Data driven value-at-risk forecasting using a SVR-GARCH-KDE hybrid
CComputational Statistics (2020) 35:947-981 https://doi.org/10.1007/s00180-019-00934-7
Data Driven Value-at-Risk Forecastingusing a SVR-GARCH-KDE Hybrid (cid:63)
Marius Lux · Wolfgang Karl H¨ardle · Stefan Lessmann
Received: 19 June 2018 / Accepted: 30 October 2019 / Published online: 13 November 2019
Abstract
Appropriate risk management is crucial to ensure the competitiveness of financial institu-tions and the stability of the economy. One widely used financial risk measure is Value-at-Risk (VaR).VaR estimates based on linear and parametric models can lead to biased results or even underesti-mation of risk due to time varying volatility, skewness and leptokurtosis of financial return series.The paper proposes a nonlinear and nonparametric framework to forecast VaR that is motivated byovercoming the disadvantages of parametric models with a purely data driven approach. Mean andvolatility are modeled via support vector regression (SVR) where the volatility model is motivated bythe standard generalized autoregressive conditional heteroscedasticity (GARCH) formulation. Basedon this, VaR is derived by applying kernel density estimation (KDE). This approach allows for flexi- (cid:63)
This is a post-peer-review, pre-copyedit version of an article published in Computational Statistics. The finalauthenticated version is available online at: http://dx.doi.org/10.1007/s00180-019-00934-7
Marius LuxSchool of Business and EconomicsHumboldt-University BerlinUnter den Linden 6D-10099 Berlin, GermanyE-mail: [email protected] Karl H¨ardleSKBI School of BusinessSingapore Management University50 Stamford Road, Singapore 178899;C.A.S.E.-Center of Applied Statistics and EconomicsHumboldt-University BerlinUnter den Linden 6D-10099 Berlin, GermanyE-mail: [email protected] Lessmann (corresponding author)School of Business and EconomicsHumboldt-University BerlinUnter den Linden 6D-10099 Berlin, GermanyORCID ID: 0000-0001-7685-262XE-mail: [email protected] a r X i v : . [ q -f i n . S T ] S e p Marius Lux et al. ble tail shapes of the profit and loss distribution, adapts for a wide class of tail events and is able tocapture complex structures regarding mean and volatility.The SVR-GARCH-KDE hybrid is compared to standard, exponential and threshold GARCHmodels coupled with different error distributions. To examine the performance in different markets,one-day-ahead and ten-days-ahead forecasts are produced for different financial indices. Model eval-uation using a likelihood ratio based test framework for interval forecasts and a test for superiorpredictive ability indicates that the SVR-GARCH-KDE hybrid performs competitive to benchmarkmodels and reduces potential losses especially for ten-days-ahead forecasts significantly. Especiallymodels that are coupled with a normal distribution are systematically outperformed.
Keywords
Value-at-Risk · Support Vector Regression · Kernel Density Estimation · GARCH
Events like the 2008 financial crisis or the outcome of the 2016 referendum in the UK came unexpectedfor many people. Yet, as these examples illustrate, unlikely events occur at times and they might havefar reaching consequences. Risk management is the practice to analyze the macro-environment of anorganization, identify possible adverse developments, and design suitable countermeasures.For financial institutions and systemically important institutions in particular, a key risk manage-ment responsibility is to sustain solvency under adverse economic conditions (e.g., Silva et al 2017;Kraus and Czado 2017). One of the most popular measures of uncertainty in financial markets isVaR (e.g., Alexander (2008)). VaR is based on the quantiles of a portfolio’s profit and loss ( P & L )distribution and can be interpreted as an upper bound on the potential loss that will not be exceededwith a given level of confidence. Its use is appealing because it summarizes the downside risk of aninstitution in one easily interpretable figure (e.g., Chen et al 2012). Regulatory frameworks for thebanking and insurance industry such as Basel III or Solvency II also rely on VaR for determiningcapital requirements. Compared to expected shortfall, an alternative risk measure with some superiormathematical properties (e.g., Kim and Lee 2016), an advantage of VaR may be seen in the fact thatits estimation is more robust due to putting less weight on tail events and large losses, which maydeteriorate the quality of statistical estimation routines (Sarykalin et al 2008).Several approaches have been proposed to estimate VaR including parametric statistical modelsand data-driven machine learning algorithms such as neural networks (NN) and SVR. In a seminalstudy, Kuester et al (2006) review several statistical methods and compare these in a forecastingbenchmark. Using more than 30 years of historical returns data, they find standard GARCH modelsto forecast VaR with the highest accuracy on average.GARCH models are also employed by Chen et al (2012) to estimate VaR for four daily series of stock market indices. More specifically, Chen et al (2012) rely on an asymmetric Laplace distributionand model volatility using a GJR-GARCH model to introduce leverage effects. They then develop atime-varying model to allow for dynamic higher moments. These extensions allow for wider applicationof the model beyond forecasting.Unlike the parametric approach of Chen et al (2012), Franke and Diagne (2006) estimate VaRfor the German stock index through fitting the mean and volatility of the return series using NNs.More specifically, they model the mean and volatility as an autoregressive (AR) and autoregressiveconditionally heteroscedastic (ARCH) process, respectively. To derive VaR and expected shortfall,Franke and Diagne (2006) use the predicted mean and variance with the normal distribution. Thismodel outperforms a standard GARCH model in terms of VaR exceedances and proofs capable of ata Driven Value-at-Risk Forecasting using a SVR-GARCH-KDE Hybrid (cid:63) quickly adjusting volatility in case of shocks with only short impact. Dunis et al (2010) also proposea NN-based approach towards forecasting VaR and expected shortfall.Khan (2011) develops a VaR-model that forecasts realized volatilities using a combination of aheterogeneous AR model and SVR. VaR is then computed based on the normal, t - and skewed t -distribution. Applying this model to 5- and 15-minutes return data, Khan (2011) is able to confirmthe suitability of the SVR component. O. Radovi´c et al (2015) provide further evidence that SVR isa useful method for VaR forecasting. Likewise, Xu et al (2016) introduce a multi-period VaR modelusing SVR in a quantile regression framework and show this approach to outperform GARCH models.The findings of Xu et al (2016) seem to disagree with prior results of Kuester et al (2006) whereGARCH models predict VaR with highest average accuracy and more accurately than quantile re-gression approaches in particular. Implementations of the quantile regression using a data-drivenSVR model might explain the results of Xu et al (2016). More specifically, the linear and paramet-ric structure of standard GARCH models might be a limiting factor in VaR forecasting. Moreover,the parameters of GARCH-type models are usually estimated via maximum likelihood estimation(Bollerslev 1986). This necessitates distributional assumptions, which might be problematic since thedistribution of financial returns is skewed and exhibits fat tails (Bali et al 2008; Harvey and Siddique2000).Noting the possible limitation of the parametric framework, Schaumburg (2012) combines extremevalue theory with nonparametric VaR estimation to forecast return distributions of four financial stockindices. A parametric conditional autoregressive value at risk (CAViaR) model serves as benchmark.The benchmark and the proposed model both circumvent the estimation of the mean and variance ofthe P & L distribution through predicting a quantile directly. In this regard, the approach of Schaum-burg (2012) can be characterized as a nonparametric CAViaR model.VaR forecasts based on CAViaR frameworks have also been considered in the benchmarking studyof Kuester et al (2006). In fact, the authors also introduce a novel CAViaR model in the paper andtest it alongside various other VaR models. However, GARCH models and models relying on the t -distribution in particular emerge as most suitable for VaR modeling.In summary, parametric GARCH models are superior to parametric quantile regression approachesfor modeling VaR. To achieve better results with quantile regression approaches, it is necessary to in-clude nonparametric parts into the model. Additionally, data driven GARCH models where the meanand variance components are modeled nonlinearly and nonparametrically lead to better results thanparametric GARCH models. However, one shortcoming of the so far proposed data driven GARCHmodels is the use of parametric residual distributions. Particularly with regard to skewness and kurto-sis, this can lead to misspecified residual distributions, resulting in wrong VaR estimates. Therefore, anovel purely nonparametric VaR model that grounds on the GARCH framework is proposed here. Weapply data driven approaches to all GARCH components, i.e. mean, variance and residual distribution.Using this approach, we can overcome the possible misspecification of the residual distribution as well as the misspecifications of mean and variance. More specifically, we estimate the mean and variance ofthe P & L distribution using SVR and employ KDE to model the density of the standardized residuals(e.g., H¨ardle et al 2004). We then integrate these components to derive a VaR forecast. In other words,we propose to start from the most effective parametric modeling approach of Kuester et al (2006) anddevelop models that estimate its components in a purely data-driven manner. In contrast to otherso far proposed GARCH based approaches to forecast VaR (e.g., Youssef et al 2015; Khosravi et al2013; Aloui and Mabrouk 2010; Huang et al 2009; Fan et al 2008; Hang Chan et al 2007; Hartz et al2006) no assumptions about process dependence structures or the distribution of residuals are madeby combining SVR and KDE in a GARCH like fashion. Training the SVR-GARCH-KDE hybrid is,therefore, mainly computationally driven. Marius Lux et al.
The use of SVR is motivated by the existence of a large body of research showing the effectivenessof SVR in forecasting financial time series (e.g., Chang et al 2016; Devi et al 2015; Tay and Cao 2001).Additionally, Sheta et al (2015) compare the forecasting performance of SVR, ANN and traditionallinear regression for the S&P 500. They find that SVR solves the task most successfully. Kazem et al(2013) also provide evidence that SVR based models outperform ANNs in the context of financialforecasting. Moreover, Chen et al (2010) use a SVR approach for predicting stock market volatilityand use a recurrent ANN as benchmark which is outperformed. The use of nonparametric densityestimation is motivated by the fact that the existence of fat tails and skewness in the distributionof financial returns can be considered as an empirically proven fact (e.g., Bali et al 2008; Harveyand Siddique 2000). Although KDE is not a new approach in VaR forecasting (e.g., Chen et al2016; Schaumburg 2012; Malec and Schienle 2014), the particular combination of data-driven VaRestimation using SVR and nonparametric density estimation, which we propose in this paper, has, tothe best of our knowledge, not been considered in prior work.We assess the performance of the proposed model in comparison to GARCH-type models withdifferent error distributions also including skewed and fat-tailed distributions. Empirical experimentsusing data from three major financial indices, namely the Euro STOXX 50, Nikkei 225 and Standard& Poor’s 500 (S&P 500), suggest that the SVR-GARCH-KDE hybrid typically outperforms modelsthat are coupled with a normal distribution and performs competitive to other benchmark models.The remainder of the paper is organized as follows. In Section 2 VaR is defined and the methodsunderlying the proposed VaR modeling framework are presented. Specifically, the standard GARCHapproach, nonparametric density estimation via KDE and SVR are introduced. The proposed SVR-GARCH-KDE hybrid is then developed based on these building blocks. After outlining the theoreticalbackground, the SVR-GARCH-KDE hybrid is compared to other models on different datasets inSection 3. Concluding remarks and suggestions for future research are provided in the last section. P & L distribution. However, since today’s portfoliovalue is usually known, it suffices to model the return distribution. For a formal description of VaR,let the portfolio returns r t in period t have the cumulative distribution function (CDF) F t . Then, theVaR in d trading days for a confidence level 1 − α is defined as V aR αt + d = − F − t + d ( α ) = − inf { x ∈ R : F t + d ( x ) ≥ α } with α ∈ (0 , . (1)In the rest of the paper, VaR refers to the negative α -quantile of the next period’s portfolio returndistribution. r t = µ t + u t = µ t + σ t z t , z t ∼ (0 ,
1) i.i.d. (2) ata Driven Value-at-Risk Forecasting using a SVR-GARCH-KDE Hybrid (cid:63) In (2), µ t is the location and σ t > r t belongs to the location-scalefamily and F z is the CDF of z , we can compute VaR as V aR αt = − (cid:110) µ t + σ t F − z ( α ) (cid:111) . (3)Autoregressive moving average (ARMA) processes and GARCH-type models are commonly used toestimate µ t and σ t in (2).2.3 Modeling Volatility Using GARCH ModelsBollerslev (1986) introduces GARCH models by generalizing the volatility modeling approach of Engle(1982). In deriving the GARCH regression model Bollerslev (1986) starts by assuming conditionalnormality of the return process r t : r t |F t − ∼ N ( β (cid:62) x t , σ t ) . (4)where x t is a vector of lagged endogenous as well as exogenous variables, β an unknown parametervector and F t − the information set available at t −
1. Rewriting (4) as linear model with conditionallyheteroscedastic and normally distributed disturbances gives: r t = β (cid:62) x t + u t , u t |F t − ∼ N (0 , σ t ) . (5)Then, the GARCH( p , q ) representation of the variance σ t is σ t = ω + q (cid:88) i =1 δ i u t − i + p (cid:88) j =1 θ j σ t − j . (6)Bollerslev (1986) notes that (6) has an ARMA representation. To see this let ν t = u t − σ t andsubstitute σ t in (6) with u t − ν t to obtain u t − ν t = ω + q (cid:88) i =1 δ i u t − i + p (cid:88) j =1 θ j ( u t − j − ν t − j ) . (7)Rearranging (7) yields an ARMA representation for u t : u t = ω + q (cid:88) i =1 δ i u t − i + p (cid:88) j =1 θ j ( u t − j − ν t − j ) + ν t (8) = ω + max( p,q ) (cid:88) i =1 ( δ i + θ i ) u t − i − p (cid:88) j =1 θ j ν t − j + ν t . (9)Based on (9) nonlinear and nonparametric volatility models can be introduced. This can be seen bynoting that the conditional expectation of u t is equal to σ . Consequently, the variance process canbe modeled solely based on the observed values without making assumptions about the distributionalform of the residuals or the structure of the variance process. Hence, the volatility model in theSVR-GARCH-KDE hybrid is motivated by the ARMA representation of σ . Marius Lux et al. X be a random variable with an absolutely continuous distribution function F . Further, denotethe corresponding density function as f and let { x , . . . , x n } be a sample of i.i.d. realizations of X .Then, the kernel density estimator ˆ f h ( x ) of f ( x ) is defined asˆ f h ( x ) = 1 hn n (cid:88) i =1 K (cid:16) x i − xh (cid:17) (10)where h is a bandwidth parameter with h > K is a so-called kernel function. Usually, a kernelfunction is assumed to be a symmetric density function, i.e. (cid:90) ∞−∞ K ( u ) du = 1 with K ( u ) ≥ (cid:90) ∞−∞ uK ( u ) du = 0 . (12)Conveniently, (11) implies that ˆ f h ( x ) is also a density. Note that ˆ f h ( x ) inherits all properties of K regarding continuity and differentiability.The KDE based quantile estimator to forecast VaR can be derived as follows. First, the estimatorfor F ( x ) that is based on KDE needs to be derived. Denote (cid:98) F h ( x ) as the KDE based estimate of F ( x ).Then, (cid:98) F h ( x ) can be derived as follows: (cid:98) F h ( x ) = (cid:90) x −∞ ˆ f h ( z ) dz (13)= (cid:90) x −∞ nh n (cid:88) i =1 K (cid:16) z − x i h (cid:17) dz (14)= 1 nh n (cid:88) i =1 (cid:90) x −∞ K (cid:16) z − x i h (cid:17) dz. (15) Since the given kernel function K is a density, let Γ denote the corresponding CDF. Moreover, usingthe substitution u = ( z − x i ) /h one obtains (cid:98) F h ( x ) = 1 n n (cid:88) i =1 (cid:90) x − xih −∞ K ( u ) du (16)= 1 n n (cid:88) i =1 Γ (cid:16) x − x i h (cid:17) . (17) ata Driven Value-at-Risk Forecasting using a SVR-GARCH-KDE Hybrid (cid:63) Thus, (cid:98) F h ( x ) is the mean of the CDF corresponding to K evaluated at ( x − x i ) /h for i = 1 , . . . , n . Then,for α ∈ (0 ,
1) the KDE based quantile function (cid:98) Q h is obtained as (cid:98) Q h ( α ) = (cid:98) F − h ( x ) . (18)2.5 Support Vector RegressionSVR can be understood as a learning method to solve nonlinear regression tasks (e.g., Smola andSch¨olkopf 2004). It shares some similarities with a three-layer feed-forward NN and is able to approx-imate arbitrarily complex functions (Chen et al 2010). However, NNs are based on minimizing theso-called empirical risk and tend to find only locally optimal solutions. In contrast, SVR minimizesthe so-called structural risk to achieve better generalization and solves a convex optimization problemleading to a globally optimal solution. To describe the SVR model, let { ( y i , x i ) | i = 1 , . . . , n ; n ∈ N } with x i ∈ R p and y i ∈ R denote the training data set. Suppose f is a linear function such that f ( x ) = ω (cid:62) x + b (19)where ω ∈ R p and b ∈ R . Then, SVR aims to find an approximation of f that deviates at most by (cid:15) from the observed target y while being as flat as possible (i.e., in the sense that weights in ω aresmall). This translates into the following convex optimization problem:minimize 12 (cid:107) ω (cid:107) subject to (cid:40) y i − ω (cid:62) x i − b ≤ (cid:15)ω (cid:62) x i + b − y i ≤ (cid:15). (20)In view that (20) might lack a feasible solution, Vapnik (1995) introduces an (cid:15) -insensitive loss function: L (cid:15) { y − f ( x ) } = (cid:40) | y − f ( x ) | ≤ (cid:15) | y − f ( x ) | − (cid:15) otherwise . (21)To measure empirical loss (and thus model fit) using (21) Vapnik (1995) reformulates (20) usingslack variables ζ and ζ ∗ that capture losses above and below the (cid:15) -tube around f ( x ), respectively.Figure 1 depicts this approach. Only points outside the gray shaded (cid:15) -tube contribute linearly to theloss function.Integrating the slack variables ζ and ζ ∗ into (20), the task to estimate a SVR model is equivalentto solving: minimize 12 (cid:107) ω (cid:107) + C n (cid:88) i =1 ( ζ i + ζ ∗ i )subject to y i − ω (cid:62) x i − b ≤ (cid:15) + ζ i ω (cid:62) x i + b − y i ≤ (cid:15) + ζ ∗ i ζ i , ζ ∗ i ≥ , (22)where C > C put more (less) weight on maximizing model fit during SVRlearning. Marius Lux et al.
Fig. 1
The (cid:15) -insensitive loss function of the SVR algorithm. Slack variable ζ captures the loss above the (cid:15) -tube.Points within the grey shaded area have no impact on the loss. In contrast, all other observations contribute linearlyto the loss. Source: Smola and Sch¨olkopf (2004). To capture nonlinear relationships between covariates and the response variable, SVR maps theinput data into a higher dimensional feature space. The linear regression is then constructed in thetransformed space, which corresponds to a nonlinear regression in the input space. The transformationis feasible from a computational point of view because SVR calculates the mapping by implicitly usinga kernel function k ( x (cid:62) i x ) = φ (cid:62) ( x i ) φ ( x ). To implement this approach, it is common practice to estimatea SVR model through solving the dual of (22), which is given as (e.g., Smola and Sch¨olkopf 2004):maximize − n (cid:88) i,j =1 ( ρ i − ρ ∗ i )( ρ j − ρ ∗ j ) x (cid:62) i x j − (cid:15) n (cid:88) i =1 ( ρ i + ρ ∗ i ) + n (cid:88) i =1 y i ( ρ i − ρ ∗ i )subject to (cid:40)(cid:80) ni =1 ( ρ i − ρ ∗ i ) = 0 ρ i , ρ ∗ i ∈ [0 , C ] . (23)The dual program (23) includes the input data only in the form of scalar products x (cid:62) i x j . Replacingthe scalar product by means of a kernel function is thus straightforward and does not affect the solver.In this work, we employ the Gaussian radial basis function (RBF) kernel (24) which is defined as k ( x (cid:62) i x ) = exp (cid:18) − (cid:107) x − x i (cid:107) γ (cid:19) (24)where the meta-parameter γ > modeler. The RBF kernel is used because it includes other kernels as special cases, possesses numericaladvantages compared to alternatives, and often performs well in practical applications. Moreover, theRBF kernel can capture nonlinear relations. Other kernels that are usually presented as potentialchoices are e.g. the linear, polynomial or sigmoid kernel (e.g., Smola and Sch¨olkopf 2004; Hastie et al2009). Keerthi and Lin (2003) show that the linear kernel is a special case of the RBF kernel. Moreover,the polynomial kernel has more parameters than the RBF kernel that make the tuning process morecostly. Another advantage of the RBF over the polynomial kernel is that the polynomial kernel canconverge to infinity which can cause numerical instability. In contrast, the domain of the RBF kernelis always between 0 and 1. Moreover, Lin and Lin (2003) show that the sigmoid kernel behaves similarto the RBF kernel for certain parameters. ata Driven Value-at-Risk Forecasting using a SVR-GARCH-KDE Hybrid (cid:63) In order to construct the regression function (19), the weight vector ω is represented as a linearcombination of observations in the training set: ω = n (cid:88) i =1 ( ρ i − ρ ∗ i ) x i . (25)More specifically, for observations x i where f ( x i ) is within the (cid:15) -tube holds that ρ i = ρ ∗ i = 0.Consequently, f ( x ) depends only on the observations outside the (cid:15) -tube. These x i are called supportvectors. Accordingly, (25) is also called the support vector expansion of ω . Rewriting the regressionfunction in terms of the support vector expansion gives the SVR forecasting model: f ( x ) = n (cid:88) i =1 ( ρ i − ρ ∗ i ) x (cid:62) i x + b. (26)In the nonlinear case, the scalar product in (26) is once again replaced by a kernel function.2.6 SVR-GARCH-KDE HybridIn the following section, we introduce a nonlinear GARCH hybrid to forecast VaR based on a combina-tion of SVR and KDE. Subsequently, we elaborate on the estimation of the corresponding forecastingmodel.We assume the distribution of the return series r t to belong to the location-scale class, such that: r t = µ t + u t = µ t + σ t z t , z t ∼ (0 ,
1) i.i.d. (27)Consider an ARMA structure for the mean model where the only assumption about the error distri-bution is a zero mean and a finite variance. In addition, recall equation (9), which shows that GARCHprocesses can also be given an ARMA representation. This leads to the following mean and variancemodel: r t = c + s (cid:88) i =1 α i r t − i + d (cid:88) j =1 κ j u t − j + u t , u t ∼ (0 , σ t ) (28) u t = ω + max( p,q ) (cid:88) i =1 ( δ i + β i ) u t − i − p (cid:88) j =1 β j ν t − j + ν t . (29) Let e = max( p, q ), r t,s = ( r t − , r t − , . . . , r t − s ), u t,k = ( u t − , u t − , . . . , u t − k ) and ν t,p = ( ν t − , ν t − , . . . , ν t − p ).Then, following Chen et al (2010), we introduce the nonlinear and nonparametric functions h and g such that the conditional mean and variance models of r t are r t = h ( r t , s , u t , d ) + u t u t ∼ (0 , σ t ) (30) u t = g ( u t , e , ν t,p ) + ν t ν t ∼ W N (0 , a t ) (31)where W N (0 , a t ) denotes white noise with expectation zero and variance a t . We propose to estimate h ( · ) and g ( · ) using SVR. The estimates for µ t and σ t in (27) are then obtained as: (cid:98) µ t = h ( r t , s , u t , d ) (32) (cid:98) σ t = (cid:113) g ( u t , e , ν t,p ) . (33)By defining the estimated residuals as (cid:98) u t = r t − (cid:98) µ t , estimates of z t are obtained as (cid:98) z t = (cid:98) u t (cid:98) σ t . (34)Then, for (cid:98) Q (cid:98) z ( α ) being the estimated quantile function of z , the VaR estimate for r t is: (cid:100) V aR αt = − (cid:26) h ( r t , s , u t , d ) + (cid:113) g ( u t , e , ν t,p ) (cid:98) Q (cid:98) z ( α ) (cid:27) . (35)whereby we estimate (cid:98) Q (cid:98) z ( α ) using KDE.We now present a procedure to estimate VaR as in (35) and describe it in the context of producingone-day-ahead VaR forecasts. A step-by-step overview is given in Algorithm 1. Let { r t } Tt =1 be thetraining set consisting of the daily returns from a portfolio where r T is the most recent observation.In the first step, we model the mean process (30). To do this, we estimate an AR( s ) model usingSVR to obtain the estimated returns { (cid:98) r t } Tt =1+ s . The set of estimated residuals { (cid:98) u t } Tt =1+ s is derived as (cid:98) u t = r t − (cid:98) r t . Then, a moving average (MA) part can be introduced to the model such that r t can bemodeled as an ARMA( s , d ) process by running SVR and including (cid:98) u t . The sets of estimated returnsand residuals from the ARMA( s , d ) model are denoted as { (cid:98) r ∗ t } Tt =1+ s + d and { (cid:98) u ∗ t } Tt =1+ s + d , respectively.We also estimate the variance process in a two step approach and start by fitting the squaredmean model residuals { (cid:98) u ∗ t } Tt =1+ s + d in the way of an AR( e ) process with SVR. Based on this, fittedvariances { (cid:98) σ t } Tt =1+ s + d + e are obtained. Then, an ARMA model for (31) is obtained in the same wayas for the mean process by using the estimated model residuals { (cid:98) ν t } Tt =1+ s + d + e where (cid:98) ν t = (cid:98) u ∗ t − (cid:98) σ t .Consequently, the final set of fitted variances is denoted as { (cid:98) σ ∗ t } Tt =1+ s + d + e + p . No assumptions aremade about the starting values of the residuals. Hence, the final estimation of (31) is done using datafor T − s − d − e − p time points. Since SVR is applied without introducing further restrictions, it is notensured that (cid:98) σ t and (cid:98) σ ∗ t are positive. Therefore, if the SVR estimate is (cid:98) σ ( ∗ )2 t ≤
0, it will be replacedby the last positive estimated variance. In case the first fitted variance is negative, it will be replacedby the first squared residual from the final mean model.The set of estimated standardized residuals { (cid:98) z t } Tt =1+ s + d + e + p can be computed by applying (34).However, (cid:98) z t does not necessarily have zero mean and unit variance. Hence, we perform the quantileestimation using scaled standardized residuals (cid:98) z ∗ t : (cid:98) z ∗ t = (cid:98) z t − (cid:98) z t (cid:113) T − (cid:80) Ti =1 ( (cid:98) z t − (cid:98) z t ) (36) where (cid:98) z t denotes the empirical mean of (cid:98) z t . The forecasted mean (cid:98) µ T +1 and standard deviation (cid:98) σ T +1 are obtained from the mean and variance model. Finally, we use KDE to estimate the α -quantile of { (cid:98) z ∗ t } Tt =1+ s + d + e + p . Then, the one-day-ahead VaR forecast is: (cid:100) V aR αT +1 = − [ (cid:98) µ T +1 + (cid:98) σ T +1 (cid:98) Q (cid:98) z ∗ ( α )] . (37)An important aspect to note is that SVR and KDE depend on hyperparameters which cannot bederived analytically but must be found computationally. Therefore, the above described estimation ata Driven Value-at-Risk Forecasting using a SVR-GARCH-KDE Hybrid (cid:63) procedure describes the estimation process only for one fixed set of hyperparameters. The hyperpa-rameters for SVR given a RBF kernel is used are (cid:15) , γ and C . Additionally, if the SVR kernel is notset beforehand, it can be seen as hyperparameter itself. In the context of KDE the bandwidth h andthe KDE specific kernel function K are hyperparameters. Since the goal is to train an efficient modelon a purely data driven basis it is advisable to derive the hyperparameters computationally. However,for some hyperparameters exist theoretical results that support fixing them beforehand. As stated inSection 2.5 the RBF kernel is a reasonable choice in SVR. Moreover, for KDE the kernel choice hasonly a low practical relevance and there exist rule-of-thumb estimators for h when a Gaussian kernelis used which are computationally inexpensive (H¨ardle et al 2004; Silverman 1986). Hence, the mostrelevant aspect in hyperparameter tuning of the SVR-GARCH-KDE hybrid are (cid:15) , γ and C for theSVR with a RBF kernel. It is important to note that the overall goal is to forecast VaR. Hence, thehyperparameters should be set with respect to the measure that is used to evaluate quantile fore-casts. There exist different approaches as grid search, random search or more advanced optimizationstrategies that can be used to automate the process of hyperparameter optimization. Algorithm 1
SVR-GRACH-KDE Estimation Algorithm for Forecasting VaR
1: AR( s ) model for { r t } Tt =1 using SVR2: Get errors from Step 1 { (cid:98) u } Tt =1+ s
3: ARMA( s , d ) model for { r t } Tt =1+ s with results from Step 2 using SVR4: Get errors from Step 3 { (cid:98) u ∗ } Tt =1+ s + d
5: AR( e ) model for { (cid:98) u ∗ t } Tt =1+ s + d using SVR6: Get errors from Step 5 { (cid:98) ν t } Tt =1+ s + d + e
7: ARMA( e , p ) model for { (cid:98) u ∗ t } Tt =1+ s + d + e with results from Step 6 using SVR8: Obtain volatility estimates { (cid:98) σ ∗ } Tt =1+ s + d + e + p from Step 79: Get standardized residuals (cid:98) z t = (cid:98) u ∗ t / (cid:98) σ t for t = 1 + s + d + e + p, . . . , T
10: Scale { (cid:98) z t } Tt =1+ s + d + e + p to zero mean and unit variance and obtain { (cid:98) z ∗ t } Tt =1+ s + d + e + p
11: Estimate the α -quantile (cid:98) Q (cid:98) z ∗ ( α ) with KDE12: Obtain (cid:98) r T +1 and (cid:98) σ T +1 by using the models from Step 3 and 713: VaR forecast: (cid:91) V aR αT +1 = − [ (cid:98) µ T +1 + (cid:98) σ T +1 (cid:98) Q (cid:98) z ∗ ( α )] After defining the SVR-GARCH-KDE hybrid the question arises how model complexity, predictivepower and the computational time for making VaR forecasts are related. Regarding the time tocompute predictions the KDE part is fixed beforehand. This is can be seen in Equation 10 andAlgorithm 1. In order to compute an estimate the sum needs to be evaluated making the computational complexity O ( n ) where n is number of observations. Regarding the SVR-GARCH-KDE hybrid thecomputational complexity of the KDE part is reduced depending on the order of the autoregressiveand moving-average part of the mean and variance process to O ( n − s − d − e − p ). With respect toSVR the complexity of making a prediction is O ( n SV d ), where d is the dimension of the feature spaceand n SV the number of SVs. For the proposed application of forecasting the mean and variance ofa financial time series the number of features is usually low. Therefore, the computational time forgenerating predictions is mainly driven by the sample size and negligible in this application.Regarding the predictive performance, the regression function of the mean and variance processis determined by the SVs which implies that the higher the number of SVs, the higher the complex-ity. Using a SVR decision function with a high number of SVs can, therefore, lead to a mediocre out-of-sample performance. However, it is not possible to make an ex ante exact statement since theVapnik-Chervonenkis dimension of the RBF kernel is infinite (Burges 1998). Moreover, the goal of theSVR-GARCH-KDE hybrid is to forecast quantiles. Evaluating the performance ex ante does, there-fore, depend on the measure that is used to evaluate forecasted quantiles rather than the statisticalproperties of KDE and potential error bounds of SVR. In Section 3 the framework of Christoffersen(1998) will be applied. For achieving a good performance it is necessary to tune the parameters ofthe SVR and KDE part appropriately with respect to the target measure. Consequently, in this studythe focus is put on the out-of-sample performance to optimize the quality of predictions and thehyperparameter tuning is done with a separate training set. P t : r t = log( P t ) − log( P t − ) . (38)The descriptive statistics of the analyzed indices are given in Table 1. Index 1st Quartile Mean Median 3rd Quartile Variance Skewness Kurtosis
EuroStoxx50 -0.74 -0.01 0.01 0.78 2.39 -0.06 5.15S&P500 -0.46 0.02 0.07 0.59 1.74 -0.33 9.94Nikkei225 -0.76 0.00 0.05 0.88 2.68 -0.51 7.49
Table 1
Descriptive statistics for the log-returns of the analyzed indices in the period from July 1, 2006 to June 30,2016. Note that the log-returns were multiplied by 100 before computing the descriptive statistics.
We forecast VaR for the quantiles α ∈ { . , . , . } , considering forecast horizons of one andten trading days. Estimating VaR for a horizon of ten trading days is especially important regardingthe applicability of a VaR model. Besides forecasting VaR for a confidence level of 99%, which isequivalent to α = 0 .
01 in our setting, a ten days forecast horizon is required in the regulations of theBasel Committee on Banking Supervision.In empirical applications, the quality of SVR depends on the kernel and parameter values whichneed to be set manually. The prevailing approach to determine parameter settings is grid search (e.g.,
Lessmann and Voß 2017), which we also apply in this study. For the density estimation via KDE, theGaussian kernel function in combination with Silverman’s rule of thumb are used to reduce computa-tional cost. The Gaussian kernel function in KDE is equivalent to the standard normal distribution: K ( u ) = 12 π e − u . (39)Silverman (1986) showed that when (39) is used, robust density estimates can be obtained withthe following estimator for the bandwidth h : h rot = 0 . { (cid:98) σ e , (cid:98) σ iqr } n − / . (40) ata Driven Value-at-Risk Forecasting using a SVR-GARCH-KDE Hybrid (cid:63) In 40 (cid:98) σ e denotes the empirical standard deviation and (cid:98) σ iqr is an estimate of the standard deviationthat is based on the interquartile range R : (cid:98) σ iqr = R . . (41)The rule-of-thumb estimator h rot is computationally inexpensive whereas other proposed ap-proaches to bandwidth estimation are more expensive since they are usually based on cross validation(H¨ardle et al 2004). Moreover, the importance of the kernel choice regarding the performance of KDEis limited. For instance, H¨ardle et al (2004) conclude that the kernel choice has almost no practicalrelevance after deriving the asymptotic mean integrated squared error for different kernel choices.The evaluation of the models is based on Christoffersen (1998) who proposes a likelihood ratio (LR)test framework, which assesses the unconditional and conditional coverage as well as the independenceof VaR exceedances. Moreover, Christoffersen (1998) shows that the test statistic for conditionalcoverage can be derived as the sum of the test statistics of the test for unconditional coverage andindependence of VaR exceedances. Hence, it is possible to test whether the performance of a VaRmodel in terms of conditional coverage is determined by its ability to achieve correct unconditionalcoverage or adjust for changing volatility. This is useful in situations where the model has relativelybad conditional coverage but only one of the test statistics for unconditional coverage or independenceis small. The hypotheses for the three tests are shown in Table 2 where α is the target quantile and F t − the information set available at t − V t is a series of VaR violations with V t = I ( r t < − V aR αt )where I ( x < c ) denotes the indicator function: I ( x < c ) = (cid:40) x < c x ≥ c. (42)For the test of independence of violations V t is assumed to be a binary first-order Markov chain. Thecorresponding transition probability matrix of V t is Π = (cid:20) − π π − π π (cid:21) (43)where π ij = P ( V t = j | V t − = i ) . (44) Test H H Unconditional Coverage E [ V t ] = α E [ V t ] (cid:54) = α Independence of Violations π = π π (cid:54) = π Conditional Coverage E [ V t |F t − ] = α E [ V t |F t − ] (cid:54) = α Table 2
Hypotheses for evaluating the appropriateness of VaR forecasts with the testing framework introduced byChristoffersen (1998).
As criterion for selecting a model from grid search and evaluate the performance of the SVR-GARCH-KDE hybrid with respect to benchmark models, the p -value of the test for conditional cover-age is used. Since the null hypothesis corresponds to correct conditional coverage which is the desiredproperty, the one with the highest p -value is considered to be the best.By using the framework of Christoffersen (1998) to evaluate and select models, the main focus isput on the statistical properties and VaR violations of the considered models. In order to measure the performance from the perspective of a loss function that also takes into account the magnitudeof violations, the quadratic and asymmetric loss function that was introduced in one of the seminalpapers for evaluating VaR models by Lopez (1998) will be used: L ( r t , V aR αt ) = (cid:40) r t − V aR αt ) if r t < V aR αt r t ≥ V aR αt . (45)The model losses will be evaluated by employing the superior predictive ability test (SPA) frameworkproposed by Hansen (2005). Following the notation of Hansen (2005), for a finite set of decision rules[ δ k,t − h , k = 0 , , . . . , m ] made h periods in advance with respect to a random variable ξ t , the relativeperformance corresponding to the benchmark δ ,t − h is measured as d k,t = L ( ξ t , δ ,t − h ) − L ( ξ t , δ k,t − h ) . (46)Based on the assumption that the alternatives are superior if and only if E [ d k,t ] >
0, Hansen (2005)formulates the hypothesis of interest for d t = ( d ,t , . . . , d m,t ) (cid:62) asH : E [ d t ] ≤ . (47)A high p -value indicates that none of the alternatives is superior to the benchmark. For furtherinformation regarding the SPA test we refer to Hansen (2005). As for the LR test of Christoffersen(1998), the SPA test will be performed for every model. This is done by using every considered modelonce as benchmark in terms of Hansen (2005) and comparing it to all other models. Note that dueto performing multiple statistical tests the p -values should be interpreted rather as an indication ofmodel performance than in the context of a fixed significance level.We perform all analyses using the statistical software R. The data has been downloaded fromYahoo Finance using the quantmod package. For SVR, we use the package e1071, which is the Rimplementation of the LIBSVM library of Chang and Lin (2011). To reduce computational time, weemploy the doParallel package for parallelization of computations. The benchmark methods introducedbelow are implemented by using the rugarch package. All codes are available on . Fordetails we refer to Borke and H¨ardle (2018) and Borke and H¨ardle (2017).3.2 Benchmark MethodsTo test the SVR-GARCH-KDE hybrid, we compare its performance to the standard GARCH modeland two of its variations. In particular, Franke et al (2015) state that the most important variations are the EGARCH and TGARCH model. Hence, they serve as benchmarks in the empirical compar-ison. The EGARCH and TGARCH models are introduced by Nelson (1991) and Zakoian (1994),respectively. In contrast to standard GARCH models, both can account for asymmetric behaviorwith respect to past positive or negative returns. The two main differences between EGARCH andTGARCH models are that the former has a multiplicative and the latter an additive model structure.Moreover, TGARCH models allow for different coefficients depending on the lags whereas EGARCHmodels capture the asymmetric behavior for all lags with one coefficient. The GARCH-type modelsthat Kuester et al (2006) analyze are coupled with different error distributions, i.e. the normal dis-tribution, t -distribution and skewed t -distribution. We adopt this approach, which implies that wecompare the SVR-GARCH-KDE hybrid to nine benchmarks. ata Driven Value-at-Risk Forecasting using a SVR-GARCH-KDE Hybrid (cid:63) We assume the mean process of r t in (32) is zero. Moreover, we assume the variance process to haveone AR and one MA part. These assumptions are imposed on both the SVR-GARCH-KDE hybridand the benchmark models. We then forecast VaR for every index trading day from 2011-07-01 until2016-06-30. The data is scaled to zero mean and unit variance in the SVR step; as suggested in thedocumentation of the e1071 package.The tuning of the hyperparameters is done in a moving window approach using 251 return ob-servations, which corresponds to approximately one trading year. The hyperparameters are tuned forevery combination of index and quantile separately. For a given set of hyperparameters the modelis trained based on the returns of the last trading year to predict the next day’s VaR. Then, thewindow is shifted by one trading day. The tuning period reaches from 2006-07-01 until 2011-06-30,where 2006-07-01 marks the date of the first VaR prediction. The considered parameter values in thegrid for SVR are – C ∈ { − , − , . . . , } – ψ ∈ { , . , . . . , . } where (cid:15) = Q u scale ( ψ ) – γ ∈ { − , − , . . . , } .Note that the second point indicates that tuning is not done over fixed values of (cid:15) . Instead, in everystep of the estimation, (cid:15) is determined based on the ψ -quantile of the squared scaled disturbances of themean model. This corresponds to the squared scaled returns because we assume zero mean of r t . Themotivation behind this is the tendency of returns to form volatility clusters. Hence, a fixed (cid:15) can leadto good results in one volatility regime but might have a poor performance after a regime change. Forinstance, in the case of a financial crisis, the right tail of the distribution of past volatilities gets thicker.Hence, an (cid:15) that depends on the quantile of the distribution will increase such that large volatilitieshave automatically a higher influence on the estimated parameters from the SVR optimization. Byusing squared values it is ensured that only positive values are obtained for (cid:15) . However, notice thatthe distribution of the scaled squared disturbances, which are used in the SVR training, is shifted tothe left of the distribution of the squared scaled disturbances. Hence, it is possible that for high valuesof ψ no observations are outside the (cid:15) -tube such that the model cannot be estimated.The parameter settings that resulted in the best models for the SVR-GARCH-KDE hybrid duringthe tuning period are shown in Table 3. It can be seen that especially for ψ only a certain range appearsamong the best models. Moreover, the optimal ψ tends to be higher for lower quantiles. Based on theobtained parameters, VaR forecasts are produced from 2011-07-01 until 2016-06-30. The results are presented for each quantile separately at the end of the section. In the Tables 5, 6 and7 are the results for the one-day-ahead forecast. The results for the ten-days-ahead forecasts can befound in the Tables 8, 9 and 10. To clearly identify the best performing models, every table is sorted indescending order for every index according to the p -value of the conditional coverage test. The p -valueof the conditional coverage test is used for sorting to focus rather on statistical properties than purelosses. The abbreviations NORM, STD and SSTD indicate the normal, t - and skewed t -distribution,respectively. Additionally, the column headers UC, ID, CC and SPA refer to the corresponding p -valueof the test for unconditional coverage, independence of violations, conditional coverage and superiorpredictive ability. C ψ γ Index Quantile Violations UC ID CC
10 0.7 0.1 S&P500 1.0 0.95 86.62 63.08 87.8410 0.7 0.01 S&P500 2.5 2.46 93.15 96.55 96.2010 0.6 0.001 S&P500 5.0 5.00 99.48 99.57 99.57100 0.8 0.01 Nikkei225 1.0 0.82 50.63 68.48 73.85100 0.7 0.01 Nikkei225 2.5 2.70 66.43 57.80 52.61100 0.7 0.10 Nikkei225 5.0 4.49 40.84 94.38 67.06100 0.7 0.01 EuroStoxx50 1.0 1.02 93.28 60.41 87.110.1 0.6 0.001 EuroStoxx50 2.5 2.52 96.42 97.76 97.6610000 0.6 10 EuroStoxx50 5.0 4.96 94.86 99.71 99.50
Table 3
The best models in the tuning period according to the p-value of the test for conditional coverage. UC, IDand CC indicate the p-value of the corresponding LR test. All values in the columns Quantile, Violations, UC, IDand CC are given in percent.
One-Step-Ahead Forecast Model Evaluation for α = 0 .
01 The SVR-GARCH-KDE hybrid is the bestmodel for the Euro STOXX 50 for α = 0 .
01. A visualization of its performance is given in Figure 2.Here, the SVR-GARCH-KDE hybrid estimates in general higher values for VaR than the other modelsand exhibits more variability. For the S&P 500 and Nikkei 225 the SVR-GARCH-KDE hybrid out-performs all models that are coupled with a normal distribution. However, all models using a skewed t -distribution perform better. Especially for the Nikkei 225 this is caused by having low unconditionalcoverage due to risk overestimation. However, since risk overestimation causes less losses the SPA test p -value is highest. In general, the models coupled with a normal distribution perform poorly for allindices. This comes as no surprise since the distribution of asset returns is usually leptokurtic. One-Step-Ahead Forecast Model Evaluation for α = 0 .
025 The performance of the SVR-GARCH-KDEhybrid at the 2.5% level is not as good as for α = 0 .
01 for the Euro STOXX 50. We observe the lowest p -value in the test for independence of violations but the third best regarding the test for unconditionalcoverage. Interestingly, although the TGARCH and EGARCH model account for asymmetries involatility, the former is the best and the latter the worst variance model. A relatively high riskoverestimation for the S&P 500 and Nikkei 225 causes the SVR-GARCH-KDE hybrid to be on thesixth and ninth rank, respectively. As seen for α = 0 .
01 this leads, however, to high p -values in theSPA test due to lower losses. The performance for the S&P 500 can be seen in Figure 3. One-Step-Ahead Forecast Model Evaluation for α = 0 .
05 The best performance of the SVR-GARCH-KDE hybrid for α = 0 .
05 is rank two for the Nikkei 225. Here, it is only beaten by the EGARCH-SSTDmodel. A visualization is given in Figure 4. For the other indices the SVR-GARCH-KDE hybrid rankson place five. This is caused by having relatively low p -values for the ID test. In terms of UC, theSVR-GARCH-KDE hybrid is the second and third best model for the S&P 500 and Nikkei 225, respectively. In comparison to the results for α = 0 .
01, the models with a normal distribution showa better performance for α ∈ { . , . } . However, using the skewed t -distribution leads also for α ∈ { . , . } to the best rankings. Ten-Steps-Ahead Forecast Model Evaluation for α = 0 .
01 As for the one-step-ahead forecasts, the SVR-GARCH-KDE hybrid is also the best model for the Euro STOXX 50. For the S&P 500 and Nikkei225 the SVR-GARCH-KDE hybrid is the second and third best model, respectively. For all indices,it is the only model not underestimating the risk. Especially the EGARCH models perform poorly.A visualization is given in Figure 5. In contrast to the one-step-ahead forecasts, the models using a ata Driven Value-at-Risk Forecasting using a SVR-GARCH-KDE Hybrid (cid:63) normal distribution are not shown but the EGARCH models due to their bad performance for ten-days-ahead forecasts. With respect to the SPA test, the SVR-GARCH-KDE hybrid causes less severelosses than any other model for all levels of the ten-steps-ahead forecast horizon. Ten-Steps-Ahead Forecast Model Evaluation for α = 0 .
025 Overall, the SVR-GARCH-KDE hybrid per-forms worse at the 2.5% than for the 1% level. In case of the the S&P 500 and Nikkei 225 this ismainly driven by overestimating the risk leading to low p -values in the UC test. In contrast, eventhough the SVR-GARCH-KDE hybrid has the highest p -value in the UC test for the EURO STOXX50, it is relatively low for the ID test. An exemplary visualization is given for the EURO STOXX 50in Figure 6. Ten-Steps-Ahead Forecast Model Evaluation for α = 0 .
05 The performance of the SVR-GARCH-KDEhybrid for α = 0 .
05 is mixed. While it is by far the best model for the EURO STOXX 50, it achievesonly rank seven and nine for the Nikkei 225 and S&P 500, respectively. As for α = 0 .
025 this is mainlydriven by risk overestimation and low p -values for the UC test. Similar to the one-day-ahead forecasts,the models using a normal distribution perform better for α = 0 .
05 than for the other quantiles. TheTGARCH-NORM model is even the best for the S&P 500. A visualization is shown in Figure 7.
Evaluation Summary
Summarizing the results observed across all indices, quantiles and the two fore-cast horizons, we conclude that the SVR-GARCH-KDE hybrid displays a competitive performance.This can be seen in Table 4 where the mean ranks per index and quantile are presented for each modelwith respect to the CC and SPA test as well as the two forecast horizons. The SVR-GARCH-KDEhybrid is overall and for the CC test p -values the third best model for both forecast horizons. Re-garding the SPA test it has rank two for the one-day-ahead forecast horizon and is the best modelfor the ten-days-ahead forecast horizon. Benchmark models coupled with a normal distribution areusually outperformed by the SVR-GARCH-KDE hybrid. Consequently, the SVR-GARCH-KDE hy-brid has a relatively high accuracy especially compared to models with a normal distribution. Theimportance of this result stems from the fact that GARCH models with a normal distribution canbe seen as a popular standard approach for modeling VaR or volatility and are, therefore, a naturalbenchmark. Additionally, excepting the case of the ten-days-ahead S&P 500 forecast for α = 0 . t -distribution show in many cases the best performance. For instance, the TGARCHmodel coupled with the skewed t -distribution is with two exceptions always among the top three interms of the CC test. This confirms the results of previous research showing usually skewed returndistributions. Unlike the benchmarks, the SVR-GARCH-KDE hybrid tends to overestimate marketrisk. This might come from the choice of time interval for SVR parameter tuning. In particular, thetuning period covers the financial crisis of 2008 where market risk was extremely high. However, in the context of risk management, risk underestimation is more critical than risk overestimation because itcan lead to bankruptcy in the short term. For instance, assume a hypothetical situation with the goalto forecast the 5% VaR, where the SVR-GARCH-KDE hybrid and a benchmark model have the same p -value regarding the independence test. Additionally, assume the p -value of the benchmark in thetest of conditional coverage is higher because it has an unconditional coverage of 5.5% whereas that ofthe SVR-GARCH-KDE hybrid is 4%. The 1% overestimation of the SVR-GARCH-KDE hybrid workslike a buffer for model risk since all estimation techniques exhibit statistical uncertainty. Hence, theuse of the SVR-GARCH-KDE hybrid may be still more appealing than the use of benchmark modelsthat tend to underestimate risk. Model Overall CC 1 Day SPA 1 Day CC 10 Days SPA 10 Days
TGARCH-SSTD 2.3 2.2 1.4 2.3 3.2GARCH-SSTD 2.6 2.3 2.6 2.7 2.7SVR-GARCH-KDE 3.2 4.9 2.2 4.8 1.0GARCH-STD 5.9 6.1 6.4 4.9 6.3TGARCH-NORM 6.0 6.0 7.7 5.1 5.1GARCH-NORM 6.6 6.1 6.8 6.3 7.3TGARCH-STD 6.7 5.3 9.0 5.1 7.4EGARCH-SSTD 6.9 5.0 5.3 9.4 7.9EGARCH-NORM 8.6 8.7 7.3 9.3 9.0EGARCH-STD 8.6 8.8 7.2 9.6 8.8
Table 4
The models are presented with their mean rank. The mean rank was computed per index and quantileusing the ties method max in the frankv function of the R package data.table. The lower the mean rank, the betterthe model.ata Driven Value-at-Risk Forecasting using a SVR-GARCH-KDE Hybrid (cid:63) Model Index Violations SPA UC ID CC
EGARCH-SSTD S&P500 1.11 4.99 69.27 57.46 79.01TGARCH-SSTD S&P500 0.72 92.22 28.52 71.87 52.94GARCH-SSTD S&P500 1.27 20.76 35.24 43.34 28.13TGARCH-STD S&P500 1.43 0.20 14.91 46.97 27.20SVR-GARCH-KDE S&P500 0.79 29.34 44.83 18.28 13.71GARCH-STD S&P500 1.67 16.17 2.95 65.77 6.15EGARCH-STD S&P500 1.75 5.79 1.58 37.62 3.67TGARCH-NORM S&P500 2.15 3.39 0.04 27.64 0.10EGARCH-NORM S&P500 2.23 2.40 0.02 25.88 0.04GARCH-NORM S&P500 2.38 0.20 0.00 94.85 0.01TGARCH-SSTD Nikkei225 1.21 4.99 47.61 54.48 64.58GARCH-SSTD Nikkei225 1.21 2.79 47.61 54.48 64.58EGARCH-SSTD Nikkei225 1.37 4.19 21.61 49.21 36.76GARCH-STD Nikkei225 1.69 1.80 2.60 66.33 5.56TGARCH-STD Nikkei225 1.77 1.00 1.37 37.30 3.23SVR-GARCH-KDE Nikkei225 0.24 59.48 0.13 90.41 0.55EGARCH-STD Nikkei225 2.09 0.80 0.07 29.17 0.19GARCH-NORM Nikkei225 2.17 0.40 0.03 88.16 0.14TGARCH-NORM Nikkei225 2.33 0.20 0.01 93.19 0.03EGARCH-NORM Nikkei225 2.42 0.20 0.00 95.21 0.01SVR-GARCH-KDE EuroStoxx50 1.53 76.25 8.17 44.23 16.36GARCH-SSTD EuroStoxx50 1.69 72.85 2.60 39.54 5.84TGARCH-SSTD EuroStoxx50 1.69 43.51 2.60 66.33 5.56TGARCH-STD EuroStoxx50 1.77 1.60 1.37 70.53 3.39GARCH-STD EuroStoxx50 1.85 1.80 0.70 35.15 1.70TGARCH-NORM EuroStoxx50 2.09 0.20 0.07 85.17 0.28GARCH-NORM EuroStoxx50 2.17 6.19 0.03 27.33 0.08EGARCH-STD EuroStoxx50 2.33 9.38 0.01 93.19 0.03EGARCH-SSTD EuroStoxx50 2.42 4.79 0.00 95.21 0.01EGARCH-NORM EuroStoxx50 2.50 3.19 0.00 96.87 0.00
Table 5
Results for one-day-ahead VaR forecasts from July 1, 2011 to June 30, 2016 for α = 0 .
01. UC, ID, CC andSPA indicate the p-value of the corresponding test. The results are in descending order with respect to CC for eachindex. All values in the columns Quantile, Violations, UC, ID, CC and SPA are given in percent.0 Marius Lux et al.
Model Index Violations SPA UC ID CC
GARCH-SSTD S&P500 2.62 34.33 78.12 98.97 95.22EGARCH-SSTD S&P500 3.02 1.40 25.18 98.94 51.31TGARCH-SSTD S&P500 2.38 80.44 79.19 22.60 46.40GARCH-STD S&P500 3.50 0.80 3.24 93.45 9.47GARCH-NORM S&P500 3.82 1.20 0.55 99.20 2.10SVR-GARCH-KDE S&P500 1.43 34.53 0.83 52.45 1.62TGARCH-STD S&P500 3.90 0.00 0.33 75.52 1.02TGARCH-NORM S&P500 4.05 1.60 0.12 69.37 0.36EGARCH-STD S&P500 4.37 0.00 0.01 96.11 0.06EGARCH-NORM S&P500 4.53 0.20 0.00 92.48 0.02GARCH-SSTD Nikkei225 2.90 43.91 38.01 99.90 67.96TGARCH-SSTD Nikkei225 2.42 71.66 84.78 22.29 46.72TGARCH-STD Nikkei225 3.14 0.40 16.44 97.67 37.15TGARCH-NORM Nikkei225 3.22 1.60 11.91 96.33 28.60EGARCH-NORM Nikkei225 3.22 0.40 11.91 96.33 28.60GARCH-NORM Nikkei225 3.38 0.00 5.87 92.85 15.55GARCH-STD Nikkei225 3.38 0.00 5.87 89.33 14.96EGARCH-SSTD Nikkei225 3.06 13.97 22.21 12.14 14.30SVR-GARCH-KDE Nikkei225 1.69 73.65 5.26 14.27 2.18EGARCH-STD Nikkei225 3.86 0.20 0.43 99.42 1.68TGARCH-SSTD EuroStoxx50 2.98 79.84 29.36 72.90 42.00TGARCH-NORM EuroStoxx50 3.22 0.00 11.91 83.46 24.78TGARCH-STD EuroStoxx50 3.38 0.00 5.87 89.33 14.96GARCH-SSTD EuroStoxx50 3.46 68.86 3.99 91.84 11.13SVR-GARCH-KDE EuroStoxx50 3.14 29.74 16.44 11.43 4.35GARCH-STD EuroStoxx50 3.70 5.19 1.11 97.42 3.88GARCH-NORM EuroStoxx50 3.95 5.59 0.26 75.92 0.81EGARCH-STD EuroStoxx50 4.27 1.20 0.03 88.73 0.12EGARCH-SSTD EuroStoxx50 4.51 1.40 0.00 95.50 0.02EGARCH-NORM EuroStoxx50 4.59 1.60 0.00 97.11 0.01
Table 6
Results for one-day-ahead VaR forecasts from July 1, 2011 to June 30, 2016 for α = 0 . (cid:63) Model Index Violations SPA UC ID CC
GARCH-SSTD S&P500 5.01 57.49 98.97 89.07 89.06GARCH-NORM S&P500 5.72 0.00 24.94 99.79 51.40TGARCH-SSTD S&P500 5.41 87.82 51.47 60.38 48.83GARCH-STD S&P500 5.80 2.20 20.21 99.23 43.99SVR-GARCH-KDE S&P500 4.37 24.95 29.68 30.25 17.55EGARCH-SSTD S&P500 6.44 1.20 2.47 83.92 6.73TGARCH-NORM S&P500 6.60 0.40 1.30 77.88 3.56EGARCH-NORM S&P500 6.76 1.00 0.65 94.35 2.33TGARCH-STD S&P500 6.76 0.00 0.65 71.36 1.77EGARCH-STD S&P500 7.00 0.80 0.21 87.46 0.78EGARCH-SSTD Nikkei225 5.23 15.17 70.78 97.26 90.66SVR-GARCH-KDE Nikkei225 4.59 27.74 50.10 97.11 77.43TGARCH-SSTD Nikkei225 4.67 94.81 58.95 89.41 77.30GARCH-SSTD Nikkei225 5.07 34.93 90.69 75.29 74.78TGARCH-NORM Nikkei225 5.23 0.20 70.78 68.85 64.18TGARCH-STD Nikkei225 5.56 0.20 37.71 89.62 60.67GARCH-NORM Nikkei225 5.88 4.99 16.68 78.41 30.16EGARCH-NORM Nikkei225 5.72 0.60 25.68 49.18 25.85GARCH-STD Nikkei225 6.04 0.00 10.34 71.98 19.11EGARCH-STD Nikkei225 6.04 0.60 10.34 71.98 19.11TGARCH-SSTD EuroStoxx50 5.15 89.62 80.55 65.27 63.32GARCH-SSTD EuroStoxx50 5.96 57.29 13.21 95.83 30.84GARCH-NORM EuroStoxx50 5.96 48.90 13.21 95.83 30.84TGARCH-NORM EuroStoxx50 6.04 2.59 10.34 78.10 20.73SVR-GARCH-KDE EuroStoxx50 5.48 33.53 44.90 26.86 20.17EGARCH-SSTD EuroStoxx50 6.12 1.40 7.99 68.65 14.82TGARCH-STD EuroStoxx50 6.20 0.00 6.11 59.29 10.26EGARCH-NORM EuroStoxx50 6.52 1.80 1.85 99.10 6.18GARCH-STD EuroStoxx50 6.60 3.99 1.33 96.58 4.52EGARCH-STD EuroStoxx50 6.84 0.80 0.46 69.51 1.27
Table 7
Results for one-day-ahead VaR forecasts from July 1, 2011 to June 30, 2016 for α = 0 .
05. UC, ID, CC andSPA indicate the p-value of the corresponding test. The results are in descending order with respect to CC for eachindex. All values in the columns Quantile, Violations, UC, ID, CC and SPA are given in percent.2 Marius Lux et al.
Model Index Violations SPA UC ID CC
TGARCH-SSTD S&P500 1.03 11.78 90.58 30.09 29.88SVR-GARCH-KDE S&P500 0.95 94.01 86.85 25.94 25.59GARCH-SSTD S&P500 1.19 10.38 50.57 3.91 3.14TGARCH-STD S&P500 1.99 1.80 0.19 25.26 0.21GARCH-STD S&P500 1.99 2.99 0.19 25.26 0.21TGARCH-NORM S&P500 2.31 3.59 0.01 39.72 0.01GARCH-NORM S&P500 2.54 3.39 0.00 51.83 0.00EGARCH-STD S&P500 2.78 0.00 0.00 64.14 0.00EGARCH-SSTD S&P500 2.70 0.80 0.00 4.50 0.00EGARCH-NORM S&P500 3.26 0.60 0.00 14.87 0.00TGARCH-SSTD Nikkei225 1.45 3.19 13.59 52.97 17.42GARCH-SSTD Nikkei225 1.45 2.40 13.59 52.97 17.42SVR-GARCH-KDE Nikkei225 0.32 57.29 0.51 87.23 1.97GARCH-STD Nikkei225 1.85 0.80 0.70 19.57 0.51TGARCH-STD Nikkei225 1.93 1.80 0.34 22.56 0.31GARCH-NORM Nikkei225 2.01 1.80 0.16 25.77 0.18TGARCH-NORM Nikkei225 2.09 2.20 0.07 5.45 0.02EGARCH-STD Nikkei225 3.95 0.20 0.00 75.92 0.00EGARCH-SSTD Nikkei225 2.50 1.20 0.00 48.53 0.00EGARCH-NORM Nikkei225 2.58 0.80 0.00 15.65 0.00SVR-GARCH-KDE EuroStoxx50 0.97 64.07 90.41 62.85 88.31GARCH-SSTD EuroStoxx50 2.01 8.78 0.16 25.77 0.18TGARCH-SSTD EuroStoxx50 2.33 5.99 0.01 40.43 0.01TGARCH-STD EuroStoxx50 2.58 4.99 0.00 15.65 0.00TGARCH-NORM EuroStoxx50 2.66 4.79 0.00 18.07 0.00GARCH-STD EuroStoxx50 2.33 5.79 0.01 9.64 0.00GARCH-NORM EuroStoxx50 2.58 0.80 0.00 15.65 0.00EGARCH-STD EuroStoxx50 4.03 1.40 0.00 17.32 0.00EGARCH-SSTD EuroStoxx50 3.70 1.80 0.00 2.49 0.00EGARCH-NORM EuroStoxx50 3.30 0.20 0.00 44.47 0.00
Table 8
Results for ten-days-ahead VaR forecasts from July 1, 2011 to June 30, 2016 for α = 0 .
01. UC, ID, CC andSPA indicate the p-value of the corresponding test. The results are in descending order with respect to CC for eachindex. All values in the columns Quantile, Violations, UC, ID, CC and SPA are given in percent.ata Driven Value-at-Risk Forecasting using a SVR-GARCH-KDE Hybrid (cid:63) Model Index Violations SPA UC ID CC
TGARCH-SSTD S&P500 2.62 13.57 78.12 55.96 53.84GARCH-SSTD S&P500 2.78 6.99 52.88 22.94 18.81GARCH-STD S&P500 3.18 3.39 13.82 39.68 13.22TGARCH-STD S&P500 3.42 0.20 4.80 51.32 7.27GARCH-NORM S&P500 3.58 2.79 2.14 59.33 4.20SVR-GARCH-KDE S&P500 1.75 95.21 7.15 16.45 3.24TGARCH-NORM S&P500 3.58 0.40 2.14 25.03 1.77EGARCH-STD S&P500 4.37 0.00 0.01 30.25 0.02EGARCH-SSTD S&P500 4.77 0.20 0.00 22.74 0.00EGARCH-NORM S&P500 4.61 0.20 0.00 17.75 0.00GARCH-STD Nikkei225 2.74 0.20 59.74 60.95 53.02GARCH-SSTD Nikkei225 2.25 8.58 57.31 36.53 31.17GARCH-NORM Nikkei225 3.30 0.00 8.45 86.53 19.54TGARCH-SSTD Nikkei225 2.74 0.00 59.74 20.72 18.02TGARCH-STD Nikkei225 3.06 0.00 22.21 33.30 15.80TGARCH-NORM Nikkei225 3.46 0.00 3.99 52.40 6.35EGARCH-SSTD Nikkei225 4.03 0.40 0.15 43.03 0.28EGARCH-NORM Nikkei225 4.11 0.00 0.09 46.87 0.19SVR-GARCH-KDE Nikkei225 1.29 48.10 0.26 5.17 0.06EGARCH-STD Nikkei225 5.31 0.20 0.00 96.40 0.00TGARCH-SSTD EuroStoxx50 3.22 50.30 11.91 40.62 12.06GARCH-SSTD EuroStoxx50 3.30 50.10 8.45 44.47 10.04TGARCH-NORM EuroStoxx50 3.38 9.38 5.87 48.40 8.11GARCH-STD EuroStoxx50 3.46 8.78 3.99 52.40 6.35SVR-GARCH-KDE EuroStoxx50 3.06 85.03 22.21 9.73 4.61TGARCH-STD EuroStoxx50 3.46 13.77 3.99 20.23 2.45GARCH-NORM EuroStoxx50 3.78 0.20 0.70 68.40 1.80EGARCH-STD EuroStoxx50 5.88 0.40 0.00 43.78 0.00EGARCH-SSTD EuroStoxx50 4.91 0.20 0.00 3.67 0.00EGARCH-NORM EuroStoxx50 4.83 0.00 0.00 23.77 0.00
Table 9
Results for ten-days-ahead VaR forecasts from July 1, 2011 to June 30, 2016 for α = 0 . Model Index Violations SPA UC ID CC
TGARCH-NORM S&P500 5.09 1.20 88.72 34.98 34.63TGARCH-STD S&P500 5.01 0.00 98.97 31.66 31.66GARCH-SSTD S&P500 4.29 2.20 23.85 57.38 28.64GARCH-STD S&P500 5.17 0.00 78.70 17.88 17.24TGARCH-SSTD S&P500 4.21 4.39 18.86 24.22 10.20GARCH-NORM S&P500 5.48 0.00 43.69 4.89 3.62EGARCH-NORM S&P500 6.52 0.00 1.80 29.69 1.81EGARCH-STD S&P500 6.60 0.00 1.30 32.87 1.50SVR-GARCH-KDE S&P500 3.34 56.89 0.41 17.11 0.28EGARCH-SSTD S&P500 7.23 0.00 0.06 62.62 0.18TGARCH-SSTD Nikkei225 5.23 10.18 70.78 94.63 88.21GARCH-SSTD Nikkei225 4.99 21.16 98.96 87.41 87.40TGARCH-STD Nikkei225 5.31 8.18 61.51 72.91 64.25TGARCH-NORM Nikkei225 5.31 12.18 61.51 72.91 64.25GARCH-NORM Nikkei225 5.72 5.99 25.68 99.95 52.55GARCH-STD Nikkei225 5.72 8.18 25.68 89.19 46.89SVR-GARCH-KDE Nikkei225 3.95 88.82 7.70 75.92 15.90EGARCH-SSTD Nikkei225 6.36 0.60 3.44 90.35 9.65EGARCH-NORM Nikkei225 6.12 3.59 7.99 31.60 6.82EGARCH-STD Nikkei225 7.97 0.00 0.00 91.62 0.00SVR-GARCH-KDE EuroStoxx50 5.07 50.10 90.69 61.34 60.92TGARCH-NORM EuroStoxx50 6.12 2.59 7.99 6.55 1.41TGARCH-SSTD EuroStoxx50 6.12 4.99 7.99 2.43 0.53TGARCH-STD EuroStoxx50 6.36 0.00 3.44 4.28 0.46GARCH-NORM EuroStoxx50 6.36 2.20 3.44 4.28 0.46GARCH-SSTD EuroStoxx50 6.52 2.99 1.85 6.06 0.38GARCH-STD EuroStoxx50 6.76 0.40 0.67 9.76 0.25EGARCH-STD EuroStoxx50 9.10 0.20 0.00 1.03 0.00EGARCH-SSTD EuroStoxx50 7.41 0.20 0.03 0.13 0.00EGARCH-NORM EuroStoxx50 7.17 0.60 0.10 0.18 0.00
Table 10
Results for ten-days-ahead VaR forecasts from July 1, 2011 to June 30, 2016 for α = 0 .
05. UC, ID, CCand SPA indicate the p-value of the corresponding test. The results are in descending order with respect to CC foreach index. All values in the columns Quantile, Violations, UC, ID, CC and SPA are given in percent.ata Driven Value-at-Risk Forecasting using a SVR-GARCH-KDE Hybrid (cid:63) lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll Jul 01 2011 Jan 02 2013 Jul 01 2014 Jan 05 2016 − − − − − − − SVR−GARCH−KDE Hybrid R e t u r n i n P e r c en t llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll l l l l l ll ll l l l l l Returns−VaRViolations ll Returns−VaRViolations lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll
Jul 01 2011 Jan 02 2013 Jul 01 2014 Jan 05 2016 − − − − − − − TGARCH−SSTD R e t u r n i n P e r c en t lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll l l l ll l l ll ll l l l lll Returns−VaRViolations ll Returns−VaRViolations llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll
Jul 01 2011 Jan 02 2013 Jul 01 2014 Jan 05 2016 − − − − − − − GARCH−SSTD R e t u r n i n P e r c en t lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll l l l l ll ll l l l l ll Returns−VaRViolations ll Returns−VaRViolations llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll
Jul 01 2011 Jan 02 2013 Jul 01 2014 Jan 05 2016 − − − − − − − TGARCH−NORM R e t u r n i n P e r c en t llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll l l l l ll l l ll ll lll lll l lll Returns−VaRViolations ll Returns−VaRViolations llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll
Jul 01 2011 Jan 02 2013 Jul 01 2014 Jan 05 2016 − − − − − − − GARCH−NORM R e t u r n i n P e r c en t lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll l ll l l l ll ll ll l lll l lll Returns−VaRViolations ll Returns−VaRViolations llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll
Jul 01 2011 Jan 02 2013 Jul 01 2014 Jan 05 2016 − − − − − − − EGARCH−NORM R e t u r n i n P e r c en t lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll l l l ll l lll lllllll l lll l lll Returns−VaRViolations ll Returns−VaRViolations
Fig. 2
VaR one-day-ahead forecast model comparison for the Euro Stoxx 50 at α = 0 .
01 in the period from July 1,2011 to June 30, 2016. The SVR-GARCH-KDE hybrid is compared to models that are overall on average better andto the models using a normal distribution.6 Marius Lux et al. llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll
Jul 01 2011 Jan 02 2013 Jul 01 2014 Jan 04 2016 − − − − − − SVR−GARCH−KDE Hybrid R e t u r n i n P e r c en t lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll l l ll l l l l ll l Returns−VaRViolations ll Returns−VaRViolations llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll
Jul 01 2011 Jan 02 2013 Jul 01 2014 Jan 04 2016 − − − − − − TGARCH−SSTD R e t u r n i n P e r c en t llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll l l ll lll lll l l ll l l ll ll l ll ll l Returns−VaRViolations ll Returns−VaRViolations llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll
Jul 01 2011 Jan 02 2013 Jul 01 2014 Jan 04 2016 − − − − − − GARCH−SSTD R e t u r n i n P e r c en t lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll l l ll ll l l l ll l l l ll lll lllll lll l Returns−VaRViolations ll Returns−VaRViolations llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll
Jul 01 2011 Jan 02 2013 Jul 01 2014 Jan 04 2016 − − − − − − TGARCH−NORM R e t u r n i n P e r c en t llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll l ll lll lll l l ll l llllll ll lll ll llllllllll l ll l Returns−VaRViolations ll Returns−VaRViolations llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll
Jul 01 2011 Jan 02 2013 Jul 01 2014 Jan 04 2016 − − − − − − GARCH−NORM R e t u r n i n P e r c en t llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll l lllll ll l lll l ll l ll llll lll llllllllllllllll l Returns−VaRViolations ll Returns−VaRViolations llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll
Jul 01 2011 Jan 02 2013 Jul 01 2014 Jan 04 2016 − − − − − − EGARCH−NORM R e t u r n i n P e r c en t lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll l ll lll llllllll l lll ll ll ll llllll lllllll ll llll l Returns−VaRViolations ll Returns−VaRViolations
Fig. 3
VaR one-day-ahead forecast model comparison for the S&P 500 at α = 0 .
025 in the period from July 1, 2011to June 30, 2016. The SVR-GARCH-KDE hybrid is compared to models that are overall on average better and tothe models using a normal distribution.ata Driven Value-at-Risk Forecasting using a SVR-GARCH-KDE Hybrid (cid:63) lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll Jul 01 2011 Jan 04 2013 Jul 01 2014 Jan 04 2016 − − − − − − SVR−GARCH−KDE Hybrid R e t u r n i n P e r c en t llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll l lllllllllll llll llllllll l l ll l lllllll l llllllllll lllll llll Returns−VaRViolations ll Returns−VaRViolations lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll
Jul 01 2011 Jan 04 2013 Jul 01 2014 Jan 04 2016 − − − − − − TGARCH−SSTD R e t u r n i n P e r c en t lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll l lllll lll lllll llllllll l lllll l l llll lllllllll llllll lllllll Returns−VaRViolations ll Returns−VaRViolations llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll
Jul 01 2011 Jan 04 2013 Jul 01 2014 Jan 04 2016 − − − − − − GARCH−SSTD R e t u r n i n P e r c en t llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll ll llllll ll lllll llllll l llllll l l lllllll llllllllllllllll llllll Returns−VaRViolations ll Returns−VaRViolations llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll
Jul 01 2011 Jan 04 2013 Jul 01 2014 Jan 04 2016 − − − − − − TGARCH−NORM R e t u r n i n P e r c en t llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll lllll lll llll llllllll ll lllll l l lllll lllllllllllllllll llllllll Returns−VaRViolations ll Returns−VaRViolations llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll
Jul 01 2011 Jan 04 2013 Jul 01 2014 Jan 04 2016 − − − − − − GARCH−NORM R e t u r n i n P e r c en t llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll lllllllll llll llllll ll llllllll l lllllll lllllllllllllllllllll lllllll Returns−VaRViolations ll Returns−VaRViolations llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll
Jul 01 2011 Jan 04 2013 Jul 01 2014 Jan 04 2016 − − − − − − EGARCH−NORM R e t u r n i n P e r c en t llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll lll llll lll llllll llllllll lllllllllll l l lllll llllllllllllllllll lllllll Returns−VaRViolations ll Returns−VaRViolations
Fig. 4
VaR one-day-ahead forecast model comparison for the Nikkei 225 at α = 0 .
05 in the period from July 1, 2011to June 30, 2016. The SVR-GARCH-KDE hybrid is compared to models that are overall on average better and tothe models using a normal distribution.8 Marius Lux et al. lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll
Jul 01 2011 Jan 04 2013 Jul 01 2014 Jan 04 2016 − − − − − − SVR−GARCH−KDE Hybrid R e t u r n i n P e r c en t lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll l l Returns−VaRViolations ll Returns−VaRViolations lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll
Jul 01 2011 Jan 04 2013 Jul 01 2014 Jan 04 2016 − − − − − − TGARCH−SSTD R e t u r n i n P e r c en t llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll ll l lll l l lll ll llll Returns−VaRViolations ll Returns−VaRViolations llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll
Jul 01 2011 Jan 04 2013 Jul 01 2014 Jan 04 2016 − − − − − − GARCH−SSTD R e t u r n i n P e r c en t lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll ll l lllll l l l ll lll l Returns−VaRViolations ll Returns−VaRViolations llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll
Jul 01 2011 Jan 04 2013 Jul 01 2014 Jan 04 2016 − − − − − − EGARCH−SSTD R e t u r n i n P e r c en t lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll lllll l ll ll lllllll l l l lll ll lllll Returns−VaRViolations ll Returns−VaRViolations llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll
Jul 01 2011 Jan 04 2013 Jul 01 2014 Jan 04 2016 − − − − − − EGARCH−NORM R e t u r n i n P e r c en t lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll l ll ll llll l l lll l lllllll llll lllll Returns−VaRViolations ll Returns−VaRViolations llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll
Jul 01 2011 Jan 04 2013 Jul 01 2014 Jan 04 2016 − − − − − − EGARCH−STD R e t u r n i n P e r c en t llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll l l ll l llllllllll ll lllllll l lll l llllll llllllllllll Returns−VaRViolations ll Returns−VaRViolations
Fig. 5
VaR ten-days-ahead forecast model comparison for the Nikkei 225 at α = 0 .
01 in the period from July 1,2011 to June 30, 2016. The SVR-GARCH-KDE hybrid is compared to models that are overall on average better andto the EGARCH models.ata Driven Value-at-Risk Forecasting using a SVR-GARCH-KDE Hybrid (cid:63) lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll Jul 01 2011 Jan 02 2013 Jul 01 2014 Jan 05 2016 − − − − − − − SVR−GARCH−KDE Hybrid R e t u r n i n P e r c en t lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll ll l l ll ll lllll l lllll lllll ll Returns−VaRViolations ll Returns−VaRViolations lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll
Jul 01 2011 Jan 02 2013 Jul 01 2014 Jan 05 2016 − − − − − − − TGARCH−SSTD R e t u r n i n P e r c en t llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll llll l l ll l l ll ll lllll l llll l l lll Returns−VaRViolations ll Returns−VaRViolations llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll
Jul 01 2011 Jan 02 2013 Jul 01 2014 Jan 05 2016 − − − − − − − GARCH−SSTD R e t u r n i n P e r c en t lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll lll l ll l l ll lll llll lll llll llll ll Returns−VaRViolations ll Returns−VaRViolations llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll
Jul 01 2011 Jan 02 2013 Jul 01 2014 Jan 05 2016 − − − − − − − EGARCH−SSTD R e t u r n i n P e r c en t lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll lllll l ll ll llll l ll ll llllll l lllllll lllllllll llll Returns−VaRViolations ll Returns−VaRViolations llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll
Jul 01 2011 Jan 02 2013 Jul 01 2014 Jan 05 2016 − − − − − − − EGARCH−NORM R e t u r n i n P e r c en t llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll llll l l ll llll l l ll llllllllllllllll lllll lllll llll Returns−VaRViolations ll Returns−VaRViolations llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll
Jul 01 2011 Jan 02 2013 Jul 01 2014 Jan 05 2016 − − − − − − − EGARCH−STD R e t u r n i n P e r c en t lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll llllll l lll ll llll l l ll lllllllllll lll lllll lllllllllllllllll Returns−VaRViolations ll Returns−VaRViolations
Fig. 6
VaR ten-days-ahead forecast model comparison for the Euro Stoxx 50 at α = 0 .
025 in the period from July1, 2011 to June 30, 2016. The SVR-GARCH-KDE hybrid is compared to models that are overall on average betterand to the EGARCH models.0 Marius Lux et al. llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll
Jul 01 2011 Jan 02 2013 Jul 01 2014 Jan 04 2016 − − − − − − SVR−GARCH−KDE Hybrid R e t u r n i n P e r c en t lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll l l l l ll l l llll lll l lllllll lll l Returns−VaRViolations ll Returns−VaRViolations llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll
Jul 01 2011 Jan 02 2013 Jul 01 2014 Jan 04 2016 − − − − − − TGARCH−SSTD R e t u r n i n P e r c en t llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll ll ll ll ll l l ll ll l ll lllll ll ll lllllllllllll ll Returns−VaRViolations ll Returns−VaRViolations llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll
Jul 01 2011 Jan 02 2013 Jul 01 2014 Jan 04 2016 − − − − − − GARCH−SSTD R e t u r n i n P e r c en t llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll l ll llll l llllll l ll ll ll lllll ll ll lllllll lllll ll Returns−VaRViolations ll Returns−VaRViolations llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll
Jul 01 2011 Jan 02 2013 Jul 01 2014 Jan 04 2016 − − − − − − EGARCH−SSTD R e t u r n i n P e r c en t llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll lllllllllllllllllll ll lllllllll ll l lll l ll lllll lllll ll llllllllllllllllll ll ll Returns−VaRViolations ll Returns−VaRViolations llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll
Jul 01 2011 Jan 02 2013 Jul 01 2014 Jan 04 2016 − − − − − − EGARCH−NORM R e t u r n i n P e r c en t lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll ll ll ll lllllllll ll l llllll ll lllll lllll ll l llllllllllllllllllllll ll ll Returns−VaRViolations ll Returns−VaRViolations llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll
Jul 01 2011 Jan 02 2013 Jul 01 2014 Jan 04 2016 − − − − − − EGARCH−STD R e t u r n i n P e r c en t lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll ll ll ll lllllllll llll llllll ll lllll lllll ll l llllllllllllllllllllll ll ll Returns−VaRViolations ll Returns−VaRViolations
Fig. 7
VaR ten-days-ahead forecast model comparison for the S&P 500 at α = 0 .
05 in the period from July 1, 2011to June 30, 2016. The SVR-GARCH-KDE hybrid is compared to models that are overall on average better and tothe EGARCH models.ata Driven Value-at-Risk Forecasting using a SVR-GARCH-KDE Hybrid (cid:63) In a large-scale empirical comparison Kuester et al (2006) find VaR models belonging to the location-scale class superior to alternative approaches. However, the location-scale models considered in theirstudy are parametric and based on distributional assumptions. Motivated by the potential shortcom-ings of a parametric approach, the paper introduces a nonparametric and nonlinear VaR forecastingframework based on the location-scale class. The mean and volatility model are modeled with SVRin an ARMA and GARCH like fashion, respectively. In addition, the VaR forecast is obtained byestimating the distribution function of the standardized residuals via KDE.To evaluate the performance of the SVR-GARCH-KDE hybrid, VaR is forecasted for three indices:Euro STOXX 50, Nikkei 225 and S&P 500, considering the different quantiles of α ∈ { . , . , . } for forecast horizons of one and ten days. GARCH, EGARCH and TGARCH models coupled with thenormal, t - and skewed t -distribution serve as benchmarks and are compared to the proposed modelusing a LR testing framework for interval forecasts (Christoffersen 1998) and the SPA test of Hansen(2005). Grid search with the goal to maximize the p -value of the LR test for conditional coverage is usedto set the SVR parameters. The SVR-GARCH-KDE hybrid delivers competitive results. For instance,in case of the one-day-ahead forecast horizon it is the best model for the Euro STOXX 50 for α = 0 . t -distribution show the best results. In contrast to the benchmark models, which usually underestimaterisk, the SVR-GARCH-KDE hybrid tends to overestimate risk. This can lead to situations where theSVR-GARCH-KDE hybrid has an average performance regarding the p -value of the test for conditionalcoverage. However, with respect to risk management, the use of the SVR-GARCH-KDE might be stillfavorable over using the benchmarks since all approaches exhibit statistical uncertainty. The tendencyto overestimate risk can, therefore, serve as a model risk buffer. This is supported by the results of theSPA test that evaluates the loss function which penalizes especially large VaR violations. For instance,the proposed model has the best performance for the ten-days-ahead forecast horizon.In general, the competitive results indicate that the proposed SVR-GARCH-KDE hybrid is apromising alternative. Despite fixing and not retuning the hyperparameters for five years, it is amongthe top three models with respect to conditional coverage and loss minimization measured by the SPAtest for a forecast horizon of one as well as ten days. Additionally, it is the best model in minimizingthe loss function for ten-days-ahead forecasts. This indicates the proposed SVR-GARCH-KDE hybridis a robust VaR modeling approach that is capable of capturing complex nonlinear structures inthe volatility process and has the flexibility to model a wide class of tail events. Moreover, furtherimprovements can be expected by refining the tuning routine. However, there exist several ways thatcan lead to an improved performance. First, tuning could be done for more parameters. For instance,in the KDE part of the estimation procedure, the kernel function and bandwidth estimator are setwithout tuning. Hence, considering different kernel functions and more flexible bandwidth estimatorsare potential ways to improve the performance further. Moreover, the kernel in the SVR part is also fixed and could be varied. Second, more recent information could be used in the parameter selectionby re-tuning the model. Here, tuning is done for a block of five years of data. Then, based on theoptimal parameters found for this data block, forecasts for five years are made and the parametersare held fixed. Thus, annual or even shorter re-tuning periods could result in parameters that aremore appropriate for the existing market risk. Additionally, refining the grid can also result in betterparameter choices.In addition to modifying the tuning routine, the SVR-GARCH-KDE hybrid could be improvedby changing the model specification. Overall, the TGARCH model with the skewed t -distributionachieves very good results. Hence, the SVR-GARCH-KDE hybrid could be modified such that italso accounts for asymmetric reactions of the volatility to past returns in a TGARCH like manner. Moreover, the proposed procedure does not ensure that the estimated variances are positive. Here,this problem is handled by replacing non-positive estimates with the last positive. However, positivevariance estimates could be ensured by modeling the logarithm of the squared mean model residualsinstead.All above mentioned adjustments are potential starting points for future research to further im-prove the proposed framework. However, although the suggested modifications of the tuning procedureare reasonable approaches to improve the model performance, they also increase the computationalcomplexity. After all, this is a slight drawback of the SVR-GARCH-KDE hybrid in comparison tostandard models. It is, however offset by potential performance gains. ata Driven Value-at-Risk Forecasting using a SVR-GARCH-KDE Hybrid (cid:63) References
Alexander C (2008) Value-at-Risk Models, Market Risk Analysis, vol 4. Wiley, ChichesterAloui C, Mabrouk S (2010) Value-at-risk estimations of energy commodities via long-memory, asymmetry and fat-tailed GARCH models. Energy Policy 5:2326 – 2339, DOI 10.1016/j.enpol.2009.12.020Bali TG, Mo H, Tang Y (2008) The role of autoregressive conditional skewness and kurtosis in the estimation ofconditional VaR. Journal of Banking & Finance 32(2):269–282, DOI 10.1016/j.jbankfin.2007.03.009Bollerslev T (1986) Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics 31(3):307–327,DOI 10.1016/0304-4076(86)90063-1Borke L, H¨ardle WK (2017) GitHub API based QuantNet mining infrastructure in R (March 6, 2017). SFB 649Discussion Paper (2017-008), DOI 10.2139/ssrn.2927901Borke L, H¨ardle WK (2018) Q3-D3-LSA. In: H¨ardle, Lu, Shen (eds) Handbook of Big data Analytics, SpringerVerlag, ISBN: 978-3-319-18284-1, DOI: 10.1007/978-3-319-18284-1Burges CJ (1998) A tutorial on support vector machines for pattern recognition. Data Mining and KnowledgeDiscovery 2(2):121–167, DOI 10.1023/A:1009715923555Chang CC, Lin CJ (2011) LIBSVM: A library for support vector machines. ACM Transactions on Intelligent Systemsand Technology 2(3):27:1–27:27Chang PC, Wu JL, Lin JJ (2016) A takagi sugeno fuzzy model combined with a support vector regression for stocktrading forecasting. Applied Soft Computing 38:831 – 842, DOI 10.1016/j.asoc.2015.10.030Chen CYH, Chiang TC, H¨ardle WK (2016) Downside risk and stock returns. Humboldt-Universit¨at zu Berlin,Wirtschaftswissenschaftliche Fakult¨at, Submitted to Journal of Banking & Finance, DOI 10.18452/4612Chen Q, Gerlach R, Lu Z (2012) Bayesian value-at-risk and expected shortfall forecasting via the asymmetric Laplacedistribution. Computational Statistics & Data Analysis 56(11):3498 – 3516, DOI 10.1016/j.csda.2010.06.018,1st issue of the Annals of Computational and Financial Econometrics Sixth Special Issue on ComputationalEconometricsChen S, H¨ardle WK, Jeong K (2010) Forecasting volatility with support vector machine-based GARCH model.Journal of Forecasting 29:406–433, DOI 10.1002/for.1134Christoffersen PF (1998) Evaluating interval forecasts. International Economic Review 39(4):841–862, DOI 10.2307/2527341Devi KN, Bhaskaran VM, Kumar GP (2015) Cuckoo optimized SVM for stock market prediction. IEEE Sponsored2nd International Conference on Innovations in Information, Embedded and Communication systems (ICJJECS)DOI 10.1109/ICIIECS.2015.7192906Dunis CL, Laws J, Sermpinis G (2010) Modelling commodity value at risk with higher order neural networks. AppliedFinancial Economics 20(7):585–600, DOI 10.1080/09603100903459873Engle RF (1982) Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdominflation. Econometrica 50(4):987–1007, DOI 10.2307/1912773Fan Y, Zhang YJ, Tsai HT, Wei YM (2008) Estimating value at risk of crude oil price and its spillover effect usingthe GED-GARCH approach. Energy Economics 30(6):3156 – 3171, DOI 10.1016/j.eneco.2008.04.002Franke J, Diagne M (2006) Estimating market risk with neural networks. Statistics & Decisions 24(2):233–253,DOI 10.1524/stnd.2006.24.2.233Franke J, H¨ardle WK, Hafner CM (2015) Statistics of Financial Markets: An Introduction, 4th edn. Universitext,Springer, Berlin, Heidelberg, DOI 10.1007/978-3-642-16521-4Hang Chan N, Deng SJ, Peng L, Xia Z (2007) Interval estimation of value-at-risk based on GARCH models withheavy-tailed innovations. Journal of Econometrics 137(2):556 – 576, DOI 10.1016/j.jeconom.2005.08.008Hansen PR (2005) A test for superior predictive ability. Journal of Business and Economic Statistics 23(4):365 380,DOI 10.1198/073500105000000063H¨ardle W, M¨uller M, Sperlich S, Werwatz A (2004) Nonparametric and Semiparametric Models. Springer Series inStatistics, Springer, Berlin, HeidelbergHartz C, Mittnik S, Paolella M (2006) Accurate value-at-risk forecasting based on the normal-GARCH model.Computational Statistics & Data Analysis 51(4):2295 – 2312, DOI 10.1016/j.csda.2006.09.017Harvey CR, Siddique A (2000) Conditional skewness in asset pricing tests. The Journal of Finance 55(3):1263–1295,DOI 10.1111/0022-1082.00247Hastie T, Tibshirani R, Friedman JH (2009) The elements of statistical learning: Data mining, inference, and pre-diction, 2nd edn. Springer Series in Statistics, Springer, New York NYHuang JJ, Lee KJ, Liang H, Lin WF (2009) Estimating value at risk of portfolio by conditional copula-GARCHmethod. Insurance: Mathematics and Economics 45(3):315 – 324, DOI 10.1016/j.insmatheco.2009.09.009Kazem A, Sharifi E, Hussain FK, Saberi M, Hussain OK (2013) Support vector regression with chaos-based fireflyalgorithm for stock market price forecasting. Applied Soft Computing 13(2):947 – 958, DOI 10.1016/j.asoc.2012.09.0244 Marius Lux et al.Keerthi SS, Lin CJ (2003) Asymptotic behaviors of support vector machines with gaussian kernel. Neural Computa-tion 15(7):1667–1689Khan AI (2011) Modelling daily value-at-risk using realized volatility, non-linear support vector machine and ARCHtype models. Journal for Economics and International Finance 3(5):305–321Khosravi A, Nahavandi S, Creighton D (2013) A neural network-garch-based method for construction of predictionintervals. Electric Power Systems Research 96:185–193, DOI 10.1016/j.epsr.2012.11.007Kim M, Lee S (2016) Nonlinear expectile regression with application to value-at-risk and expected shortfall estimation.Computational Statistics & Data Analysis 94(Supplement C):1–19, DOI 10.1016/j.csda.2015.07.011Kraus D, Czado C (2017) D-vine copula based quantile regression. Computational Statistics & Data Analysis 110(Sup-plement C):1–18, DOI 10.1016/j.csda.2016.12.009Kuester K, Mittnik S, Paolella MS (2006) Value-at-risk prediction: A comparison of alternative strategies. Journalof Financial Econometrics 4(1):53–89, DOI 10.1093/jjfinec/nbj002Lessmann S, Voß S (2017) Car resale price forecasting: The impact of regression method, private information, andheterogeneity on forecast accuracy. International Journal of Forecasting 33(4):864877, DOI 10.1016/j.ijforecast.2017.04.003Lin HT, Lin CJ (2003) A study on sigmoid kernels for svm and the training of non-psd kernels by smo-type methods.Technical report, Department of Computer Science, National Taiwan University URL