The 33 reference contexts in paper Christopher F Baum, Paola Zerilli, Liyuan Chen (2018) “Stochastic volatility, jumps and leverage in energy and stock markets: evidence from high frequency data” / RePEc:boc:bocoec:952

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    of York, York YO10 5DD, UK, paola.zerilli@york.ac.uk and lc844.york@gmail.com 1 Introduction The study of volatility in crude oil and natural gas markets and its interaction with returns (leverage) has a broad range of Önancial impacts both from an hedging point of view and also for forecasting purposes. The main limitation of using daily data is that volatility is not observable (see
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    Bollerslev and Zhou (2002) and Baum and Zerilli (2016)).
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    As suggested by Zhou (1996), the availability of high-frequency data has opened up new possibilities in estimating volatility. Tick-by-tick data provide an almost continuous observation of the return series, making the daily volatility observable so that it can be studied in great detail.
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    york@gmail.com 1 Introduction The study of volatility in crude oil and natural gas markets and its interaction with returns (leverage) has a broad range of Önancial impacts both from an hedging point of view and also for forecasting purposes. The main limitation of using daily data is that volatility is not observable (see Bollerslev and Zhou (2002) and Baum and Zerilli (2016)). As suggested by
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    Zhou (1996),
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    the availability of high-frequency data has opened up new possibilities in estimating volatility. Tick-by-tick data provide an almost continuous observation of the return series, making the daily volatility observable so that it can be studied in great detail.
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    From an econometric point of view, the employment of intraday data leads to the estimation of the structural parameters of the stochastic volatility models using simple moment conditions tailored to Öt all the relevant empirical features of energy and stock index returns. In terms of hedging strategies in the crude oil market, it has been shown (see
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    Chen, Zerilli and Baum (2018))
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    that the introduction of the leverage e§ect in the traditional SV model with normally distributed errors is capable of adequately estimating risk in a VaR and CVaR sense for conservative oil suppliers in both the WTI and Brent spot markets while it tends to overestimate risk for more speculative oil suppliers.
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    We also show that introducing jumps improves only the ability of modeling the behaviour of the volatility for the crude oil futures market. 2 2 Literature review Traditionally, the termleverage e§ectindicates the negative correlation between asset returns and changes in their volatility (see
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    Ait-Sahalia et al. (2013)
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    for an extensive literature review). The interpretation of this e§ect is very intuitive if we think that events that have a negative impact on Önancial markets would eventually cause an increase in volatility.
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    There are two possible economic interpretations for this empirical property: 1) when asset returns fall, the Örm becomes more levered as the market value of their debt increases compared to the market value of their equity, making the stock riskier and therefore increasing its volatility (see
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    Black (1976));
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    2) when volatility increases, future prices fall (see French et al. (1987)). Bekaert and Wu (2000) and Ait-Sahalia et al. (2013) study di§erent possible interpretations. The latter authors Önd that the leverage e§ect in high frequency data is not statistically signiÖcant over short periods, but become negative and statistically signiÖcant over long periods.
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    There are two possible economic interpretations for this empirical property: 1) when asset returns fall, the Örm becomes more levered as the market value of their debt increases compared to the market value of their equity, making the stock riskier and therefore increasing its volatility (see Black (1976)); 2) when volatility increases, future prices fall (see
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    French et al. (1987)). Bekaert and Wu (2000) and Ait-Sahalia et al. (2013)
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    study di§erent possible interpretations. The latter authors Önd that the leverage e§ect in high frequency data is not statistically signiÖcant over short periods, but become negative and statistically signiÖcant over long periods.
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    The latter authors Önd that the leverage e§ect in high frequency data is not statistically signiÖcant over short periods, but become negative and statistically signiÖcant over long periods. In recent years, the commodity price literature has shown that there is evidence of leverage e§ects in various energy markets. More speciÖcally,
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    Chan and Grant (2016),
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    considering lower frequency (weekly) commodity returns conclude that SV models (with an MA component) are able to replicate the main features of the data more e¢ ciently than GARCH models. At the same time, they Önd a signiÖcant negative leverage e§ect in crude oil spot markets.
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    More speciÖcally, Chan and Grant (2016), considering lower frequency (weekly) commodity returns conclude that SV models (with an MA component) are able to replicate the main features of the data more e¢ ciently than GARCH models. At the same time, they Önd a signiÖcant negative leverage e§ect in crude oil spot markets.
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    Kristoufek (2014)
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    focuses on the leverage e§ect in commodity futures markets and provides an extensive literature review in this area. Fan et al. (2008) estimate VaR of crude oil prices using a GED-GARCH approach with daily WTI and Brent prices spanning from 1987 to 2006.
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    At the same time, they Önd a signiÖcant negative leverage e§ect in crude oil spot markets. Kristoufek (2014) focuses on the leverage e§ect in commodity futures markets and provides an extensive literature review in this area.
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    Fan et al. (2008)
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    estimate VaR of crude oil prices using a GED-GARCH approach with daily WTI and Brent prices spanning from 1987 to 2006. They Önd that this type of model speciÖcation does as well as the standard normal distribution at a 95% conÖdence level.
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    They Önd that this type of model speciÖcation does as well as the standard normal distribution at a 95% conÖdence level. They also test and Önd evidence for asymmetric leverage e§ects without modelling them directly.
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    Youssef et al. (2015)
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    3 evaluate VaR and CVar for crude oil and gasoline markets using a long memory GARCH-EVT approach. Their Öndings and backtesting exercise show that crude oil markets are characterized by asymmetry, fat tails and long range memory.
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    Youssef et al. (2015) 3 evaluate VaR and CVar for crude oil and gasoline markets using a long memory GARCH-EVT approach. Their Öndings and backtesting exercise show that crude oil markets are characterized by asymmetry, fat tails and long range memory. In the commodity price literature,
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    Kristoufek (2014) and Nomikos and Andriosopoulos (2012),
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    using daily data, Önd aninverse leverage e§ectin the natural gas market: a positive correlation coe¢ cient. Larsson and Nossman (2011) Önd evidence for stochastic volatility and jumps in both returns and volatility daily spot prices of WTI crude oil from 1989 to 2009.
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    In the commodity price literature, Kristoufek (2014) and Nomikos and Andriosopoulos (2012), using daily data, Önd aninverse leverage e§ectin the natural gas market: a positive correlation coe¢ cient.
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    Larsson and Nossman (2011)
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    Önd evidence for stochastic volatility and jumps in both returns and volatility daily spot prices of WTI crude oil from 1989 to 2009. See Kristoufek (2014) for an extensive literature review on the leverage e§ect in commodity markets.
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    In the commodity price literature, Kristoufek (2014) and Nomikos and Andriosopoulos (2012), using daily data, Önd aninverse leverage e§ectin the natural gas market: a positive correlation coe¢ cient. Larsson and Nossman (2011) Önd evidence for stochastic volatility and jumps in both returns and volatility daily spot prices of WTI crude oil from 1989 to 2009. See
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    Kristoufek (2014)
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    for an extensive literature review on the leverage e§ect in commodity markets. We contribute to the current debate by testing for the existence of the leverage e§ect when considering a near-continuous observation of the processes with the ability to study their volatility in great detail because of the use of high frequency futures returns in the S&P500, natural gas and crude oil markets and by st
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    In this paper, we employ a stochastic volatility model in order to 4 analyze the uncertainty of S&P500, natural gas and crude oil futures returns. The empirical analysis makes use of high-frequency (tick-by-tick) data from the futures markets, aggregated to 10-minute intervals during the trading day.
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    Schwartz (1997), Schwartz and Smith (2000), and Casassus and Collin-Dufresne (2005)
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    propose multi-factor models for energy prices where returns are only a§ected by Gaussian shocks, but they constrain volatility to be constant. Pindyck (2004) examines the volatility of energy spot and futures prices, estimating the standard deviation of their Örst di§erences.
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    Schwartz (1997), Schwartz and Smith (2000), and Casassus and Collin-Dufresne (2005) propose multi-factor models for energy prices where returns are only a§ected by Gaussian shocks, but they constrain volatility to be constant.
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    Pindyck (2004)
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    examines the volatility of energy spot and futures prices, estimating the standard deviation of their Örst di§erences. Mason and Wilmot (2014) investigate the potential presence of jumps in two key daily natural gas prices: the spot price at the Henry Hub in the US, and the spot price for natural gas at the National Balancing Point in the UK.
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    Schwartz (1997), Schwartz and Smith (2000), and Casassus and Collin-Dufresne (2005) propose multi-factor models for energy prices where returns are only a§ected by Gaussian shocks, but they constrain volatility to be constant. Pindyck (2004) examines the volatility of energy spot and futures prices, estimating the standard deviation of their Örst di§erences.
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    Mason and Wilmot (2014)
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    investigate the potential presence of jumps in two key daily natural gas prices: the spot price at the Henry Hub in the US, and the spot price for natural gas at the National Balancing Point in the UK.
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    We focus on the role of volatility as a measure of variability and uncertainty of futures returns and we examine the impact of the leverage e§ect between futures returns and their variance. In our investigation, we use the moment conditions derived from the stochastic variance as proposed by
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    Bollerslev and Zhou (2002) and Garcia et al. (2011).
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    3 Data The raw data used in this study are 10-minute aggregations of natural gas, crude oil and S&P500 futures contract transactions-level data provided by TickData, Inc. Industry analysts have noted that to avoid market disruptions, major participants 5 in the natural gas futures market roll over their positions from the near contract to the next-near contract over several days before the near c
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    ret WTI 0.00 0.01 -1.64 1.73 0.23 -0.09 7.33 Table 2:Descriptive Statistics for the daily futures realised variance 20012016 mean median min max stdev skew kurt fut rv SP500 0.00 0.00 0.00 0.32 0.01 12.75 258.67 fut rv NG0.06 0.02 0.00 5.86 0.19 19.15 495.16 fut rv WTI 0.03 0.01 0.00 2.68 0.08 20.95 648.12 4 Estimation method Following
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    Bollerslev and Zhou (2002),
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    who use continuously observed futures prices on oil, we build a conditional moment estimator for stochastic variance models based on matching the sample moments ofRealized Variancewith population moments of 10 Table 3: Test Statistics and P-values for daily futures returns KSmirnov p-val SFrancia p-valQp-val fut ret SP5000.0870.000 14.078 0.000 156.575 0.000 fut ret NG0.0560.000 11.640
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    X t-k-s-t (x(s)dN(-s)) 2 12 Barndor§-Nielsen and Shephard (2004) proposed the Realized Bipower Variation as a consistent estimate of integrated volatility component in the presence of jumps: BV(t;k;n) =-2 Pn-k i=2 \f \fr t-k+ikn;1n -\f \f \f \f \fr t-k+ (i-1)k n; 1 n <\f \f \f RV(t;k;n)-BV(t;k;n)-!QV(t;k)-IV(t;k) QV(t;k)-IV(t;k) = X t-k-s-t (x(s)dN(-s)) 2 asn-!1 A similar approach was used in
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    Baum and Zerilli (2016).
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    In this section, we are going to introduce the three models that are going to be estimated and compared in terms of ability to Öt the data, risk measurement and out of sample performance: the Stochastic Volatility model (SV), the Stochastic Volatility model with jumps (SVJ) and the Stochastic Volatility model with leverage (SVL). 13 4.1 Stochastic Volatility model (SV) We model the returns on cru
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    models that are going to be estimated and compared in terms of ability to Öt the data, risk measurement and out of sample performance: the Stochastic Volatility model (SV), the Stochastic Volatility model with jumps (SVJ) and the Stochastic Volatility model with leverage (SVL). 13 4.1 Stochastic Volatility model (SV) We model the returns on crude oil, natural gas and S&P500 futures using the
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    Heston (1993)
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    model, setting the leverage e§ect equal to zero at the beginning. dpt=dln(Ft) = p VtdW1t dVt=-(>-Vt)dt+p VtdW2t dW1tdW2t= 0 In this model, there are two orthogonal Wiener processes,dW1tanddW2t, driving the evolution of returns and volatility.
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    It is the parameter of the Poisson counting process that takes values: 1when an extreme event happens 0otherwise The main moment conditions used for the SV speciÖcation, are augumented using two additional moment conditions. From
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    Bollerslev and Zhou (2002)
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    Appendix Bpage 62(B:14)a§ects all the moment conditions (impact of jumps) At time(t;t+ 1) e1Jt;t+1=E[RVt;t+1jGt]-RVt;t+1 =E[BPt;t+1jGt] +--2xdt-RVt;t+1 since E[RVt;t+1jGt] =E[BPt;t+1jGt] +--2xdt 18 and e2Jt;t+1=E RV2t;t+1 \f \fGt -RV2t;t+1 =E BP2t;t+1 \f \fGt + 2--2xE RV2t;t+1 \f \fGt dt---4xdt-RV2t;t+1 They are then augmented using appropriate functions of the past values of the realised variance
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    This main moment condition is then augmented using past realizations of the returns and the variance as instruments (see the Appendix for more details). Additional details about the moment conditions and more speciÖcally about the equations for the leverage e§ect can be found in
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    Garcia et al. (2011).
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    As shown in Tables 20, 22, 24 all the moment conditions are in accordance with the data and the overall HansenísJstatistic indicates that the overidentifying restrictions are valid. As shown in Tables 19, 21, 23 all estimated parameters of the model are very precisely estimated (-is signiÖcant at 10% level for WTI) and take on sensible values from an analytical perspective. 26 We Önd that stochast
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    More speciÖcally, we Önd signiÖcant evidence of aleverage e§ectfor S&P500 and crude oil markets: a negative shock to returns increases volatility in these markets. In contrast, we Önd evidence ofinverse leverage e§ectfor the natural gas market (in line with
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    Kristoufek (2014)).
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    27 Table 19:GMM estimates for SVL model for the S&P500 futures: 09/2001ñ06/2016 0.0424--(2.92) > 0.00649--(5.55) 0.249--(17.96) -0.379--(-11.29) N3704 t statistics in parentheses -p<0:10;--p<0:05;---p<0:01 28 29 Table 21:GMM estimates for SVL model for the Natural Gas futures: 09/2001ñ06/2016 0.760--(3.45) > 0.0460--(5.60) 0.925--(3.49) 0.201--(4.57) N3708 t statistics in parentheses -p<0:10;--p<0
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    Despite its popularity, an obvious and distinctive limitation of theV aRapproach is that it only speciÖes the maximum one can lose at a given risk level, but provides no indication for how much more thanV aRone can lose if extreme tail events happen. This may lead to an equivalentV aRestimate for two di§erent positions, though they have completely di§erent risk exposures.
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    Artzner et al. (1999)
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    proposed the concept of coherent risk measure, which has become the paradigm of risk measurement. A good alternative is conditional Value-at-Risk (CV aR), which is a coherent risk measure and retains the beneÖts ofV aRin terms of the capability to deÖne quantiles of the loss distribution.
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    Although theCV aRapproach has been widely used for risk analysis, the implementation of backtesting forCV aRmodels is much harder than forV aRmodels. Nevertheless, formal backtesting methods can be found in literature, such as the most commonly used approach zero-mean residual test by
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    McNeil and Frey (2000)
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    which rely on bootstrapping or one sampletprinciple, censored Gaussian method by Berkowitz (2001) and the functional delta approach by Kerkhol and Melenverg 40 (2004)2. However, applying these methods tend to be di¢ cult and overly complex.
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    Nevertheless, formal backtesting methods can be found in literature, such as the most commonly used approach zero-mean residual test by McNeil and Frey (2000) which rely on bootstrapping or one sampletprinciple, censored Gaussian method by
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    Berkowitz (2001) and
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    the functional delta approach by Kerkhol and Melenverg 40 (2004)2. However, applying these methods tend to be di¢ cult and overly complex. The application of these methods is based upon the realization of speciÖc conditions, hence it is possible to backtestCV aRonly under speciÖc circumstances.
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    aRs andFRV aRdas follows: FRV aRd= 1 T XT t=1 It(yt<-V aRd;t) FRV aRu= 1 T XT t=1 It(yt> V aRu;t) whereV aRd;tandV aRu;tare the estimatedV aRsfor downside and upside risk at time t for a given conÖdence interval,Tis the number of observations andIt(-)is 2A comprehensive discussion of variousCV aRbacktesting methodologies as well as their implementations at di§erent circumstances can refer to
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    Wimmerstedt (2015).
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    41 the indicator function which is deÖned as: Downside:It= 1if yt<-V aRd;t 0if yt-V aRu;t Upside:It= 1if yt> V aRu;t 0if yt-V aRd;t Furthermore, there are three formal tests based on the above criteria to backtest theV aRestimates.
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    methodologies as well as their implementations at di§erent circumstances can refer to Wimmerstedt (2015). 41 the indicator function which is deÖned as: Downside:It= 1if yt<-V aRd;t 0if yt-V aRu;t Upside:It= 1if yt> V aRu;t 0if yt-V aRd;t Furthermore, there are three formal tests based on the above criteria to backtest theV aRestimates. The unconditional coverage test (LRuc), proposed by
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    Kupiec (1995),
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    is to examine whether the null hypothesisH0:FR=-can be satisÖed. A good performance of theV aRmodel should be accompanied by an accurate unconditional coverage, that is, the failure rate is statistically expected to be equal to the prescribedV aRlevel-.
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    A good performance of theV aRmodel should be accompanied by an accurate unconditional coverage, that is, the failure rate is statistically expected to be equal to the prescribedV aRlevel-. The method proposed by
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    Kupiec (1995)
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    is capable to test the overestimates or underestimates of aV aRmodel. It does not, however, consider whether the exceptions are scattered or if they appear in clusters3. In order to examine whether the V aRviolations are serially uncorrelated over time, Christo§ersen (1998) proposes the independent likelihood ratio test (LRind).
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    This test investigates if the failure rate is equal to the expected prescribed risk level and if the exceptions are independently distributed over time. The null hypothesis for this test is that the exceptions are independent and that the expected failure rate is equal to prescribed risk level. 6.1 Out-of-sample VaR and CVaR estimations As in
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    Fan et al. (2008) and
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    Youssef at al. (2015), in this subsection, we test the forecasting ability of the SV and SVL models by computing the out-of-sample VaR 3Kupiecís (1995) approach is an unconditional test. On the other hand, we need to conditionally examine theV aRperformance under the time-varying volatility framework.
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    futures and speculative investors who are short futures would be better o§ not considering the leverage e§ect in order to have more precise forecasts of risk in a VaR sense. 45 46 47 48 7 Conclusions In this paper we estimate a Stochastic Volatility model using high frequency data on crude oil, natural gas and stock index (S&P500) futures. In terms of hedging strategies, it has been shown (see
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    Chen, Zerilli and Baum 2018),
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    at lower frequencies and for crude oil spot returns, that the introduction of the leverage e§ect in the traditional SV model with Normally distributed errors is capable of adequately estimating risk (in a VaR and CVaR sense) for conservative oil suppliers in both the WTI and Brent markets while it tends to overestimate risk for more speculative oil suppliers.
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    In this Appendix we also show the expressions of 2 additional moment conditions that have the main purpose of modelling the leverage e§ect: we have not used them in our results as, when tested, these moment conditions did not add any relevant information to the available set and therefore they revelead to be redundant in this case. 8.1 Residual 1 From
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    Bollerslev and Zhou (2002),
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    page 56Appendix A.1equation(A:3) 51 e1t+1;t+2=E[Vt+1;t+2jGt]-Vt+1;t+2 =-E[Vt;t+1jGt] +\f-Vt+1;t+2 whereVt+1;t+2is therealised varianceandGtis the information set. dt=T-t=t+ 1-t= 1 =t+ 2-(t+ 1) = 1 -=e-\f=>(1--) a= 1 (1--) = \f -> b=>(1-a) =>\f 8.2 Residual 2 E V2t+1;t+2 \f \fGt =H1-E V2t;t+1 \f \fGt +I-E[Vt;t+1jGt] +J e2t+1;t+2=E V2t+1;t+2 \f \fGt -V2t+1;t+2 =H1-E V2t;t+1 \f \fGt +I-E[Vt;t+1jGt]
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    2-1 e-2-B= -<2 > >(1-2-) + > 2(-+ 5) (--1) C=--2 (1--) =--a--2 D= -2> 2(1--) 2 =a2----2-> 52 H1 =-2 G1 =\f2 I= 1 a2(C+ 2-\f) + ---2 (2ab+A) a N= a2 D+\f2 +\f(2ab+A) + 1--2 -b2+B -J=-bI+N 8.3 Residual 3 Residuals 3 to 5 are built in order to deal with the leverage aspect as they focus on the relationship between futures returns and their variance. This moment condition derives from the paper by
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    Garcia et al. (2011)
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    page 32 top equation in Box I: e3t+1;t+2= E[pt;t+1Vt+1;t+2jGt]-b a -pt;t+1Vt;t+1 Considering the relationship between the population varianceVtand the realised varianceVt;t+1 E[Vt;t+1jFt] =aVt+b Vt= E[Vt;t+1jFt]-b a Ab+ab2-aB-(A-2ab)E[Vt;t+1jFt] +aE V2t;t+1 \f \fFt V2t= a3 E[pt;t+1Vt;t+1jGt] = --e-(E[Vt;t+1jGt]-+>(e--k-1)) = --e-> E[Vt;t+1jGt]-b a -+>(e--k-1) E[pt;t+1Vt+1;t+2jGt]-b a 53 e3t+1;t+2=
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