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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
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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
1592 M. Marzo and P. Zagaglia
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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.
Volatility forecasting for crude oil futures 1593
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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.
1594 M. Marzo and P. Zagaglia
Dow
nloa
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by [
Ston
y B
rook
Uni
vers
ity]
at 0
1:34
02
Nov
embe
r 20
14
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.
Volatility forecasting for crude oil futures 1595
Dow
nloa
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by [
Ston
y B
rook
Uni
vers
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embe
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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
Benchmark MSE1 MSE2 QLIKE R2LOG MAD1 MAD2 HMSE
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
c 1 0.975 1 0.931 1 1 0.686GARCH-G RC 1 0.998 1 0.935 1 1 1EGARCH-N SPA0
l 0 0.005 0 0 0 0 0.008EGARCH-N SPA0
c 0 0.005 0 0 0 0 0.008EGARCH-N RC 0 0.005 0 0 0 0 0.008EGARCH-t SPA0
l 0 0.001 0 0 0 0 0.001EGARCH-t SPA0
c 0 0.001 0 0 0 0 0.266EGARCH-t RC 0 0.001 0 0 0 0 0.266EGARCH-G SPA0
l 0 0.002 0 0 0 0 0.001EGARCH-G SPA0
c 0 0.04 0.05 0 0 0 0.433EGARCH-G RC 0 0.04 0.05 0 0 0 0.44GJR-N 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
1596 M. Marzo and P. Zagaglia
Dow
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by [
Ston
y B
rook
Uni
vers
ity]
at 0
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02
Nov
embe
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14
Table 10. Reality check and SPA tests (horizon: 2 weeks)
Loss function
Benchmark MSE1 MSE2 QLIKE R2LOG MAD1 MAD2 HMSE
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
c 1 0.961 1 0.315 1 1 0.986GARCH-G RC 1 0.996 1 0.376 1 1 1EGARCH-N SPA0
l 0 0.003 0 0 0 0 0.004EGARCH-N SPA0
c 0 0.003 0 0 0 0 0.004EGARCH-N RC 0 0.003 0 0 0 0 0.004EGARCH-t SPA0
l 0 0.002 0 0 0 0 0EGARCH-t SPA0
c 0 0.002 0 0 0 0 0.001EGARCH-t RC 0 0.002 0 0 0 0 0.001EGARCH-G SPA0
l 0 0.004 0 0 0 0 0.001EGARCH-G SPA0
c 0 0.004 0.023 0 0 0 0.297EGARCH-G RC 0 0.004 0.023 0 0 0 0.301GJR-N 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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