ARTICLE IN PRESS

Energy Policy 37 (2009) 2346–2355

Contents lists available at ScienceDirect

Energy Policy

0301-42

doi:10.1

� Tel.:

E-m

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

Modeling and forecasting crude oil markets using ARCH-type models

Chin Wen Cheong �

Research Centre of Mathematical Sciences, Faculty of Information Technology, Multimedia University, 63100 Cyberjaya, Selangor, Malaysia

a r t i c l e i n f o

Article history:

Received 9 December 2007

Accepted 11 February 2009Available online 21 March 2009

Keywords:

Financial time series

ARCH model

Forecasting evaluations

15/$ - see front matter & 2009 Elsevier Ltd. A

016/j.enpol.2009.02.026

+60 3 83125249; fax: +60 3 83125264.

ail address: wcchin@mmu.edu.my

a b s t r a c t

This study investigates the time-varying volatility of two major crude oil markets, the West Texas

Intermediate (WTI) and Europe Brent. A flexible autoregressive conditional heteroskedasticity (ARCH)

model is used to take into account the stylized volatility facts such as clustering volatility, asymmetric

news impact and long memory volatility among others. The empirical results indicate that the intensity

of long-persistence volatility in the WTI is greater than in the Brent. It is also found that for the WTI, the

appreciation and depreciation shocks of the WTI have similar impact on the resulting volatility.

However, a leverage effect is found in Brent. Although both the estimation and diagnostic evaluations

are in favor of an asymmetric long memory ARCH model, only the WTI models provide superior in the

out-of-sample forecasts. On the other hand, from the empirical out-of-sample forecasts, it appears that

the simplest parsimonious generalized ARCH provides the best forecasted evaluations for the Brent

crude oil data.

& 2009 Elsevier Ltd. All rights reserved.

1. Introduction

In recent years, numerous studies have been devoted toworldwide crude oil prices, since this energy commodity is afundamental driver of most economic activities. Higher oil priceshave a direct impact on macroeconomic issues including inflation(Chen, 2009; Cologni and Manera, 2008; Doroodian and Boyd,2003; Farzanegan and Markwardt, 2009; Mussa, 2000), grossdomestic product (Cologni and Manera, 2009; Doroodian andBoyd, 2003; Gronwald, 2008; Jones et al., 2004; Lardic andMignon, 2006; Narayan and Smyth, 2007; Prasad et al., 2007),reduction of investment (Abel, 1990; Hamilton, 2003; Rafiq et al.,2008) and recession (Gisser and Goodwin, 1986; Hamilton, 1983;Jones et al., 2004) among others.

Oil price shocks are also closely related to the global financialmarkets, including futures contracts, options, risk managementand other related financial derivatives. Nandha and Faff (2008)investigated the adverse effect of oil price shocks on globalindustry indices for a period of approximately 22 years, andconcluded that oil price increases have a negative impact on mostof the industry returns besides just mining, oil and gas industries.Ciner (2001) found a non-linear relationship between oil pricefutures and real stock returns. Park and Ratti (2008) studied theconnection between oil price shocks and the stock market forthe US and 13 European countries. They found that for most of theEuropean countries, but not for the US, increased oil pricevolatility significantly depressed their real stock returns. Other

ll rights reserved.

empirical studies related to the relationships between the crudeoil and stock markets can also be found in studies by Boyer andFilion (2007), El-Sharif et al. (2005) and Miller and Ratti, (2009).For risk management analysis, Cabedo and Moya (2003) and Giotand Laurent (2003) estimated the value-at-risk for daily spotBrent and West Texas Intermediate (WTI) prices using autore-gressive and moving average (ARMA) and autoregressive condi-tional heteroskedasticity (ARCH) models. To date, Huang et al.(2009) introduced an improved CAViaR approach to forecast theWTI oil price risk. Some studies also reported the impact of crudeoil price fluctuations on foreign exchange rates such as theSpanish peseta’s real exchange rate (Camarero and Tamarit, 2002),the G7 real exchange rate (Chen and Chen, 2007), the US exchangerate (Benassy-Quere et al., 2007), and the Fiji exchange rate(Narayan et al., 2008).

Besides the macroeconomic issues and relationships withfinancial markets, the uncertainty in crude oil prices can leavecrude oil market participants with heavy potential losses. Thereare ample studies (Cabedo and Moya, 2003; Giot and Laurent,2003) addressing the impact of crude oil price volatility on thegeneral energy market risks. Fluctuations of crude oil prices havebeen profoundly influenced by not only the supply and demandcondition in oil markets, but also by political events, speculations,military conflicts and natural disasters. For example, when OPECdecided to curtail crude oil production by 4.2 million barrels dailyin December 2000, the price rose to approximately USD 36 perbarrel in the WTI crude oil spot markets (Fig. 1). The priceslumped to around USD 18 per barrel in December 2001 after theUS invasion of Iraq. Recently, the so-called third crude oil crisisoccurred when prices reached rocket-high levels of USD 100 and145 per barrel at the end of 2007 and middle of 2008, respectively.

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Important events: 1996-1998: Asian financial crisis; 2000-2001: OPEC cuts 4.2 million barrels per day; 2001-2003: 9/11 and Iraq War; 2002-2007: Depreciation of the USD increases oil demand; 2005 : Iraq nuclear program; 2006-2007: Demands from emerging markets such as Republic of China and India;

Fig. 1. Brent and WTI daily prices between 4th January 1993 to 31st December 2007.

C.W. Cheong / Energy Policy 37 (2009) 2346–2355 2347

However, when the crude oil bubble burst mostly due tospeculations, the price plunged to USD 30 per barrel at the endof 2008.

As exemplified in the aforementioned studies, the fluctuationof crude oil prices is closely tied to global macroevents, financialmarkets movements and strategy in risk management. Therefore,understanding the stochastic process that lies beneath crude oilprices is important for energy researchers, econometricians andpolicy makers. Among the many issues related to oil pricefluctuations, the task of modeling and forecasting oil pricevolatility is one of the hot topics that have attracted the interestsof global energy-related researchers and investors. There areample studies addressing the accuracy of crude oil volatilitymodeling and forecasting. These include ARCH-type models (Fongand See, 2002; Giot and Laurent, 2003), asymmetric thresholdautoregressive (TAR) model (Godby et al., 2000), and artificial-based forecast methods (Fan et al., 2008; Moshiri, 2004), amongothers. However, the complexity of the model specification doesnot guarantee high performance on out-performed out-of-sampleforecasts. Sadorsky (2006) found that the out-of-sample forecastsof a single equation generalized ARCH model are more superior tothose of state space, vector autoregression and bivariate GARCHmodels in predicting the price of petroleum futures.

In the aforementioned literature, the ARCH-type modelintroduced by Engle (1982) is one of the most promising andreliable models that specializes in financial time series volatilitymodeling. ARCH-type models have a long history (approximatelythree decades) in the applications of financial markets, especiallyin pricing financial derivatives (Baillie et al., 1996; Bollerslev,1986; Nelson, 1991) and measuring investment risk (Engle et al,1987; Giot and Laurent, 2004; Jorion, 2000). In addition, ARCH-type models also play an important role in the literature on crudeoil market volatility analysis (Alizadeh et al., 2008; Fong and See,2002; Narayan and Narayan, 2007). There are ample investiga-tions in the literature on the extension of ARCH models. Most ofthe extension-ARCH models are intended to relate the empiricalstylized facts that are often observed in the worldwide crude oilmarkets, including the leverage effects (Godby et al., 2000; Lienand Yang, 2008), long memory clustering volatility (Tabak and

Gajueiro, 2007; Serletis and Andreadis, 2004), and the riskpremium effect (Doran and Ronn, 2008; Kretzschmar et al.,2008). The asymmetric power ARCH (APGARCH) model (Dinget al., 1993) and the fractionally integrated ARCH (FIGARCH)model are the two most flexible ARCH-family models, which takeinto account various relevant stylized facts. The APGARCH, forexample, contains seven specific ARCH-family models. Since theearly study of fractional difference operators (1�B)d by Grangerand Joyeux (1980) and Hosking (1981), fractionally integratedARMA models have been extended to essentially equivalentmodels for the volatility modeling in FIGARCH (Baillie et al.,1996), FIEGARCH (Bollerslev and Mikkelsen, 1996), and FIAPARCH(Tse, 1998). It is worth noting that there is still a lack of researchon how the aforementioned stylized facts are combined in thecrude oil prices.

In this study, we adopted the asymmetric power ARCH model(Ding et al., 1993) with the fractional difference operator (1�B)d

for the conditional volatility modeling. However, only four specificmodels, which include the asymmetric effect, volatility clustering,long memory, and heavy-tailed innovations, are considered in theBrent and WTI crude oil markets. Finally, the out-of-sampleforecasts are constructed on several forecast horizons andevaluated under numerous loss functions.

2. Data source

Two crude oil spot prices datasets, namely the WTI and EuropeBrent, are considered as the energy commodities by the US EnergyInformation Administration (EIA). The WTI and Brent are themajor international oil benchmarks due to their low sulfur andgeographical location. The datasets consisted of 3761 and 3805points for the WTI and Brent, respectively, from 4th January 1993to 31st December 2008. The latest 2008 dataset, until December2008, is reserved for the out-of-sample forecasting evaluations.Fig. 1 illustrates the way oil price dynamics are often stimulatedby economic or political events. It is found that the price levelsbecame more volatile and developed a strong upward driftstarting from the year 2002 to 2007 for both of the crude oilmarkets.

ARTICLE IN PRESS

C.W. Cheong / Energy Policy 37 (2009) 2346–23552348

The percentage continuously compounded inter-day returnseries rt is defined in terms of the close-to-close prices (USD perbarrel) on consecutive trading days

rt ¼ 100ðln Pt;close � ln Pt�1;closeÞ (1)

This return is mainly used in estimating the conditional meanand volatility.

3. Methodology

Let {rt} be a general compounded return stochastic process.This series is usually serially uncorrelated, but not independent inARCH. For the given information set It�1, available at time t�1, rt isdefined as

rt ¼ mt � at

at ¼ st�t ; �t �i:i:d

f ð�Þ; (2)

where f( � )1 is the density function of et, and the conditional mean,E(rt|It�1) ¼ Et�1(rt) ¼ mt, normally follows a stationary ARMA(m, n)model specified by mt ¼ a0+

Pi ¼ 1m airt�i�

Pj ¼ 1n bjat�j. The condi-

tional variance model is specified in the ARCH framework.

3.1. Volatility modeling and estimation

Using the power asymmetric generalized ARCH model pro-posed by Ding et al. (1993) and incorporating the fractionallyintegrated filter introduced by Granger and Joyeux (1980) andHosking (1981) results in the following specification:

sdt ¼

a0

1� bðBÞþ 1�

jðBÞð1� BÞd

1� bðBÞ

( )ðjatj � gatÞ

d, (3)

where

�

Gau

a(B) ¼ (a1B+?+aqBq), b(B) ¼ (b1B+?+bpBp) and j(B) ¼(1�a(B)�b(B))/1�B represent the lag polynomials;

� (1�B)d

¼P

n ¼ 0N (�1)d((d(d�1)y(d�k+1))/k!)Bd with dA[0,1],

denotes the fractional integrated operator;

� g A(�1,1) denotes the asymmetric effect. For example, a

negative g means that negative shocks give rise to strongervolatility than positive shocks;

� And d indicates the Box–Cox transformation.

In this specific study, we only considered four specialized ARCHmodels under the following conditions:

�

Bollerslev (1986) GARCH model with g ¼ 0, d ¼ 2 and d ¼ 0; � Ding et al (1993) Asymmetric Power (AP) GARCH with g, d and

d ¼ 0;

� Baillie, Bollerslev and Mikkelsen (BBM) Fractionally Integrated

(FI) GARCH (Baillie et al., 1996) with g ¼ 0, d ¼ 2;

� Tse (1998) FIAPGARCH with g, d and d.

For the maximum likelihood estimation, the joint estimations(vector parameter, w) involve the conditional mean parametersx0 ¼ (a0, a1,y, b1, b2,y) and the density function parameters g0,as well as the conditional variance parameters, h0 ¼ (a0, a1,y, g, d,d, b0, b1,y), all set at time t and et ¼ (rt�mt)/st. Given a normally

1 The density function is taken as one of the standard ones such as the

ssian, student-t, etc.

distributed et, the log-likelihood function can be expressed as

lt ¼ ln1ffiffiffiffiffiffi2pp

� ��

1

2

�t

st

� �2

�1

2s2

t . (4)

Similarly, the log-likelihood function for N samples is defined as

LN ¼1

2N lnð2pÞ þ

XN

t¼1

�t

st

� �2

þXN

t¼1

s2t

!. (5)

The information matrix (gradients) corresponding to x and hcan be derived from the second derivatives of the log-likelihoodfor the tth observation

Ixx0 ¼ �1

N

XN

t¼1

E@2ltðrt;x; hÞ@x@x0

" # !,

Ihh0 ¼ �1

N

XN

t¼1

E@2ltðrt;x; hÞ

@h@h0

" # !,

Ixh ¼ �1

N

XN

t¼1

E@2ltðrt;x; hÞ@x@h0

" # !, (6)

where Ixh represents the elements in the off-diagonal blockmatrix. For FI models only, the truncation lag of the slow decayingfractional difference operator (1�B)d is set to 1000 to account forthe possible long-run dependencies in line with the common BBMsetting. Or more precisely, (1�B)d

¼P

n ¼ 01000(�1)n((d(d�1)y

(d�k+1))/k!) �Bd. Due to the large number of estimated para-meters, the iterative optimization algorithm is used instead of theanalytical derivative approach. Consider a log-likelihood functionLN with overall joint estimator vector

@LN

@w�

@LN

@wð0Þþ ðw�wð0ÞÞ

@2LN

@wð0Þ@w0 ð0Þ, (7)

where w(0)denotes the trial values of the estimates. Rearrangingthe terms as in the Newton–Raphson algorithm, the (k+1)th vectorset of parameter values is defined as

wðkþ1Þ¼ wðkÞ �

@2LN

@wð0Þ@w0ð0Þ

!�1@LðkÞN

@w. (8)

For a faster computational process, one may use the methods ofBerndt et al. (1974), Marquardt (1963) or Bollerslev and Wool-dridge (1992) to improve the speed of the estimations. For heavy-tailed et, we used the standardized Student-t distribution withbetween three to six degrees freedom, with the followingrepresentation:

f ð�t;vÞ ¼G½ðvþ 1Þ=2�

G½v=2�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipðv� 2Þ

p 1þ�2

t

v� 2

� ��ððvþ1Þ=2Þ

, (9)

where G[ � ] is the gamma function. The Student-t distributionwith v42 exceeded the normal kurtosis which indicates fat-tailedbehavior. By replacing the log-likelihood function of the normaldistribution in Eq. (5), the Student-t log-likelihood is defined as

LN ¼ N ln Gvþ 1

2

� �� ln G

v

2

h i�

1

2lnðpðv� 2ÞÞ

� �

�1

2

XN

t¼1

lnðs2t Þ þ ð1þ vÞ ln 1þ

�2t

v� 2

� � !(10)

3.2. Diagnostics and forecasting evaluations

The adequacy of the models is tested using Ljung-Box statisticsfor the residuals, standardized residuals and squared standardizedresiduals. The Ljung-Box Q statistic at lag n, a test statistic for thenull hypothesis that there is no autocorrelation up to order n, is

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Table 1Descriptive statistics for crude oil returns.

Statistic Brent WTI

Mean 0.018415 0.019187

Median 0.042474 0.031213

Maximum 6.680608 7.059866

Minimum �7.422868 �8.638399

Standard Devation 1.004979 0.958789

Skewness �0.335130 �0.152718

Kurtosis 7.620049 7.370128

C.W. Cheong / Energy Policy 37 (2009) 2346–2355 2349

defined by

Q ¼ TðT þ 2ÞXn

i¼1

t2i

T � i, (11)

where T is the total number of observations and the ti2is the i-th

autocorrelation, which can be represented by the residual,standardized residual or squared standardized residual series.Under the null hypothesis, the Q statistic is asymptoticallydistributed as a w2 distribution with a number of degrees offreedom equivalent to the number of calculated autocorrelations.Furthermore, the Engle LM ARCH test is implemented to check thepresence of ARCH effects. The LM test statistic is determined froman auxiliary test regression. The null hypothesis states that thereis no ARCH effect up to order q in the at (standardized or squaredstandardized residual) with the regression model as follows:

a2t ¼ y0 þ

Xq

j¼1

yja2t�j

0@

1Aþ vt . (12)

The LM test statistic is asymptotically distributed as w2(q) undergeneral conditions.

For forecasting evaluations, consider a mean adjusted condi-tional mean rðhÞ;t � m ¼ ~rðhÞ;t ¼ sðhÞ;t�ðhÞ;t where s(h),t

2 is the latent

volatility and e(h),t is an i.i.d. process with constant variance. Sincethe s(h),t

2 is not directly unobservable, normally r(h),t2 (or |r(h),t|) is

used as an unbiased estimator of the actual volatility, given that

E½~r2ðhÞ;t� ¼ E½s2

ðhÞ;t�2ðhÞ;t� ¼ E½s2

ðhÞ;t�E½�2ðhÞ;t� ¼ ks2

ðhÞ;t , where k is a con-

stant. It is noted by Ebens (1999) and Andersen et al. (1999) thatalthough the squared return is an unbiased estimator, it can besomewhat noisy with respect to s(h),t.

2 There is a wide range ofstandard statistical loss functions used in forecasting accuracyevaluations. The loss function works in such a way that a penaltyis assigned when there is a deviation between the predicted andactual values. Therefore, the objective of forecasting evaluation issimply to minimize the expected loss. Since the comparativeforecasting performance is not invariant for all the loss functions(e.g., the mean square forecast error is sensitive to extremevalues), a collection of loss functions evaluations is necessary tofind the best model. In this study, a series of loss functions areconsidered as evaluation criteria. Each volatility model isestimated H-times based on the fixed intervals of 3086 and3761 observations for Brent and WTI, respectively. A rollingparameter estimation is implemented in which the first one-day-ahead forecast for Brent at t ¼ 3086 is using the estimation oft ¼ 1–3086, and the estimation from t ¼ 2 to 3087 is used toforecast the volatility at t ¼ 3088. Therefore, the H one-day-aheadvolatility forecasts can be obtained by using the rolling estima-

tions procedures for s2ðhÞ;t, where h ¼ 1,y, H. Three different loss

functions are used to evaluate the predictive accuracy of avolatility model

Mean Squared ErrorðMSEÞ ¼1

H

XmþH

h¼mþ1

ðactualh � forecasthÞ2,

Mean Absolute ErrorðMAEÞ ¼1

H

XmþH

h¼mþ1

jactualh � forecasthj,

Log L ¼1

H

XmþH

h¼mþ1

lnactualh

forecasth

� �2

, (13)

2 Alternatively, one may use the sum of high frequency intraday squared

returns (Andersen,1999) to approximate the latent volatility. However due to the

availability of tick-to-tick datasets, only the squared return is considers as the

proxy for the actual volatility.

where the variables actualh and forecasth represent the proxysquared returns and forecasted volatility, respectively. MSE, MAEand Log L are the most commonly used loss functions inforecasting evaluations. The loss functions reported the evalua-tions based directly on the deviations among the forecasts andrealizations. MAE is preferable to MSE, due to its robustnessagainst extreme values. The Log L (Pagan and Schwert, 1990)statistic conveys the influence of low volatility of the models.

4. Empirical results

Table 1 shows that both the Brent and WTI returns series havesimilar descriptive statistics. It is interesting to note that for bothof the returns series, which are very close to an i.i.d. standardizeddistribution, N (0,1), the unconditional mean and standard-deviation are close to zero and unity, respectively. However, thenon-zero skewness and excess kurtosis cause both of the returnsseries to deviate from a N (0,1). For the normality test, the jointtests of skewness (0 for normal distribution) and kurtosis (3 fornormal distribution) using Jacque–Bera statistics are both sig-nificantly different from a normal distribution. Figs. 1 and 2 showthe kernel density plots and QQ-plots of the returns series againstthe standard normal distribution and heavy-tailed Student-t

distribution. Both the returns series have high peaks at the mean,but are almost normally distributed in the tails. In addition, theStudent-t with three degrees of freedom is much heavier thanboth of the returns series. For the QQ-plots, the tail distributionfor WTI return series seems to be somewhat heavier than theBrent against the normal distribution. Especially for the Brentseries, the QQ-plot appeared very close to linear against theStudent-t distribution. In conclusion, the suitability of heavy-tailed innovations in the model specification is still questionable,until formal statistical analysis is conducts in the next section.

4.1. Estimates and diagnostics

Under the restricted predefine specifications, the generalizedheavy-tail ARCH model become GARCH (d ¼ 2, g ¼ 0), APARCH(d, g), FIGARCH (d ¼ 2, g ¼ 0,d) and FIAPARCH (d, g, d).

4.1.1. Brent crude oil market

In Table 2, all the models exhibit heavy tails, with the range ofdegrees of freedom from approximately 7.46431 to 7.99874 for thedensity function. However, this finding is quite different fromcommon equity markets (Cheong, 2008; Giot and Laurent, 2003)that often exhibit heavier tails around three to six degrees offreedom. The joint estimated AR(1) parameter, a1, in the condi-tional mean is statistically significant in all the models which may

Jarque-Bera 3415.317� 3043.422�

observation 3761 3806

Note:� Indicates 5% significance level.

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0.0

0.1

0.2

0.3

0.4

0.5

0.6

-30 -20 -10 0 10 20 30 40

normal distributionBrent returnstudent t (v = 3)

Den

sity

kernel

0.0

0.1

0.2

0.3

0.4

0.5

-30 -20 -10 0 10 20 30 40

WTI returnNormal distributionstudent t-(v = 3)

Den

sity

kernel

Fig. 2. Kernel density plots.

C.W. Cheong / Energy Policy 37 (2009) 2346–23552350

be caused by the first autocorrelation or upward drift. First, for theARCH estimates, the power coefficients, d, are 1.88774 and1.57269 for the normal and Student-t APARCH, respectively. Thenull hypothesis is rejects for d ¼ 1 but not for d ¼ 2, at the 5%significance level, which indicates that the Brent series are betteroff in conditional variance modeling.

Second, the asymmetric news impact coefficients, g, haveindicated positive values of 0.12224 and 0.107269, respectively, forthe normal and Student-t APARCH. This implies that the down-ward movements (shocks) in the Brent crude oil market arefollowed by greater volatilities than upward movements of thesame magnitude. In short, the leverage effect is present in themarkets. From an economic viewpoint, this is an expectedphenomenon since ‘bad news’ such as the 9/11 attack and theIraq war caused higher fluctuations in the crude oil market.

Third, the fractional difference parameters, d, are all significantin the FIGARCH and FIAPGARCH models, with a range from0.30109 to 0.36697. As a comparison, the FIAPARCH has a higherintensity in the long-persistence volatility. This phenomenon maybe caused by the cascading volatility (Mullier et al., 1997) createdby heterogeneous market participants (Dacorogna et al., 2001)with different endowments, reaction times and investmentstrategies.

In model selection, three indicators, namely, the log-likelihood,Akaike information and Schwarz information criteria (AIC andSIC), are used to evaluate the most appropriate models. In Table 2,FIAPARCH-Student-t model shows superior results (smallest forlog-likelihood and AIC) to the rest. However, the GARCH-normalmodel was superior in SIC with a value of 2.62102. This result isexpected because the SIC imposes additional penalties foradditional estimated parameters. This outcome is acceptablesince FIAPARCH provides the richest flexibility in capturing thestylized facts such as the clustering volatility, leverage effect, longmemory and Cox–Box transformation, in the volatility.

In diagnostic checking tests, two tests, the serial correlationtests and Engle’s Lag range multiplier tests, are performed on allthe models. All the Ljung-Box tests fail to reject the nullhypothesis of no serial correlation in the standardized residuals.However, conditional volatility is still found in the GARCH-normaland Student-t, the APARCH-normal and Student-t, and theFIGARCH-Student-t models, at the 5% significance level in thesquared standardized residual. For heteroskedasticity test, the LMtest with six lags show ARCH effects only in GARCH-student-t andAPARCH-Student-t models. Overall, the FIGARCH-type modelsseem to model the dependence in the conditional volatility moreadequately than the standard short memory GARCH.

4.1.2. WTI crude oil market

Although both of the prices display strong cointegration inFig. 1, the estimation analysis shows slightly different results in theWTI. Table 3 indicates stronger heavy-tailed with degrees offreedom around 6 as compared to 7.5 in Brent. These findings arealso illustrate in the QQ-plot in Fig. 3. For the parameter a1, all themodels fail to reject the null hypothesis of the non-existence of thisparameter. For the power coefficients, d, values are 1.60227 and1.28159 for the normal and Student-t APARCH, respectively. The nullhypotheses of d ¼ 1 and d ¼ 2 are not rejected at 5% significancelevel for the APARCH-normal, which indicates the WTI series isindifferent in either the conditional standard-deviation or theconditional variance model. For APARCH-Student-t, the test indi-cates that the series is better in the conditional standard-deviationmodel. On the other hand, both the FIGARCH and FIAPARCH modelsshow a better fit in the conditional variance specifications.

Next, the asymmetric news impact coefficients, g, are all notsignificantly different from zero under the 5% significance level. Inanother words, there is no asymmetric effect of so-called ‘bad’ or‘good’ news on the WTI crude oil market. These results contrastwith those of the leverage effect in the Brent market. Finally, thefractional difference parameters, d, are all significant in theFIGARCH and FIAPGARCH models with a range from 0.25159 to0.28709. For comparison, the FIAPARCH models show higherintensity in the long-persistence volatility. In addition, the WTIshows longer persistence volatility than the Brent crude oilmarket. (Table 4)

For model selection, none of the specific ARCH models isuniformly superior to the others. However, in general, the ARCH-Student-t models have lower L, AIC and SIC results. In diagnosticchecking tests, both the short memory ARCH models fail to rejectthe serial correlations in both the standardized and squaredstandardized residuals tests. Similarly, they also fail to reject thenull hypothesis of conditional heteroskedasticity variance in theLM test. On the other hand, the FIGARCH-type models indicate noserial correlation and heteroskedastic effect in the models.

4.2. Forecasting evaluation results

As indicated in the estimation and diagnostic evaluations, theFIGARCH-type models appear to fit well for both the Brent andWTI. However, there is no guarantee that they will perform betterin actual forecasting tests. A rolling out-of-sample forecast isimplements in both the Brent and WTI for 5-, 20-, 60- and 100-day horizons.

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Table 2BRENT maximum likelihood estimations.

Model specification GARCH APRARCH FIGARCH FIAPGARCH

Normal Student-t Normal Student-t Normal Student-t Normal Student-t

Conditional mean

a0 0.02677* (0.0142) 0.02935* (0.0136) 0.02013 (0.0145) 0.02692* (0.0137) 0.03024* (0.0142) 0.03343* (0.0137) 0.02282 (0.0143) 0.03121* (0.0138)

a1 0.03719* (0.0170) 0.03209* (0.0160 ) 0.03709* (0.0169) 0.03156* (0.0159) 0.03627* (0.0168) 0.03382* (0.0161) 0.03520* (0.0168) 0.03400* (0.0162)

Conditional variance:

a0 0.01099* (0.0039) 0.00607* (0.0024) 0.01222* (0.0043) 0.00621* (0.0026) 0.04163* (0.0130) 0.04405*(0.0160) 0.04212* (0.0185) 0.04292* (0.0215)

a1 0.05909* (0.0091) 0.03570* (0.0069) 0.05941* (0.0104) 0.03942* (0.0079) 0.23630* (0.0432) 0.29876* (0.0552) 0.25040* (0.0482) 0.31234* (0.0599)

b1 0.93083* (0.0116) 0.95811* (0.0082) 0.93059* (0.0122) 0.95957 *(0.0082) 0.55508* (0.0635) 0.58260* (0.0685) 0.52215* (0.0710) 0.56308* (0.0754)

d 0.36697* (0.0494) 0.32940* (0.0503) 0.32218* (0.0591) 0.30109* (0.0648)

g 0.12224* (0.0510) 0.10753 (0.0818) 0.16099* (0.0547) 0.13810* (0.0703)

d 1.88774* (0.2453) 1.57269* (0.2742) 2.01360* (0.1679) 2.01438* (0.2138)

Density function

v 7.46131* (0.8254) 7.56871* (0.8428) 7.84600* (0.8871) 7.99874* (0.9252)

Model selection

L �5035.982 �4964.391 �5032.58 �4962.32 �5025.33 �4961.03 �5020.01 �4958.81AIC 2.64827 2.61118 2.64753 2.61114 2.64320 2.60994 2.64145 2.60983SIC 2.65647 2.62102 2.65902 2.62427 2.65304 2.62142 2.65458 2.62459

Diagnostic

(1) Q-(6) on at 2.1092 [0.8338] 2.1730 [0.8247] 2.0365 [0.8440] 2.1946 [0.8216] 2.0210 [0.8462] 2.1190 [0.8324] 1.9808 [0.8517] 2.0435 [0.8430]

(2) Q-(6) on ~at2 10.6483* [0.0308] 27.5439* [0.0000] 10.0886*[0.0389] 30.9778*[0.0000] 5.4590 [0.2433] 10.5376 *[0.0322] 4.4399 [0.3497] 8.1480 [0.0863]

(3) ARCH (6) test 1.7840 [0.0983] 4.5041* [0.0001] 1.6878 [0.1198] 5.0468* [0.0000] 0.9163 [0.4818] 1.7645 [0.1023] 0.7381 [0.6188] 1.3660 [0.2244]

Notes:

1. For FI-models, the positive constraints are observed in all the models according to the BBM specifications;

2. ~atrepresents the standardized residual. Ljung Box Serial Correlation Test (Q-statistics) on ~at and ~a2t : Null hypothesis – No serial correlation;LM ARCH test: Null hypothesis-No ARCH effect;

3. ** and * denote 5% and 1% levels of significance. The values in the parentheses represent the standard-error and p-values for the estimation and the diagnostic test respectively.

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Table 3WTI maximum likelihood estimation.

Model specification GARCH APRARCH FIGARCH FIAPGARCH

Normal Student-t Normal Student-t Normal Student-t Normal Student-t

Conditional mean

a0 0.01423 (0.0141) 0.03224* (0.0135) 0.01434 (0.0145) 0.03072* (0.0136) 0.01858 (0.0143) 0.03300* (0.0136) 0.01959 (0.0145) 0.03266* (0.0137)

a1 �0.01505 (0.0170) �0.01455 (0.0159) �0.01468 (0.0168) �0.01342 (0.0155) �0.00924 (0.0176) �0.01099 (0.0161) �0.00883 (0.0177) �0.01088 (0.0161)

Conditional variance

a0 0.01309*(0.0037) 0.01246* (0.0038) 0.01327* (0.0041) 0.01127* (0.0039) 0.08890* (0.0244) 0.09827* (0.0343) 0.07668* (0.0316) 0.10357* (0.0381)

a1 0.05314* (0.0077) 0.03896* (0.0070) 0.06267* (0.0097) 0.04829* (0.0085) 0.18042* (0.0790) 0.20544* (0.1058) 0.17093* (0.0862) 0.20851* (0.1031)

b1 0.93543* (0.0095) 0.94858* (0.0091) 0.93437* (0.0099) 0.95072* (0.0091) 0.38063* (0.0889) 0.40743* (0.1146) 0.35463* (0.1070) 0.42133* (0.1283)

d 0.28709* (0.0367) 0.25159* (0.0388) 0.26638* (0.0529) 0.26434* (0.0689)

g �0.00134 (0.0551) 0.06767 (0.0948) �0.02191 (0.0504) 0.02081 (0.0751)

d 1.60227* (0.2020) 1.28159* (0.2402) 2.12172* (0.2068) 1.91804* (0.3011)

Density function

v 6.15158* (0.5803) 6.15032* (0.5764) 6.44371(0.62121) 6.44461* (0.6204)

Model selection

L �5156.90 �5042.80 �5154.99 �5038.96 �5146.31 �5040.42 �5146.11 �5040.36AIC 2.74423 2.68410 2.74428 2.68312 2.73913 2.68337 2.74009 2.68440

SIC 2.75252 2.69404 2.75588 2.69638 2.74907 2.69497 2.75334 2.69931

Diagnostic

(1) Q-(6) on at 9.0805 [0.1058] 9.9027 [0.0780] 9.8599 [0.0793] 11.5991* [0.0407] 9.3255 [0.0967] 9.9157 [0.0776] 9.1858 [0.1018] 10.0023 [0.0751]

(2) Q-(6) on ~at2 12.5123* [0.0139] 21.0272* [0.0003] 16.1419* [0.0028] 32.0454* [0.0000] 4.6174 [0.3288] 8.9573 [0.0621] 4.4265 [0.3513] 9.1959 [0.0563]

(3) ARCH (6) test 2.0620 [0.0544] 3.3265* [0.0029] 2.6632* [0.0140] 5.0773* [0.0000] 0.7796 [0.5858] 1.4981 [0.1745] 0.7479 [0.6110] 1.5389 [0.1612]

Notes:

1. For FI-models, the positive constraints are observed in all the models according to the BBM specifications;

2. ~atrepresents the standardized residual. Ljung Box Serial Correlation Test (Q-statistics) on ~at and ~a2t : Null hypothesis – No serial correlation;LM ARCH test: Null hypothesis-No ARCH effect;

3. ** and * denote 5% and 1% levels of significance. The values in the parentheses represent the standard-error and p-values for the estimation and the diagnostic test respectively

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-4

-3

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-1

0

1

2

3

4

-12 -8 -4 0 4 8Quantiles of BRENT return

Qua

ntile

s of

Nor

mal

-4

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1

2

3

4

-8 -4 0 4 8Quantiles of WTI return

Qua

ntile

s of

Nor

mal

-8

-6

-4

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2

4

6

8

-12 -8 -4 0 4 8Quantiles of BRENT return

Qua

ntile

s of

Stu

dent

's t

-8

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2

4

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-8 -4 0 4 8Quantiles of WTI return

Qua

ntile

s of

Stu

dent

's t

Fig. 3. Quantile–quantile plots.

C.W. Cheong / Energy Policy 37 (2009) 2346–2355 2353

For the Brent forecast evaluations, the short memory GARCH-normal and Student-t models have the lowest losses for theshorter 5- and 20-day horizon forecasts. However, for the 60- and100-day horizons, the APARCH-normal model is superior to allother models. These results imply that models with highercomplexity do not always perform the best in actual forecastingtests. In line with this result, Sadorsky (2006) found that the out-of-sample forecasts of a single equation GARCH model are moresuperior to the state space, vector autoregression and bivariateGARCH models in the petroleum futures prices returns. On theother hand, for the WTI crude oil market, the FIAPARCH andFIGARCH dominate the forecast evaluations for all the timehorizon forecasts. It is also found that the FIAPARCH-Student-t

models have the highest accuracy of all the models in the out-of-sample volatility forecasts.

As a conclusion, the parsimonious principle in volatilitymodeling appears to be very useful in the Brent crude oil marketvolatility forecasts. The standard short memory GARCH withnormal and Student-t distribution models demonstrates super-iority in the various time horizon forecasts. However, thecomplexity with the inclusion of asymmetric news impact, long-persistence volatility and power transformation of conditionalstandard-deviation in the WTI model specification has producedfruitful improvement in forecasting at specific time horizons.

5. Conclusion and implications

This study evaluates the volatility behavior of the two majorcrude oil markets particularly in the empirical stylized facts suchas the asymmetric news impact, long-persistence volatility and

tail behavior in the crude oil series. A very powerful and flexibleARCH model is used to evaluate the aforementioned stylized factsfor both the Brent and WTI markets.

In short, there are three important stylized facts that mayattract the interest of investors and policy makers. First, for theBrent series, the specification of conditional variance gainsadditional accuracy over the conditional standard-deviation inthe time-varying volatility modeling. One of the direct applica-tions of conditional variance is the quantitative market riskmeasurement such as value-at-risk (Jorion, 1997). From theeconomic point of view, the crude oil market risk is a vital issuefor financial institutes (including private or government invest-ments) because a large amount of wealth can be lost due to failureof supervising and controlling the financial risks. On the otherhand, the WTI series indicates indifferent results in both theconditional standard-deviation and the conditional variancemodels.

Second, the Brent series encounters the leverage effect oncrude oil price shocks. This implies that the downward move-ments (shocks) in the Brent crude oil market are follow by greatervolatilities than upward movements of the same magnitude. ForWTI series, there is no asymmetric effect of so-called ‘bad’ or‘good’ news to on the WTI crude oil market. These results contrastwith those of the leverage effect in the Brent market. However, theWTI series may have other forms of interrelationships (positively/negatively correlated) with the macroeconomic indicators andfinancial derivatives prices which are not included in this study.

Third, the crude oil price volatility persistence is observed inboth the Brent and WTI series. This implies that the fluctuation ofprices has permanent effects on its volatility. The long-persistencevolatility can be explained in the micromanner where the

ARTICLE IN PRESS

Table 4Empirical forecast evaluations for H one-day ahead forecasts.

GARCH APRARCH FIGARCH FIAPGARCH

Normal Student-

t

Normal Student-

t

Normal Student-

t

Normal Student-

t

BRENT

5-day Horizon

MSE 0.9632 0.9381 1.055 1.003 1.177 1.123 1.365 1.262

MAE 0.8722 0.8621 0.914 0.892 0.990 0.963 1.065 1.023

LL 12.6902 12.6201 12.980 12.830 13.330 13.140 13.750 13.480

20-day Horizon

MSE 1.4152 1.3911 1.424 1.4152 1.452 1.427 1.470 1.443

MAE 0.9242 0.9161 0.944 0.936 0.972 0.957 0.999 0.980

LL 5.1461 5.1702 5.197 5.268 5.335 5.305 5.384 5.348

60-day Horizon

MSE 0.7192 0.742 0.7151 0.757 0.779 0.782 0.781 0.781

MAE 0.6871 0.7072 0.6871 0.717 0.735 0.741 0.744 0.744

LL 10.5002 10.800 10.4701 10.880 11.030 11.100 11.060 11.100

100-day Horizon

MSE 0.5632 0.578 0.5591 0.587 0.613 0.619 0.613 0.616

MAE 0.6112 0.631 0.6081 0.636 0.663 0.671 0.666 0.670

LL 7.1092 7.316 7.0801 7.359 7.638 7.716 7.660 7.715

WTI

5-day Horizon

MSE 0.882 0.960 0.832 0.918 0.5781 0.613 0.6042 0.599

MAE 0.804 0.854 0.771 0.828 0.6542 0.670 0.666 0.6631

LL 9.113 9.295 8.989 9.201 8.1941 8.3232 8.294 8.272

20-day Horizon

MSE 2.716 2.713 2.715 2.714 2.6222 2.6041 2.6222 2.6041

MAE 0.999 1.020 0.995 1.017 0.963 0.9552 0.970 0.9521

LL 2.629 2.731 2.614 2.721 2.4881 2.511 2.516 2.4942

60-day Horizon

MSE 17.750 17.640 17.6002 17.4301 17.970 17.710 18.010 17.690

MAE 1.987 1.934 1.952 1.899 1.936 1.8852 1.949 1.8801

LL 5.287 5.191 5.213 5.105 5.119 4.9952 5.155 4.9771

100-day Horizon

MSE 10.950 10.900 10.8702 10.7801 11.070 10.910 11.100 10.900

MAE 1.566 1.548 1.550 1.535 1.534 1.5032 1.545 1.4981

LL 6.749 6.761 6.721 6.766 6.682 6.6132 6.713 6.6001

Note: The superscripts 1 and 2 denote the lowest and 2nd lowest error statistics.

C.W. Cheong / Energy Policy 37 (2009) 2346–23552354

heterogeneous market participants (Dacorogna et al., 2001;Mullier et al., 1997) interpret the same information differentlyaccording to their trading opportunities. Each time horizontrading activities create a unique volatility under the fluctuatingprice movements. Thus, the crude oil markets, which arecomposed by participants with different reaction times to news,have created a volatility cascade ranging from low to highfrequencies. These combinations of dissimilar volatilities arebelieved to produce the very slow decaying volatility in the crudeoil market. Therefore, investors or policy makers should take intoaccount this behavioral economic in their future investment orpolicy-making.

Finally, although both the estimation and diagnostic evalua-tions are in favor the FIAPARCH model, from empirical out-of-sample forecast it appears that the simplest parsimonious GARCHfits the Brent crude oil data better than the other models. On theother hand, the FIAPARCH out-of-sample WTI forecasts providesuperior performance. These findings suggest that energy econ-

omists and financial analysts should consider not only thecomplexity, but also the parsimonious principle and actualperformance of out-of-sample forecasts, in choosing a crude oilvolatility model.

Acknowledgements

The author would like to thank the anonymous referees fortheir helpful comments on an earlier version of the paper.

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