Improving Value-at-Risk Estimation from the Normal Egarch Model

Returns in financial assets display consistent excess kurtosis and skewness, implying the presence of large fluctuations not forecasted by Gaussian models. This paper applies a resampling method based on the bootstrap and a bias-correction step to improve Value-at-Risk (VaR) forecasting ability of the n-EGARCH (normal EGARCH) model and correct the VaR for both long and short positions. Our aim is to utilize the advantages of this model, but still use the bootstrap resampling method to accurate for the tendency of the model tomiscalculate the VaR. Empirical results indicate that the bias-correction method can improve the n-GARCH and n-EGARCH VaR forecasts so much that the acquired VaR predictions are different from the proposed probability. Additionally, allowing asymmetry in the conditional variance using the EGARCH model with normal distribution instead of GARCH improves the performance of the bias-correction method in forecasting the VaR for almost all considered indices. Moreover, the bias-corrected n-EGARCH model performs better than the simple t-EGARCH model. Thus, it seems that this model can take account of both the asymmetry in the conditional variance and leptokurtosis in returns distribution. However, we find that the superiority of the bias-corrected n-EGARCH model over the t-EGARCH model is not completely confirmed for short positions based on the censored likelihood scoring rule.


Introduction
Since the Basle Committee (1995; began allowing banks to implement internal VaR models for calculating capital requirements, various methods have been proposed to achieve this purpose. However, the theoretical and computational complexity of these methods has also been raised. Furthermore, various methods have been suggested for modeling conditional variance, and a large number of candidate distributions have been considered for modeling empirical features of the returns (Alexander, 2001;Bao, Lee, & Saltoğlu, 2007). Although more complex shapes of the tails have the potential preference of increased abilities to describe the VaR, they may lead to more uncertainty in the parameters and hence in the VaR estimate itself (Bams, Lehnert, & Wolff, 2005).
The simple method that can consider two characteristics of financial asset returns, namely time-varying volatility and excess kurtosis, is the GARCH model by Engle (1982) and Bollerslev (1986). Researchers beginning with Black (1976) have demonstrated that stock returns are negatively correlated with changes in return volatility. Nevertheless, symmetric models such as the GARCH model have difficulties in correctly modeling the tails of the returns distribution (Giot & Laurent, 2003) due to leverage effects. To overcome this shortcoming, various studies have proposed the inclusion different asymmetric terms in the conditional variance equation (Ding, Granger, & Engle, 1993;Engle & Ng, 1993;Glosten, Jagannathan, & Runkle, 1993). Nelson (1991) also proposed the exponential GARCH model, which was re-expressed by Bollerslev and Mikkelsen (1996).
Hartz‚ Mittnik and Paolella (2006) have developed a resampling method based on the bootstrap and bias-correction for improving the VaR forecasting ability of the n-GARCH (normal GARCH) model. Their proposed method has improved the VaR forecasts of the n-GARCH model. Our main objective is to extend their study by allowing asymmetry in conditional variance. To this end, we apply the n-EGARCH in addition to the n-GARCH model to consider certain theoretical advantages of this model over the n-GARCH. However, Fama (1965) has demonstrated that return distributions of financial instruments are more leptokurtic than normal distributions and tend to be exhibit "fat tails". In addition, empirical studies of high-frequency financial time series demonstrate that the tail behavior of GARCH models remains too short even with standardized Student's t innovations (Tsay, 2005). Therefore, we implement the bias-correction procedure based on the bootstrap method to remove the deficiencies of the n-GARCH and n-EGARCH models with respect to appropriate VaR forecasts. We try to conserve the simplicity of these methods, but still use the bootstrap resampling method to accurate for the tendency of these models to miscalculate the VaR. While Hartz et al. (2006), model the long VaR only, we try to extend their analysis by correcting the VaR for both long and short positions based on the aforementioned GARCH models (n-GARCH and n-EGARCH). Additionally, we evaluate models based on the censored likelihood (csl) scoring rule proposed by Diks, Panchenko, and Van Dijk (2011) in addition to the well-known Christoffersen's LR test. Empirical validation shows that considering asymmetry in conditional variance generally leads to improvements in accurately forecasting oneday-ahead VaR based on the bias-correction method for long and short positions, which is somewhat confirmed by the csl scoring rule.
The rest of the paper is organized as follows. Section 2 explains the methodology for estimating the distribution of the VaR point forecast and how it can be used to improve its accuracy. The empirical analysis and evaluation the performance of competing models in forecasting VaR are presented in Section 3. Finally, Section 4 concludes the paper.

Forecasting VaR
In general, for the set of equally spaced asset returns, t r , t = 1, . . . , T , the class of ARMA(p, q)-EGARCH (r, s) models is given by where j d represents the magnitude effect that indicates how much volatility increases autonomously of the direction of the shock. The j θ is the sign effect.
The log of the conditional variance guarantees that forecasts of the conditional variance are non-negative (Nelson, 1991). Some additional properties of the EGARCH model can be found in Nelson (1991).
For a given return series and a chosen model from the n-ARMA-EGARCH class in (1) and (2), the usual conditional VaR forecast is acquired by estimating the unknown parameter vector 0 1 0 1 1 ( , , , , , , , , , , , , , , ) We also define the set of estimated standardized re- The VaR for a short position is similarly computed where the same definition is used for the right tail of the distribution function, i.e., 1 λ − substitutes for λ .

VaR forecast distribution
For implementing the bias-correction method we need to estimate VaR forecast distribution in addition to a VaR point estimator (ˆ) v λ . Therefore, as Hartz et al. (2006) proposed, we apply the bootstrap method to this end. This method coincides with that described in Pascual, Romo and Ruiz (2006), Reeves (2005) and Trucíos and Hotta (2016). It is connected to the filtered historical simulation method proposed by Barone-Adesi, Giannopoulos and Vosper (1999;2002) and the bootstrap methodology described by Dowd (2005).
Despite that the sampling distribution of the VaR point forecast is unknown and intangible, the bootstrap method provides the possibility for its approximation. Because, assuming the true data generating process is constant over time, the bias caused by the use of the inappropriate but simple n-GARCH model will display certain regularities and thus can be corrected based on a set of past VaR bootstrap distributions (Hartz et al., 2006). This is the assumption used in the bias-correction method described in Section 2.2. Step 0: For a chosen set of values p, q, r, s (for which p = r = s = 1 and q = 0 is most common), obtain QML parameter vector estimate θ , estimated standardized residuals {ˆ} t z , then forecast VaR for a certain h (we consider h=1) by (5).
Step 1: Simulate the ( ' b )th of B, n-ARMA(p, q)-EGARCH(r, s) time series, To eliminate the effect of initial values, we simulate T+  series and then discard first  observations (  =T).
Step 2: Obtain the QML parameter vector estimate Step 3: Estimate a resampled VaR estimate, ( ) using the original series { } t r , and the bootstrap param- where (.) ψ is the indicator function. This function could be used to construct a bootstrap confidence interval for the VaR; for instance Christoffersen and Goncalves (2005) propose to use the bootstrap for constructing confidence intervals of a conditional VaR estimator. Nieto and Ruiz (2010) suggest a new bootstrap procedure to obtain prediction intervals of future VaR and Expected Shortfall (ES), as well. We use the bootstrap method as described in the previous section to obtain a more accurate VaR based on n-GARCH models (n-GARCH and n-EGARCH) as much as possible.

Bias-correcting VaR forecasts
The corrective method uses the VaR distributions (approximated based on the illustrated bootstrapped method) and an objective function, which is described by definition of the VaR. The VaR is defined to be the worst possible loss from an investment over a target horizon and for a given probability level (Crouhy, Galai, & Mark, 2001). Therefore, the evident criterion to construct this function is the observed frequency of exceptions, or past realized returns that are less (higher) than or equal to the predicted VaR for long (short) positions.
For a given probability level λ , the observed frequency of exceptions, denoted f , for a set of successive VaR predictions for long positions obtained from the usual n-GARCH models between times, say, 1 τ and 2 τ , and the equivalent realized returns is given by The observed frequency of exceptions for a short position is similarly computed where the same definition is used for the right tail of the distribution function, i.e., The observed frequency of exceptions is less (higher) than the real risk level λ , if the VaR forecasts calculated by the n-GARCH models tend to overestimate (underestimate) the risk. Therefore, the logic behind the bias-correction method is to find the quantile of past VaR distributions which causes observed frequency of exceptions conforms to (as close as possible) a given risk level.
= be the sorted VaR predictions, with length (B+1) produced by the resampling algorithm with the original n-GARCH models forecast, Calculating the correct quantile of the VaR distribution k F is equivalent to find the largest index b for the long VaR (the smallest for the short VaR), denoted * b , such that for the conforming series ( ) , the observed frequency of exceptions is less than or equal to the given risk level. Therefore, we need a cer- L is the fixed number of preceding VaR forecast distributions that we consider for finding the proper quantile for the h-step-ahead prediction for the downfall risk made at time T. Accordingly, the optimal quantile is described as (10) where * b denotes the greatest quantile of the last L feasible VaR distributions for which the conforming series of VaR predictions, , leads to an observed frequency of VaR exceptions that is equal to (or just smaller) than the given risk level. The * b for a short position is similarly computed where it determines the smallest quantile of the VaR distributions and the [ ] ( ) As the data generating process is not constant over time, calculating an optimal L would be reliable only for particular segment of a particular data set (Hartz et al., 2006). Thus, following the suggestion of Hartz et al.
(2006) concerning the choice of L, we assume two sizes of the moving window (L) to examine whether this criterion could affect the results of our study.
We consider two values, L=250 and 500, (one and two years of trading data, respectively) to perform the bias-correction method.

Empirical analysis
We consider the daily percentage log-return series, defined by  For each series, we use p =2000 out-of-sample values, and the last forecast is made for July 16, 2013 for  (h=1) where Γ( ) x is the usual gamma function.
The corresponding VaR prediction for the t-AR(1)-EGARCH(1,1) model is given by where ( ) 1 t F λ; υ − is the inverse of the cumulative distribution function of the t-distribution with υ degrees of freedom and a standardized variance of one.
In the remainder of the paper, we first examine the VaR forecast distributions generated by the bootstrap algorithm. Then, we compare the VaR estimations of the competing models.

Performance of Models in Forecasting VaR
For assessing the accuracy of competing methods, we define the Boolean sequence as is one-step-ahead VaR predictions for λ , a given probability level, and 1 t r + is the observed return. Christoffersen (1998) showed that evaluating interval forecasts can be reduced to examining whether the Boolean sequence, For un P below the desired significance level, the null hypothesis is rejected.
The cc LR test adapted from Christoffersen (1998) is used to test the conditional coverage.
We compare models based on the p-values, and models with higher p-values are preferred.      In addition to unconditional and conditional tests we consider a csl scoring rule suggested by Diks et al. (2011) to assess the performance of the bias-correction method. They have shown that this scoring rule is handy as the main interest lies in comparing the accuracy of density forecasts for a specific region, such as the left tail in financial risk management. While they have considered long VaR only, we also evaluate our models for both long and short positions. The censored likelihood (csl) score function is given by We first examine the capability of the bias-correction procedure to improve the VaR forecasts of the usual n-GARCH and n-EGARCH models. Comparing the forecasts from the usual models with the biascorrected forecasts, we find that the uc P -values for at least five out of the 6 specified probabilities for L = 500 are superior (larger) to those for the usual n-GARCH model, while for the n-EGARCH model the p-values for at least four out of the 6 cases for both window lengths are superior to those for the usual n-EGARCH model.
We found no uc P -values below the 5% significance level for the bias-corrected VaR predictions of the n-EGARCH model with L = 500 for all series except the NIKKEI at 99%, while for the usual VaR forecasts, we     Comparing the unconditional coverage results for the bias-correction method based on the n-EGARCH model with those based on the n-GARCH model, we : average score difference based on n-AR(1)-GARCH(1,1) (n-AR(1)-EGARCH(1,1)) model relative to calibrated n-AR(1)-GARCH(1,1) (n-AR(1)-EGARCH(1,1)) model with L=500; , 500 N EGARCH d − : average score difference for calibrated n-AR(1)-GARCH(1,1) relative to calibrated n-AR(1)-EGARCH(1,1) model with L=500; _ ,500 EGARCH t d : average score difference for t-AR(1)-EGARCH(1,1) model relative to calibrated n-AR(1)-EGARCH(1,1) model with L=500. Additionally, the corresponding test statistics are shown. The results for both long and short positions are reported in the left and right panel of the ing to empirical results, the bias-correction method based on the n-EGARCH model performs better than the t-EGARCH model. Therefore, it seems that there is no need to consider a fat-tailed distribution to describe the returns' conditional distribution. Table 5 presents the average score differences d with the accompanying tests of equal predictive accuracy for models relative to the bias-corrected n-GARCH and n-EGARCH models. The score difference d is computed by subtracting the score of the bias-corrected n-EGARCH from the score of n-EGARCH, bias-corrected n-GARCH, and t-EGARCH models, such that negative values of d indicate better predictive ability of the bias-corrected n-EGARCH model. Additionally, the score difference d for the n-GARCH and bias-corrected n-GARCH models is calculated and interpreted in a similar way. For all series, the csl scoring rule suggests superior or equal predictive ability of the bias-correction method in comparison with usual n-GARCH models. Comparing the results for the bias-correction method based on the n-EGARCH model with those based on the n-GARCH model, we conclude that the n-EGARCH model outperforms the n-GARCH model for both the FTSE and NIKKEI indexes (except the NIK-KEI at 99%), but evidence is weaker when we consider the CAC index. On the other hand, the bias-correction method based on the n-EGARCH model performs better than the t-EGARCH model for the NIKKEI, while this is not true for short FTSE and CAC positions.

Conclusions
In this paper, we extend the study by Hartz et al. (2006) to take account of asymmetry in conditional variance and correct the VaR for both long and short positions.
We focus on three extreme percentiles α = 0.5%, 1% and 5% in the empirical study. Our results are robust to the chosen bias-correction window length, with a slight preference for the longer window length of L=500 for the three real return series investigated.
Our findings support those found in the study by Hartz et al. (2006). They obtain similar results with the other data, finding that the bias-correction method based on the n-GARCH model performs better than the usual n-GARCH model. This is also confirmed by the csl scoring rule for both GARCH models which has not been considered by Hartz et al. (2006). Our empirical study shows that the bias-correction method based on the n-EGARCH model instead of n-GARCH leads to improvements in correctly forecasting oneday-ahead VaR for long and short positions of almost all real return series investigated based on the three performance tests of Christoffersen (1998), while the independence of the VaR violations is unaffected by this method. We found that the bias-corrected n-EGARCH model is the only model never rejected by any of the three performance tests for all the specified probabilities and real return series investigated. Overall, it seems that allowing for an asymmetric response of the conditional variance to positive and negative shocks yields an improvement in the VaR performance based on the bias-correction method in terms of the three performance tests, but the improvement is not highly confirmed based on the csl scoring rule.
Moreover, the bias-correction method based on the n-EGARCH model performs better than the t-EGARCH model based on three performance tests. This is also true even for the NIKKEI with the most skewness and kurtosis in our data investigated. Thus, it seems that the bias-corrected n-EGARCH model can take account of both the asymmetry in the conditional variance and leptokurtosis in return distribution. However, we observe that the superiority of this model is not maintained in terms of predictive power according to the csl scoring rule for short positions.
Thus, further research is advisable to adjust the standard VaR predictions of the n-GARCH models based on csl scoring rule rather than the observed frequency  Diks et al. (2011).