# Forecasting crude oil market volatility: Further evidence using GARCH-class models

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<ul><li><p>rt</p><p>rtheoad</p><p>rkconarkeforss ftherH-crecas ve or twenty days.</p><p> 2010 Elsevier B.V. All rights reserved.</p><p>yed byes volans andn pricus, mre criti</p><p>Energy Economics 32 (2010) 14771484</p><p>Contents lists available at ScienceDirect</p><p>Energy Ec</p><p>.e lvolatility models in which a time-varying volatility process isextracted from nancial return data. Most such models are variantsof generalized autoregressive conditional heteroskedasticity (GARCH)models (Bollerslev, 1986). Much research has been done to evaluatethe forecasting performance of different volatility models, especiallyGARCH-class ones, regarding oil markets (Adrangi et al., 2001;Agnolucci, 2009; Aloui and Mabrouk, 2010; Cabedo and Moya,2003; Cheong, 2009; Fong and See, 2002; Giot and Laurent, 2003;</p><p>than the other models examined. On the other hand, the FIAPARCHout-of-sample WTI forecasts provided superior performance. In arecent investigation using weekly crude oil spot prices in eleveninternational markets, Mohammadi and Su (2010) compared theforecasting accuracy of four GARCH-class models (GARCH, EGARCH,APARCH, and FIGARCH) under two loss functions. The DM testshowed that the APARCH model provided the best performance. Theforegoing summary of the extant literature clearly indicates that priorKang et al., 2009; Kolos and Ronn, 2008; MoMorana, 2001; Narayan and Narayan, 2007; S2006; Sadorsky, 2006).</p><p>Investigating the volatility forecasting ofKang et al. (2009) evaluated the out-of-sampl</p><p> Corresponding author.E-mail address: weiyusy@126.com (Y. Wei).</p><p>0140-9883/$ see front matter 2010 Elsevier B.V. Aldoi:10.1016/j.eneco.2010.07.009cal for businesses andl paper of Engle (1982)r of so-called historical</p><p>mance of four GARCH-class models (GARCH, APARCH, FIGARCH, andFIAPARCH) under three loss functions, nding that the simplest andmost parsimonious GARCH model tted Brent crude oil data bettergovernments around the world. The seminaled to the development of a large numbeKeywords:Crude oil marketVolatility forecastingGARCHSPA test</p><p>1. Introduction</p><p>Because of the important role plaeconomy, recent increases in oil pricconcern among consumers, corporatiovolatility is a key input into optioallocation and risk measurement. Ththe volatility of crude oil prices acrude oil in the worldtility have caused greatgovernments. Oil priceing formulas, portfolioodeling and forecasting</p><p>four GARCH-class (GARCH, IGARCH, CGARCH, FIGARCH) models usingthe DM test of Diebold and Mariano (1995) under two loss functions.They found that the CGARCH and FIGARCH models could capture thelong-memory volatility of crude oil markets and obtain superiorperformance compared to that of the GARCH and IGARCH ones (Kanget al., 2009). In a study complementing that of Kang et al. (2009),Cheong (2009) investigated the out-of-sample forecasting perfor-hammadi and Su, 2010;adeghi and Shavvalpour,</p><p>crude oil prices, recentlye forecasting accuracy of</p><p>empirical resultsobvious which levaluation of vodifferent roles innew technique f(SPA) test, whicapproaches, sucreality check (RC</p><p>l rights reserved.C32C52</p><p>longer time horizons, suchForecasting crude oil market volatility: Fu</p><p>Yu Wei a,, Yudong Wang b, Dengshi Huang a</p><p>a School of Economics and Management, Southwest Jiaotong University, First Section of Nob Antai College of Economics and Management, Shanghai Jiaotong University, Fahuazhen R</p><p>a b s t r a c ta r t i c l e i n f o</p><p>Article history:Received 23 May 2010Received in revised form 18 July 2010Accepted 18 July 2010Available online 24 July 2010</p><p>JEL classication:Q40E30</p><p>This paper extends the wogeneralized autoregressivefeatures of two crude oil mday out-of-sample volatilityability test and with more loof the other models for eigeneral, the nonlinear GARCvolatility, exhibit greater fo</p><p>j ourna l homepage: wwwher evidence using GARCH-class models</p><p>rn Second Ring Road, Chengdu, Sichuan Province, China535, Shanghai, China</p><p>of Kang et al. (2009). We use a greater number of linear and nonlinearditional heteroskedasticity (GARCH) class models to capture the volatilityts Brent and West Texas Intermediate (WTI). The one-, ve- and twenty-ecasts of the GARCH-class models are evaluated using the superior predictiveunctions. Unlike Kang et al. (2009), we nd that no model can outperform allthe Brent or the WTI market across different loss functions. However, inlass models, which are capable of capturing long-memory and/or asymmetricasting accuracy than the linear ones, especially in volatility forecasting over</p><p>onomics</p><p>sev ie r.com/ locate /enecoare mixed. As discussed by Lopez (2001), it is notoss function or criterion is more appropriate for thelatility models, and different loss functions may playpractical applications. Hansen (2005) proposed a</p><p>or model comparison, the superior predictive abilityh has been proven to be more robust than similarh as the DM test (Diebold and Mariano, 1995) or) test of White (2000). Another important advantage</p></li><li><p>of the SPA test is that it enables the comparison of the performance ofmore than two models at one time under a specic loss function,whereas the DM test can only be utilized in pairwise testing of twomodels.</p><p>Based on the aforementioned considerations, this paper extendsthe work of Kang et al. (2009) and related research in three ways.First, we use a greater number of linear and nonlinear GARCH-classmodels to depict the important stylized facts about volatility,including clustering volatility, long-memory volatility and theasymmetric leverage effect in volatility, among others (Cont, 2001).In contrast, CGARCH and FIGARCH models used in Kang et al. (2009)can capture only long-memory volatility and not the asymmetricleverage effect. We estimate nine GARCH-class models: RiskMetrics,GARCH, IGARCH, GJR, EGARCH, APARCH, FIGARCH, FIAPARCH andHYGARCH models. Second, we adopt six loss functions as theforecasting criteria to reexamine the conclusions of previous research,in which the mean square error (MSE) and mean absolute error(MAE) are applied mostly. Regarding forecasting models andevaluation criteria, this paper incorporates those adopted by Kang etal. (2009), Cheong (2009), and Mohammadi and Su (2010). Lastly, weemploy the SPA test (Hansen, 2005) to get more robust results. Incontrast to the conclusions reached by Kang et al. (2009), Cheong(2009), and Mohammadi and Su (2010), we nd that none of theGARCH-type models considered here is superior to the others. Inaddition, our results show that the nonlinear GARCH-class models aremore effective than the linear ones in capturing the long-rundynamics of crude oil price volatility.</p><p>The rest of this paper is organized as follows. Section 2 introducesthe sample data and the statistical characteristics. Section 3 discussesthe nine linear and nonlinear GARCH-type models used in this paper.</p><p>Section 44 presents the forecasting methodology and SPA test.Section 5 shows the empirical results of out-of-sample volatilityforecasting using the GARCH-class models and the SPA test. Section 6contains concluding remarks.</p><p>2. Data</p><p>We use the daily price data (in US dollars per barrel) of Brent andWest Texas Intermediate (WTI) from January 6, 1992, to December 31,2009. The data of the last three years, i.e., 2007 to 2009, are used toevaluate the out-of-sample volatility forecasts. During the 20072009period, the global nancial crisis greatly affected the world economy,and the price of crude oil uctuated tremendously from about USD 30to USD 145 per barrel. Therefore, this period is a good testing timehorizon to evaluate the performance of different volatilitymodels. Thedata are obtained from the Energy Information Administration of theUS Department of Energy. Let Pt denote the price of crude oil on day t.All daily sample prices are converted into a daily nominal percentagereturn series for crude oil, i.e., rt=100 ln (Pt/Pt1) for t=1,2,, T, inwhich rt is the returns for crude oil at time t. Following Sadorsky(2006) and Kang et al. (2009), daily actual volatility (variance) isassessed by daily squared returns (rt2). The graphical representation ofthe prices, returns and volatility for Brent andWTI crude oil is given inFig. 1. The left panel of the gure gives the data for the Brent marketand the right panel those for the WTI market.</p><p>Table 1 provides the descriptive statistics of the two return series.The Brent and WTI sample returns display similar statisticalcharacteristics. The sample means of the two return series are quitesmall in comparison to the standard deviations. The JarqueBerastatistic shows that the null hypothesis of normality is rejected at the</p><p>1478 Y. Wei et al. / Energy Economics 32 (2010) 14771484Fig. 1. Daily prices, returns and volatility for Brent and WTI crude oil during the period from January 6, 1992, to December 31, 2009.</p></li><li><p>1479Y. Wei et al. / Energy Economics 32 (2010) 147714841% level of signicance, also as evidenced by a high excess kurtosisand negative skewness. The LjungBox statistic for serial correlationshows that the null hypothesis of no autocorrelation up to the 20thorder is rejected and conrms serial autocorrelation in the crude oilreturns. The augmented DickeyFuller and PhillipsPerron unit roottests both support the rejection of the null hypothesis of a unit root atthe 1% signicance level, implying that the two return series arestationary and may be modeled directly without further transforms.</p><p>3. Model framework</p><p>3.1. Linear GARCH-class models</p><p>Based on the work of Engle (1982), the most popular volatilitymodel is the GARCH model proposed by Bollerslev (1986). Bollerslevet al. (1994) showed that the GARCH(1,1) specication worked wellin most applied situations, and Sadorsky (2006) also demonstratedthat the GARCH(1,1) model was a good t for crude oil volatility. Thestandard GARCH(1,1) model for daily returns is given by</p><p>rt = t + t = t + tzt ; zteNID 0;1 ;2t = + </p><p>2t1 + </p><p>2t1;</p><p>1</p><p>where t denotes the conditional mean and t2 is the conditionalvariance with parameter restrictions N0, N0, N0 and +b1. Asshown in Table 1, the sample mean of crude oil returns is quite smallin comparisonwith its standard deviation (volatility); thus, we set the</p><p>Table 1Descriptive statistics for oil price returns.</p><p>Brent WTI</p><p>Number of observations 4474 4446Mean (%) 0.032 0.032Standard deviation (%) 2.324 2.480Minimum 19.891 17.092Maximum 18.130 16.414Skewness 0.087 0.195Excess kurtosis 5.072 4.874JarqueBera 4801.221 4429.502</p><p>Q (20) 45.150 49.208</p><p>ADF 16.510 26.585PP 65.450 67.556</p><p>Note: The JarqueBera statistic tests for the null hypothesis of normality in the samplereturns distribution. Q (20) is the LjungBox statistic of the return series for up to the20th order serial correlation. ADF and PP are the statistics of the augmented DickeyFuller and PhillipsPerron unit root tests, respectively, based on the lowest AIC value. indicates rejection at the 1% signicance level.conditional mean t to equal 0 in this paper, following Koopman et al.(2005), among others.</p><p>Another linear GARCH-class model is the IGARCH model devel-oped by Engle and Bollerslev (1986), which can capture innitepersistence in the conditional variance. The model setting of theIGARCH(1,1) model is similar to that of the GARCH(1,1) but with theparameter restriction +=1. The RiskMetrics volatility specica-tion of JP Morgan is also popular among market practitioners. In itsmost simple form, the RiskMetrics model is equivalent to a normalIGARCH (1,1) model, where the autoregressive parameter is set at apre-specied value of 0.94 and the coefcient of t12 is equal to 0.06.In this specication, the conditional variance is dened as</p><p>2t = 0:062t1 + 0:94</p><p>2t1: 2</p><p>Therefore, the RiskMetrics specication does not require theestimation of unknown parameters in the volatility equation as allparameters are present at given values. Although this is a crudeway tomodel volatility, it is widely used by practitioners as it often givesacceptable short-term volatility forecasts.In summary, for the linear model setting of the conditionalvariance, we use these three linear GARCH-class models, i.e., theGARCH, IGARCH and RiskMetrics models.</p><p>3.2. Nonlinear GARCH-class models</p><p>To take the stylized facts of nancial markets into account (Cont,2001), other GARCH-class models have been developed to capturelong-memory and short-memory volatility effects, asymmetric lever-age effects and so forth. Because of the nonlinear model setting ofthese newly developed models, we call them nonlinear GARCH-classmodels. The following nonlinear GARCH-class models are used in thispaper to forecast the volatility of crude oil prices.</p><p>The GJRmodel developed by Glosten et al. (1993) is constructed tocapture the potential larger impact of negative shocks on returnvolatility, which is usually named as asymmetric leverage volatilityeffect. Specication for the conditional variance of GJR(1,1) model is</p><p>2t = + + I t1b0 2t1 + 2t1; 3</p><p>where I(.) is an indicator function; i.e., when the condition in (.) ismet, I(.)=1, and 0 otherwise. is the asymmetric leveragecoefcient, which describes the volatility leverage effect.</p><p>Another popular nonlinear GARCH-class model, which can alsodepict the volatility leverage effect, is the exponential GARCH(EGARCH) one proposed by Nelson (1991). Nelson argued that thenonnegative constraints in the linear GARCHmodel are too restrictive.The GARCH model imposes nonnegative constraints on parameters and , whereas no restrictions are placed on these parameters in theEGARCH model. The EGARCH(1,1) model is given as</p><p>log 2t </p><p>= + zt1 + jzt1jEjzt1j </p><p>+ log 2t1 </p><p>; 4</p><p>where is again the asymmetric leverage coefcient to describe thevolatility leverage effect.</p><p>The asymmetric power ARCH (APARCH) model of Ding et al.(1993) is one of the most promising ARCH-type models. This modelnests several ARCH-type models and has been found to be particularlyrelevant in many recent applications (see, for example, Giot andLaurent, 2003; Mittnik and Paolella, 2000). The APARCH(1,1) model isdened as follows:</p><p>t = + jt1 jt1 + t1; 5</p><p>where parameter (N0) plays the role of a BoxCox transformationof the conditional standard deviation t, while reects the so-calledleverage effect. The APARCH model includes several ARCH extensionsas special cases, including the GARCH(1,1) model when =2 and=0, and the GJR(1,1) one when =2.</p><p>As standard GARCH model focuses only on short-term volatilityspecication and forecasting, some authors argue that long-rundependencies (memories) in nancial market volatility may be bettercharacterized by a fractionally integrated ARCH (FIGARCH) model(e.g., Andersen and Bollerslev, 1997; Baillie et al., 1996). The FIGARCHmodel implies the nite persistence of volatility shocks (no suchpersistence exists in the GARCH framework), i.e., long-memorybehavior and a slow rate of decay after a volatility shock. An IGARCHmodel implies the complete persistence of a shock, and apparentlyquickly fell out of favor. Interestingly, the FIGARCH(1,d,1) nests aGARCH(1,1) with d=0 and the IGARCH(1,1) for d=1. The FIGARCH(1,d,1) model can be written as follows:</p><p>2t = + 2t1 + 1 1L 1 1L 1L d</p><p>h i2t ; 6</p><p>whe...</p></li></ul>

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