827]

[0.1

432]

[0.1

356]

[0.0

139]

[0.0

528]

[0.0

944]

[0.0

915]

Not

es: *a

nd

**

den

ote

5%

an

d 1

% s

ign

ifi c

ian

t le

vel

res

pec

tiv

aly

. Ja

rqu

e-B

era s

tati

stic

s te

st f

or

the

nu

ll h

yp

oth

esis

of

no

rmali

ty i

n t

he

sam

pel

ret

urn

dis

trib

uti

on

. T

he

nu

mb

er i

np

are

nth

eses

are

erro

r o

f th

e es

tim

ati

on

Lo

g (

L)i

s th

e lo

gari

tham

max

imu

m l

ick

lih

oo

d f

un

ctio

n v

alu

e. A

IC i

s th

e av

erag

e A

kaik

e in

form

ati

on

cri

teri

on

an

d S

IC i

s th

e av

erag

e S

chw

arz

in

form

ati

on

cri

teri

on

.

Q(1

0)

an

d Q

2 (

10)

are

th

e L

jun

g-B

ox

Q-s

tati

stic

s o

f o

rder

10 c

om

pu

ted

on

th

e st

an

dard

ized

res

idu

als

an

d s

qu

red

sta

nd

rad

ized

res

idu

als

res

pec

tiv

ely

AR

CH

(10)

is t

he

no

n -

het

ero

sces

dast

icit

y

stati

stic

of

ord

er 1

0 P

-valu

es o

f th

e st

ati

stic

are

rep

ort

ed i

n s

qu

are

bra

cket

s. T

he

AR

CH

-in

Mea

n c

oeff

ici

ent

is e

xclu

ded

fro

m t

he

esti

mati

on

mo

del

s.

E

ner

gy

Mark

et

Sto

ck M

ark

et

B

ren

t K

ero

sen

e W

TI

FT

SE

100

KL

CI

Nasd

aq

100

N

orm

al

GE

D

No

rmal

GE

D

No

rmal

GE

D

No

rmal

GE

D

No

rmal

GE

D

No

rmal

GE

D

Tabl

e 2

(Con

tinu

ed)

Dow

nloa

ded

by [

Kim

Len

g T

an]

at 1

9:32

20

Mar

ch 2

015

Tabl

e 3

GJR

-GA

RC

H M

odel

Est

imat

ion

Res

ults

E

ner

gy

Mark

et

Sto

ck M

ark

et

B

ren

t K

ero

sen

e W

TI

FT

SE

100

KL

CI

Nasd

aq

100

N

orm

al

GE

D

No

rmal

GE

D

No

rmal

GE

D

No

rmal

GE

D

No

rmal

GE

D

No

rmal

GE

D

Con

ditio

nal M

ean

0

a

0.1

10970**

0.1

39370**

0.1

53594**

0.1

73140**

0.0

97487

0.1

51366**

0.0

06590

0.0

13137

0.0

32519

0.0

28979

0.0

22619

0.0

26517

(0

.051681)

(0.0

49630)

(0.0

57961)

(0.0

57132)

(0.0

53057)

(0.0

54309)

(0.0

16843)

(0.0

17244)

(0.0

19177)

(0.0

17136)

(0.0

29813)

(0.0

30308)

a1

0.0

16

342

0.0

07532

-0.0

22373

-0.0

32968

-0.0

32624

-0.0

42445

-0.0

64726**

-0

.059776**

0.1

85002**

0

.135093**

-0

.051741*

-0.0

48796*

(0

.023

097)

(0.0

20997)

(0.0

23634)

(0.0

22523)

(0.0

24485)

(0.0

22862)

(0.0

23536)

(0.0

22931)

(0.0

26264)

(0.0

13515)

(0.0

22109)

(0.0

21560)

AR

CH

-M

-0.2

99292

-0.3

48755

-0.3

03561*

-0.2

95049*

0.0

08290

-0.0

04208

-0.0

01824

0.0

00465

0.0

43187

-0.0

02268

-0.0

05148

-0.0

06760

(0

.18282)

(0.1

9117)

(0.1

3524)

(0.1

3260)

(0.2

0792)

(0.0

47946)

(0.0

59715)

(0.0

60400)

(0.0

66846)

(0.0

67246)

(0.0

47285)

(0.0

47098)

Con

ditio

nal V

aria

nce

0a

0.5

220

85**

0.4

72385**

0.3

05784**

0.5

96861

0.2

62579

0.0

12116**

0.0

11647**

0.0

04892

0.0

059570

.005006

0.0

04749

0.3

04337**

(0

.18

944)

(0.1

9300)

(0.1

0465)

(0.1

1069)

(0.7

6285)

(0.3

6061))

(0

.0039528)

(0.0

037548)

(0.0

040523)

(0.0

046490)

(0.0

039725)

(0.0

038976)

a1

0.0

25

081

0.0

27091

0.1

04529**

0.1

00526**

0.0

53981

0.0

25790

0.0

07595

0.0

04351

0.0

54482*

0.0

64381*

0.0

18829*

0.0

17944*

(0

.017

777)

(0.0

16364)

(0.0

23581)

(0.0

22834)

(0.0

35465)

(0.0

27923)

(0.0

15020)

(0.0

13127)

(0.0

21988)

(0.0

26245)

(0.0

093910)

(0.0

090365)

b1

0.8

28795**

0.8

44345**

0.8

79621**

0.8

82243**

0.8

22430**

0.9

10957**

0.9

19679**

0.9

21113**

0.9

21972**

0.9

12386**

0.9

53929**

0.9

54364**

(0

.045588)

(0.0

47538)

(0.0

23746)

(0.0

25236)

(0.1

7639)

(0.0

96346)

(0.0

14849)

(0.0

14384)

(0.0

32668)

(0.0

36480)

(0.0

085716)

(0.0

083990)

c

0.1

05354*

0.0

83619

-0.0

41126

-0.0

39546

0.0

46717

0.0

32740

0.1

17280**

0.1

20783**

0.0

45784

0.0

43017

0.0

51827**

0.0

52767**

(0

.051653)

(0.0

43007)

(0.0

25171)

(0.0

23828)

(0.0

65263)

(0.0

33196)

(0.0

26385)

(0.0

24719)

(0.0

30327)

(0.0

28707)

(0.0

16872)

(0.0

16507

v

1.5

43084**

1.6

21099**

1.4

15141**

1.7

99136**

1.3

01592**

1.8

85177**

(0.0

78735)

(0

.081370)

(0

.074339)

(0

.11151)

(0

.057871)

(0

.10510)

(Con

tinue

d )

Dow

nloa

ded

by [

Kim

Len

g T

an]

at 1

9:32

20

Mar

ch 2

015

E

ner

gy

Mark

et

Sto

ck M

ark

et

B

ren

t K

ero

sen

e W

TI

FT

SE

100

KL

CI

Nasd

aq

100

N

orm

al

GE

D

No

rmal

GE

D

No

rmal

GE

D

No

rmal

GE

D

No

rmal

GE

D

No

rmal

GE

D

Tabl

e 3

(Con

tinu

ed)

Mod

el S

elec

tion

Lo

g (

L)

-4580.6

97

-4560.9

65

-4716.5

84

-4705.5

80

-4532.6

46

-4494.3

55

-2749.6

49

-2746.8

81

-2486.2

01

-2430.1

06

-4014.6

03

-4013.8

71

AIC

4.5

30071

4.5

11570

4.8

43676

4.8

33415

4.5

93771

4.5

56027

2.7

28366

2.7

26614

2.5

31438

2.4

75476

4.0

08577

4.0

08845

SIC

4.5

46703

4.5

30974

4.8

60832

4.8

53430

4.6

10741

4.5

75826

2.7

45031

2.7

46058

2.5

48458

2.4

95332

4.0

25339

4.0

28399

Dia

gnos

tic

Q(1

0)

6.6

7097

6.6

0776

4.5

1500

5.0

6105

6.0

5973

6.6

3592

3.2

0185

2.9

6727

15.3

269

24.8

722**

8.3

6791

8.2

8251

[0

.6713323]

[0.6

778826]

[0.8

743756]

[0.8

289544]

[0.7

339256]

[0.6

749656]

[0.9

557514]

[0.9

655784]

[0.0

823400]

[0.0

031158]

[0.4

97

5369]

[0.5

059462]

Q2(

10)

14.9

621

16.0

488

2.8

5882

2.8

7003

5.2

7124

9.0

5208

12.4

721

12.3

612

17.0

493*

14.4

452

18.1

232*

18.3

033*

[0

.0598861]

[0.0

416875]

[0.9

429024]

[0.9

422468]

[0.7

282316]

[0.3

379215]

[0.1

313503]

[0.1

358028]

[0.0

296002]

[0.0

708745]

[0.0

20

3216]

[0.0

190639]

AR

CH

(10)

1.4

302

1.5

322

0.2

8396

0.2

8283

0.4

8535

0.8

4177

1.1

965

1.1

856

1.6

424

1.3

892

1.8

089

1.8

324

[0

.1606]

[0.1

217]

[0.9

849]

[0.9

851]

[0.9

005]

[0.5

882]

[0.2

882][

0.2

956]

[0.0

890]

[0.1

790]

[0.0

542]

[0.0

505]

Not

es :

* an

d *

* d

eno

te 5

% a

nd

1%

sig

nifi

can

ce l

evel

res

pec

tiv

ely.

Jarq

ue-

Ber

a s

tati

stic

s te

sts

for

the

nu

ll h

yp

oth

esis

of

no

rmali

ty i

n t

he

sam

ple

ret

urn

dis

trib

uti

on

.T

he

nu

mb

ers

in p

are

nth

eses

are

sta

nd

ard

err

or

of

the

esti

mati

on

. L

og

(L)

is t

he

log

ari

thm

max

imu

m l

ikel

iho

od

fu

nct

ion

valu

e. A

IC i

s th

e av

erag

e A

kaik

e in

form

ati

on

cri

teri

on

an

d S

IC i

s th

e av

erag

e S

chw

arz

in

form

ati

on

crit

erio

n. Q

(10)

an

d Q

2 (1

0)

are

th

e L

jun

g-

Bo

x Q

-sta

tist

ics

of

ord

er 1

0 c

om

pu

ted

on

th

e st

an

dard

ized

res

idu

als

an

d s

qu

are

d s

tan

dard

ized

res

idu

als

res

pec

tiv

ely.

AR

CH

(10)

is t

he

no

n-

het

ero

sces

dast

icit

y s

tati

stic

of

ord

er 1

0. P

-valu

es o

f th

e st

ati

stic

s are

rep

ort

ed i

n s

qu

are

bra

cket

s. T

he

AR

CH

-in

Mea

n c

oeff

ici

ent

is e

xclu

ded

fro

m t

he

esti

mati

on

mo

del

s.

Dow

nloa

ded

by [

Kim

Len

g T

an]

at 1

9:32

20

Mar

ch 2

015

Tabl

e 4

FIG

AR

CH

Mod

el E

stim

atio

n R

esul

ts

E

ner

gy

Mark

et

Sto

ck M

ark

et

B

ren

t K

ero

sen

e W

TI

FT

SE

100

KL

CI

Nasd

aq

100

N

orm

al

GE

D

No

rmal

GE

D

No

rmal

GE

D

No

rmal

GE

D

No

rmal

GE

D

No

rmal

GE

D

Con

ditio

nal M

ean

0a

0.1

23563*

0.1

46168**

0.1

29052*

0.1

57

037**

0.1

18795

0.1

66676**

0.0

35947*

0.0

41501*

0.0

37990*

0.0

33521

0.0

43984

0.0

47697

(0

.053

219)

(0.0

50877)

(0.0

57080)

(0.0

55727)

(0.0

54043)

(0.0

53928)

(0.0

16816)

(0.0

16895)

(0.0

17940)

(0.0

18427)

(0.0

31756)

(0.0

31812)

a1

0.0

11131

0.0

04763

.-0.0

18962

-0.0

31596

-0.0

23626

-0.0

37194

-0.0

59840**

-0

.055530*

0.1

81909**

0

.136291**

-0

.045961*

-0.0

43353*

(0

.022

342)

(0.0

21131)

(0.0

23988)

(0.0

22646)

(0.0

25623)

(0.0

22466)

(0.0

22974)

(0.0

22714)

(0.0

27468)

(0.0

27261)

(0.0

21666)

(0.0

21175)

AR

CH

-M

-0.1

37806

-0.1

89733

-0.2

37220*

-0.2

32897*

-0.0

86780

-0.0

72437

0.0

01304

-0.0

5978

0.0

67494

0.0

01848

0.0

18578

0.0

14074

(0

.15288)

(0.1

6862)

(0.1

0636)

(0.1

0858)

(0.1

8087)

(0.1

9371)

(0.0

57174)

(0.0

58760)

(0.0

77380)

(0.0

77366)

(0.0

52004)

(0.0

51356)

Con

ditio

nal V

aria

nce

0a

0.6

22

446

0.7

03181*

0.4

20934

0.4

97432

0.8

69976

0.8

59074

0.0

13390

0.0

13267

0.0

47403

0.0

45011

0.0

06576

0.0

05945

(0

.32

737)

(0.3

3416)

(0.3

8878)

(0.4

5086)

(0.5

6989)

(0.4

6620)

(0.0

10619)

(0.0

10290)

(0.0

26906)

(0.0

30844)

(0.0

17772)

(0.0

18062)

a1

0.0

81337

0.1

03600

0.2

39290

0.2

07171

0.3

74929

0.3

49225

0.1

07179

0.1

04599

-0.0

98200

-0.0

82978

0.1

35570**

0.1

32399**

(0

.082

777)

(0.0

84812)

(0.1

5151)

(0.1

6700)

(0.2

9663)

(0.2

2518)

(0.0

76091)

(0.0

70408)

(0.3

2336)

(0.4

0587)

(0.0

48540)

(0.0

47430)

b1

0.3

45532**

0.3

39562**

0.5

09212

0.4

58823

0.4

50339

0.4

50972*

0.5

47468**

0.5

55668**

0.0

50850

0.0

59926

0.6

96360**

0.6

88611**

(0

.11278)

(0.1

0746)

(0.2

7261)

(0.2

7045)

(0.2

7857)

(0.2

1369)

(0.1

0320)

(0.0

96175)

(0.3

3721)

(0.4

2755)

(0.1

1235)

(0.1

0786)

ab

+1

1

0.2

820

92**

0.2

52930**

0.3

74040*

0.3

43007*

0.1

85571**

0.1

85382**

0.5

18939**

0.5

25280**

0.2

87813**

0.2

87895**

0.5

52571*

0.5

45574**

d (0

.10159)

(0.0

75710)

(0.1

7493)

(0.1

4409)

(0.0

64980)

(0.0

52378)

(0.0

76902)

(0.0

75318)

(0.0

31480)

(0.0

36702)

(0.1

2435)

(0.1

1511)

v

1.5

21736**

1.5

83298**

1.4

34976**

1.7

70924**

1.3

51930**

1.8

53716**

(0.0

78333)

(0

.080723)

(0

.077467)

(0

.092407)

(0

.060660)

(0

.098700)

(Con

tinue

d )

Dow

nloa

ded

by [

Kim

Len

g T

an]

at 1

9:32

20

Mar

ch 2

015

E

ner

gy

Mark

et

Sto

ck M

ark

et

B

ren

t K

ero

sen

e W

TI

FT

SE

100

KL

CI

Nasd

aq

100

N

orm

al

GE

D

No

rmal

GE

D

No

rmal

GE

D

No

rmal

GE

D

No

rmal

GE

D

No

rmal

GE

D

Tabl

e 4

(Con

tinu

ed)

Mod

el S

elec

tion

Lo

g (

L)

-4584.8

65

-4562.4

07

-4726.7

95

-4713.1

90

-4528.1

90

-4493.2

11

-2766.2

30

-2762.8

41

-2463.9

13

-2415.1

17

-4022.9

67

-4021.7

65

AIC

4.5

34188

4.5

12995

4.8

54149

4.8

41220

4.5

89261

4.5

54869

2.7

44782

2.7

42417

2.5

08800

2.4

60250

4.0

16916

4.0

16714

SIC

4.5

50820

4.5

32399

4.8

71304

4.8

61235

4.6

06231

4.5

74668

2.7

61448

2.7

61861

2.5

25819

2.4

80107

4.0

33677

4.0

36269

Dia

gnos

tic

Q(1

0)

7.0

1565

7.3

5293

4.1

6894

4.7

6615

5.9

8324

6.6

6511

3.9

1710

30.7

1023

3.7

10273

9.5

133*

10.1

370

10.0

565

[0

.6354895]

[0.6

004256]

[0.8

999453]

[0.8

541986]

[0.7

415944]

[0.6

719404]

[0.9

167836]

[0.9

294294]

[0.1

654627]

[0.0

211652]

[0.3

39

5116]

[0.3

459279]

Q2(

10)

12.8

788

14.6

534

3.1

4120

3.0

4444

4.8

5688

5.4

9848

11.2

527

17.0

207

4.5

0056

4.5

0056

10.9

917

10.7

674

[0

.1160900]

[0.0

662458]

[0.9

251866]

[0.9

315376]

[0.7

727574]

[0.7

032080]

[0.1

877875]

[0.5

216832]

[0.8

086767]

[0.8

093771]

[0.2

02

1671]

[0.2

152277]

AR

CH

(10)

1.2

946

1.4

616

0.3

1503

0.3

0060

0.4

6636

0.5

2223

1.1

459

1.1

511

0.4

4263

0.4

4789

1.0

883

1.0

655

[0

.2276]

[0.1

477]

[0.9

776]

[0.9

812]

[0.9

123]

[0.8

756]

[0.3

236]

[0.3

199]

[0.9

258]

[0.9

229]

[0.3

673]

[0.3

856]

Not

es : *

an

d *

* d

eno

te 5

% a

nd

1%

sig

nifi

can

ce l

evel

res

pec

tiv

ely.

Jarq

ue-

Ber

a s

tati

stic

s te

sts

for

the

nu

ll h

yp

oth

esis

of

no

rmali

ty i

n t

he

sam

ple

ret

urn

dis

trib

uti

on

.T

he

nu

mb

ers

in p

are

nth

eses

are

sta

nd

ard

err

or

of

the

esti

mati

on

. L

og

(L)

is t

he

log

ari

thm

max

imu

m l

ikel

iho

od

fu

nct

ion

valu

e. A

IC i

s th

e av

erag

e A

kaik

e in

form

ati

on

cri

teri

on

an

d S

IC i

s th

e av

erag

e S

chw

arz

in

form

ati

on

crit

erio

n. Q

(10)

an

d Q

2 (1

0)

are

th

e L

jun

g-B

ox

Q-s

tati

stic

s o

f o

rder

10 c

om

pu

ted

on

th

e st

an

dard

ized

res

idu

als

an

d s

qu

are

d s

tan

dard

ized

res

idu

als

res

pec

tiv

ely.

AR

CH

(10)

is t

he

no

n-

het

ero

sces

dast

icit

y s

tati

stic

of

ord

er 1

0. P

-valu

es o

f th

e st

ati

stic

s are

rep

ort

ed i

n s

qu

are

bra

cket

s. T

he

AR

CH

-in

Mea

n c

oeff

ici

ent

is e

xclu

ded

fro

m t

he

esti

mati

on

mo

del

s.

Dow

nloa

ded

by [

Kim

Len

g T

an]

at 1

9:32

20

Mar

ch 2

015

Tabl

e 5

FIA

PAR

CH

Mod

el E

stim

atio

n R

esul

ts

E

ner

gy

Mark

et

Sto

ck M

ark

et

B

ren

t K

ero

sen

e W

TI

FT

SE

100

KL

CI

Nasd

aq

100

N

orm

al

GE

D

No

rmal

GE

D

No

rmal

GE

D

No

rmal

GE

D

No

rmal

GE

D

No

rmal

GE

D

Con

ditio

nal M

ean

0a

0.0

95552

0.1

29335**

0.1

60610**

0.1

82187**

0.0

96871

0.1

50089**

0.0

04688

0.0

10265

0.0

29156

0.0

28127

0.0

17879

0.0

22581

(0

.051460)

[0.0

49804]

(0.0

59387)

(0.0

57429)

(0.0

50891)

(0.0

53416)

(0.0

16754)

(0.0

16655

)(0.0

18803)

(0.0

17433)

(0.0

30788)

(0.0

31326)

a1

0.0

12787

0.0

06026

-0.0

30157

-0.0

39128

-0.0

32113

-0.0

41596

-0.0

55740*

-0.0

52412*

0.1

83791**

0.1

38235**

-0

.042786

-0.0

40266

(0

.022608)

(0.0

21073)

(0.0

24646)

(0.0

22376)

(0.0

24708)

(0.0

23104)

(0.0

23919)

(0.0

21970)

(0.0

25862)

(0.0

18392)

(0.0

22166)

(0.0

21471)

AR

CH

-M

-0.3

07552*

-0.3

24834*

-0.1

87688

-0.2

36957*

-0.0

78619

-0.0

69242

-0.0

71416

-0.0

65029

0.0

19353)

-0.0

19470

-0.0

05721

-0.0

08139

(0

.12722)

(0.1

2811)

(0.1

1468)

(0.1

1507)

(0.1

4681)

(0.1

6369)

(0.0

72877)

(0.0

69922)

(0.0

70624

(0.0

53566)

(0.0

51034)

(0.0

50440)

Con

ditio

nal V

aria

nce

0a

0.7

98144*

0.8

36679*

0.0

86491

0.2

50967

0.2

11353

1.0

02495

0.0

35380

0.0

34380

0.0

10474

0.0

14566

0.0

10725

0.0

07207

(0

.38340)

(0.3

6367)

(0.0

45102)

(0.3

4851)

(0.1

4078)

(0.9

0210)

(0.0

19504)

(0.0

18852)

(0.0

27176)

(0.0

27292)

(0.0

36838)

(0.0

36928)

a1

0.1

00753

0.1

19302

-0.0

15887

0.1

34005

0.8

08363**

0.1

89941

0.2

18021**

0.2

20033**

-0

.286172

-0.3

12920

0.1

37738

0.1

34315

(0

.10018)

(0.0

97372)

(0.1

3633)

(0.1

7278)

(0.0

82996)

(0.5

0317)

(0.0

50947)

(0.0

49600)

(0.2

4296)

(0.3

3197)

(0.0

78108)

(0.0

77428)

b1

0.2

55362**

0.2

66506**

0.8

95822**

0.7

09322*

0.8

50106**

0.2

69911

0.5

63604**

0.5

67868**

-0

.178917

-0.2

11612

0.5

17672**

0.5

13778**

(0

.096730)

(0.1

0031)

(0.0

55828)

(0.3

5658)

(0.0

71852)

(0.5

3264)

(0.0

81655)

(0.0

77896)

(0.2

5707)

(0.3

5578)

(0.1

5590)

(0.1

5342)

c

0.3

77546

0.3

57260

-0.1

50743

-0.1

66482

0.2

76093

0.3

80641

0.6

48098**

0.6

72313**

0.1

33471*

0.1

26123*

0.2

97065*

0.3

00837*

(0

.22571)

(0.2

1291)

(0.0

86694)

(0.0

85928)

(0.1

6679)

(0.2

8824)

(0.2

0463)

(0.1

9497)

(0.0

64076)

(0.0

60093)

(0.1

3125)

(0.1

3030)

d

1.8

46652**

1.8

49915**

1.3

06489**

1.3

64452**

2.0

27061**

1.5

24998**

1.4

27385**

1.4

06893**

2.2

75997**

2.2

50136**

1.8

76705**

1.8

78107**

(0

.25633)

(0.2

6748)

(0.2

2024)

(0.3

7789)

(0.3

2667)

(0.4

8282)

(0.1

8957)

(0.1

7409)

(0.1

5697)

(0.1

5214)

(0.1

3340)

(0.1

3320)

d 0.1

95027

0.1

83568*

0.9

95968**

0.6

45854

0.1

74405**

0.1

64713*

0.4

05104**

0.4

06932**

0.2

33097**

0.2

36462**

0.3

93974**

0.3

91947**

(0

.10468)

(0.0

86835)

(0.1

3633)

(0.5

1533)

(0.0

61285)

(0.0

81144)

(0.0

70082)

(0.0

69250)

(0.0

44531)

(0.0

46555)

(0.0

98749)

(0.0

9666

3)

v

1.5

43080**

1.6

03356**

1.4

40074**

1.8

25344*

1.3

50898**

(0.0

96663

(0.0

79771)

(0

.077434)

(0

.076243)*

(0.1

1472)

(0

.060305)

1.8

82829**

d =

1

3.3

02977

3.1

77490

1.3

91614

0.9

64439

3.1

44032

1.0

87358

2.2

54497

2.3

37257

8.1

28923

8.2

18483

6.5

7200

6.5

92395

d =

2

-0.5

98244

-0.5

61107

-3.1

48888

-1.6

81833

-0.0

82839

-0.9

83808

-3.0

20599

-3.4

06899

1.7

58279

1.6

45590

-0.9

24250

-0.9

15113

(Con

tinue

d )

Dow

nloa

ded

by [

Kim

Len

g T

an]

at 1

9:32

20

Mar

ch 2

015

E

ner

gy

Mark

et

Sto

ck M

ark

et

B

ren

t K

ero

sen

e W

TI

FT

SE

100

KL

CI

Nasd

aq

100

N

orm

al

GE

D

No

rmal

GE

D

No

rmal

GE

D

No

rmal

GE

D

No

rmal

GE

D

No

rmal

GE

D

Tabl

e 5

(Con

tinu

ed)

Mod

el S

elec

tion

Lo

g (

L)

-4577.8

89

-4558.0

12

-4719.6

67

-4708.6

25

-4523.3

66

-4489.9

51

-2743.7

98

-2741.8

11

-2456.4

59

-2410.6

75

-4013.8

40

-4013.0

78

AIC

4.5

29273

4.5

10629

4.8

48890

4.8

38590

4.5

86402

4.5

53594

2.7

24553

2.7

23575

2.5

03260

2.4

57770

4.0

09811

4.0

10048

SIC

4.5

51449

4.5

35578

4.8

71764

4.8

64323

4.6

09029

4.5

79049

2.7

46774

2.7

48574

2.5

25953

2.4

83300

4.0

32159

4.0

35190

Dia

gnos

tic

Q(1

0)

7.0

6318

7.1

2535

4.8

7439]

5.1

7838

5.4

9338

6.2

6573

3.1

8132

3.1

7688

13.3

508

19.4

627*

8.8

0815

8.7

9923

[0

.6305417]

[0.6

240708]

[0.8

451174

[0.8

184898]

[0.7

893554]

[0.7

130564]

[0.9

566702]

[0.9

568672]

[0.1

473634]

[0.0

215340]

[0.4

551

690][

0.4

560102]

Q2(

10)

13.8

679

14.8

395

4.6

1440

5.6

0394

6.6

6737

5.2

9262

11.4

487

11.3

946

3.4

0134

3.4

3883

8.1

4139

8.0

5227

[0

.0852751]

[0.0

623432]

[0.7

978816]

[0.6

914994]

[0.5

729081]

[0.7

258911]

[0.1

775490]

[0.1

803253]

[0.9

067100]

[0.9

038846]

[0.4

19

7822]

[0.4

283810]

AR

CH

(10)

1.3

442

1.4

395

0.4

5644

0.5

2717

0.6

5417

0.4

8406

1.2

071

1.2

039

0.3

3791

0.3

4429

0.8

1910

0.8

0991

[0

.2009]

[0.1

567]

[0.9

181]

[0.8

720]

[0.7

677]

[0.9

013]

[0.2

811]

[0.2

833]

[0.9

709]

[0.9

689]

[0.6

102]

[0.6

192]

Not

es : *

an

d *

* d

eno

te 5

% a

nd

1%

sig

nifi

can

ce l

evel

res

pec

tiv

ely.

Jarq

ue-

Ber

a s

tati

stic

s te

sts

for

the

nu

ll h

yp

oth

esis

of

no

rmali

ty i

n t

he

sam

ple

ret

urn

dis

trib

uti

on

.T

he

nu

mb

ers

in p

are

nth

eses

are

sta

nd

ard

err

or

of

the

esti

mati

on

. L

og

(L)

is t

he

log

ari

thm

max

imu

m l

ikel

iho

od

fu

nct

ion

valu

e. A

IC i

s th

e av

erag

e A

kaik

e in

form

ati

on

cri

teri

on

an

d S

IC i

s th

e av

erag

e S

chw

arz

in

form

ati

on

crit

erio

n. Q

(10)

an

d Q

2 (1

0)

are

th

e L

jun

g -

Bo

x Q

-sta

tist

ics

of

ord

er 1

0 c

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and stock indices show a positive degrees of freedom at 1 percent level of

signifi cant that vary from 1.414800 to 1.621099 and 1.292311 to 1.885177

for energy markets and stock markets respectively. Anyhow, as according

to Chin (2009) and Giot and Sebastian (2003), common energy markets

often exhibit heavy tails around three to six degrees of freedom. There-

fore, a normal distribution assumption will fi ts well for higher degrees of

freedom.

Reponses of market volatility to price movements are examined

by the asymmetric news impact coeff icient, γ where only GJR and FI-

APARCH are capable to capture this feature. According to Table-3 and

Table 5, energy markets show no response on ‘good’ or ‘bad’ news ex-

cept for Brent crude oil which presents a signifi cant positive coeff icient

for γ under normal distribution of GJR GARCH model. Meanwhile, stock

markets display a signifi cant positive of γ under normal and GED distri-

bution of GJR GARCH and FIAPARCH model. However, KLCI denotes

an insignifi cant positive coeff icient for both normal and GED distribution

under GJR GARCH model. Since γ is positive and signifi cant for stock

markets, this implies that there are leverage eff ects in stock market, where

negative innovation (news) has a greater impact on volatility than a posi-

tive innovation (news). From the economics perspective, the above results

are expected as events like the Asian fi nancial crisis, Sub-prime mortgage

crisis and gain tax rates will cause the stock or commodity prices to drop

and subsequently increase the volatility. However, not all events will have

similar impacts on both energy and stock markets. For example, crude

oil crisis might have a direct impact on energy markets rather than stock

markets.

Phenomenon of volatility clustering has attracted the attention from

researchers and inspired numerous debates as to whether there is mem-

ory in volatility. This phenomenon is often attributed to traders’ switch-

ing between chartist and fundamentalist strategies (Kirchler and Huber,

2007). According to Cont (2005), although returns are uncorrelated, abso-

lute returns or their squares often display positive, signifi cant and slowly

decaying autocorrelation functions. The persistence of absolute returns

autocorrelation is a sign for volatility clustering. Both energy and stock

markets show persistence, positive, signifi cant and slowly decaying to

zero over long lags for the autocorrelation function of absolute returns

and their squares. Meanwhile, the estimated fractional diff erence param-

eter, d, as show in Table 4 and Table 5 are all statistically signifi cant dif-

ferent from zero. However, Brent crude oil under normal distribution and

Kerosene under GED distribution of FIAPARCH model are insignifi cant

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in values. Value of d indicates that impact of shocks to the conditional

volatility displays a hyperbolic rather than exponential rate of decay. The

results also showing that long memory will have a direct impact on the

market informationally eff iciency with predictability component and

can pose a serious challenge to the proponents of random walk behavior

of the stock returns (Bhattacharya and Bhattacharya, 2012). In order to

improve the level of information eff iciency, sources of long memory in

volatility of returns will be the concerns of fi nancial market regulators

especially in emerging markets.

On the other hand, the coeff icients of power transformation, d in

Table 5 are all signifi cantly diff erent from unity (Taylor/Schwert model)

or two (Bollerslev GARCH) under both normal and GED distributions at

one percent level of signifi cance. Based on the statistics test of SE

1d -` j and

SE2d -

` j, we found that all stock indices are more favourable to conditional

variance. For energy commodities, Brent crude oil and WTI under normal

distribution are more favourable to conditional variance while Kerosene

under both normal and GED distribution and WTI under GED distribu-

tion, which cannot be distinguished between neither the Taylor’s nor the

Bollerslev’s representation.

For risk premium analysis, expected returns of the markets are posi-

tively and proportionately related to conditional volatility theoretically.

Positive risk-return tradeoff coeff icients indicate that risk adverse inves-

tors require higher expected returns (a higher risk premium) as the risk

increases (Henri, 2010). Our results denote that only Kerosene and Brent

crude oil display a negative and statistically signifi cant risk-return trad-

eoff . Salisu and Fasanya (2012) argued that negative and signifi cant risk-

return tradeoff implying high volatility in energy commodities prices and

risk-adverse investors will thereafter shifted to the less risky commodities.

Consequently, it reduces the prices of energy commodities. Meanwhile,

all stock indices denote positive insignifi cant results. For stock market,

positive and insignifi cant values suggest that investors are satisfi ed with

relatively lower returns as the investment has relatively lower level of ex-

pected risk (Suliman Abdalla, 2012).

The lower parts of Table 2 to Table 5 show the results of both diagnos-

tic and residuals tests. Generally, values of log likelihood, Akaike informa-

tion criteria (AIC) and Schwarz information criteria (SIC) are very close to

each other, which are similar to that reported by Wei, Wang and Huang

(2010). Besides that, models with GED distribution obtained the superior

results (smallest log-likelihood, AIC and SIC), which have some advan-

tages as compared to normally distributed models.

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Next, the results for Ljung-Box Q tests and ARCH tests are mixed.

Overall, the Ljung-Box tests have denoted that there is no serial correlation

in standardized residuals for all energy commodities and stock market,

although there is still a serial correlation in standard residuals over lags

of tenth for KLCI under GED distribution. Meanwhile, squared residuals

with lags of tenth indicate that there is still serial correlation in condi-

tional volatility for KLCI at 1 percent or 5 percent signifi cance level under

normal and GED distributions for GARCH and GJR GARCH models. As

comparison, energy markets in overall show no presence of conditional

volatility under both normal and GED distributions for all models. Lastly,

both energy and stock markets indicate that there are no ARCH eff ects

under both normal and GED distributions over ten lags.

3.3. Forecasting Results

Various time horizon forecasts have been conducted to obtain the

volatility forecasts performance evaluations. Table 6 and 7 show the eval-

uations of 5-day, 20-days, 60-days, 120-days and 240-days ahead out of

sample forecasts for energy and stock markets. For energy market, Brent

crude oil displays the lowest losses over short term horizon forecasts un-

der GJR GARCH model. Meanwhile, the FIAPARCH model has the lowest

losses for Brent and WTI crude oil over 240-days ahead horizon forecasts.

For Kerosene, the FIAPARCH model denotes the lowest losses for 5-days

ahead horizon forecasts and the GARCH model presents the lowest losses

over 240-days ahead horizon forecasts. Overall, our results suggest that

the FIAPARCH model fi ts better over the long horizon forecast and the

results are mixed over the short period horizon forecast for energy market.

For stock market, FTSE100 demonstrates the lowest losses over

5-days horizon forecasts under FIAPARCH model. However, FIGARCH

model displays the lowest losses over long horizon forecasts. For KLCI,

FIAPARCH model indicates the lowest losses over short and long term

horizon forecasts. Nevertheless, FIGARCH model denotes the lowest

losses over short and long term horizon forecasts for Nasdaq 100. Thus,

the results suggest that FIAPARCH model suits better over short period

forecast and FIGARCH model performs better for long term forecasts for

stock markets. As a conclusion, we have found that in general, the FI-

APARCH model provides superior out of sample forecasting results over

shorter and/or longer horizon forecasts for stock and energy markets

respectively. However, our results also agreed with Cheong (2009) that

models with higher complexity do not always perform the best in the

actual forecasting.

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Table 6 Forecasting Evaluation Results for Energy Markets

GARCH GJR GARCH FIGARCH FIAPARCH

Normal GED Normal GED Normal GED Normal GED

Brent5-Day

MSE 45.1900 45.0800 40.95001 41.60002 48.5100 48.0000 44.5000 44.5000

MAE 6.1690 6.1670 6.04001 6.06202 6.2890 6.2720 6.1750 6.1710

LL 1.74701 1.74802 1.7640 1.7600 1.7970 1.7880 1.7740 1.7700

20-Day

MSE 28.5900 28.4500 28.0200 27.9300 28.8400 28.710- 27.91002 27.88001

MAE 4.8180 4.7970 4.8690 4.8370 4.66902 4.66501 4.6900 4.6710

LL 4.5230 4.4910 4.6710 4.6090 4.22202 4.22001 4.3240 4.2950

60-Day

MSE 25.0000 24.5600 24.6000 24.1700 22.8200 22.6500 22.52002 22.30001

MAE 4.4210 4.3580 4.4110 4.3450 4.02102 3.99901 4.0390 3.9930

LL 6.6030 6.5350 6.6320 6.5490 6.0640 6.03201 6.1140 6.04502

120-Day

MSE 23.4200 22.7000 22.5600 21.9000 19.6500 19.2400 19.20002 18.66001

MAE 4.3880 4.3010 4.3100 4.2270 3.8710 3.81602 3.8400 3.75401

LL 9.0880 8.9940 9.0220 8.9210 8.4380 8.36102 8.4070 8.28501

240-Day

MSE 26.5700 25.8000 25.5800 24.8900 22.9900 22.4000 22.28002 21.58001

MAE 4.5090 4.4150 4.4000 4.3110 4.0280 3.9380 3.93302 3.81601

LL 8.6420 8.5450 8.5400 8.4420 8.0870 7.98002 7.9830 7.83301

Kerosene5-Day

MSE 22.6900 22.8800 21.3400 21.5300 21.8700 22.1800 20.04002 19.58001

MAE 4.2660 4.2920 4.1860 4.1950 4.2230 4.2470 4.11802 4.10001

LL 5.9130 5.9410 5.7110 5.7410 5.7810 5.8140 5.48302 5.41801

20-Day

MSE 33.5000 33.6100 32.5700 32.6900 33.2800 33.4800 31.4402 31.25001

MAE 4.5930 4.6020 4.5150 4.5230 4.5830 4.5970 4.38402 4.36101

LL 6.4580 6.4700 6.3260 6.3430 6.4180 6.4370 6.11602 6.06201

60-Day

MSE 36.5000 36.2100 36.5400 36.3800 36.3000 35.86002 37.4800 34.34001

MAE 5.4090 5.3870 5.4060 5.3940 5.3960 5.36302 5.4400 5.21501

LL 9.2720 9.250 9.2520 9.2420 9.2480 9.21402 9.2590 9.01301

(Continued )

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120-Day

MSE 38.8600 38.07001 40.4900 39.9200 39.9700 38.3102 51.6500 38.7200

MAE 5.7350 5.67601 5.8500 5.8090 5.8150 5.69402 6.5820 5.7050

LL 10.5700 10.52002 10.6500 10.6200 10.6300 10.5300 11.1300 10.50001

240-Day

MSE 42.01002 40.97001 44.6800 43.8600 46.7000 43.5200 69.0100 45.6900

MAE 6.00302 5.92001 6.2080 6.1460 6.3610 6.1240 7.7730 6.2780

LL 9.70602 9.64801 9.8390 9.7970 9.9320 9.7760 10.750 9.8500

WTI5-Day

MSE 114.50002 115.8000 112.80001 114.50002 117.8000 118.2000 115.7000 115.3000

MAE 8.6670 8.6000 8.7100 8.6260 8.5390 8.53501 8.53702 8.6120

LL 7.1580 7.0670 7.2720 7.1440 6.86301 6.87302 6.8860 7.0660

20-Day

MSE 86.26002 91.0300 84.91001 90.2600 93.6600 94.6500 94.2300 91.6900

MAE 6.94802 7.0320 6.93901 7.0310 7.0630 7.0800 7.0880 7.0570

LL 2.58301 2.6120 2.6060 2.6250 2.59202 2.6100 2.6190 2.6240

60-Day

MSE 50.3400 50.420 49.7500 49.9700 50.0500 50.1700 49.83002 49.46001

MAE 5.2940 5.1010 5.2720 5.0660 4.8550 4.80202 4.73601 4.8670

LL 10.5800 10.3400 10.5700 10.3000 10.0300 9.96602 9.88501 10.0600

120-Day

MSE 37.3500 36.4700 36.6900 35.6000 34.1100 33.7600 33.05001 33.60002

MAE 4.8680 4.6590 4.8140 4.55900 4.2410 4.15602 4.04401 4.2170

LL 10.0800 9.8580 10.0400 9.7680 9.4150 9.31402 9.17401 9.3980

240-Day

MSE 32.5600 31.5000 31.9300 30.4800 28.7700 28.2300 27.40001 28.1902

MAE 4.7870 4.6080 4.7240 4.4820 4.1910 4.09302 3.95301 4.1290

LL 8.0560 7.8800 8.0030 7.7660 7.4610 7.34902 7.18601 7.3970

Note: The superscripts 1 and 2 represent the lowest and second lowest error statistics.

Table 6 (Continued)



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Table 7 Forecasting Evaluation Results for Stock Market



FTSE 1005-Day

MSE 0.77061 0.77162 0.7730 0.7747 0.7907 0.7910 0.8083 0.8086

MAE 0.6197 0.6193 0.6187 0.6179 0.6113 0.6111 0.60932 0.60891

LL 2.9280 2.9260 2.9240 2.9210 2.88401 2.88401 2.8880 2.88702

20-Day

MSE 0.39981 0.39992 0.4001 0.4005 0.4049 0.4050 0.4096 0.4100

MAE 0.4442 0.4427 0.4402 0.4386 0.4317 0.4316 0.42982 0.42891

LL 5.2030 5.1830 5.1420 5.1080 4.9550 4.9510 4.89902 4.87001

60-Day

MSE 1.74601 1.75002 1.7770 1.7860 1.8470 1.8460 1.8600 1.8700

MAE 0.7101 0.7090 0.7001 0.6965 0.6828 0.6826 0.68002 0.67791

LL 7.6060 7.5760 7.4610 7.4000 7.1600 7.1600 7.1042 7.0611

120-Day

MSE 1.1000 1.0960 1.0600 1.05702 1.05401 1.05401 1.0580 1.0620

MAE 0.6501 0.6431 0.5917 0.5812 0.5369 0.5373 0.53172 0.52761

LL 9.9250 9.8700 9.4400 9.3370 8.6620 8.6730 8.52802 8.42401

240-Day

MSE 4.94801 4.95802 5.1920 5.2250 5.4760 5.4680 5.5460 5.5920

MAE 1.1120 1.1070 1.0740 1.0690 1.05501 1.05501 1.05702 1.0580

LL 6.9370 6.9000 6.5430 6.4790 6.01002 6.1060 6.0440 6.00401

KLCI5-Day

MSE 1.72701 1.72802 1.7310 1.7320 1.8750 1.8770 1.9010 1.9060

MAE 0.9809 0.9775 0.9785 0.9769 0.8205 0.8193 0.81561 0.81632

LL 8.1960 8.1690 8.1710 8.1570 6.8290 6.8050 6.68801 6.69002

20-Day

MSE 0.6007 0.5994 0.6047 0.6049 0.59231 0.59362 0.6029 0.6031

MAE 0.5587 0.5559 0.5612 0.5606 0.4318 0.4305 0.42991 0.43002

LL 5.4200 5.4040 5.4290 5.4250 4.4310 4.4120 4.34201 4.35502

60-Day

MSE 13.0900 13.1000 13.02002 13.01001 14.0600 14.0800 14.1700 14.1300

MAE 1.48902 1.48801 1.4900 1.4900 1.4960 1.4970 1.5060 1.5010

LL 4.9700 4.9630 5.0100 5.0110 4.52201 4.52201 4.53602 4.52201

(Continued )

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120-Day

MSE 6.90302 6.90101 6.9280 6.9260 7.2550 7.2650 7.3170 7.2920

MAE 1.1280 1.1250 1.1660 1.1670 0.9710 0.9708 0.97072 0.97061

LL 6.3140 6.2980 6.5220 6.5280 4.7390 4.7200 4.61101 4.66402

240-Day

MSE 6.60302 6.59201 6.6890 6.6700 7.0090 7.0200 7.0950 7.0560

MAE 1.2580 1.2460 1.3450 1.3370 1.00202 1.00202 1.00101 1.00101

LL 7.7880 7.7320 8.2050 8.1720 5.4840 5.4590 5.29701 5.38102

Nasdaq100

5-Day

MSE 1.96302 1.96302 1.94501 1.94501 2.0030 2.0060 2.0010 2.0030

MAE 1.0050 1.0050 1.0470 1.0460 0.95122 0.95021 0.9796 0.9773

LL 6.7420 6.7390 7.0830 7.0820 6.30902 6.30001 6.5390 6.5210

20-Day

MSE 1.34102 1.3420 1.31601 1.31601 1.3810 1.3830 1.3580 1.3610

MAE 0.8331 0.8326 0.8563 0.8556 0.82322 0.8231 0.8284 0.8269

LL 4.9350 4.9290 5.1800 5.1740 4.71502 4.70601 4.8700 4.8460

60-Day

MSE 5.8720 5.8760 5.81301 5.81702 5.9130 5.9180 5.8930 5.9050

MAE 1.0960 1.0940 1.1360 1.1330 1.07702 1.07501 1.0860 1.0810

LL 6.2000 6.1840 6.5320 6.5110 5.99502 5.97601 6.1180 6.0690

120-Day

MSE 3.2160 3.2140 3.2420 3.2360 3.2070 3.2080 3.19601 3.19802

MAE 0.8683 0.8646 0.9327 0.9247 0.8327 0.82972 0.8358 0.82931

LL 8.7360 8.7020 9.2160 9.1650 8.3480 8.30702 8.3810 8.29501

240-Day

MSE 5.5770 5.5970 5.43101 5.45302 5.9560 5.9830 6.0220 6.0710

MAE 1.2790 1.2760 1.3260 1.3190 1.25702 1.25601 1.2600 1.2580

LL 7.3120 7.2800 7.7010 7.6510 6.9280 6.89802 6.9240 6.86701

Note : The superscripts 1 and 2 represent the lowest and second lowest error statistics.

Table 7 (Continued)



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After that, the SPA test has been conducted in order to confi rm the

reliability and robustness of the forecasts. Table 8 and 9 display the SPA

results for the various volatility models for short period forecasts (5-day

and 20-days) while Table 10 to 12 shows the longer horizon forecasts (60-

days, 120-days and 240-days). The fi rst column of each table is represented

by the benchmark model of SPA tests while the remaining three models

are treated as competitive models.

For energy market analysis, long memory GARCH models display su-

perior out of sample forecasts results and better forecasting accuracy over

short term forecasts. For example in Table 8, WTI and Brent for 5 days ho-

rizon forecast indicate that at least four out of six of the best model is long

memory GARCH model. However, the best model across six loss func-

tions is standard GARCH model for 240 days horizon as demonstrated in

Table 12. This result is in contrast with Wei, Wang and Huang (2010) as the

best model across six models for longer horizon is long memory GARCH

model. The main reason of this contrary is due to the consideration of 20

days as long horizon forecast while this paper extended the long horizon

forecasts to 240 days in order to study the impact of longer horizon fore-

casts. On the other hand, Kerosene denotes that standard GARCH model

displays superior out of sample forecasts results and better forecasting

accuracy over shorter horizon forecasts. For long horizon forecasts, the

result is mixed for normal and GED distributions whereby the best model

when tested under normal distribution is long memory GARCH model

while the GJR-GARCH model becomes the best model when it is tested

under GED distribution.

For stock markets, the SPA test results show that there are similarities

between short and long horizon forecasts. As indicated in Table 8 and Ta-

ble 12, four out of six among the loss functions indicate that long memory

GARCH model is the best model for 5 days and 240 days horizon forecasts

respectively. In short, p-values of long memory GARCH models which

are close to unity suggested that long memory GARCH models display

higher forecasting accuracy than standard GARCH models in this study.

However, none of the ARCH family models are absolutely outperformed

others. Therefore, researchers, economists and practitioners should be

cautious when choosing the ARCH family models for volatility forecast-

ing. They should take into consideration also the possible complexity in

model specifi cation, parsimonious principles and actual performance of

forecast evaluations.

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EMPIRICAL STYLIZED FACTS MODELLING 337Ta

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FIG

AR

CH

0.4

0310

0.4

0360

0.0

0000

0.0

0000

0.4

1810

0.4

1810

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.4

2330

0.4

2330

FIA

PAR

CH

0.0

0000

0.0

0000

0.4

2330

0.4

2350

0.0

0000

0.0

0000

0.4

3940

0.4

3940

0.4

3980

0.4

3980

0.0

0000

0.0

0000

H

MA

EG

AR

CH

0.0

0000

0.0

0000

0.0

0070

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0050

0.0

0000

0.0

0000

0.0

0000

GJR

-GA

RC

H

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0050

0.0

0050

0.0

0000

0.0

0000

FIG

AR

CH

0.4

1020

0.4

1020

0.0

0000

0.0

0000

0.4

1370

0.4

1370

0.0

0000

0.0

0000

0.0

0050

0.0

0050

0.4

2580

0.4

2580

FIA

PAR

CH

0.0

0000

0.0

0000

0.4

2560

0.4

1380

0.0

0000

0.0

0000

0.4

3940

0.4

3420

0.4

3950

0.4

3950

0.0

0000

0.0

0000

Q

LG

AR

CH

0.0

0110

0.0

0110

0.4

4330

0.4

6730

0.0

0970

0.0

1270

0.0

0000

0.0

0000

0.0

0450

0.0

0110

0.0

0490

0.0

0490

GJR

-GA

RC

H

0.0

0000

0.0

0000

0.0

0700

0.0

0000

0.0

0100

0.0

0100

0.0

0000

0.0

0000

0.0

0450

0.0

0110

0.0

1400

0.0

1400

FIG

AR

CH

0.4

0210

0.4

0110

0.0

0700

0.0

1910

0.4

1380

0.4

1380

0.0

0000

0.0

0000

0.0

0110

0.0

0110

0.4

2770

0.4

2630

FIA

PAR

CH

0.0

1390

0.0

2970

0.0

1950

0.0

4820

0.0

0090

0.0

0000

0.4

5580

0.4

5580

0.4

4050

0.4

4050

0.0

7980

0.0

7980

R

2 LO

GG

AR

CH

0.0

0450

0.0

0450

0.4

5530

0.4

5530

0.0

0110

0.0

0110

0.2

7820

0.2

7820

0.4

1780

0.7

4860

0.0

0000

0.0

0000

GJR

-GA

RC

H

0.6

9030

0.4

3780

0.0

0000

0.0

0000

0.4

3160

0.4

3990

0.5

2660

0.5

2660

0.1

2830

0.2

7750

0.4

2480

0.4

2480

FIG

AR

CH

0.1

6800

0.1

5360

0.0

0000

0.0

0000

0.0

0110

0.0

0110

0.3

0100

0.3

0100

0.2

8130

0.3

0930

0.0

0000

0.0

0000

FIA

PAR

CH

0.4

2960

0.7

4940

0.0

0000

0.0

0000

0.0

0050

0.0

0050

0.7

6340

0.7

5570

0.2

2840

0.2

4110

0.0

0000

0.0

0000

Not

e: T

he

nu

mb

ers

in b

old

in

dic

ate

s th

e b

est

mo

del

fo

r S

up

erio

r P

red

icti

ve

Ab

ilit

y T

est.

Dow

nloa

ded

by [

Kim

Len

g T

an]

at 1

9:32

20

Mar

ch 2

015


Tabl

e 9

SPA

Res

ults

for 2

0 D

ays

Out

-of-

sam

ple

Vol

atili

ty F

orec

asts

E

ner

gy

Mark

et

Sto

ck M

ark

et

B

ren

t K

ero

sen

e W

TI

FT

SE

100

KL

CI

Nasd

aq

100

N

orm

al

GE

D

No

rmal

GE

D

No

rmal

GE

D

No

rmal

GE

D

No

rmal

GE

D

No

rmal

GE

D

M

SEG

AR

CH

0.0

0030

0.0

0030

0.5

2780

0.5

3250

0.0

0000

0.0

0000

0.0

0160

0.0

0160

0.0

1910

0.0

0320

0.0

0010

0.0

0010

GJR

-GA

RC

H

0.0

0110

0.0

0070

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0130

0.0

0100

0.0

3380

0.0

3350

0.0

0010

0.0

0010

FIG

AR

CH

0.5

1450

0.4

0110

0.0

0000

0.0

0030

0.5

1940

0.4

8540

0.0

0190

0.0

0170

0.0

0680

0.0

0730

0.5

7860

0.5

5660

FIA

PAR

CH

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0350

0.0

0000

0.5

4580

0.5

4260

0.5

3890

0.5

3910

0.0

0000

0.0

0000

M

AE

GA

RC

H

0.6

9840

0.7

0910

0.5

0790

0.5

1250

0.1

8710

0.1

1900

0.4

7300

0.4

7180

0.4

6140

0.0

0120

0.0

0000

0.0

0000

GJR

-GA

RC

H

0.6

5680

0.6

5810

0.0

0000

0.0

0000

0.9

0100

0.9

1630

0.0

0000

0.0

0000

0.5

3880

0.6

1930

0.5

0910

0.5

0970

FIG

AR

CH

0.5

8250

0.5

9020

0.0

0000

0.0

0000

0.2

2090

0.1

3710

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

FIA

PAR

CH

0.7

8460

0.7

6930

0.0

0000

0.0

0000

0.2

5690

0.1

8590

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

H

MS

E

GA

RC

H

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0330

0.0

0330

0.0

0000

0.0

0000

GJR

-GA

RC

H

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0330

0.0

0330

0.0

0000

0.0

0000

FIG

AR

CH

0.5

1490

0.5

1570

0.0

0000

0.0

0000

0.5

1890

0.5

2450

0.0

0150

0.0

0120

0.0

0310

0.0

0380

0.5

3490

0.5

3530

FIA

PAR

CH

0.0

0000

0.0

0000

0.5

2860

0.5

2880

0.0

0020

0.0

0000

0.5

3500

0.5

3460

0.5

3480

0.5

3500

0.0

0000

0.0

0000

H

MA

E

GA

RC

H

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0500

0.0

0540

0.0

0000

0.0

0000

GJR

-GA

RC

H

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0450

0.0

0500

0.0

0000

0.0

0000

FIG

AR

CH

0.5

1600

0.5

1520

0.0

0000

0.0

0000

0.5

1850

0.5

4650

0.0

0040

0.0

0010

0.0

0220

0.0

0360

0.5

3640

0.5

3460

FIA

PAR

CH

0.0

0000

0.0

0000

0.5

2960

0.5

2690

0.0

0050

0.0

0000

0.5

3490

0.5

3460

0.5

3530

0.5

3590

0.0

0000

0.0

0000

Q

L

GA

RC

H

0.0

0020

0.0

0020

0.5

3490

0.5

3760

0.0

0000

0.0

0000

0.0

0170

0.0

0150

0.0

1890

0.0

0370

0.0

0010

0.0

0010

GJR

-GA

RC

H

0.0

0070

0.0

0050

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0150

0.0

0140

0.0

2130

0.0

2100

0.0

0010

0.0

0010

FIG

AR

CH

0.5

1590

0.5

1930

0.0

0010

0.0

0070

0.5

2160

0.4

9450

0.0

0230

0.0

0200

0.0

0630

0.0

0710

0.5

4450

0.5

3990

FIA

PAR

CH

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0160

0.0

0000

0.5

3730

0.5

3530

0.5

3810

0.5

3830

0.0

0000

0.0

0000

R

2 L

OG

GA

RC

H

0.0

0000

0.0

0000

0.5

2560

0.5

2510

0.0

0100

0.0

0300

0.4

7900

0.4

7800

0.8

5310

0.1

3850

0.0

0000

0.0

0000

GJR

-GA

RC

H

0.4

8520

0.4

8610

0.0

0000

0.0

0000

0.5

2970

0.5

3590

0.0

0000

0.0

0000

0.1

4690

0.8

6150

0.5

3450

0.5

3400

FIG

AR

CH

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0360

0.0

0290

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

FIA

PAR

CH

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0310

0.0

1360

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

Not

e : T

he

nu

mb

ers

in b

old

in

dic

ate

s th

e b

est

mo

del

fo

r S

up

erio

r P

red

icti

ve

Ab

ilit

y T

est.

Dow

nloa

ded

by [

Kim

Len

g T

an]

at 1

9:32

20

Mar

ch 2

015


ble

10SP

A R

esul

ts fo

r 60

Day

s O

ut-o

f-sa

mpl

e V

olat

ility

For

ecas

ts

E

ner

gy

Mark

et

Sto

ck M

ark

et

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ren

t K

ero

sen

e W

TI

FT

SE

100

KL

CI

Nasd

aq

100

N

orm

al

GE

D

No

rmal

GE

D

No

rmal

GE

D

No

rmal

GE

D

No

rmal

GE

D

No

rmal

GE

D

M

SEG

AR

CH

0.7

2470

0.6

7700

0.6

3620

0.6

5020

0.0

0000

0.0

0000

0.0

0090

0.0

0080

0.0

0000

0.0

0000

0.0

0000

0.0

0000

GJR

-GA

RC

H

0.0

0940

0.0

0450

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0020

0.0

0020

0.0

0000

0.0

0000

0.0

0000

0.0

0000

FIG

AR

CH

0.3

1930

0.5

1770

0.0

0000

0.0

0000

0.9

3650

0.4

9210

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.5

0880

0.5

1190

FIA

PAR

CH

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

6350

0.0

0000

0.5

2790

0.5

2310

0.5

1330

0.5

1290

0.0

0000

0.0

0000

M

AE

GA

RC

H

0.0

3280

0.0

5430

0.5

2220

0.5

2360

0.5

7420

0.9

4550

0.5

0380

0.5

0610

0.0

1940

0.0

0010

0.0

0000

0.0

0000

GJR

-GA

RC

H

0.5

0300

0.9

4570

0.0

0000

0.0

0000

0.4

2580

0.0

5450

0.0

0000

0.0

0000

0.5

3290

0.6

7040

0.4

9130

0.5

1330

FIG

AR

CH

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

FIA

PAR

CH

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0030

0.0

0010

0.0

0000

0.0

0000

H

MSE

GA

RC

H

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

GJR

-GA

RC

H

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

FIG

AR

CH

0.5

2290

0.5

2250

0.0

0000

0.0

0000

0.9

0350

0.5

2280

0.0

0000

0.0

0000

0.0

0010

0.0

0000

0.5

0660

0.5

0710

FIA

PAR

CH

0.0

0000

0.0

0000

0.5

2290

0.5

2260

0.0

9650

0.0

0000

0.5

4250

0.5

5570

0.5

1330

0.5

1210

0.0

0000

0.0

0000

H

MA

E G

AR

CH

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

GJR

-GA

RC

H

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

FIG

AR

CH

0.5

2030

0.5

2040

0.0

0000

0.0

0000

0.8

4950

0.5

0780

0.0

0040

0.0

0000

0.0

0000

0.0

0000

0.5

2670

0.5

2600

FIA

PAR

CH

0.0

0000

0.0

0000

0.5

2560

0.5

3200

0.1

5050

0.0

0000

0.5

3960

0.5

3880

0.5

1250

0.5

1150

0.0

0000

0.0

0000

Q

L G

AR

CH

0.1

7990

0.1

9160

0.5

2490

0.5

3340

0.0

0000

0.0

0000

0.0

0020

0.0

0010

0.0

0000

0.0

0000

0.0

0000

0.0

0000

GJR

-GA

RC

H

0.0

0540

0.0

0340

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0010

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

FIG

AR

CH

0.8

2010

0.8

0840

0.0

0000

0.0

0000

0.5

2800

0.4

8820

0.0

0070

0.0

0010

0.0

0000

0.0

0000

0.5

5460

0.5

5140

FIA

PAR

CH

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

2640

0.0

0000

0.5

5990

0.5

8510

0.5

1210

0.5

1080

0.0

0000

0.0

0000

R

2 LO

G

GA

RC

H

0.0

0000

0.0

0000

0.5

2630

0.5

3320

0.0

0220

0.0

3110

0.5

0090

0.4

9940

0.0

0380

0.0

0000

0.0

0000

0.0

0000

GJR

-GA

RC

H

0.5

4300

0.5

3100

0.0

0000

0.0

0000

0.5

2890

0.5

2530

0.0

0000

0.0

0000

0.4

8930

0.5

6360

0.4

8960

0.4

9120

FIG

AR

CH

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

FIA

PAR

CH

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

0.0

0000

No

te: T

he

nu

mb

ers

in b

old

in

dic

ate

s th

e b

est

mo

del

fo

r S

up

erio

r P

red

icti

ve

Ab

ilit

y T

est

Dow

nloa

ded

by [

Kim

Len

g T

an]

at 1

9:32

20

Mar

ch 2

015


Tabl

e 11

SPA

Res

ults

for 1

20 D

ays

Out

-of-

sam

ple

Vol

atili

ty F

orec

asts

E

ner

gy

Mark

et

Sto

ck M

ark

et

B

ren

t K

ero

sen

e W

TI

FT

SE

100

KL

CI

Nasd

aq

100

N

orm

al

GE

D

No

rmal

GE

D

No

rmal

GE

D

No

rmal

GE

D

No

rmal

GE

D

No

rmal

GE

D

M

SEG

AR

CH

0.7

8910

0.6

5880

0.0

0090

0.1

1400

0.1

7460

0.5

6010

0.0

0110

0.0

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4. Conclusion

There are two important contributions in this study: fi rst is to exam-

ine and compare the similarities and diff erences of stylized facts behav-

iour between energy and stock markets; second, is to compare the perfor-

mances of two or more forecasting models and subsequently highlight the

best forecast models.

The stylized facts of energy markets and stock markets are examined

using some ARCH family models. Both energy markets and stock markets

show a persistence and positively correlated absolute returns, which is a

main sign for long memory and clustering volatility. Besides that, the two

markets also display a hyperbolic rather than exponential rate of decay

of volatility shocks. In contrary for leverage eff ect, energy markets show

no response on ‘good’ or ‘bad’ news while stock markets indicate a lever-

age eff ect. For risk and return relationship analysis, the ARCH-in-Mean

models have found that only the energy commodities exhibit a signifi cant

risk-return tradeoff while none of the stock indices indicate the eff ect on

risk-return tradeoff .

For out of sample forecasting performance, the simplest GARCH

model fi ts better over short time horizon whereas the FIAPARCH model

performs better over long time horizon in stock markets. Meanwhile for

energy markets, we found that the FIGARCH model suits better in short

horizon forecasts as compared to long term. Nevertheless, the reliability

and robustness of the forecasts are tested using SPA test. The results high-

light that long memory GARCH model is a better choice than the standard

GARCH for both energy and stock markets. However, there is no single

model displays superior results in all markets. Hence, in order to choose

the best forecast model, economists and fi nancial analysts should take into

consideration the complexity, parsimonious principles and actual perfor-

mance of out of sample forecasts.

In future, analysis can be extended to determine market risks using

Value at Risk (VaR) for a single or cross markets. This further study al-

lows us to identify market risk exposures and sensitivities to the diff er-

ent risk factors which could be the guidance for portfolio analysis and

hedging.

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Appendix

Figure 1Graph for Return Series

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Empirical Stylized Facts Modelling and Forecast Evaluations for Energy and Stock Markets

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