The 31 reference contexts in paper Liyuan Chen, Paola Zerilli, Christopher F Baum (2018) “Leverage effects and stochastic volatility in spot oil returns: A Bayesian approach with VaR and CVaR applications” / RePEc:boc:bocoec:953

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    In addition, as a primary source of energy in the power industry, industrial production and transportation, volatile oil prices may lead to cost uncertainties for other markets, thus extensively affecting the development of the economy. A large number of studies have shown that oil price fluctuations could have considerable impact on economic activities.
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    Papapetrou (2001)
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    argues that the variability of oil prices plays a critical role in affecting real economic activity and employment. Lardic and Mignon (2008) explore the long-term relationship between oil prices and GDP, and find evidence that aggregate economic activity seems to slow down particularly when oil prices increase.
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    A large number of studies have shown that oil price fluctuations could have considerable impact on economic activities. Papapetrou (2001) argues that the variability of oil prices plays a critical role in affecting real economic activity and employment.
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    Lardic and Mignon (2008)
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    explore the long-term relationship between oil prices and GDP, and find evidence that aggregate economic activity seems to slow down particularly when oil prices increase. This asymmetry is found in both the U.
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    Although VaR is recommended by Basel II and III and has been widely adopted by financial institutions, it has been challenged by the Bank of International Settlements (BIS) Committee, who pointed out that VaR cannot measure market risk as it fails to consider the extreme tail events of a return distribution (see,
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    Chen et al., 2012).
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    In addition, Artzner et al. (1999) argue that VaR does not meet the requirements of sub-additivity and thus is not a coherent measure of risk. As an alternative, they proposed a conservative, but more coherent measure, called Conditional VaR at risk (CVaR) or expected shortfall (ES), which considers the average loss as that exceeding the VaR threshold.
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    Although VaR is recommended by Basel II and III and has been widely adopted by financial institutions, it has been challenged by the Bank of International Settlements (BIS) Committee, who pointed out that VaR cannot measure market risk as it fails to consider the extreme tail events of a return distribution (see, Chen et al., 2012). In addition,
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    Artzner et al. (1999)
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    argue that VaR does not meet the requirements of sub-additivity and thus is not a coherent measure of risk. As an alternative, they proposed a conservative, but more coherent measure, called Conditional VaR at risk (CVaR) or expected shortfall (ES), which considers the average loss as that exceeding the VaR threshold.
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    Therefore, in this paper we consider risks affecting both oil supply and oil demand. In recent years, the commodity price literature has shown that there is evidence of leverage effects in various energy markets. More specifically,
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    Chan and Grant (2016a),
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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 efficiently than GARCH models.
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    More specifically, Chan and Grant (2016a), 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 efficiently than GARCH models. At the same time, they find a significant negative leverage effect in crude oil spot markets.
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    Kristoufek (2014)
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    focuses on the leverage effect 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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    They find that this type of model specification does as well as the standard normal distribution at a 95% confidence level. They also test and find evidence for asymmetric leverage effects without modelling them directly.
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    Youssef et al. (2015)
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    evaluate VaR and CVar for crude oil and gasoline markets using a long memory GARCH-EVT approach. Their findings and backtesting exercise show that crude oil markets are characterized by asymmetry, fat tails and long range memory.
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    Using the same logic, firms who are on the supply side, would be better off not considering the leverage effect (SV-N model). 2. Stochastic volatility models We use a general SV model to capture the volatility features for oil markets which has been studied recently by
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    Takahashi et al. (2009),
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    Chai et al. (2011) and Chan et al. (2016a): yt=μ+σtzt(1) lnσ2t=ht=δ+β(lnσ2t−1−δ) +ηtηt∼N(0,σ2η)(2) whereytdenotes stock returns at timetwitht= 1,2,...,T,μdenotes the conditional mean,σt is the stochastic volatility,lnσ2tfollows a stationary AR(1) process with persistence parameter βhaving|β|<1,ztandηtrepresent a series of independent identical (i.i.d.) random errors in the return and volatility
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    of t-distribution. (2) Standard Normal errors zt∼N(0,1) (3) Standard Asymmetric Laplace errors zt∼ALD(0,κ,1) whereσ= 1 andκis the coefficient driving the skewness of the distribution, is related toμ andσas follows: μ= σ √ 2 ( 1 κ −κ ) ht 2>0 (Symmetric Laplace Distribution).3 as a special caseκ= 1 forμ'0 andσ=e 2A number of original empirical works via extended SV models can be found from
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    Breidt et al. (1998), So et al. (1998), Yu and Yang (2002),
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    Koopman and Uspensky (2002), Cappuccio et al. (2004), Chan (2013), Chan and Hsiao (2013), Chan and Grant (2016c), Chan (2017). 3See appendix for the density of ALD. (4) Standard Student t errors with leverage effect yt=μ+σtzt zt∼tν lnσ2t=ht=δ+β ( lnσ2t−1−δ ) +ξt ξt=ρzt+ √ 1−ρ2ηt ηt∼N ( 0,σ2η ) where the coefficientρdrives the so calledleverage effect.
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    errors zt∼ALD(0,κ,1) whereσ= 1 andκis the coefficient driving the skewness of the distribution, is related toμ andσas follows: μ= σ √ 2 ( 1 κ −κ ) ht 2>0 (Symmetric Laplace Distribution).3 as a special caseκ= 1 forμ'0 andσ=e 2A number of original empirical works via extended SV models can be found from Breidt et al. (1998), So et al. (1998), Yu and Yang (2002), Koopman and Uspensky (2002),
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    Cappuccio et al. (2004), Chan (2013),
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    Chan and Hsiao (2013), Chan and Grant (2016c), Chan (2017). 3See appendix for the density of ALD. (4) Standard Student t errors with leverage effect yt=μ+σtzt zt∼tν lnσ2t=ht=δ+β ( lnσ2t−1−δ ) +ξt ξt=ρzt+ √ 1−ρ2ηt ηt∼N ( 0,σ2η ) where the coefficientρdrives the so calledleverage effect.
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    the skewness of the distribution, is related toμ andσas follows: μ= σ √ 2 ( 1 κ −κ ) ht 2>0 (Symmetric Laplace Distribution).3 as a special caseκ= 1 forμ'0 andσ=e 2A number of original empirical works via extended SV models can be found from Breidt et al. (1998), So et al. (1998), Yu and Yang (2002), Koopman and Uspensky (2002), Cappuccio et al. (2004), Chan (2013), Chan and Hsiao (2013),
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    Chan and Grant (2016c), Chan (2017).
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    3See appendix for the density of ALD. (4) Standard Student t errors with leverage effect yt=μ+σtzt zt∼tν lnσ2t=ht=δ+β ( lnσ2t−1−δ ) +ξt ξt=ρzt+ √ 1−ρ2ηt ηt∼N ( 0,σ2η ) where the coefficientρdrives the so calledleverage effect.
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    Estimation methodology of Bayesian MCMC In order to improve the tractability of the ALD model, we introduce a modified Bayesian MCMC method. That is, a new Scale Mixture of Uniform (SMU) representation for the AL density (following
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    Kotz et al., 2001)
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    is proposed to facilitate the estimation of the SV-ALD model. 4See appendix B for the derivations ofV aRs,t,V aRd,t,CV aRs,tandCV aRd,t. 4.1. Scale mixture of uniform representation of ALD Expressing the ALD via the representation can alleviate the computational burden when using the Gibbs sampling algorithm in the MCMC approach and thus can simplify the estimation method in Bayesian
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    of ALD Expressing the ALD via the representation can alleviate the computational burden when using the Gibbs sampling algorithm in the MCMC approach and thus can simplify the estimation method in Bayesian analysis. To estimate the latent variables in the SV model, we use the scaled ALD (SALD) which means that the ALD random variable is scaled by its standard deviation (See
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    Chen et al., 2009 and Wichitaksorn et al., 2015).
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    Proposition 1.Letztbe the ALD random variable withzt∼ALD(0,κ,1), then the random variableεt=ztS.D.[z]has SALD withp.d.f.given by: f(εt|κ,σt) =      √ 1 +κ4 1 +κ2 1 σt exp( − √ 1 +κ4 σt εt)εt≥0 √ 1 +κ4 1 +κ2 1 σt exp( √ 1 +κ4 κ2σt εt)εt<0 (7) whereκis skewness parameter andσtis the standard deviation (or the time-varying volatility) ofzt.5 Hence, the corresponding SMU of SALD can be obtained
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    If the ratio is greater thanα, we conclude that the model underestimate the risks, and vice versa. In addition, three likelihood ratio backtesting criteria are implemented to test the statistical accuracy of the methods. These criteria include unconditional coverage test (LRucby
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    Kupiec, 1995),
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    independent test (LRind by Christoffersen, 1998) and conditional coverage test (LRccby Christoffersen, 1998). For backtesting CVaR in the SV model with Normal errors, we select the nominal risk level at 5% and 1% as 1.96% and 0.38%, following the work by Chen et al. (2012).
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    In addition, three likelihood ratio backtesting criteria are implemented to test the statistical accuracy of the methods. These criteria include unconditional coverage test (LRucby Kupiec, 1995), independent test (LRind by
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    Christoffersen, 1998) and
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    conditional coverage test (LRccby Christoffersen, 1998). For backtesting CVaR in the SV model with Normal errors, we select the nominal risk level at 5% and 1% as 1.96% and 0.38%, following the work by Chen et al. (2012).
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    In addition, three likelihood ratio backtesting criteria are implemented to test the statistical accuracy of the methods. These criteria include unconditional coverage test (LRucby Kupiec, 1995), independent test (LRind by Christoffersen, 1998) and conditional coverage test (LRccby
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    Christoffersen, 1998).
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    For backtesting CVaR in the SV model with Normal errors, we select the nominal risk level at 5% and 1% as 1.96% and 0.38%, following the work by Chen et al. (2012). To identify the nominal risk level for ALD, we use a cumulative distribution function (c.d.f.) of 7Note that for the prior setting in the Bayesian inference, the WinBugs software uses a nonstandard par
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    These criteria include unconditional coverage test (LRucby Kupiec, 1995), independent test (LRind by Christoffersen, 1998) and conditional coverage test (LRccby Christoffersen, 1998). For backtesting CVaR in the SV model with Normal errors, we select the nominal risk level at 5% and 1% as 1.96% and 0.38%, following the work by
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    Chen et al. (2012). To
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    identify the nominal risk level for ALD, we use a cumulative distribution function (c.d.f.) of 7Note that for the prior setting in the Bayesian inference, the WinBugs software uses a nonstandard parametrization of Normal distribution in terms of the precision (1/variance) instead of the variance. 8See appendix E for the derivation of the parameters of full conditional posterior dist
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    ALD, we use a cumulative distribution function (c.d.f.) of 7Note that for the prior setting in the Bayesian inference, the WinBugs software uses a nonstandard parametrization of Normal distribution in terms of the precision (1/variance) instead of the variance. 8See appendix E for the derivation of the parameters of full conditional posterior distributions. ALD which, according to
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    Kotz et al. (2001),
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    is given by: F(z|κ,θ,σ) =      1− 1 1 +κ2 exp ( − √ 2κ σ z ) z≥0 κ2 exp (√ 2 σκ z ) z <0 (13) 1 +κ2 Then, thisc.d.f.is evaluated at the point that equates to the CVaR level.As a consequence, the probability ( ̃α) that CVaR occurs under ALD for oil supply and demand can be mathematically expressed as: Supply: ̃α=F(CV aRs|α) = α e (14) Demand: ̃α=1−F(CV aRd|α) = α e (15) whereeis
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    For both of oil supply and demand, the quantile level of CVaR under ALD is simply a function ofαandeand does not depend on other parameters in the AL density. This surprising finding is consistent with results of
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    Chen et al. (2012)
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    although different ALD forms have been studied. Hence, according to formula (14) and (15), the nominal risk level ̃αfor CVaR under ALD at 5% and 1% are obtained as 1.84% and 0.37%, respectively.
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    Efficiency measures Adhering to the Basel Committee’s guidelines, supervisors are not only concerned with the quantities of violations in a VaR model but also with the magnitude of those violations (Basel Committee on Banking Supervision, 1996a, 1996b). Hence, we employ the regulatory loss function (RLF) and firm’s loss function (FLF) of
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    Sarma et al. (2003),
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    which considers both the number of violations and their magnitude. This is a two-stage model evaluation procedure where the first stage aims to test the models in terms of statistical accuracy, while in the second stage the models surviving the statistical accuracy tests are then evaluated for efficiency (details see Sarma et al., 2003). 6.
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    This is a two-stage model evaluation procedure where the first stage aims to test the models in terms of statistical accuracy, while in the second stage the models surviving the statistical accuracy tests are then evaluated for efficiency (details see
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    Sarma et al., 2003).
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    6. Simulation experiment To examine the effectiveness of the proposed MCMC sampling procedure, we estimate the SV-ALD model using simulated data. We generate 2874 observations from the SVALD model given by (1) and (2) with ALD errors by fixing parameter valuesδ=−7.587, β= 0.9947,ση= 0.0889 andκ= 0.9956.
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    The posterior means and medians forβ,σηand κare very close to the true values with small MC errors, which are all located inside the 95% credible intervals. The posterior means and medians forδvary more than others as it would be expected since the variance parameter is quantile-dependent (similar findings also see
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    Chen et al., 2009).
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    7. Empirical analysis 7.1. Data We consider two major crude oil markets: West Texas Intermediate crude oil (WTI) and Europe Brent crude oil (Brent). Daily closing spot prices, which are quoted in US dollars per barrel, are obtained from the U.
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    that the estimated posterior mean ofσηfor the SV-ALD model is lower comparing to the correspondingσηin the SV-N and SV-t model in both of the two oil markets, andσηin the SV-t model is lower than thatσηin the SV-N model. The estimate for theσηparameter from the SV-ALD-L and SV-t-L model is lower than the estimate coming from the SV-N-L model. These results are consistent with the findings from
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    Chib et al. (2002) and Abanto-Valle et al. (2010),
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    indicating that the introduction of heavy tailed error distribution in the mean equation appears to explain excess returns, thus decreasing the variance of the volatility process. More importantly, we find statistically significant negative correlation (ρ <0) between shocks affecting oil returns and shocks affecting volatility in the SV-t-L, SV-N-L and the SV-ALD-L s
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    More importantly, we find statistically significant negative correlation (ρ <0) between shocks affecting oil returns and shocks affecting volatility in the SV-t-L, SV-N-L and the SV-ALD-L specifications. Although WinBugs can generate deviance information criterion (DIC) values straightforwardly, as pointed out by
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    Chan and Grant (2016),
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    conditional DIC typically favors over-fitted models in a series of Monte Carlo experiments. Therefore this cannot be used as a reliable criterion to compare across models. For this reason, we use various comparison criteria.
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    The Shapiro Francia test (1972) for normality concurs with those judgements for the standard errors coming from all the models. The Box Pierce portmanteau (or Q) test for white noise rejects its null for both series of standard errors. Diebold Mariano test.This test calculates a measure of predictive accuracy proposed by
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    Diebold and Mariano (1995).
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    We ran the test for each of 500 simulations per model Table 10: WTI: Engle’s Lagrange multiplier test for autoregressive conditional heteroskedasticity for standardised residuals and squared standardised residuals for SV-t and SV-t-L models 1 lag p-val 5 lags p-val 10 lags p-val 30 lags p-val SV-t res2.040.158.980.1113.750.1847.150.02 SV-t res squ0.190.660.820.981.
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    The model is said to be correctly specified if the calculated ratio is equal to the pre-specified VaR levelα(i.e. 5% and 1%). If the Failure Rate is greater thanα, we can conclude that the model underestimates the risk, and vice versa (see
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    Marimoutou et al., 2009; Aloui and Mabrouk, 2010; Louzis et al., 2014).
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    A criterion for evaluating our results comes from the consideration that a conservative investor (see for example, Zhao et al., 2015 and Hung et al., 2008) might choose a greater confidence level and estimate a relatively greater risk (corresponding toα= 5% in the VaR definition), while a more speculative investor might estimate a smaller risk and face a relatively smaller confidence l
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    Modelling the data using SV-N and SVN-L models, the testLRucandLRccare passed, which indicates the capability of the models of estimating tail risks. The SV-ALD and SV-ALD-L models overestimate the tail risk and the null hypothesis of testsLRucandLRccare rejected in WTI market and partly rejected in 12For details see
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    Sarma et al. (2003).
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    Table 23: VaR backtesting results for WTI and Brent markets αRisk Failure timesFailure rateLRucLRindLRcc WTI BrentWTIBrentWTIBrentWTIBrentWTIBrent SV-N 5% VaRst1071224.248%4.839%0.07570.70990.8269 0.21140.19300.3869 VaRdt1111164.407%4.720%0.16340.37540.9598 0.49680.36150.9944 1% VaRst22240.873%0.952%0.51380.80710.5334 0.53340.65980.7559 VaRdt16220.635%0.873%0.04860.51130.6510 0.659
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    % corresponds to 5% and 1% risk level of ALD.LRuccolumns show p-values of Kupiec’s (1995) unconditional coverage test,LRindcolumns are p-values of Christoffersen’s (1998) independent test andLRcc columns are p-values of Christoffersen’s (1998) conditional coverage test, * denotes significance. leads to overestimating risk for both type of investors. Stage 2: Efficiency Measures.
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    Lopez (1998, 1999)
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    was the first to propose the comparison between VaR models on the basis of their ability to minimise some specific loss function which reflected a specific objective of the risk manager. Adhering to the Basel Committee’s guidelines, supervisors are not only concerned with the number of violations in a VaR model but also with the magnitude of those violations (Basel Committee on Banking Superv
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    Adhering to the Basel Committee’s guidelines, supervisors are not only concerned with the number of violations in a VaR model but also with the magnitude of those violations (Basel Committee on Banking Supervision, 1996a, 1996b). In order to address this aspect, following
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    Sarma et al. (2003),
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    we compare the relevant models in terms of theRegulatory Loss Function(RLF) which focuses on the magnitude of the failure and in terms of theFirm’s Loss Function(FLF) which, while giving relevance to the magnitude of failures, imposes an Table 25: RLF and FLF Loss function approach applied to the models surviving the VaR backtesting stage Volatility models and VaR methods RLFFLF 5%1%5%1% WTIBrent
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    Considering the RLF criterion, this test shows that the competing models (leverage vs no-leverage models) are not significantly different from 13This criterion penalizes large failures more than small failures (See
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    Sarma et al., 2003).
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    14For the sign test see Lehmann (1974), Diebold and Mariano (1995), Hollander and Wolfe (1999) and Sarma et al. (2003). Table 26: RLF and FLF Loss function approach applied to the models surviving the CVaR backtesting stage Volatility models and CVaR methods RLFFLF 1.84%/1.96%0.37%/0.38%1.84%/1.96%0.37%/0.38% WTIBrentWTIBrentWTIBrentWTIBrent Panel A: Average loss values SV-N Suppl
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    Considering the RLF criterion, this test shows that the competing models (leverage vs no-leverage models) are not significantly different from 13This criterion penalizes large failures more than small failures (See Sarma et al., 2003). 14For the sign test see
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    Lehmann (1974), Diebold and Mariano (1995), Hollander and Wolfe (1999) and Sarma et al. (2003).
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    Table 26: RLF and FLF Loss function approach applied to the models surviving the CVaR backtesting stage Volatility models and CVaR methods RLFFLF 1.84%/1.96%0.37%/0.38%1.84%/1.96%0.37%/0.38% WTIBrentWTIBrentWTIBrentWTIBrent Panel A: Average loss values SV-N Supply0.0001670.000201 0.0001430.000136 0.0008770.0007960.0002220.000204 Demand--0.0002840.000167---0.001221-0.000203 SV-N-L Supply0.00
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