volatility forecasting for crude oil futures

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Volatility forecasting for crude oil futuresMassimiliano Marzo a & Paolo Zagaglia ba Department of Economics , Università di Bologna , Piazza Scaravilli 2, Bologna, Italyb Department of Economics , Stockholm University, Universitetsvägen 10A , SE-106 91,Stockholm, SwedenPublished online: 22 Jan 2010.

To cite this article: Massimiliano Marzo & Paolo Zagaglia (2010) Volatility forecasting for crude oil futures, Applied EconomicsLetters, 17:16, 1587-1599, DOI: 10.1080/13504850903084996

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Volatility forecasting for crude

oil futures

Massimiliano Marzoa,* and Paolo Zagagliab

aDepartment of Economics, Universita di Bologna, Piazza Scaravilli 2,Bologna, ItalybDepartment of Economics, Stockholm University, Universitetsvagen 10A,SE-106 91 Stockholm, Sweden

This article studies the forecasting properties of linear GARCH models forclosing-day futures prices on crude oil, first position, traded in the NewYorkMercantile Exchange from January 1995 toNovember 2005. To account forfat tails in the empirical distribution of the series, we compare models basedon the normal, Student’s t and generalized exponential distribution. Wefocus on out-of-sample predictability by ranking the models according to alarge array of statistical loss functions. The results from the tests forpredictive ability show that the GARCH-G model fares best for shorthorizons from 1 to 3 days ahead. For horizons from 1 week ahead, nosuperior model can be identified. We also consider out-of-sample lossfunctions based on value-at-risk that mimic portfolio managers andregulators’ preferences. Exponential GARCH models display the bestperformance in this case.

The swings in oil prices that gave investors and traders whiplash in 2004 arenot preventing new investors from rushing into oil and other energy-relatedcommodities this year. (. . .)Ultimately, the rising number of speculator could lead to even more price

volatility in 2005, pushing the highs higher and the lows lower. (. . .)After a generation in the wilderness, the oil futures that are used to make a

bet on oil prices have become a bona fide investment, said Charles O’Donnell,who manages Lake Asset Management, a small energy fund based in London.

Heather Timmons, The New York Times1

I. Introduction

Futures contracts are one of the key instruments used

to trade oil products in international financial markets

(see Edison et al., 1999). Hence, the evolution of the

daily volatility of oil futures prices conveys key infor-

mation for understanding the functioning of oil

markets.Various studies analyse the usefulness of volatility

models for the prediction of oil prices. In particular,

Sadorsky (2006) considers univariate, bivariate and

state-space models. He finds that single-equation

GARCH overperforms more sophisticated models

for forecasting petroleum futures prices. Fong and

See (2002) study a Markov switching model of the

conditional volatility of crude oil futures prices and

show that the regimes identified by their model capture

major oil-related events. A related strand of literature

investigates the transmission of volatility between

energy markets. For instance, Ewing et al. (2002)

*Corresponding author. E-mail: [email protected] money pumps up volatility of oil prices, 7 January 2005.

Applied Economics Letters ISSN 1350–4851 print/ISSN 1466–4291 online � 2010 Taylor & Francishttp://www.informaworld.com

DOI: 10.1080/13504850903084996

1587

Applied Economics Letters, 2010, 17, 1587–1599

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show that there are significant patterns of volatilityspillovers between the markets for oil and natural gas.This article evaluates the predictive performance of

linear GARCH models for closing-day futures priceson crude oil traded in the New York MercantileExchange. To account for fat tails typical of financialseries (see Bollerslev, 1987), we compare models basedon the normal, Student’s t and generalized exponentialdistribution.We focus on out-of-sample predictabilityover short (1–3 days ahead) and long (1–3 weeksahead). Our empirical application ranks the modelsaccording to a large array of statistical loss functions.The results from the tests for predictive ability show

that the GARCH-G model fares best for short hori-zons from 1 to 3 days ahead. For horizons from 1weekahead, no superior model can be identified. We alsoconsider out-of-sample loss functions based on Value-at-Risk (VaR). FollowingMarcucci (2005), we introduceVaR-based functions that mimic portfolio managersand regulators’ preferences for penalizing large forecastfailures, as well as opportunity costs from over-investments. In this case, models of the ExponentialGARCH (EGARCH) type display the best perfor-mance, followed closely by the GARCH-G.The outline of this article is as follows. Section II

proposes an overview of univariate GARCH modelsSection III outlines the forecast evaluation methods,including the statistical loss functions used in thisarticle, the tests for predictive ability and the VaRstrategies. Section IV presents the data set. The resultsare discussed in Section V. Section VI proposes someconcluding remarks.

II. An overview of Garch Models

Let the model for the conditional mean of the return rttake the form

rt ¼ � þ �tffiffiffiffiht

pð1Þ

where �t is an independent and identically distributed.process with variance ht. In the standard GARCH(1,1)model, the model for the conditional variance is

ht ¼ �0 þ �1�2t�1 þ �ht�1 ð2Þ

with �0>0; �1 � 0 and �1 � 0 in order to ensure apositive conditional variance. The presence of skewnessin financial data has motivated the introduction of theEGARCH model:

logðhtÞ ¼ �0 þ �1�t�1ht�1

��þ � �t�1ht�1

þ � logðht�1Þ ð3Þ

The GJR model, instead, deals with the asymmetric

reaction of the conditional variance depending on the

sign of the shock:

ht ¼�0 þ �1�2t�1½1� If�t�1>0g�

þ ��2t�1If�t�1>0g þ �ht�1 ð4Þ

Bollerslev (1987) shows that financial times series

are typically characterized by high kurtosis. To

model the fat tails of the empirical distribution of thereturns, we assume that the error term �t follows eithera Student’s t-distributionwith v degrees of freedom or a

generalized error distribution. The probability density

function of �t then takes the form

fð�tÞ ¼�ðð�þ 1Þ=2Þffiffiffi

�p

�ð�=2Þ ð�� 2Þ-1=2ðhtÞ-1=2

· 1þ �2thtð�� 2Þ

� �� þ12ð5Þ

where �ð�Þ indicates the gamma function with the

shape parameter �>2. Under the generalized errordistribution (G), the model errors follow the pdf

fð�tÞ ¼� exp 1=2 �t

h1=2t

��

h1=2t 2ð2þ1=vÞ�ð1=�Þ

ð6Þ

with :¼ ½ð2�2=��ð1=�ÞÞ=�ð3=�Þ�1=2

III. Forecast Evaluation

The m-step ahead volatility forecast, indicated by

mt;tþm; is computed as the aggregated sum of the fore-casts for the following m steps made at time t. We

consider the volatility forecast over three horizons,

namely 1 day, 1 week and 3 weeks ahead.

Statistical loss functions

There exists no unique criterion capable of selecting

the ‘best’ forecasting model. Hence, this article eval-

uates the predictive performance of the GARCHmodels through an array of statistical loss functions.

These criteria are listed in Table 1. The functions

named MSE1 and MSE2 are typical mean squarederror metrics. The R2LOG function penalizes the

volatility forecasts for low volatility periods in a way

different from high volatility periods. Finally, theMean Absolute Deviation (MAD) criterion is robust

to the presence of outliers. Bollerslev and Ghysels

1588 M. Marzo and P. Zagaglia

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(1996) have proposed the Heteroskedasticity-AdjustedMSE (HMSE).It is instructive to report the so-called Success Ratio

(SR). These statistics indicate the fraction of the vola-tility forecasts that have the same direction of changeas the realized volatility. For an actual volatility proxy�tþm at time tþm and a volatility forecast �ht;tþm, theSR can be written as

SR :¼ ð1=nÞXn�1j¼0If�tþmþj �htþj;tþmþj>0g ð7Þ

where I is an indicator function.The Directional Accuracy (DA) test of Pesaran and

Timmermann (1992) is based on the statistics

DA :¼ SR� SRIffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffivarðSRÞ � varðSRIÞ

p ð8Þ

where SRI :=PPþ ð1� PÞð1� PÞ; P indicates thefraction of times such that �tþmþj and P is the fractionof times for which �htþmþj>0.

Test of predictive ability

Diebold and Mariano (1995) propose a test of equalpredictive ability between two competing models anddenote by fei;tgnt¼1 and fej;tgnt¼1 the forecast errors oftwo models i and j. The loss differential between thetwo forecasts can be written as

dt :¼ ½gðei;tÞ � gðej;tÞ� ð9Þ

where g(�) is the loss function. If fdtgnt¼1 is covariancestationary and has no long memory, the sample mean

loss differential �d ¼ ð1=nÞ�nt¼1dt is asymptotically dis-

tributed asffiffiffinpð�d� �Þ �!d Nð0;Vð�dÞÞ. Under the null

of equal predictive ability, Diebold andMariano (1995)

propose the test statistics DM :¼ �dffiffiffiffiffiffiffiffiffiffiVð�dÞ

q,Nð0; 1Þ.

Harvey et al. (1997) suggest modified DM statistics

(MDM) that tackle the oversize problem that arises insmall samples. The modified test statistics are obtainedby multiplying the standard statistics by a factor ofcorrection.White (1980) introduces a test for superior predic-

tive ability – the RC test – that checks whether aspecific forecasting model is outperformed by an alter-native set of models according to a loss function. LetLð2t ; hk;tÞ denote the loss function for the predictionwith model k, with k ¼ 1; . . . l. The relative predictiveperformance of model 0 can be computed as

fk;t ¼ Lt;0 � Lt;k ð10Þ

If fk;t is stationary, we can define the expected relativeperformance E½fk;t�. The testing procedure amounts tochecking that none of the competing models out-perform the benchmark:

H0 : maxk¼1;...l

E½fk;t� � 0 ð11Þ

The rejection of the null implies that at least onecompeting model is better than the benchmark. Thetest statistics are

maxk¼1;...l

n1=2�fk;n ð12Þ

Hansen (2005) stresses that the distribution of the teststatistics is not unique under the null and that it issensitive to the inclusion of poor models. Hence, heproposes a way of obtaining a consistent estimate ofthe p-value of a modified test statistics, along with anupper and a lower bound. The resulting SPAu yieldsthe p-value of a conservative test where all the compet-ing models are assumed to be as good as the bench-mark in terms of expected loss. The SPAl test is insteadbased on p-values that assume that the models withbad performance are poor models.

Value-at-risk

As suggested by Brooks and Persand (2003), loss func-tions based on VaR are a natural alternative to thestandard statistical loss functions while evaluating thepredictive performance of a model estimated on finan-cial data. The VaR measures the market risk of a port-folio quantified in monetary terms and arising frommarket fluctuations at a given significance level.Several statistical tests can be computed to assess

the forecasting ability of the GARCH models for theVaR. The TimeUntil First Failure (TUFF) is based onthe failure process, namely the number of exceptionsof the VaR from model k� I rt<VaRk

t. For a signifi-

cance level �, the null hypothesis isH0 : � ¼ �0 and thelikelihood-ratio test statistics are

Table 1. Statistical loss functions

MSE1 1=n�nt¼1 tþm � h

1=2t;tþm

� 2MSE2 1=n�n

t¼1 2tþm � ht;tþm

� 2QLIKE 1=n�n

t¼1 log ht;tþm þ 2h�1t;tþm

� R2LOG

1=n�nt¼1 log 2h�1t;tþm

h i� 2MAD1 1=n�n

t¼1 tþm � h1=2t;tþm

�� MAD2 1=n�n

t¼1 2tþm � ht;tþm

�� HMSE

1=n�nt¼1 2h�1t;tþm � 1� 2

Volatility forecasting for crude oil futures 1589

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LRTUFFð ~T; �Þ ¼ � 2 log �ð1� �Þ ~T�1h i

þ 2 log ~T�1ð1� ~T�1Þ ~T�1h i

ð13Þ

with the number of observations ~T before the firstexception. The statistic LRTUFF is distributed as a�2(1) under the null. The 95% confidence intervalis (3, 514) for the 99% VaR and (1, 101) for the95% VaR.A VaR can be insufficient to cover the losses that a

portfolio incurs. In this sense, a model can be judgedadequate when the proportion of failures out of sampleis close to the nominal value. The unconditional criter-ion suggests that the VaR is adequate if E½I t� ¼ �. Asthe number of failures is independent and identicallydistributed as a binomial, the likelihood-ratio test sta-tistics can be written as follows:

LRPF ¼ �2 log�n1ð1� �Þn0�n1ð1� �Þn0� �

, �2ð1Þ ð14Þ

where n1 is the number of failures, � is the level of theVaR and � :¼ n1=ðn1 þ n0Þ.As financial data are characterized by volatility

clustering, good interval forecasts from a VaRmodel should be narrow in periods of low volatilityand wide in periods of high volatility. Christoffersen(1998) proposes a test of independence. The null ofindependent failure rates is tested against a first-order Markov failure process. The test statisticstake the form

LRIND¼�2logð1� �Þðn00þn10Þð1� �Þðn01þn11Þ

ð1� �01Þn00 �n0101 ð1� �11Þn10 �n1111

" #,�2ð1Þ

ð15Þ

where �ij¼PrfI t¼ ijI t�1¼ jg. Finally, we consider aconditional test of correct coverage where the nullof independent failures with a probability � is testedagainst the first-order Markov failure:

LRCC¼�2logð1��Þn0�n10

ð1� �01Þn00 �n0101 ð1� �11Þn10 �n1111

� �,�2ð2Þ

ð16Þ

We follow Marcucci (2005) and evaluate the com-peting models through VaR loss functions that mimicthe utility functions of risk managers. In particular,the Regulator Loss Function (RLF) introduces anasymmetric penalty for the large losses. The RLFtakes the form

L1t :¼ ðrt � VaRk

t Þ2Ifrt<VaRk

t g ð17Þ

The Firm Loss Function (FLF), instead, penalizes

the models that require an excessive investment of

capital and that bear larger opportunity costs. This

function is defined as

L2t :¼ðrt�VaRk

t Þ2Ifrt<VaRk

t g��VaRkt Ifrt>VaRk

t g ð18Þ

IV. Data

The data set consists of daily observations of closing-

day futures prices on crude oil traded in the NewYork

Mercantile Exchange. We focus on futures on the first

position. The series span from 2 January 1995 to 22

November 2005, for a total of 2842 observations. We

use 2080 observations for in-sample analysis and the

remaining 762 for out-of-sample forecasts. The

GARCH models are estimated on the percentage

returns rt :¼ 100 logðpt=pt�1Þ.Table 2 reports the main properties of the data. The

kurtosis coefficient is larger than 3 and supports the

hypothesis of fat-tailed distribution. The Jarque–Bera

statistics suggest that the returns are consistent with a

strong deviation from normality. Table 2 includes the

results from the normality test of Anderson and

Darling (1952). This is a modification of the

Kolmogorov–Smirnov test and gives more weight to

the tails than the Kolmogorov–Smirnov test itself.

Also in this case, there is a rejection of the null of

normality. The significance of the Ljung–Box statis-

tics up to the 12th order points towards the presence of

ARCH effects in the returns (Table 2).

Table 2. Descriptive statistics of the returns

Maximum 32.43Minimum -37.57Mean 6.7e-2SD 3.82Kurtosis 13.50Skewness -0.433JB 1.7e+3Anderson–Darling 17.4557

[0.0]LJB(12) 16.13

Notes: Brackets report the marginal probability. The LJB(12) is the Ljung–Box test statistics on the squared residualsfrom the regression of the conditional mean. Under the nullof no serial correlation, it is distributed as a �2ðqÞ distribu-tion with q lags. Like for the LM test, the critical value is21.03. JB is the Jarque–Bera test of normality. It has a �2

distribution with two degrees of freedom. The critical valueat the 5% level is 5.99.


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The GARCH models are estimated through quasi-

maximum likelihood by maximizing the log-likelihood

function obtained as the logarithm of the product of the

conditional densities of the prediction errors. The max-

imization step is carried out by the Broyden, Fletcher,

Goldfarb and Shanno Newton algorithm.A measure of ‘true volatility’ is required for the

evaluation of the forecasting performance of the that

arises form the use of daily data. However, intra-daily

returns removes most of the type of futures considered

in this article. Hence, we approximate the true volati-

lity through the actual volatility at each point in time.

V. Results

Estimated models

The estimates of the parameters of the GARCHmod-

els are reported in Table 3.2 The SEs are robustified

again for heteroskedasticity through a Sandwich

formula. The first point of interest concerns the fact

that not all the conditional means are statistically

significant at standard confidence levels (see, e.g. the

EGARCH-G). Most of the parameters of the condi-

tional variance retain statistical validity. For the mod-

els based on the t-distribution, the conditional

kurtosis is equal to 3ðv� 2Þ=ðv� 4Þ. The resulting

estimates of conditional kurtosis are all larger than 6

for all the specifications. This confirms the importance

of modelling fat-tailed distributions for oil futures.

Also the models based on the GED support the evi-

dence for fat tails. In this case, the conditional kurtosis

takes a value of ð�ð1=vÞ�ð5=vÞÞ=ðð�=vÞ2Þ, which gives

6.6127 for the GARCH-G, 6.3802 for the EGARCH-

G and 6.5451 for the GJR-G.

In-sample forecast evaluation

Table 4 reports some descriptive statistics for in-sampleevaluation. These tests can be used for model selection.The maximized log-likelihood suggests that the GJRmodel with t errors provides the most accurate descrip-tion of the data. Also according to the Akaike andSchwartz information criteria, the GRJ-t model fitsthe best. However, there is no unique best alternativeemerging from the use of the statistical loss functions ofTable 3. Except for the HMSE, the main pattern con-cerns the fact that the GARCH models estimated witht-distribution obtain the highest ranking. This suggeststhat the estimates are capable of capturing the lepto-kurtosis of the empirical distribution of the returns.

Out-of-sample forecast evaluation

A good in-sample fit provides no indication for theforcasting performance of a model out-of-sample.Table 5 reports the evaluation for out-of-sample fore-casts over 1 day, 1 and 2 weeks. The proxy for the truevolatility is the realized (daily) volatility. All but one ofthe DA test statistics are statistically significant. TheGARCH-G model provides the best forecasts for 1, 2and 3 days ahead. For forecasts 1 week ahead, theGARCH-G and EGARCH-G models are competi-tors. Instead, the EGARCH-G model is the best per-former for 2 and 3 weeks ahead.Table 6 and 7 report the results from both the DM

and the modified DM tests. As benchmarks, we usethe models that perform best in the DM tests. Again,all the statistical loss functions of Table 3 are used forthe comparison. Table 6 used, respectively, theGARCH-G and EGARCH-G as benchmark. Theresults indicate that the null of equal predictive abilityis rejected strongly, suggesting that the benchmarkoutperforms the competing models. Furthermore,

Table 3. Estimates of GARCH models

GARCH-N GARCH-t GARCH-G EGARCH-N EGARCH-t EGARCH-G GJR-N GJR-t GJR-G

� 0.096 0.114 0.059 0.059 0.098 0.046 0.072 0.103 0.050[0.051] [0.047] [0.044] [0.052] [0.048] [0.044] [0.052] [0.048] [0.044]

�0 0.735 0.765 0.770 0.015 0.030 0.029 0.707 0.768 0.763[0.156] [0.308] [0.303] [0.015] [0.028] [0.030] [0.136] [0.272] [0.267]

�1 0.081 0.048 0.059 0.081 0.060 0.073 0.109 0.077 0.089[0.011] [0.016] [0.017] [0.013] [0.022] [0.025] [0.015] [0.023] [0.025]

� 0.794 0.822 0.808 -0.068 -0.053 -0.059 0.801 0.822 0.811[0.033] [0.062] [0.062] [0.008] [0.016] [0.016] [0.028] [0.055] [0.055]

� – – – 0.956 0.956 0.952 0.046 0.013 0.024[0.008] [0.018] [0.019] [0.011] [0.017] [0.018]

v – 5.245 1.199 – 5.345 1.206 – 5.299 1.201[0.648] [0.046] – [0.656] [0.046] – [0.650] [0.047]

Note: Brackets report SE.2As the focus of this article is on predictability and risk management, we do not conduct any specification test.


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the sign of the test statistics is negative, indicating thatthe loss is lower under the benchmark than under thealternative model for all pairwise comparisons.Table 6 shows that, when the GARCH-G model is

the benchmark, the EGARCH-G fares better foralmost all the loss function. The reserve happenswhen the EGARCH-G is the benchmark. Finally,for a predictive horizon of 2 and 3 weeks ahead, theEGARCH-G model does not outperform two modelsthat do not rank well in terms of DM test. Results notreported here suggest that these alternative competi-tors generate higher statistical losses when used asbenchmark with respect to the EGARCH-G model.Overall, theGARCH-G appears to be themost appro-priate model for short-term forecasts. At longer hor-izons, a suitable benchmark cannot be found.Tables 8–10 report the results from the reality check

and super-predictive ability for short horizons. Eachmodel is evaluated against all the others. For everymodel, the rows indicate the p-values of the RC tests.SPA0

l and SPA0c refer to the p-values of Hansen (2005)

computed through a stationary bootstrap with 3000re-samples. The main result concerns the fact that,when the GARCH-G is the benchmark, the null ofSPA is not rejected for all the loss functions at shorthorizons. There are also occasional rejections whenthe GARCH-t and GJR-G are the benchmark, albeitwith lower p-values. There are also occasional rejec-tions when the GARCH-t and GJR-G are the bench-mark, albeit with lower p-values. These results shouldno be striking as they are obtained also by Marcucci(2005) on stock market data. For instance, Hansenand Lunde (2005) suggest that the GARCH(1,1) isnot the best specification when compared with othermodels. form the predictive horizon. These are rele-vant result, as they cast doubts on the lack of predic-tive power of the GARCH-G for long horizonsemphasized by the DM tests. Occasional acceptancesare also displayed by the GARCH-t, the EGARCH-Gand the GJR-G models.In the following step, we compare the models with

measures of conditional and unconditional coverageof VaR estimates. FollowingMarcucci (2005), we alsointroduce subjectives loss functions that are meant tomimic the preferences of risk managers. The RLF andFLF penalize large failures in the VaR forecast. Table11 presents the VaR estimates at the 95 and 99% forshort and long horizons, respectively. The table showsthe results from the test of correct coverage (LRPF) tocheck whether PE is significantly higher than the nom-inal rate, the LRIND test of independence and the testof correct conditional coverage LRCC. Numbers inbold identify the minima for each evaluation criterion.The theoreticalTUFF at 5 and 1%are, respectively, 20and 100.T

able4.In-samplepredictability

Model

Pers

AIC

Rank

BIC

Rank

Log(L)

Rank

MSE1Rank

MSE2

Rank

QLIK

ERank

R2LOG

Rank

MAD

2Rank

MAD

1Rank

HMSE

Rank

GARCH-N

0.944

4.43

74.44

7-4

605.24

83.03

7181.88

82.59

310.08

46.11

71.39

54.29

3GARCH-t

0.974

4.43

64.44

6-4

601.20

62.97

2179.52

12.59

110.06

26.00

31.38

24.65

7GARCH-G

0.944

4.43

84.45

8-4

605.24

73.03

6181.87

72.59

210.08

36.11

61.39

44.29

4EGARCH-N

0.951

4.36

44.37

4-4

526.95

53.05

9181.40

32.59

710.24

96.14

81.41

84.19

1EGARCH-t

0.727

5.00

95.01

9-5

190.18

92.83

1198.86

94.11

97.13

14.91

11.12

186.67

9EGARCH-G

0.952

4.36

54.37

5-4

526.41

43.05

8181.52

42.59

810.23

86.15

91.41

94.24

2GJR

-N0.949

4.34

24.36

2-4

510.73

33.03

5181.53

52.59

510.12

76.10

51.39

74.34

5GJR

-t0.989

4.33

14.35

1-4

502.21

12.98

3180.18

22.59

410.10

56.00

21.38

35.21

8GJR

-G0.949

4.34

34.36

3-4

510.61

23.02

4181.54

62.59

610.12

66.10

41.39

64.36

6


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Table5.Out-of-samplepredictability

Model

MSE1

Rank

MSE2

Rank

QLIK

ERank

R2LOG

Rank

MAD2

Rank

MAD1

Rank

HMSE

Rank

SR

DA

One-step

aheadvolatility

forecast

GARCH-N

2.5341

795.6489

92.6486

97.9849

41.3143

65.484

64.3277

90.58

0.0702

GARCH-t

2.2847

282.6025

32.4891

27.8479

21.274

25.2946

21.462

30.68

6.6520**

GARCH-G

2.2034

177.997

12.4614

17.8447

11.2535

15.1799

11.1987

10.71

8.1032**

EGARCH-N

2.584

995.0834

72.6299

78.1556

91.3402

95.595

93.3602

70.59

-0.2702

EGARCH-t

2.4975

689.2458

62.5531

68.1168

81.3316

85.5739

81.8151

60.65

4.2484**

EGARCH-G

2.3745

484.1185

42.5158

48.0643

71.3032

45.4091

41.4671

40.7

7.3769**

GJR

-N2.5528

895.2812

82.6361

88.0632

61.3259

75.5325

73.7637

80.57

-0.8742

GJR

-t2.4107

587.1772

52.5303

57.9888

51.3063

55.4563

51.7387

50.66

5.1195**

GJR

-G2.2966

381.8286

22.4935

37.9496

31.2799

35.3034

31.3813

20.71

7.7374**

Seven-stepaheadvolatility

forecast

GARCH-N

14.4585

51774.7723

57.8519

53.1028

63.3319

630.5326

639.1514

50.81

15.6381**

GARCH-t

18.3184

91929.3805

911.4239

95.1143

93.8675

932.7361

9114.5564

90.75

13.1472**

GARCH-G

13.9793

11742.9475

17.5519

12.949

23.2864

230.3018

232.0057

10.84

17.5793**

EGARCH-N

14.3148

41770.9453

47.7573

43.0263

43.3007

430.392

438.0017

40.79

14.1349**

EGARCH-t

16.4193

71859.0518

79.4063

74.0483

73.6146

731.7511

768.3559

70.72

10.7186**

EGARCH-G

14.0432

21753.4072

37.5817

22.9391

13.2755

130.2646

133.6106

20.82

16.0067**

GJR

-N14.4688

61776.2451

67.8585

63.1007

53.3289

530.5209

539.4156

60.8

14.7728**

GJR

-t17.0239

81880.718

89.9887

84.387

83.7034

832.1067

879.6431

80.74

12.3028**

GJR

-G14.1045

31752.4542

27.6237

32.9828

33.2947

330.3486

333.6277

30.82

16.2786**

Fourteen-stepaheadvolatility

forecast

GARCH-N

38.3654

56647.3113

513.9559

55.9558

55.7811

566.2684

5164.7149

60.76

13.1921**

GARCH-t

50.0413

97159.2939

933.8145

911.699

96.7059

969.8064

91342.1712

90.66

8.4546**

GARCH-G

38.0254

26620.0351

213.68

25.8662

25.761

266.1657

2152.3708

20.79

14.6833**

EGARCH-N

38.1732

36634.0569

413.7952

45.8956

35.7663

366.1971

3157.7935

40.77

13.3317**

EGARCH-t

45.7174

76991.4793

723.0183

79.2003

76.3988

768.8086

7524.6104

70.7

10.5994**

EGARCH-G

37.9858

16618.1606

113.6432

15.8489

15.756

166.1438

1150.6241

10.77

13.2131**

GJR

-N38.4603

66649.7002

614.0261

65.9976

65.7919

666.3183

6164.7069

50.78

13.8105**

GJR

-t47.2432

87053.0348

826.1577

810.0395

86.5128

869.1974

8708.7607

80.67

9.5743**

GJR

-G38.1763

46627.2827

313.7936

35.9209

45.7747

466.2313

4154.4992

30.78

13.7930**

Note:**indicatessignificance

at5%

level.


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Table6.Diebold–Marianotests(benchmark:GARCH-G

)

Diebold–Mariano

ModifiedDiebold–Mariano

Model

MSE1

MSE2

QLIK

ER2LOG

MAD2

MAD1

HMSE

MSE1

MSE2

QLIK

ER2LOG

MAD2

MAD1

HMSE

One-step

ahead

GARCH-N

-4.70**

-2.95**

-4.84**

-6.93**

-6.97**

-8.52**

-2.89**

-4.70**

-2.95**

-4.83**

-6.92**

-6.96**

-8.51**

-2.89**

GARCH-t

-3.61**

-2.78**

-4.73**

-0.08

-3.35**

-2.97**

-3.82**

-3.61**

-2.77**

-4.73**

-0.08

-3.34**

-2.96**

-3.81**

EGARCH-N

-6.10**

-3.01**

-5.52**

-12.73**

-10.09**

-13.82**

-2.77**

-6.09**

-3.00**

-5.51**

-12.71**

-10.08**

-13.80**

-2.77**

EGARCH-t

-5.75**

-3.23**

- 6.81**

-4.78**

-5.57**

-6.20**

-3.31**

-5.74**

-3.22**

-6.80**

-4.78**

-5.57**

-6.19**

-3.31**

EGARCH-G

-8.64**

-3.12**

-9.91**

-10.85**

-9.15**

-12.18**

-3.53**

-8.63**

-3.12**

-9.89**

-10.83**

-9.14**

-12.16**

-3.52**

GJR

-N-5

.29**

-2.97**

-5.13**

-10.60**

-8.34**

-11.11**

-2.83**

-5.29**

-2.96**

-5.13**

-10.58**

-8.32**

-11.09**

-2.83**

GJR

-t-5

.02**

-3.08**

-6.31**

-2.99**

-4.74**

-5.06**

-3.65**

-5.02**

-3.07**

-6.30**

-2.98**

-4.74**

-5.05**

-3.65**

GJR

-G-7

.67**

-3.14**

-9.74**

-9.16**

-7.78**

-10.75**

-4.23**

-7.66**

-3.14**

-9.73**

-9.15**

-7.77**

-10.74**

-4.22**

Seven-stepahead

GARCH-N

-4.83**

-3.24**

-5.24**

-7.08**

-6.68**

-7.66**

-3.50**

-4.75**

-3.18**

-5.15**

-6.96**

-6.56**

-7.53**

-3.44**

GARCH-t

-14.39**

-6.17**

-17.75**

-35.03**

-35.54**

-49.29**

-8.62**

-14.14**

-6.06**

-17.44**

-34.42**

-34.92**

-48.43**

-8.47**

EGARCH-N

- 3.07**

-2.55*

-3.38**

-3.16**

-2.20*

-1.97*

-2.93**

-3.02**

-2.51*

-3.32**

-3.10**

-2.16*

-1.93

-2.88**

EGARCH-t

-11.08**

-5.07**

-12.65**

-19.51**

-20.66**

-22.59**

-6.36**

-10.89**

-4.98**

-12.42**

-19.16**

-20.30**

-22.19**

-6.25**

EGARCH-G

-1.13

-1.73

-1.09

0.76

1.49

2.41+

-2.11*

-1.11

-1.7

-1.07

0.74

1.47

2.36+

-2.08*

GJR

-N-4

.67**

-3.19**

-5.10**

-6.51**

-5.89**

-6.54**

-3.55**

-4.59**

-3.13**

-5.02**

-6.39**

-5.79**

-6.43**

-3.49**

GJR

-t-1

3.43**

-5.78**

-15.51**

-24.89**

-28.65**

-31.31**

-7.87**

-13.20**

-5.68**

-15.24**

-24.46**

-28.15**

-30.76**

-7.73**

GJR

-G-3

.80**

-2.78**

-4.44**

-4.29**

-3.28**

-3.20**

-3.90**

-3.73**

-2.73**

-4.36**

-4.21**

-3.23**

-3.15**

-3.83**

Note:*and**indicate

rejectionofthenullofequalpredictiveaccuracy

at5and1%

levels,respectively.


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Table7.Diebold–Marianotests(benchmark:EGARCH-G

)

Model

MSE1

MSE2

QLIK

ER2LOG

MAD2

MAD1

HMSE

MSE1

MSE2

QLIK

ER2LOG

MAD2

MAD1

HMSE

Seven-stepahead

GARCH-N

-6.78**

-4.43**

-6.60**

-9.52**

-10.42**

-11.20**

-3.76**

-6.66**

-4.35**

-6.49**

-9.35**

-10.24**

-11.00**

-3.70**

GARCH-t

-16.32**

-6.99**

-19.00**

-36.99**

-40.61**

-48.52**

-9.00**

-16.03**

-6.87**

-18.66**

-36.35**

-39.90**

-47.67**

-8.84**

GARCH-G

1.13

1.73

1.09

-0.76

-1.49

-2.41*

2.11+

1.11

1.7

1.07

-0.74

-1.47

-2.36*

2.08+

EGARCH-N

-4.44**

-3.29**

-4.48**

-5.51**

-5.46**

-5.71**

-3.17**

-4.36**

-3.23**

-4.40**

-5.42**

-5.36**

-5.61**

-3.12**

EGARCH-t

-13.57**

-6.08**

-14.23**

-21.45**

-24.73**

-24.50**

-6.87**

-13.34**

-5.97**

-13.98**

-21.07**

-24.30**

-24.07**

-6.75**

GJR

-N-7

.11**

-4.58**

-6.88**

-10.00**

-11.21**

-12.03**

-3.98**

-6.99**

-4.50**

-6.76**

-9.82**

-11.02**

-11.82**

-3.91**

GJR

-t-1

6.20**

-6.90**

-17.02**

-26.23**

-32.84**

-31.47**

-8.47**

-15.91**

-6.78**

-16.73**

-25.77**

-32.26**

-30.92**

-8.32**

GJR

-G-2

.36*

0.35

-3.30**

-7.13**

- 7.09**

-8.61**

-0.05

-2.32*

0.34

-3.24**

-7.00**

-6.97**

-8.46**

-0.04

Fourteen-stepahead

GARCH-N

-2.29*

-2.27*

-2.31*

-2.15*

-2.10*

-2.04*

-2.20*

-2.17*

-2.15*

-2.18*

-2.03*

-1.98*

-1.92

-2.08*

GARCH-t

-20.99**

-10.73**

-12.43**

-35.11**

-62.75**

-60.80**

-4.96**

-19.85**

-10.15**

-11.75**

-33.19**

-59.32**

-57.48**

-4.69**

GARCH-G

-1.26

-1.45

-1.35

-0.98

-0.96

-0.9

-1.58

-1.19

-1.37

-1.28

-0.93

- 0.91

-0.85

-1.5

EGARCH-N

-1.35

-1.75

-1.47

-0.78

-0.69

-0.54

-1.9

-1.27

-1.65

-1.39

-0.74

-0.66

-0.51

-1.8

EGARCH-t

-25.86**

-11.84**

-19.65**

-40.23**

-57.52**

-52.29**

-8.33**

-24.45**

-11.20**

-18.58**

-38.03**

-54.38**

-49.43**

-7.88**

GJR

-N-5

.99**

-4.35**

-5.44**

-7.57**

-7.84**

-8.10**

-3.50**

-5.66**

-4.11**

-5.14**

-7.16**

-7.41**

-7.66**

-3.31**

GJR

-t-2

3.54**

-11.30**

-16.48**

-36.38**

-57.61**

-52.58**

-7.03**

-22.25**

-10.68**

-15.58**

-34.39**

-54.46**

-49.70**

-6.65**

GJR

-G-8

.40**

-5.94**

-7.29**

-9.86**

-10.17**

-10.24**

-4.01**

-7.94**

-5.62**

-6.89**

-9.33**

-9.62**

-9.68**

-3.79**

Note:*and**indicate

rejectionofthenullofequalpredictiveaccuracy

at5and1%

levels,respectively.


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Table 8. Reality check and SPA tests (horizon: 1 day)

Loss function

Benchmark MSE1 MSE2 QLIKE R2LOG MAD1 MAD2 HMSE

GARCH-N SPA0l 0 0.003 0 0.005 0 0 0.005

GARCH-N SPA0c 0 0.003 0 0.006 0 0 0.005

GARCH-N RC 0 0.003 0 0.013 0 0 0.005GARCH-t SPA0

l 0.001 0.01 0 0.491 0.003 0.002 0.001GARCH-t SPA0

c 0.088 0.156 0.234 0.774 0.033 0.018 0.419GARCH-t RC 0.09 0.158 0.235 0.789 0.033 0.018 0.426GARCH-G SPA0

l 0.551 0.553 0.524 0.495 0.504 0.632 0.557GARCH-G SPA0

c 1 0.977 1 0.949 1 1 0.689GARCH-G RC 1 0.994 1 0.953 1 1 1EGARCH-N SPA0

l 0 0.007 0 0 0 0 0.007EGARCH-N SPA0

c 0 0.007 0 0 0 0 0.007EGARCH-N RC 0 0.007 0 0 0 0 0.007EGARCH-t SPA0

l 0 0.002 0 0 0 0 0.004EGARCH-t SPA0

c 0 0.002 0 0 0 0 0.271EGARCH-t RC 0 0.002 0 0 0 0 0.272EGARCH-G SPA0

l 0 0.006 0 0 0 0 0.003EGARCH-G SPA0

c 0 0.05 0.044 0 0 0 0.413EGARCH-G RC 0 0.05 0.044 0 0 0 0.419GJR-N SPA0

l 0 0.005 0 0 0 0 0.004GJR-N SPA0

c 0 0.005 0 0 0 0 0.004GJR-N RC 0 0.005 0 0 0 0 0.004GJR-t SPA0

l 0 0.003 0 0.008 0 0 0.002GJR-t SPA0

c 0 0.003 0.005 0.008 0 0 0.295GJR-t RC 0 0.003 0.005 0.008 0 0 0.296GJR-G SPA0

l 0 0.006 0 0.006 0 0 0GJR-G SPA0

c 0.057 0.239 0.196 0.023 0.007 0.017 0.449GJR-G RC 0.057 0.242 0.197 0.023 0.007 0.017 0.461

Table 9. Reality check and SPA tests (horizon: 1 week)

Loss function


GARCH-N SPA0l 0 0.006 0 0 0 0 0.006

GARCH-N SPA0c 0 0.006 0 0 0 0 0.006

GARCH-N RC 0 0.006 0 0 0 0 0.006GARCH-t SPA0

l 0 0.007 0 0.459 0 0 0.002GARCH-t SPA0

c 0.075 0.137 0.218 0.768 0.014 0.006 0.454GARCH-t RC 0.076 0.138 0.218 0.781 0.014 0.006 0.464GARCH-G SPA0

l 0.55 0.562 0.524 0.512 0.507 0.594 0.549GARCH-G SPA0


l 0 0.005 0 0 0 0 0.008EGARCH-N SPA0


l 0 0.001 0 0 0 0 0.001EGARCH-t SPA0


l 0 0.002 0 0 0 0 0.001EGARCH-G SPA0


l 0 0.007 0 0 0 0 0.005GJR-N SPA0

c 0 0.007 0 0 0 0 0.005GJR-N RC 0 0.007 0 0 0 0 0.005GJR-t SPA0

l 0 0.003 0 0 0 0 0.001GJR-t SPA0

c 0 0.003 0.002 0 0 0 0.299GJR-t RC 0 0.003 0.002 0 0 0 0.299GJR-G SPA0

l 0 0.002 0 0.001 0 0 0GJR-G SPA0

c 0.045 0.233 0.2 0.002 0 0.001 0.446GJR-G RC 0.045 0.236 0.2 0.002 0 0.001 0.457


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Table 10. Reality check and SPA tests (horizon: 2 weeks)

Loss function


GARCH-N SPA0l 0 0.003 0 0 0 0 0.004

GARCH-N SPA0c 0 0.003 0 0 0 0 0.004

GARCH-N RC 0 0.003 0 0 0 0 0.004GARCH-t SPA0

l 0 0.002 0 0.501 0.006 0 0GARCH-t SPA0

c 0 0.002 0 0.981 0.033 0.003 0.022GARCH-t RC 0 0.002 0 0.981 0.033 0.003 0.022GARCH-G SPA0

l 0.532 0.519 0.521 0.16 0.508 0.55 0.521GARCH-G SPA0


l 0 0.003 0 0 0 0 0.004EGARCH-N SPA0


l 0 0.002 0 0 0 0 0EGARCH-t SPA0


l 0 0.004 0 0 0 0 0.001EGARCH-G SPA0


l 0 0.003 0 0 0 0 0.003GJR-N SPA0

c 0 0.003 0 0 0 0 0.003GJR-N RC 0 0.003 0 0 0 0 0.003GJR-t SPA0

l 0 0.002 0 0 0 0 0GJR-t SPA0

c 0 0.002 0 0 0 0 0GJR-t RC 0 0.002 0 0 0 0 0GJR-G SPA0

l 0 0.002 0 0 0 0 0GJR-G SPA0

c 0.011 0.002 0.121 0 0.001 0.001 0.128GJR-G RC 0.012 0.065 0.122 0 0.001 0.001 0.382

Table 11. Out-of-sample evaluation of risk management

95%VaR 99%VaR

Model TUFF PF (%) LRPF LRIND LRCC FLF RLF TUFF PF(%) LRPF LRIND LRCC FLF RLF

One-step aheadGARCH-N 4 5.676 0.683 4.648* 5.331 0.3206 0.288 21 2.162 7.577* 5.166* 12.743* 0.1844 0.1533GARCH-t 4 2.568 11.131* 2.796 13.927* 0.194 0.1343 60 0.541 1.894 0.558 2.452 0.125 0.0098GARCH-G 4 4.595 0.263 3.69 3.953 0.284 0.2453 60 0.811 0.286 0.718 1.004 0.1473 0.0771EGARCH-N 4 4.865 0.029 4.971* 4.999 0.2972 0.2607 21 2.162 7.577* 5.166* 12.743* 0.1707 0.1306EGARCH-t 21 2.568 11.131* 2.45 13.580* 0.173 0.1107 60 0.541 1.894 0.558 2.452 0.1218 0.002

EGARCH-G 4 4.459 0.472 4.971* 5.442 0.2699 0.2296 60 0.676 0.887 0.558 1.445 0.1432 0.0694GJR-N 4 5.405 0.25 4.648* 4.898 0.3062 0.2714 21 2.162 7.577* 5.166* 12.743* 0.1761 0.1398GJR-t 4 2.568 11.131* 2.796 13.927* 0.1818 0.1212 60 0.541 1.894 0.558 2.452 0.1213 0.0045GJR-G 4 4.73 0.116 4.851* 4.967 0.2776 0.2385 60 0.676 0.887 0.558 1.445 0.1424 0.0697Seven-step aheadGARCH-N 15 4.459 0.472 9.917* 10.388* 0.2874 0.2517 54 1.757 3.493 2.687 6.180* 0.162 0.1185GARCH-t 4 5.541 0.44 7.913* 8.353* 0.3234 0.2947 54 1.081 0.048 1.064 1.112 0.1418 0.075GARCH-G 15 4.324 0.744 10.438* 11.182* 0.2745 0.236 54 0.811 0.286 0.887 1.173 0.1424 0.0683EGARCH-N 15 4.324 0.744 10.438* 11.182* 0.2803 0.243 54 1.486 1.539 3.035 4.574 0.158 0.1108EGARCH-t 15 4.189 1.081 11.621* 12.703* 0.2368 0.1922 54 0.541 1.894 0.558 2.452 0.1193 0.023

EGARCH-G 15 4.324 0.744 10.438* 11.182* 0.2696 0.2304 54 0.541 1.894 0.558 2.452 0.1361 0.0582GJR-N 15 4.324 0.744 10.438* 11.182* 0.2852 0.2491 54 1.622 2.431 2.838 5.27 0.1604 0.1156GJR-t 15 4.595 0.263 9.917* 10.180* 0.2639 0.2248 54 0.676 0.887 0.718 1.606 0.1245 0.038GJR-G 15 4.189 1.081 11.005* 12.086* 0.2723 0.2337 54 0.811 0.286 0.887 1.173 0.1377 0.0621GARCH-N 8 4.459 0.472 9.917* 10.388* 0.2893 0.2454 44 1.622 2.431 2.838 5.27 0.1649 0.1217

(continued )


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At both short and long horizons, all the models butone display failures with respect to the theoreticalTUFF for the 95 and 99% VaR. In terms of probabil-ity of failure, for short horizons, all the models with t-distributed disturbances are inadequate for the 95%VaR, as they are rejected for a too high PF. However,there are no rejections for the GARCH-G model,which fares best in terms of statistical criteria of fore-cast evaluation. Table 11 shows that the three tests ofcorrect unconditional and conditional coverage donot reject the GARCH-G. However, when the aim isthat of covering 99% of losses, there are more modelsthat perform equally well for each test of conditionaland unconditional coverage. The last two columns ofTable 11 report the average RLF and FLF. TheGARCH-G model never yields the lowest values forshort-horizon forecasts. However, for the 99% VaR,average losses closer to the lowest values are delivered.At long horizons, the GARCH model delivers betteraverage RLF and FLF. An overall look at the resultsshows that EGARCH models – the EGARCH-t forshort horizons and the EGARCH-G for long horizons– fare better than both GARCH and GJR models interms of VaR loss functions.

VI. Conclusion

This article studies the forecasting properties of linear.GARCH models for closing-day futures prices oncrude oil, first position, traded in the NYMEX. Wecompare volatility models based on the normal,Student’s t and generalized exponential distribution.Our focus is on out-of-sample predictability. To thatend, we rank the models according to a large array ofstatistical loss functions.

The results from the tests for predictive abilitiyshow that the GARCH-G model fares best for shorthorizons from 1 to 3 days ahead. For horizons from 1week ahead, no superior model can be identified. Wealso consider out-of-sample loss functions based onVaR that mimic portfolio managers and regulators’preferences for penalizing large forecast failures andopportunity costs from over-investments. In this case,EGARCH models exhibit the best performance, fol-lowed closely by the GARCH-G.

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Table 11. Continued

95%VaR 99%VaR

Model TUFF PF (%) LRPF LRIND LRCC FLF RLF TUFF PF(%) LRPF LRIND LRCC FLF RLF

GARCH-t 8 8.919 19.606* 7.657* 27.263* 0.4849 0.4675 8 2.162 7.577* 5.166* 12.743* 0.2054 0.1943GARCH-G 8 4.324 0.744 7.800* 8.544* 0.2719 0.226 47 0.676 0.887 0.718 1.606 0.1309 0.0513EGARCH-N 8 4.459 0.472 9.917* 10.388* 0.2845 0.24 47 1.486 1.539 3.035 4.574 0.162 0.1164EGARCH-t 8 6.081 1.708 7.913* 9.621* 0.3332 0.2977 47 1.081 0.048 1.25 1.298 0.1372 0.0713EGARCH-G 8 4.324 0.744 7.800* 8.544* 0.2698 0.2236 47 0.676 0.887 0.718 1.606 0.1292 0.0483

GJR-N 8 4.595 0.263 9.917* 10.180* 0.2893 0.2456 44 1.622 2.431 2.838 5.27 0.164 0.1207GJR-t 8 7.162 6.461* 6.372* 12.833* 0.3814 0.3524 8 1.622 2.431 2.838 5.27 0.1549 0.1067GJR-G 8 4.459 0.472 10.438* 10.909* 0.2731 0.2276 47 0.676 0.887 0.718 1.606 0.1308 0.0518

Note: * indicates significance at the 5% level.


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volatility forecasting for crude oil futures

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