Figure 1.
Time series plots for the prices of NASDAQ index and returns of NASDAQ index.
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A collaborative EPQ inventory model for a three-echelon supply chain with multiple products considering the effect of marketing effort on demand
July
2020, 16(4): 1635-1654.
doi: 10.3934/jimo.2019021
Model selection based on value-at-risk backtesting approach for GARCH-Type models
| 1. | Institute of Mathematical Sciences, Faculty of Science, University of Malaya, Kuala Lumpur, Malaysia |
| 2. | Department of Mathematical and Actuarial Sciences, Lee Kong Chian Faculty of Engineering and Science, Universiti Tunku Abdul Rahman, Malaysia |
This paper aims to investigate the efficiency of the value-at-risk (VaR) backtests in the model selection from different types of generalised autoregressive conditional heteroskedasticity (GARCH) models with skewed and non-skewed innovation distributions. Extensive simulation is carried out to compare the model selection based on VaR backtests and Akaike Information Criteria (AIC). When the model is given but the innovation distribution is one of the six selected distributions which may be skewed or non-skewed, the simulation results show that both AIC and the VaR backtests succeed in selecting the correct innovation distribution from the set of six distributions under consideration. This indicates that both AIC and the VaR backtests are able to distinguish between skewed and non-skewed distributions when the innovation distribution is misspecified. Using an empirical data from NASDAQ index, we observe that the selected combination of model and innovation distribution based on the smallest AIC does not agree with that selected by using the in-sample VaR backtests. Examination of confidence limits for VaR and the expected shortfall forecasts under various loss functions provides evidence that the selected combination of model and innovation distribution using the VaR backtests tends to possess smaller mean absolute percentage error and logarithmic loss.
Mathematics Subject Classification:
Primary: 62P05; Secondary: 91B84, 37M10, 62M10.
Citation:
Hao-Zhe Tay, Kok-Haur Ng, You-Beng Koh, Kooi-Huat Ng. Model selection based on value-at-risk backtesting approach for GARCH-Type models. Journal of Industrial and Management Optimization,
2020, 16
(4)
: 1635-1654.
doi: 10.3934/jimo.2019021
![]()
References:
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D. Alberg, H. Shalit and R. Yosef,
Estimating stock market volatility using asymmetric GARCH models, Applied Financial Economics, 18 (2008), 1201-1208.
doi: 10.1080/09603100701604225. |
| [2] |
T. Angelidis, A. Benos and S. Degiannakis,
The use of GARCH models in VaR estimation, Statistical Methodology, 1 (2004), 105-128.
doi: 10.1016/j.stamet.2004.08.004. |
| [3] |
A. Azzalini,
A class of distributions which includes the normal ones, Scandinavian Journal of Statistics, 12 (1985), 171-178.
|
| [4] |
T. G. Bali,
Modeling the dynamics of interest rate volatility with skew fat-tailed distributions, Annals of Operations Research, 151 (2007), 151-178.
doi: 10.1007/s10479-006-0116-6. |
| [5] |
R. Ballie, T. Bollerslev and H. Mikkelsen,
Fractionally integrated generalized autoregressive conditional heteroskedasticity, Jouranl of Econometrics, 74 (1996), 3-30.
doi: 10.1016/S0304-4076(95)01749-6. |
| [6] |
F. Black, Studies of stock price volatility changes, In: Proceedings of the 1976 Meeting of the Business and Economic Statistics Section, American Statistical Association, Washington DC, (1976), 177–181. |
| [7] |
T. Bollerslev,
Generalized autoregressive conditional heteroskedasticity, Journal of Econometrics, 31 (1986), 307-327.
doi: 10.1016/0304-4076(86)90063-1. |
| [8] |
T. Bollerslev,
A conditionally heteroskedastic time series model for speculative prices and rates of return, Review of Economics and Statistics, 69 (1987), 542-547.
doi: 10.2307/1925546. |
| [9] |
T. Bollerslev and E. Ghysels,
Periodic autoregressive conditional heteroscedasticity, Journal of Business and Economic Statisticss, 14 (1996), 139-151.
|
| [10] |
M. Braione and N. K. Scholtes,
Forecasting value-at-risk under different distributional assumptions, Econometrics, 4 (2016), 1-27.
doi: 10.3390/econometrics4010003. |
| [11] |
J. Y. Campbell and L. Hentschel,
No news is good news: An asymmetric model of changing volatility in stock returns, Journal of Financial Economics, 31 (1992), 281-318.
doi: 10.3386/w3742. |
| [12] |
M. S. Choi, J. A. Park and S. J. Hwang,
Asymmetric GARCH processes featuring both threshold effect and bilinear structure, Statistics and Probability Letters, 82 (2012), 419-426.
doi: 10.1016/j.spl.2011.11.023. |
| [13] |
P. F. Christoffersen,
Evaluating interval forecasts, International Economic Review, 39 (1998), 841-862.
doi: 10.2307/2527341. |
| [14] |
Z. Ding, C. W. Granger and R. F. Engle,
A long memory property of stock market returns and a new model, Journal of Empirical Finance, 1 (1993), 83-106.
|
| [15] |
R. F. Engle,
Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation, Econometrica, 50 (1982), 987-1007.
doi: 10.2307/1912773. |
| [16] |
R. F. Engle and V. K. Ng,
Measuring and testing the impact of new on volatility, Journal of Finance, 48 (1993), 1749-1778.
doi: 10.3386/w3681. |
| [17] |
R. F. Engle and S. Manganelli,
CAViaR: Conditional Autoregressive value at risk by regression quantiles, Journal of Business and Economic Statistics, 22 (2004), 367-381.
doi: 10.1198/073500104000000370. |
| [18] |
C. Fernández and M. F. Steel,
On Bayesian modelling of fat tails and skewness, Journal of the American Statistical Association, 93 (1998), 359-371.
doi: 10.2307/2669632. |
| [19] |
P. Giot and S. Laurent,
Value-at-risk for long and short trading positions, Journal of Applied Econometrics, 18 (2003), 641-663.
doi: 10.1002/jae.710. |
| [20] |
L. R. Glosten, R. Jagannathan and D. E. Runkle,
On the relation between the expected value and the volatility of the nominal excess return on stocks, Journal of Finance, 12 (1993), 1779-1801.
doi: 10.1111/j.1540-6261.1993.tb05128.x. |
| [21] |
C. Kosapattarapim, Y. X. Lin and M. McCrae,
Evaluating the volatility forecasting performance of best fitting GARCH models in emerging Asian stock markets, International Journal of Mathematics and Statistics, 12 (2012), 1-15.
|
| [22] |
K. Kuester, S. Mittnik and M. S. Paolella,
Value-at-risk prediction: A comparison of alternative strategies, Journal of Financial Econometrics, 4 (2006), 53-89.
doi: 10.1093/jjfinec/nbj002. |
| [23] |
P. H. Kupiec,
Techniques for verifying the accuracy of risk measurement models, Journal of Derivatives, 3 (1995), 73-84.
doi: 10.3905/jod.1995.407942. |
| [24] |
P. Lambert and S. Laurent, Modelling financial time series using GARCH-Type models and a skew Student distribution for the innovations, Discussion Paper No.0125. Institute de Statisque, Universite Catholique de Louvain, Louvain-la-Neuve, Belgium. |
| [25] |
H. C. Liu and J. C. Hung,
Forecasting S & P-100 stock index volatility: The role of volatility asymmetry and distributional assumption in GARCH models, Expert Systems with Applications, 37 (2010), 4928-4934.
doi: 10.1016/j.eswa.2009.12.022. |
| [26] |
J. A. Lopez,
Evaluating the predictive accuracy of volatility models, Journal of Forecasting, 20 (2001), 87-109.
doi: 10.1002/1099-131X(200103)20:2<87::AID-FOR782>3.0.CO;2-7. |
| [27] |
Y. Lyu, P. Wang, Y. Wei and R. Ke,
Forecasting the VaR of crude oil market: Do alternative distribution help?, Energy Economics, 66 (2017), 523-534.
doi: 10.1016/j.eneco.2017.06.015. |
| [28] |
B. Mandelbrot,
The variation of certain speculative prices, Journal of Business, 36 (1963), 394-419.
|
| [29] |
D. McMillan, S. Alan and A. Owain,
Forecasting UK stock market volatility, Applied Financial Economics, 10 (2000), 435-448.
doi: 10.1080/09603100050031561. |
| [30] |
S. Nadarajah, E. Afuecheta and S. Chan,
GARCH modeling of five popular commodities, Empirical Economics, 48 (2015), 1691-1712.
doi: 10.1007/s00181-014-0845-3. |
| [31] |
D. B. Nelson,
Conditional heteroskedasticity in asset returns: A new approach, Econometrica, 59 (1991), 347-370.
doi: 10.2307/2938260. |
| [32] |
A. R. Pagan and G. W. Schwert,
Alternative models for conditional stock volatility, Journal of Econometrics, 45 (1990), 267-90.
doi: 10.3386/w2955. |
| [33] |
R. Rabemananjara and J. M. Zakoian,
Threshold ARCH models and asymmetries in volatility, Journal of Applied Econometrics, 8 (1993), 31-49.
doi: 10.1002/jae.3950080104. |
| [34] |
E. Sentana,
Quadratic ARCH models, Review of Economic Studies, 62 (1995), 639-661.
doi: 10.2307/2298081. |
| [35] |
O. Scaillet,
Nonparametric estimation of conditional expected shortfall, Insurance and Risk Management Journal, 74 (2005), 639-660.
|
| [36] |
A. Shamiri and Z. Isa,
Modeling and Forecasting Volatility of the Malaysian Stock Markets, Journal of Mathematics and Statistics, 5 (2009), 234-240.
doi: 10.3844/jmssp.2009.234.240. |
| [37] |
J. C. Smolović, M. Lipovina-Božvić and S. Vujošević,
GARCH models in value at risk estimation: Empirical evidence from the Montenegrin stock exchange, Economic Research-Ekonomska Istraživanja, 30 (2017), 477-498.
|
| [38] |
G. Storti and C. Vitale,
BL-GARCH models and asymmetries in volatility, Statistical Methods and Applications, 12 (2003), 19-39.
doi: 10.1007/BF02511581. |
| [39] |
R. S. Tsay, Analysis of Financial Time Series, 3$ ^{rd} $ edition, John Wiley & Sons, Hoboken, New Jersey, 2010.
doi: 10.1002/9780470644560. |
| [40] |
Y. Zhang and S. Nadarajah,
A review of backtesting for value at risk, Communications in Statistics-Theory and Methods, 47 (2018), 3616-3639.
doi: 10.1080/03610926.2017.1361984. |
| [41] |
D. Zhu and J. W. Galbraith,
A generalized asymmetric student-t distribution with application to financial econometrics, Journal of Econometrics, 157 (2010), 297-305.
doi: 10.1016/j.jeconom.2010.01.013. |
| [42] |
D. Zhu and V. Zinde-Walsh,
Properties and estimation of asymmetric exponential power distribution, Journal of Econometrics, 148 (2009), 86-99.
doi: 10.1016/j.jeconom.2008.09.038. |
show all references
References:
| [1] |
D. Alberg, H. Shalit and R. Yosef,
Estimating stock market volatility using asymmetric GARCH models, Applied Financial Economics, 18 (2008), 1201-1208.
doi: 10.1080/09603100701604225. |
| [2] |
T. Angelidis, A. Benos and S. Degiannakis,
The use of GARCH models in VaR estimation, Statistical Methodology, 1 (2004), 105-128.
doi: 10.1016/j.stamet.2004.08.004. |
| [3] |
A. Azzalini,
A class of distributions which includes the normal ones, Scandinavian Journal of Statistics, 12 (1985), 171-178.
|
| [4] |
T. G. Bali,
Modeling the dynamics of interest rate volatility with skew fat-tailed distributions, Annals of Operations Research, 151 (2007), 151-178.
doi: 10.1007/s10479-006-0116-6. |
| [5] |
R. Ballie, T. Bollerslev and H. Mikkelsen,
Fractionally integrated generalized autoregressive conditional heteroskedasticity, Jouranl of Econometrics, 74 (1996), 3-30.
doi: 10.1016/S0304-4076(95)01749-6. |
| [6] |
F. Black, Studies of stock price volatility changes, In: Proceedings of the 1976 Meeting of the Business and Economic Statistics Section, American Statistical Association, Washington DC, (1976), 177–181. |
| [7] |
T. Bollerslev,
Generalized autoregressive conditional heteroskedasticity, Journal of Econometrics, 31 (1986), 307-327.
doi: 10.1016/0304-4076(86)90063-1. |
| [8] |
T. Bollerslev,
A conditionally heteroskedastic time series model for speculative prices and rates of return, Review of Economics and Statistics, 69 (1987), 542-547.
doi: 10.2307/1925546. |
| [9] |
T. Bollerslev and E. Ghysels,
Periodic autoregressive conditional heteroscedasticity, Journal of Business and Economic Statisticss, 14 (1996), 139-151.
|
| [10] |
M. Braione and N. K. Scholtes,
Forecasting value-at-risk under different distributional assumptions, Econometrics, 4 (2016), 1-27.
doi: 10.3390/econometrics4010003. |
| [11] |
J. Y. Campbell and L. Hentschel,
No news is good news: An asymmetric model of changing volatility in stock returns, Journal of Financial Economics, 31 (1992), 281-318.
doi: 10.3386/w3742. |
| [12] |
M. S. Choi, J. A. Park and S. J. Hwang,
Asymmetric GARCH processes featuring both threshold effect and bilinear structure, Statistics and Probability Letters, 82 (2012), 419-426.
doi: 10.1016/j.spl.2011.11.023. |
| [13] |
P. F. Christoffersen,
Evaluating interval forecasts, International Economic Review, 39 (1998), 841-862.
doi: 10.2307/2527341. |
| [14] |
Z. Ding, C. W. Granger and R. F. Engle,
A long memory property of stock market returns and a new model, Journal of Empirical Finance, 1 (1993), 83-106.
|
| [15] |
R. F. Engle,
Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation, Econometrica, 50 (1982), 987-1007.
doi: 10.2307/1912773. |
| [16] |
R. F. Engle and V. K. Ng,
Measuring and testing the impact of new on volatility, Journal of Finance, 48 (1993), 1749-1778.
doi: 10.3386/w3681. |
| [17] |
R. F. Engle and S. Manganelli,
CAViaR: Conditional Autoregressive value at risk by regression quantiles, Journal of Business and Economic Statistics, 22 (2004), 367-381.
doi: 10.1198/073500104000000370. |
| [18] |
C. Fernández and M. F. Steel,
On Bayesian modelling of fat tails and skewness, Journal of the American Statistical Association, 93 (1998), 359-371.
doi: 10.2307/2669632. |
| [19] |
P. Giot and S. Laurent,
Value-at-risk for long and short trading positions, Journal of Applied Econometrics, 18 (2003), 641-663.
doi: 10.1002/jae.710. |
| [20] |
L. R. Glosten, R. Jagannathan and D. E. Runkle,
On the relation between the expected value and the volatility of the nominal excess return on stocks, Journal of Finance, 12 (1993), 1779-1801.
doi: 10.1111/j.1540-6261.1993.tb05128.x. |
| [21] |
C. Kosapattarapim, Y. X. Lin and M. McCrae,
Evaluating the volatility forecasting performance of best fitting GARCH models in emerging Asian stock markets, International Journal of Mathematics and Statistics, 12 (2012), 1-15.
|
| [22] |
K. Kuester, S. Mittnik and M. S. Paolella,
Value-at-risk prediction: A comparison of alternative strategies, Journal of Financial Econometrics, 4 (2006), 53-89.
doi: 10.1093/jjfinec/nbj002. |
| [23] |
P. H. Kupiec,
Techniques for verifying the accuracy of risk measurement models, Journal of Derivatives, 3 (1995), 73-84.
doi: 10.3905/jod.1995.407942. |
| [24] |
P. Lambert and S. Laurent, Modelling financial time series using GARCH-Type models and a skew Student distribution for the innovations, Discussion Paper No.0125. Institute de Statisque, Universite Catholique de Louvain, Louvain-la-Neuve, Belgium. |
| [25] |
H. C. Liu and J. C. Hung,
Forecasting S & P-100 stock index volatility: The role of volatility asymmetry and distributional assumption in GARCH models, Expert Systems with Applications, 37 (2010), 4928-4934.
doi: 10.1016/j.eswa.2009.12.022. |
| [26] |
J. A. Lopez,
Evaluating the predictive accuracy of volatility models, Journal of Forecasting, 20 (2001), 87-109.
doi: 10.1002/1099-131X(200103)20:2<87::AID-FOR782>3.0.CO;2-7. |
| [27] |
Y. Lyu, P. Wang, Y. Wei and R. Ke,
Forecasting the VaR of crude oil market: Do alternative distribution help?, Energy Economics, 66 (2017), 523-534.
doi: 10.1016/j.eneco.2017.06.015. |
| [28] |
B. Mandelbrot,
The variation of certain speculative prices, Journal of Business, 36 (1963), 394-419.
|
| [29] |
D. McMillan, S. Alan and A. Owain,
Forecasting UK stock market volatility, Applied Financial Economics, 10 (2000), 435-448.
doi: 10.1080/09603100050031561. |
| [30] |
S. Nadarajah, E. Afuecheta and S. Chan,
GARCH modeling of five popular commodities, Empirical Economics, 48 (2015), 1691-1712.
doi: 10.1007/s00181-014-0845-3. |
| [31] |
D. B. Nelson,
Conditional heteroskedasticity in asset returns: A new approach, Econometrica, 59 (1991), 347-370.
doi: 10.2307/2938260. |
| [32] |
A. R. Pagan and G. W. Schwert,
Alternative models for conditional stock volatility, Journal of Econometrics, 45 (1990), 267-90.
doi: 10.3386/w2955. |
| [33] |
R. Rabemananjara and J. M. Zakoian,
Threshold ARCH models and asymmetries in volatility, Journal of Applied Econometrics, 8 (1993), 31-49.
doi: 10.1002/jae.3950080104. |
| [34] |
E. Sentana,
Quadratic ARCH models, Review of Economic Studies, 62 (1995), 639-661.
doi: 10.2307/2298081. |
| [35] |
O. Scaillet,
Nonparametric estimation of conditional expected shortfall, Insurance and Risk Management Journal, 74 (2005), 639-660.
|
| [36] |
A. Shamiri and Z. Isa,
Modeling and Forecasting Volatility of the Malaysian Stock Markets, Journal of Mathematics and Statistics, 5 (2009), 234-240.
doi: 10.3844/jmssp.2009.234.240. |
| [37] |
J. C. Smolović, M. Lipovina-Božvić and S. Vujošević,
GARCH models in value at risk estimation: Empirical evidence from the Montenegrin stock exchange, Economic Research-Ekonomska Istraživanja, 30 (2017), 477-498.
|
| [38] |
G. Storti and C. Vitale,
BL-GARCH models and asymmetries in volatility, Statistical Methods and Applications, 12 (2003), 19-39.
doi: 10.1007/BF02511581. |
| [39] |
R. S. Tsay, Analysis of Financial Time Series, 3$ ^{rd} $ edition, John Wiley & Sons, Hoboken, New Jersey, 2010.
doi: 10.1002/9780470644560. |
| [40] |
Y. Zhang and S. Nadarajah,
A review of backtesting for value at risk, Communications in Statistics-Theory and Methods, 47 (2018), 3616-3639.
doi: 10.1080/03610926.2017.1361984. |
| [41] |
D. Zhu and J. W. Galbraith,
A generalized asymmetric student-t distribution with application to financial econometrics, Journal of Econometrics, 157 (2010), 297-305.
doi: 10.1016/j.jeconom.2010.01.013. |
| [42] |
D. Zhu and V. Zinde-Walsh,
Properties and estimation of asymmetric exponential power distribution, Journal of Econometrics, 148 (2009), 86-99.
doi: 10.1016/j.jeconom.2008.09.038. |
Figure 2.
Histogram and QQ-plot for the returns of NASDAQ index (Solid curve is the normal distribution curve with corresponding mean and standard deviation)
Figure 3.
Out-of-sample VaR and the expected shortfall plots for the returns of NASDAQ index under TGARCH $ (1, 1) $ -SSTD (AIC best model)
Figure 4.
Out-of-sample VaR and the expected shortfall plots for the returns of NASDAQ index under BLGARCH $ (1, 1) $ -SSTD (VaR best model)
Table 1.
Average values of AIC and the number of non-rejections in null hypothesis at 5$ % $ significance level for various backtests out of 100 runs for the fitted GARCH $ (1, 1) $ model with parameters $ \alpha_{0} = 0.2, \alpha_{1} = 0.1 $ and $ \beta_{1} = 0.6 $
| Fitted Model with Related Innovation Distribution | |||||||
| True Dis-tribution | Measure | NORMD | STD |
GED |
SNORMD |
SSTD |
SGED |
| NORMD | AIC | 4839.88 | 4842.51 | 4841.12 | 4840.79 | 4843.42 | 4842.05 |
| UCK-long | 95 | 95 | 95 | 97 | 97 | 98 | |
| UCK-short | 99 | 98 | 99 | 99 | 99 | 99 | |
| CCC-long | 96 | 96 | 96 | 99 | 98 | 98 | |
| CCC-short | 97 | 97 | 97 | 99 | 99 | 99 | |
| DQ-long | 95 | 94 | 95 | 93 | 93 | 94 | |
| DQ-short | 97 | 96 | 97 | 98 | 98 | 97 | |
| STD | AIC | 4794.80 | 4615.59 | 4639.27 | 4794.10 | 4616.55 | 4639.77 |
| UCK-long | 75 | 99 | 83 | 80 | 98 | 87 | |
| UCK-short | 70 | 94 | 83 | 74 | 98 | 83 | |
| CCC-long | 86 | 96 | 89 | 86 | 97 | 89 | |
| CCC-short | 78 | 95 | 85 | 84 | 97 | 89 | |
| DQ-long | 97 | 100 | 98 | 99 | 100 | 98 | |
| DQ-short | 88 | 88 | 89 | 90 | 91 | 90 | |
| GED | AIC | 4826.53 | 4795.85 | 4790.65 | 4827.24 | 4796.84 | 4791.49 |
| UCK-long | 96 | 92 | 97 | 100 | 95 | 100 | |
| UCK-short | 99 | 94 | 98 | 98 | 96 | 100 | |
| CCC-long | 93 | 89 | 97 | 96 | 95 | 97 | |
| CCC-short | 98 | 96 | 98 | 99 | 99 | 98 | |
| DQ-long | 93 | 89 | 93 | 94 | 91 | 96 | |
| DQ-short | 97 | 95 | 97 | 98 | 96 | 97 | |
| SNORMD | AIC | 4820.81 | 4809.73 | 4820.76 | 4397.94 | 4400.95 | 4398.99 |
| UCK-long | 0 | 0 | 0 | 99 | 99 | 100 | |
| UCK-short | 1 | 0 | 1 | 93 | 91 | 95 | |
| CCC-long | 0 | 0 | 0 | 98 | 98 | 99 | |
| CCC-short | 2 | 1 | 2 | 93 | 92 | 95 | |
| DQ-long | 0 | 0 | 0 | 99 | 99 | 99 | |
| DQ-short | 5 | 4 | 6 | 97 | 96 | 96 | |
| SSTD | AIC | 4784.12 | 4446.89 | 4559.11 | 4071.87 | 3865.32 | 3889.15 |
| UCK-long | 0 | 0 | 0 | 4 | 100 | 100 | |
| UCK-short | 26 | 0 | 8 | 47 | 99 | 81 | |
| CCC-long | 0 | 0 | 0 | 11 | 99 | 98 | |
| CCC-short | 38 | 0 | 17 | 55 | 96 | 87 | |
| DQ-long | 0 | 0 | 0 | 45 | 96 | 97 | |
| DQ-short | 60 | 3 | 35 | 72 | 98 | 97 | |
| SGED | AIC | 4804.63 | 4727.06 | 4770.63 | 4224.61 | 4189.49 | 4183.04 |
| UCK-long | 0 | 0 | 0 | 70 | 100 | 100 | |
| UCK-short | 0 | 0 | 0 | 44 | 86 | 95 | |
| CCC-long | 0 | 0 | 0 | 84 | 99 | 99 | |
| CCC-short | 1 | 0 | 1 | 52 | 84 | 96 | |
| DQ-long | 0 | 0 | 0 | 97 | 97 | 96 | |
| DQ-short | 3 | 0 | 3 | 59 | 92 | 97 | |
| Fitted Model with Related Innovation Distribution | |||||||
| True Dis-tribution | Measure | NORMD | STD |
GED |
SNORMD |
SSTD |
SGED |
| NORMD | AIC | 4839.88 | 4842.51 | 4841.12 | 4840.79 | 4843.42 | 4842.05 |
| UCK-long | 95 | 95 | 95 | 97 | 97 | 98 | |
| UCK-short | 99 | 98 | 99 | 99 | 99 | 99 | |
| CCC-long | 96 | 96 | 96 | 99 | 98 | 98 | |
| CCC-short | 97 | 97 | 97 | 99 | 99 | 99 | |
| DQ-long | 95 | 94 | 95 | 93 | 93 | 94 | |
| DQ-short | 97 | 96 | 97 | 98 | 98 | 97 | |
| STD | AIC | 4794.80 | 4615.59 | 4639.27 | 4794.10 | 4616.55 | 4639.77 |
| UCK-long | 75 | 99 | 83 | 80 | 98 | 87 | |
| UCK-short | 70 | 94 | 83 | 74 | 98 | 83 | |
| CCC-long | 86 | 96 | 89 | 86 | 97 | 89 | |
| CCC-short | 78 | 95 | 85 | 84 | 97 | 89 | |
| DQ-long | 97 | 100 | 98 | 99 | 100 | 98 | |
| DQ-short | 88 | 88 | 89 | 90 | 91 | 90 | |
| GED | AIC | 4826.53 | 4795.85 | 4790.65 | 4827.24 | 4796.84 | 4791.49 |
| UCK-long | 96 | 92 | 97 | 100 | 95 | 100 | |
| UCK-short | 99 | 94 | 98 | 98 | 96 | 100 | |
| CCC-long | 93 | 89 | 97 | 96 | 95 | 97 | |
| CCC-short | 98 | 96 | 98 | 99 | 99 | 98 | |
| DQ-long | 93 | 89 | 93 | 94 | 91 | 96 | |
| DQ-short | 97 | 95 | 97 | 98 | 96 | 97 | |
| SNORMD | AIC | 4820.81 | 4809.73 | 4820.76 | 4397.94 | 4400.95 | 4398.99 |
| UCK-long | 0 | 0 | 0 | 99 | 99 | 100 | |
| UCK-short | 1 | 0 | 1 | 93 | 91 | 95 | |
| CCC-long | 0 | 0 | 0 | 98 | 98 | 99 | |
| CCC-short | 2 | 1 | 2 | 93 | 92 | 95 | |
| DQ-long | 0 | 0 | 0 | 99 | 99 | 99 | |
| DQ-short | 5 | 4 | 6 | 97 | 96 | 96 | |
| SSTD | AIC | 4784.12 | 4446.89 | 4559.11 | 4071.87 | 3865.32 | 3889.15 |
| UCK-long | 0 | 0 | 0 | 4 | 100 | 100 | |
| UCK-short | 26 | 0 | 8 | 47 | 99 | 81 | |
| CCC-long | 0 | 0 | 0 | 11 | 99 | 98 | |
| CCC-short | 38 | 0 | 17 | 55 | 96 | 87 | |
| DQ-long | 0 | 0 | 0 | 45 | 96 | 97 | |
| DQ-short | 60 | 3 | 35 | 72 | 98 | 97 | |
| SGED | AIC | 4804.63 | 4727.06 | 4770.63 | 4224.61 | 4189.49 | 4183.04 |
| UCK-long | 0 | 0 | 0 | 70 | 100 | 100 | |
| UCK-short | 0 | 0 | 0 | 44 | 86 | 95 | |
| CCC-long | 0 | 0 | 0 | 84 | 99 | 99 | |
| CCC-short | 1 | 0 | 1 | 52 | 84 | 96 | |
| DQ-long | 0 | 0 | 0 | 97 | 97 | 96 | |
| DQ-short | 3 | 0 | 3 | 59 | 92 | 97 | |
Table 2.
Average values of AIC and the number of non-rejections in null hypothesis at 5$ % $ significance level for various backtests out of 100 runs for the fitted GJRGARCH $ (1, 1) $ model with parameters $ \alpha_{0} = 0.2, \alpha_{1} = 0.1, \gamma_{1} = 0.2 $ and $ \beta_{1} = 0.6 $
| Fitted Model with Related Innovation Distribution | |||||||
| True Distribution | Measure | NORMD | STD |
GED |
SNORMD |
SSTD |
SGED |
| NORMD | AIC | 5480.34 | 5482.90 | 5481.25 | 5481.46 | 5484.05 | 5482.36 |
| UCK-long | 99 | 99 | 99 | 100 | 100 | 100 | |
| UCK-short | 99 | 99 | 100 | 99 | 99 | 99 | |
| CCC-long | 99 | 99 | 99 | 99 | 99 | 99 | |
| CCC-short | 99 | 99 | 99 | 99 | 99 | 99 | |
| DQ-long | 95 | 96 | 96 | 95 | 96 | 95 | |
| DQ-short | 99 | 99 | 99 | 97 | 97 | 98 | |
| STD | AIC | 5387.99 | 5201.69 | 5226.05 | 5386.65 | 5202.52 | 5226.47 |
| UCK-long | 73 | 99 | 77 | 76 | 98 | 79 | |
| UCK-short | 72 | 97 | 79 | 76 | 98 | 82 | |
| CCC-long | 79 | 98 | 83 | 80 | 98 | 85 | |
| CCC-short | 78 | 98 | 87 | 81 | 97 | 92 | |
| DQ-long | 94 | 95 | 95 | 94 | 95 | 97 | |
| DQ-short | 96 | 98 | 98 | 96 | 99 | 98 | |
| GED | AIC | 5123.43 | 5093.04 | 5087.91 | 5124.13 | 5094.03 | 5088.75 |
| UCK-long | 96 | 91 | 97 | 100 | 95 | 100 | |
| UCK-short | 99 | 95 | 99 | 99 | 98 | 100 | |
| CCC-long | 95 | 92 | 94 | 96 | 94 | 98 | |
| CCC-short | 97 | 97 | 99 | 99 | 98 | 99 | |
| DQ-long | 94 | 90 | 93 | 94 | 92 | 97 | |
| DQ-short | 98 | 97 | 100 | 98 | 99 | 97 | |
| SNORMD | AIC | 5352.98 | 5341.66 | 5352.87 | 4928.49 | 4931.46 | 4929.62 |
| UCK-long | 0 | 0 | 0 | 100 | 100 | 100 | |
| UCK-short | 0 | 0 | 0 | 97 | 96 | 99 | |
| CCC-long | 0 | 0 | 0 | 99 | 97 | 97 | |
| CCC-short | 0 | 0 | 0 | 95 | 96 | 98 | |
| DQ-long | 0 | 0 | 0 | 98 | 96 | 97 | |
| DQ-short | 5 | 3 | 5 | 98 | 98 | 97 | |
| SSTD | AIC | 5159.29 | 4834.74 | 4945.27 | 4466.38 | 4261.63 | 4285.05 |
| UCK-long | 0 | 0 | 0 | 5 | 100 | 99 | |
| UCK-short | 18 | 0 | 7 | 50 | 98 | 86 | |
| CCC-long | 0 | 0 | 0 | 9 | 96 | 95 | |
| CCC-short | 31 | 0 | 14 | 63 | 95 | 88 | |
| DQ-long | 0 | 0 | 0 | 52 | 99 | 97 | |
| DQ-short | 47 | 1 | 32 | 74 | 93 | 95 | |
| SGED | AIC | 5270.80 | 5194.20 | 5237.44 | 4691.87 | 4656.93 | 4650.46 |
| UCK-long | 0 | 0 | 0 | 75 | 100 | 100 | |
| UCK-short | 0 | 0 | 0 | 44 | 86 | 96 | |
| CCC-long | 0 | 0 | 0 | 82 | 99 | 100 | |
| CCC-short | 1 | 0 | 1 | 54 | 83 | 97 | |
| DQ-long | 0 | 0 | 0 | 97 | 98 | 99 | |
| DQ-short | 2 | 0 | 2 | 64 | 91 | 96 | |
| Fitted Model with Related Innovation Distribution | |||||||
| True Distribution | Measure | NORMD | STD |
GED |
SNORMD |
SSTD |
SGED |
| NORMD | AIC | 5480.34 | 5482.90 | 5481.25 | 5481.46 | 5484.05 | 5482.36 |
| UCK-long | 99 | 99 | 99 | 100 | 100 | 100 | |
| UCK-short | 99 | 99 | 100 | 99 | 99 | 99 | |
| CCC-long | 99 | 99 | 99 | 99 | 99 | 99 | |
| CCC-short | 99 | 99 | 99 | 99 | 99 | 99 | |
| DQ-long | 95 | 96 | 96 | 95 | 96 | 95 | |
| DQ-short | 99 | 99 | 99 | 97 | 97 | 98 | |
| STD | AIC | 5387.99 | 5201.69 | 5226.05 | 5386.65 | 5202.52 | 5226.47 |
| UCK-long | 73 | 99 | 77 | 76 | 98 | 79 | |
| UCK-short | 72 | 97 | 79 | 76 | 98 | 82 | |
| CCC-long | 79 | 98 | 83 | 80 | 98 | 85 | |
| CCC-short | 78 | 98 | 87 | 81 | 97 | 92 | |
| DQ-long | 94 | 95 | 95 | 94 | 95 | 97 | |
| DQ-short | 96 | 98 | 98 | 96 | 99 | 98 | |
| GED | AIC | 5123.43 | 5093.04 | 5087.91 | 5124.13 | 5094.03 | 5088.75 |
| UCK-long | 96 | 91 | 97 | 100 | 95 | 100 | |
| UCK-short | 99 | 95 | 99 | 99 | 98 | 100 | |
| CCC-long | 95 | 92 | 94 | 96 | 94 | 98 | |
| CCC-short | 97 | 97 | 99 | 99 | 98 | 99 | |
| DQ-long | 94 | 90 | 93 | 94 | 92 | 97 | |
| DQ-short | 98 | 97 | 100 | 98 | 99 | 97 | |
| SNORMD | AIC | 5352.98 | 5341.66 | 5352.87 | 4928.49 | 4931.46 | 4929.62 |
| UCK-long | 0 | 0 | 0 | 100 | 100 | 100 | |
| UCK-short | 0 | 0 | 0 | 97 | 96 | 99 | |
| CCC-long | 0 | 0 | 0 | 99 | 97 | 97 | |
| CCC-short | 0 | 0 | 0 | 95 | 96 | 98 | |
| DQ-long | 0 | 0 | 0 | 98 | 96 | 97 | |
| DQ-short | 5 | 3 | 5 | 98 | 98 | 97 | |
| SSTD | AIC | 5159.29 | 4834.74 | 4945.27 | 4466.38 | 4261.63 | 4285.05 |
| UCK-long | 0 | 0 | 0 | 5 | 100 | 99 | |
| UCK-short | 18 | 0 | 7 | 50 | 98 | 86 | |
| CCC-long | 0 | 0 | 0 | 9 | 96 | 95 | |
| CCC-short | 31 | 0 | 14 | 63 | 95 | 88 | |
| DQ-long | 0 | 0 | 0 | 52 | 99 | 97 | |
| DQ-short | 47 | 1 | 32 | 74 | 93 | 95 | |
| SGED | AIC | 5270.80 | 5194.20 | 5237.44 | 4691.87 | 4656.93 | 4650.46 |
| UCK-long | 0 | 0 | 0 | 75 | 100 | 100 | |
| UCK-short | 0 | 0 | 0 | 44 | 86 | 96 | |
| CCC-long | 0 | 0 | 0 | 82 | 99 | 100 | |
| CCC-short | 1 | 0 | 1 | 54 | 83 | 97 | |
| DQ-long | 0 | 0 | 0 | 97 | 98 | 99 | |
| DQ-short | 2 | 0 | 2 | 64 | 91 | 96 | |
Table 3.
Average values of AIC and the number of non-rejections in null hypothesis at 5% significance level for various backtests out of 100 runs for the fitted TGARCH $ (1, 1) $ model with parameters $ \alpha_{0} = 0.2, \alpha_{1}^{+} = 0.1, \alpha_{1}^{-} = 0.3 $ and $ \beta_{1} = 0.6 $
| Fitted Model with Related Innovation Distribution | |||||||
| True Distribution | Measure | NORMD | STD |
GED |
SNORMD |
SSTD |
SGED |
| NORMD | AIC | 4848.75 | 4851.33 | 4849.66 | 4849.86 | 4852.48 | 4850.76 |
| UCK-long | 99 | 99 | 99 | 100 | 100 | 100 | |
| UCK-short | 99 | 98 | 100 | 98 | 98 | 98 | |
| CCC-long | 99 | 99 | 99 | 99 | 99 | 99 | |
| CCC-short | 99 | 99 | 99 | 99 | 99 | 99 | |
| DQ-long | 95 | 95 | 95 | 94 | 94 | 94 | |
| DQ-short | 97 | 98 | 96 | 96 | 96 | 95 | |
| STD | AIC | 4661.30 | 4475.00 | 4499.38 | 4660.01 | 4475.84 | 4499.81 |
| UCK-long | 73 | 98 | 78 | 75 | 98 | 81 | |
| UCK-short | 74 | 97 | 81 | 72 | 98 | 82 | |
| CCC-long | 75 | 97 | 87 | 77 | 98 | 87 | |
| CCC-short | 77 | 98 | 85 | 80 | 99 | 89 | |
| DQ-long | 94 | 93 | 94 | 93 | 94 | 96 | |
| DQ-short | 94 | 97 | 96 | 93 | 98 | 97 | |
| GED | AIC | 4743.45 | 4712.98 | 4707.84 | 4744.16 | 4713.97 | 4780.68 |
| UCK-long | 98 | 91 | 98 | 99 | 95 | 100 | |
| UCK-short | 100 | 97 | 100 | 99 | 97 | 100 | |
| CCC-long | 97 | 89 | 97 | 97 | 94 | 98 | |
| CCC-short | 97 | 96 | 98 | 99 | 97 | 99 | |
| DQ-long | 93 | 90 | 93 | 94 | 91 | 95 | |
| DQ-short | 97 | 95 | 97 | 97 | 97 | 97 | |
| SNORMD | AIC | 4308.67 | 4297.26 | 4308.55 | 3884.16 | 3887.12 | 3885.27 |
| UCK-long | 0 | 0 | 0 | 100 | 100 | 100 | |
| UCK-short | 0 | 0 | 0 | 97 | 98 | 100 | |
| CCC-long | 0 | 0 | 0 | 100 | 98 | 98 | |
| CCC-short | 0 | 0 | 0 | 95 | 95 | 98 | |
| DQ-long | 0 | 0 | 0 | 97 | 98 | 97 | |
| DQ-short | 4 | 4 | 6 | 95 | 98 | 96 | |
| SSTD | AIC | 4650.27 | 4326.84 | 4437.44 | 3958.36 | 3753.78 | 3777.25 |
| UCK-long | 0 | 0 | 0 | 1 | 100 | 100 | |
| UCK-short | 21 | 0 | 8 | 49 | 98 | 87 | |
| CCC-long | 0 | 0 | 0 | 8 | 97 | 96 | |
| CCC-short | 33 | 0 | 14 | 61 | 95 | 88 | |
| DQ-long | 0 | 0 | 0 | 49 | 98 | 99 | |
| DQ-short | 47 | 1 | 28 | 70 | 93 | 93 | |
| SGED | AIC | 4812.02 | 4735.54 | 4778.74 | 4233.28 | 4198.40 | 4191.91 |
| UCK-long | 0 | 0 | 0 | 73 | 100 | 100 | |
| UCK-short | 0 | 0 | 0 | 45 | 85 | 96 | |
| CCC-long | 0 | 0 | 0 | 82 | 98 | 98 | |
| CCC-short | 0 | 0 | 0 | 52 | 94 | 94 | |
| DQ-long | 0 | 0 | 0 | 98 | 98 | 98 | |
| DQ-short | 2 | 0 | 2 | 63 | 89 | 97 | |
| Fitted Model with Related Innovation Distribution | |||||||
| True Distribution | Measure | NORMD | STD |
GED |
SNORMD |
SSTD |
SGED |
| NORMD | AIC | 4848.75 | 4851.33 | 4849.66 | 4849.86 | 4852.48 | 4850.76 |
| UCK-long | 99 | 99 | 99 | 100 | 100 | 100 | |
| UCK-short | 99 | 98 | 100 | 98 | 98 | 98 | |
| CCC-long | 99 | 99 | 99 | 99 | 99 | 99 | |
| CCC-short | 99 | 99 | 99 | 99 | 99 | 99 | |
| DQ-long | 95 | 95 | 95 | 94 | 94 | 94 | |
| DQ-short | 97 | 98 | 96 | 96 | 96 | 95 | |
| STD | AIC | 4661.30 | 4475.00 | 4499.38 | 4660.01 | 4475.84 | 4499.81 |
| UCK-long | 73 | 98 | 78 | 75 | 98 | 81 | |
| UCK-short | 74 | 97 | 81 | 72 | 98 | 82 | |
| CCC-long | 75 | 97 | 87 | 77 | 98 | 87 | |
| CCC-short | 77 | 98 | 85 | 80 | 99 | 89 | |
| DQ-long | 94 | 93 | 94 | 93 | 94 | 96 | |
| DQ-short | 94 | 97 | 96 | 93 | 98 | 97 | |
| GED | AIC | 4743.45 | 4712.98 | 4707.84 | 4744.16 | 4713.97 | 4780.68 |
| UCK-long | 98 | 91 | 98 | 99 | 95 | 100 | |
| UCK-short | 100 | 97 | 100 | 99 | 97 | 100 | |
| CCC-long | 97 | 89 | 97 | 97 | 94 | 98 | |
| CCC-short | 97 | 96 | 98 | 99 | 97 | 99 | |
| DQ-long | 93 | 90 | 93 | 94 | 91 | 95 | |
| DQ-short | 97 | 95 | 97 | 97 | 97 | 97 | |
| SNORMD | AIC | 4308.67 | 4297.26 | 4308.55 | 3884.16 | 3887.12 | 3885.27 |
| UCK-long | 0 | 0 | 0 | 100 | 100 | 100 | |
| UCK-short | 0 | 0 | 0 | 97 | 98 | 100 | |
| CCC-long | 0 | 0 | 0 | 100 | 98 | 98 | |
| CCC-short | 0 | 0 | 0 | 95 | 95 | 98 | |
| DQ-long | 0 | 0 | 0 | 97 | 98 | 97 | |
| DQ-short | 4 | 4 | 6 | 95 | 98 | 96 | |
| SSTD | AIC | 4650.27 | 4326.84 | 4437.44 | 3958.36 | 3753.78 | 3777.25 |
| UCK-long | 0 | 0 | 0 | 1 | 100 | 100 | |
| UCK-short | 21 | 0 | 8 | 49 | 98 | 87 | |
| CCC-long | 0 | 0 | 0 | 8 | 97 | 96 | |
| CCC-short | 33 | 0 | 14 | 61 | 95 | 88 | |
| DQ-long | 0 | 0 | 0 | 49 | 98 | 99 | |
| DQ-short | 47 | 1 | 28 | 70 | 93 | 93 | |
| SGED | AIC | 4812.02 | 4735.54 | 4778.74 | 4233.28 | 4198.40 | 4191.91 |
| UCK-long | 0 | 0 | 0 | 73 | 100 | 100 | |
| UCK-short | 0 | 0 | 0 | 45 | 85 | 96 | |
| CCC-long | 0 | 0 | 0 | 82 | 98 | 98 | |
| CCC-short | 0 | 0 | 0 | 52 | 94 | 94 | |
| DQ-long | 0 | 0 | 0 | 98 | 98 | 98 | |
| DQ-short | 2 | 0 | 2 | 63 | 89 | 97 | |
Table 4.
Average values of AIC and the number of non-rejections in null hypothesis at 5% significance level for various backtests out of 100 runs for the fitted BLGARCH $ (1, 1) $ model with parameters $ \alpha_{0} = 0.2, \alpha_{1} = 0.1, $ $ \beta_{1} = 0.6$ and $ \gamma_{1} = 0.2$
| Fitted Model with Related Innovation Distribution | |||||||
| True Distribution | Measure | NORMD | STD |
GED |
SNORMD |
SSTD |
SGED |
| NORMD | AIC | 4775.04 | 4777.6 | 4775.96 | 4776.16 | 4778.76 | 4777.07 |
| UCK-long | 99 | 99 | 99 | 99 | 99 | 99 | |
| UCK-short | 100 | 100 | 100 | 99 | 99 | 99 | |
| CCC-long | 99 | 99 | 99 | 99 | 100 | 99 | |
| CCC-short | 99 | 99 | 99 | 99 | 99 | 99 | |
| DQ-long | 97 | 98 | 97 | 98 | 97 | 97 | |
| DQ-short | 96 | 97 | 96 | 97 | 97 | 97 | |
| STD | AIC | 4773.34 | 4587.04 | 4611.49 | 4772.06 | 4587.89 | 4611.93 |
| UCK-long | 75 | 98 | 77 | 75 | 100 | 78 | |
| UCK-short | 72 | 97 | 78 | 74 | 97 | 77 | |
| CCC-long | 81 | 99 | 86 | 83 | 98 | 89 | |
| CCC-short | 78 | 96 | 86 | 83 | 98 | 85 | |
| DQ-long | 97 | 97 | 98 | 94 | 96 | 96 | |
| DQ-short | 93 | 97 | 95 | 93 | 97 | 94 | |
| GED | AIC | 4765.93 | 4735.55 | 4730.36 | 4766.63 | 4736.53 | 4731.20 |
| UCK-long | 97 | 92 | 97 | 100 | 94 | 100 | |
| UCK-short | 99 | 95 | 97 | 99 | 95 | 100 | |
| CCC-long | 95 | 92 | 96 | 97 | 96 | 98 | |
| CCC-short | 97 | 97 | 99 | 98 | 98 | 100 | |
| DQ-long | 93 | 89 | 93 | 91 | 91 | 93 | |
| DQ-short | 98 | 98 | 96 | 100 | 98 | 98 | |
| SNORMD | AIC | 4718.43 | 4707.04 | 4718.33 | 4293.51 | 4296.50 | 4294.63 |
| UCK-long | 0 | 0 | 0 | 100 | 100 | 100 | |
| UCK-short | 0 | 0 | 0 | 97 | 94 | 100 | |
| CCC-long | 0 | 0 | 0 | 99 | 99 | 99 | |
| CCC-short | 0 | 0 | 0 | 96 | 94 | 98 | |
| DQ-long | 0 | 0 | 0 | 99 | 99 | 99 | |
| DQ-short | 4 | 3 | 5 | 97 | 98 | 98 | |
| SSTD | AIC | 4663.37 | 4336.06 | 4446.73 | 3968.73 | 3762.91 | 3786.37 |
| UCK-long | 0 | 0 | 0 | 4 | 100 | 100 | |
| UCK-short | 19 | 0 | 8 | 51 | 96 | 85 | |
| CCC-long | 0 | 0 | 0 | 9 | 97 | 96 | |
| CCC-short | 28 | 0 | 15 | 63 | 95 | 88 | |
| DQ-long | 0 | 0 | 0 | 43 | 97 | 97 | |
| DQ-short | 43 | 1 | 29 | 69 | 92 | 95 | |
| SGED | AIC | 4692.89 | 4616.31 | 4659.56 | 4113.75 | 4078.82 | 4072.33 |
| UCK-long | 0 | 0 | 0 | 72 | 100 | 100 | |
| UCK-short | 0 | 0 | 0 | 45 | 83 | 93 | |
| CCC-long | 0 | 0 | 0 | 83 | 100 | 100 | |
| CCC-short | 0 | 0 | 0 | 53 | 84 | 94 | |
| DQ-long | 0 | 0 | 0 | 96 | 98 | 98 | |
| DQ-short | 2 | 0 | 2 | 63 | 93 | 97 | |
| Fitted Model with Related Innovation Distribution | |||||||
| True Distribution | Measure | NORMD | STD |
GED |
SNORMD |
SSTD |
SGED |
| NORMD | AIC | 4775.04 | 4777.6 | 4775.96 | 4776.16 | 4778.76 | 4777.07 |
| UCK-long | 99 | 99 | 99 | 99 | 99 | 99 | |
| UCK-short | 100 | 100 | 100 | 99 | 99 | 99 | |
| CCC-long | 99 | 99 | 99 | 99 | 100 | 99 | |
| CCC-short | 99 | 99 | 99 | 99 | 99 | 99 | |
| DQ-long | 97 | 98 | 97 | 98 | 97 | 97 | |
| DQ-short | 96 | 97 | 96 | 97 | 97 | 97 | |
| STD | AIC | 4773.34 | 4587.04 | 4611.49 | 4772.06 | 4587.89 | 4611.93 |
| UCK-long | 75 | 98 | 77 | 75 | 100 | 78 | |
| UCK-short | 72 | 97 | 78 | 74 | 97 | 77 | |
| CCC-long | 81 | 99 | 86 | 83 | 98 | 89 | |
| CCC-short | 78 | 96 | 86 | 83 | 98 | 85 | |
| DQ-long | 97 | 97 | 98 | 94 | 96 | 96 | |
| DQ-short | 93 | 97 | 95 | 93 | 97 | 94 | |
| GED | AIC | 4765.93 | 4735.55 | 4730.36 | 4766.63 | 4736.53 | 4731.20 |
| UCK-long | 97 | 92 | 97 | 100 | 94 | 100 | |
| UCK-short | 99 | 95 | 97 | 99 | 95 | 100 | |
| CCC-long | 95 | 92 | 96 | 97 | 96 | 98 | |
| CCC-short | 97 | 97 | 99 | 98 | 98 | 100 | |
| DQ-long | 93 | 89 | 93 | 91 | 91 | 93 | |
| DQ-short | 98 | 98 | 96 | 100 | 98 | 98 | |
| SNORMD | AIC | 4718.43 | 4707.04 | 4718.33 | 4293.51 | 4296.50 | 4294.63 |
| UCK-long | 0 | 0 | 0 | 100 | 100 | 100 | |
| UCK-short | 0 | 0 | 0 | 97 | 94 | 100 | |
| CCC-long | 0 | 0 | 0 | 99 | 99 | 99 | |
| CCC-short | 0 | 0 | 0 | 96 | 94 | 98 | |
| DQ-long | 0 | 0 | 0 | 99 | 99 | 99 | |
| DQ-short | 4 | 3 | 5 | 97 | 98 | 98 | |
| SSTD | AIC | 4663.37 | 4336.06 | 4446.73 | 3968.73 | 3762.91 | 3786.37 |
| UCK-long | 0 | 0 | 0 | 4 | 100 | 100 | |
| UCK-short | 19 | 0 | 8 | 51 | 96 | 85 | |
| CCC-long | 0 | 0 | 0 | 9 | 97 | 96 | |
| CCC-short | 28 | 0 | 15 | 63 | 95 | 88 | |
| DQ-long | 0 | 0 | 0 | 43 | 97 | 97 | |
| DQ-short | 43 | 1 | 29 | 69 | 92 | 95 | |
| SGED | AIC | 4692.89 | 4616.31 | 4659.56 | 4113.75 | 4078.82 | 4072.33 |
| UCK-long | 0 | 0 | 0 | 72 | 100 | 100 | |
| UCK-short | 0 | 0 | 0 | 45 | 83 | 93 | |
| CCC-long | 0 | 0 | 0 | 83 | 100 | 100 | |
| CCC-short | 0 | 0 | 0 | 53 | 84 | 94 | |
| DQ-long | 0 | 0 | 0 | 96 | 98 | 98 | |
| DQ-short | 2 | 0 | 2 | 63 | 93 | 97 | |
Table 5.
Summary statistics of returns for NASDAQ index
| Minimum | 1st Quartile |
Median | Mean | 3rd Quartile |
Maximum |
| -12.04323 | -0.50381 | 0.10549 | 0.03624 | 0.65116 | 13.25464 |
| Standard deviation |
Skewness | Excess Kurtosis |
Jarque- Bera |
Q(10) | ARCH(10) |
| 1.376815 | -0.2339677 | 8.771298 | 18264*** (<2.2e-16) |
34.191*** (0.0001714) |
1137.7*** (<2.2e-16) |
| Notes: Q(10) is the Ljung and Box statistics of order 10 on the returns. ARCH(10) is the Lagrange Multiplier (LM) test of orders 10 (Engle, 1982). P-values of the statistics are reported in the parentheses. *Denote rejection of the null hypothesis at the 10% significance level. **Denote rejection of the null hypothesis at the 5% significance level. ***Denote rejection of the null hypothesis at the 1% significance level. |
|||||
| Minimum | 1st Quartile |
Median | Mean | 3rd Quartile |
Maximum |
| -12.04323 | -0.50381 | 0.10549 | 0.03624 | 0.65116 | 13.25464 |
| Standard deviation |
Skewness | Excess Kurtosis |
Jarque- Bera |
Q(10) | ARCH(10) |
| 1.376815 | -0.2339677 | 8.771298 | 18264*** (<2.2e-16) |
34.191*** (0.0001714) |
1137.7*** (<2.2e-16) |
| Notes: Q(10) is the Ljung and Box statistics of order 10 on the returns. ARCH(10) is the Lagrange Multiplier (LM) test of orders 10 (Engle, 1982). P-values of the statistics are reported in the parentheses. *Denote rejection of the null hypothesis at the 10% significance level. **Denote rejection of the null hypothesis at the 5% significance level. ***Denote rejection of the null hypothesis at the 1% significance level. |
|||||
Table 6.
AIC values of various GARCH-Type models for the returns of NASDAQ index
| NORMD | STD | GED | SNORMD | SSTD | SGED | |
| GARCH (1, 1) | 15814.45 | 15559.17 | 15602.50 | 15720.81 | 15509.92 | 15545.09 |
| GJRGARCH (1, 1) | 15762.01 | 15534.35 | 15574.46 | 15671.88 | 15482.61 | 15513.74 |
| TGARCH (1, 1) | 15737.59 | 15506.22 | 15552.22 | 15641.02 | 15452.94 | 15487.44 |
| BLGARCH (1, 1) | 15734.02 | 15512.67 | 15554.82 | 15641.02 | 15461.47 | 15492.78 |
| NORMD | STD | GED | SNORMD | SSTD | SGED | |
| GARCH (1, 1) | 15814.45 | 15559.17 | 15602.50 | 15720.81 | 15509.92 | 15545.09 |
| GJRGARCH (1, 1) | 15762.01 | 15534.35 | 15574.46 | 15671.88 | 15482.61 | 15513.74 |
| TGARCH (1, 1) | 15737.59 | 15506.22 | 15552.22 | 15641.02 | 15452.94 | 15487.44 |
| BLGARCH (1, 1) | 15734.02 | 15512.67 | 15554.82 | 15641.02 | 15461.47 | 15492.78 |
Table 7.
$ P $ -values for different backtests for the returns of NASDAQ index with varying $ \alpha $ levels under various models
| DIST | Model | GARCH(1, 1) | GJRGARCH (1, 1) |
TGARCH(1, 1) | BLGARCH (1, 1) |
||||||||
| Measure | 5% | 2.5% | 1% | 5% | 2.5% | 1% | 5% | 2.5% | 1% | 5% | 2.5% | 1% | |
| NORM | UCK-long | 0.008 | 0.000 | 0.000 | 0.065 | 0.001 | 0.000 | 0.074 | 0.000 | 0.000 | 0.172 | 0.004 | 0.001 |
| UCK-short | 0.000 | 0.003 | 0.028 | 0.000 | 0.004 | 0.028 | 0.000 | 0.004 | 0.079 | 0.000 | 0.006 | 0.057 | |
| CCC-long | 0.027 | 0.000 | 0.000 | 0.163 | 0.004 | 0.000 | 0.197 | 0.000 | 0.000 | 0.386 | 0.008 | 0.004 | |
| CCC-short | 0.000 | 0.006 | 0.051 | 0.001 | 0.014 | 0.051 | 0.000 | 0.008 | 0.137 | 0.001 | 0.011 | 0.101 | |
| DQ-long | 0.018 | 0.000 | 0.000 | 0.078 | 0.002 | 0.000 | 0.130 | 0.000 | 0.000 | 0.129 | 0.008 | 0.000 | |
| DQ-short | 0.000 | 0.121 | 0.360 | 0.019 | 0.175 | 0.056 | 0.010 | 0.062 | 0.495 | 0.049 | 0.071 | 0.612 | |
| STD | UCK-long | 0.000 | 0.001 | 0.045 | 0.001 | 0.001 | 0.264 | 0.001 | 0.001 | 0.133 | 0.003 | 0.002 | 0.133 |
| UCK-short | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.001 | 0.000 | 0.000 | |
| CCC-long | 0.000 | 0.001 | 0.081 | 0.003 | 0.002 | 0.513 | 0.003 | 0.001 | 0.011 | 0.012 | 0.004 | 0.317 | |
| CCC-short | 0.000 | 0.000 | 0.000 | 0.001 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.002 | 0.000 | 0.000 | |
| DQ-long | 0.000 | 0.000 | 0.000 | 0.004 | 0.000 | 0.157 | 0.000 | 0.000 | 0.000 | 0.016 | 0.001 | 0.084 | |
| DQ-short | 0.000 | 0.001 | 0.000 | 0.027 | 0.002 | 0.000 | 0.006 | 0.000 | 0.001 | 0.035 | 0.005 | 0.002 | |
| GED | UCK-long | 0.001 | 0.023 | 0.264 | 0.032 | 0.019 | 0.390 | 0.032 | 0.036 | 0.390 | 0.043 | 0.015 | 0.323 |
| UCK-short | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | |
| CCC-long | 0.005 | 0.033 | 0.251 | 0.085 | 0.057 | 0.650 | 0.095 | 0.044 | 0.298 | 0.105 | 0.024 | 0.582 | |
| CCC-short | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | |
| DQ-long | 0.003 | 0.000 | 0.003 | 0.076 | 0.019 | 0.240 | 0.016 | 0.009 | 0.053 | 0.085 | 0.017 | 0.186 | |
| DQ-short | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.002 | 0.002 | 0.000 | 0.000 | |
| SNORM | UCK-long | 0.563 | 0.125 | 0.045 | 0.641 | 0.263 | 0.390 | 0.597 | 0.595 | 0.550 | 0.334 | 0.341 | 0.550 |
| UCK-short | 0.159 | 0.782 | 0.080 | 0.474 | 0.341 | 0.013 | 0.554 | 0.263 | 0.080 | 0.779 | 0.229 | 0.104 | |
| CCC-long | 0.715 | 0.237 | 0.081 | 0.779 | 0.532 | 0.650 | 0.613 | 0.272 | 0.329 | 0.559 | 0.570 | 0.769 | |
| CCC-short | 0.363 | 0.486 | 0.040 | 0.772 | 0.249 | 0.030 | 0.839 | 0.227 | 0.213 | 0.881 | 0.214 | 0.145 | |
| DQ-long | 0.150 | 0.017 | 0.001 | 0.431 | 0.162 | 0.247 | 0.085 | 0.141 | 0.060 | 0.397 | 0.102 | 0.260 | |
| DQ-short | 0.539 | 0.870 | 0.044 | 0.930 | 0.310 | 0.045 | 0.629 | 0.248 | 0.297 | 0.962 | 0.311 | 0.289 | |
| SSTD | UCK-long | 0.213 | 0.744 | 0.300 | 0.832 | 0.849 | 0.079 | 0.605 | 0.811 | 0.300 | 0.832 | 0.878 | 0.079 |
| UCK-short | 0.554 | 0.348 | 0.057 | 0.738 | 0.744 | 0.143 | 0.785 | 0.617 | 0.057 | 0.785 | 0.849 | 0.143 | |
| CCC-long | 0.403 | 0.364 | 0.116 | 0.736 | 0.958 | 0.137 | 0.825 | 0.389 | 0.116 | 0.584 | 0.412 | 0.137 | |
| CCC-short | 0.839 | 0.540 | 0.021 | 0.934 | 0.630 | 0.055 | 0.741 | 0.555 | 0.101 | 0.947 | 0.754 | 0.233 | |
| DQ-long | 0.003 | 0.007 | 0.022 | 0.233 | 0.050 | 0.345 | 0.089 | 0.014 | 0.168 | 0.307 | 0.051 | 0.336 | |
| DQ-short | 0.227 | 0.897 | 0.082 | 0.715 | 0.765 | 0.030 | 0.267 | 0.241 | 0.140 | 0.807 | 0.865 | 0.692 | |
| SGED | UCK-long | 0.832 | 0.229 | 0.537 | 0.436 | 0.348 | 0.040 | 0.436 | 0.196 | 0.186 | 0.334 | 0.141 | 0.057 |
| UCK-short | 0.097 | 0.082 | 0.028 | 0.474 | 0.265 | 0.107 | 0.400 | 0.196 | 0.186 | 0.597 | 0.500 | 0.143 | |
| CCC-long | 0.736 | 0.234 | 0.200 | 0.588 | 0.540 | 0.073 | 0.588 | 0.097 | 0.072 | 0.616 | 0.332 | 0.101 | |
| CCC-short | 0.247 | 0.152 | 0.009 | 0.725 | 0.269 | 0.041 | 0.643 | 0.337 | 0.293 | 0.830 | 0.472 | 0.233 | |
| DQ-long | 0.017 | 0.056 | 0.037 | 0.311 | 0.175 | 0.293 | 0.043 | 0.009 | 0.127 | 0.408 | 0.342 | 0.315 | |
| DQ-short | 0.436 | 0.723 | 0.042 | 0.923 | 0.538 | 0.021 | 0.617 | 0.518 | 0.351 | 0.868 | 0.536 | 0.074 | |
| DIST | Model | GARCH(1, 1) | GJRGARCH (1, 1) |
TGARCH(1, 1) | BLGARCH (1, 1) |
||||||||
| Measure | 5% | 2.5% | 1% | 5% | 2.5% | 1% | 5% | 2.5% | 1% | 5% | 2.5% | 1% | |
| NORM | UCK-long | 0.008 | 0.000 | 0.000 | 0.065 | 0.001 | 0.000 | 0.074 | 0.000 | 0.000 | 0.172 | 0.004 | 0.001 |
| UCK-short | 0.000 | 0.003 | 0.028 | 0.000 | 0.004 | 0.028 | 0.000 | 0.004 | 0.079 | 0.000 | 0.006 | 0.057 | |
| CCC-long | 0.027 | 0.000 | 0.000 | 0.163 | 0.004 | 0.000 | 0.197 | 0.000 | 0.000 | 0.386 | 0.008 | 0.004 | |
| CCC-short | 0.000 | 0.006 | 0.051 | 0.001 | 0.014 | 0.051 | 0.000 | 0.008 | 0.137 | 0.001 | 0.011 | 0.101 | |
| DQ-long | 0.018 | 0.000 | 0.000 | 0.078 | 0.002 | 0.000 | 0.130 | 0.000 | 0.000 | 0.129 | 0.008 | 0.000 | |
| DQ-short | 0.000 | 0.121 | 0.360 | 0.019 | 0.175 | 0.056 | 0.010 | 0.062 | 0.495 | 0.049 | 0.071 | 0.612 | |
| STD | UCK-long | 0.000 | 0.001 | 0.045 | 0.001 | 0.001 | 0.264 | 0.001 | 0.001 | 0.133 | 0.003 | 0.002 | 0.133 |
| UCK-short | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.001 | 0.000 | 0.000 | |
| CCC-long | 0.000 | 0.001 | 0.081 | 0.003 | 0.002 | 0.513 | 0.003 | 0.001 | 0.011 | 0.012 | 0.004 | 0.317 | |
| CCC-short | 0.000 | 0.000 | 0.000 | 0.001 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.002 | 0.000 | 0.000 | |
| DQ-long | 0.000 | 0.000 | 0.000 | 0.004 | 0.000 | 0.157 | 0.000 | 0.000 | 0.000 | 0.016 | 0.001 | 0.084 | |
| DQ-short | 0.000 | 0.001 | 0.000 | 0.027 | 0.002 | 0.000 | 0.006 | 0.000 | 0.001 | 0.035 | 0.005 | 0.002 | |
| GED | UCK-long | 0.001 | 0.023 | 0.264 | 0.032 | 0.019 | 0.390 | 0.032 | 0.036 | 0.390 | 0.043 | 0.015 | 0.323 |
| UCK-short | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | |
| CCC-long | 0.005 | 0.033 | 0.251 | 0.085 | 0.057 | 0.650 | 0.095 | 0.044 | 0.298 | 0.105 | 0.024 | 0.582 | |
| CCC-short | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | |
| DQ-long | 0.003 | 0.000 | 0.003 | 0.076 | 0.019 | 0.240 | 0.016 | 0.009 | 0.053 | 0.085 | 0.017 | 0.186 | |
| DQ-short | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.002 | 0.002 | 0.000 | 0.000 | |
| SNORM | UCK-long | 0.563 | 0.125 | 0.045 | 0.641 | 0.263 | 0.390 | 0.597 | 0.595 | 0.550 | 0.334 | 0.341 | 0.550 |
| UCK-short | 0.159 | 0.782 | 0.080 | 0.474 | 0.341 | 0.013 | 0.554 | 0.263 | 0.080 | 0.779 | 0.229 | 0.104 | |
| CCC-long | 0.715 | 0.237 | 0.081 | 0.779 | 0.532 | 0.650 | 0.613 | 0.272 | 0.329 | 0.559 | 0.570 | 0.769 | |
| CCC-short | 0.363 | 0.486 | 0.040 | 0.772 | 0.249 | 0.030 | 0.839 | 0.227 | 0.213 | 0.881 | 0.214 | 0.145 | |
| DQ-long | 0.150 | 0.017 | 0.001 | 0.431 | 0.162 | 0.247 | 0.085 | 0.141 | 0.060 | 0.397 | 0.102 | 0.260 | |
| DQ-short | 0.539 | 0.870 | 0.044 | 0.930 | 0.310 | 0.045 | 0.629 | 0.248 | 0.297 | 0.962 | 0.311 | 0.289 | |
| SSTD | UCK-long | 0.213 | 0.744 | 0.300 | 0.832 | 0.849 | 0.079 | 0.605 | 0.811 | 0.300 | 0.832 | 0.878 | 0.079 |
| UCK-short | 0.554 | 0.348 | 0.057 | 0.738 | 0.744 | 0.143 | 0.785 | 0.617 | 0.057 | 0.785 | 0.849 | 0.143 | |
| CCC-long | 0.403 | 0.364 | 0.116 | 0.736 | 0.958 | 0.137 | 0.825 | 0.389 | 0.116 | 0.584 | 0.412 | 0.137 | |
| CCC-short | 0.839 | 0.540 | 0.021 | 0.934 | 0.630 | 0.055 | 0.741 | 0.555 | 0.101 | 0.947 | 0.754 | 0.233 | |
| DQ-long | 0.003 | 0.007 | 0.022 | 0.233 | 0.050 | 0.345 | 0.089 | 0.014 | 0.168 | 0.307 | 0.051 | 0.336 | |
| DQ-short | 0.227 | 0.897 | 0.082 | 0.715 | 0.765 | 0.030 | 0.267 | 0.241 | 0.140 | 0.807 | 0.865 | 0.692 | |
| SGED | UCK-long | 0.832 | 0.229 | 0.537 | 0.436 | 0.348 | 0.040 | 0.436 | 0.196 | 0.186 | 0.334 | 0.141 | 0.057 |
| UCK-short | 0.097 | 0.082 | 0.028 | 0.474 | 0.265 | 0.107 | 0.400 | 0.196 | 0.186 | 0.597 | 0.500 | 0.143 | |
| CCC-long | 0.736 | 0.234 | 0.200 | 0.588 | 0.540 | 0.073 | 0.588 | 0.097 | 0.072 | 0.616 | 0.332 | 0.101 | |
| CCC-short | 0.247 | 0.152 | 0.009 | 0.725 | 0.269 | 0.041 | 0.643 | 0.337 | 0.293 | 0.830 | 0.472 | 0.233 | |
| DQ-long | 0.017 | 0.056 | 0.037 | 0.311 | 0.175 | 0.293 | 0.043 | 0.009 | 0.127 | 0.408 | 0.342 | 0.315 | |
| DQ-short | 0.436 | 0.723 | 0.042 | 0.923 | 0.538 | 0.021 | 0.617 | 0.518 | 0.351 | 0.868 | 0.536 | 0.074 | |
Table 8.
VaR measures for the returns of NASDAQ index under various models (in-sample)
| Model | GARCH (1, 1) |
GJRGARCH (1, 1) |
TGARCH (1, 1) |
BLGARCH (1, 1) |
||||
| α% | 5% Long |
5% Short |
5% Long |
5% Short |
5% Long |
5% Short |
5% Long |
5% Short |
| NORM | 322 | 204 | 308 | 218 | 307 | 217 | 300 | 220 |
| Diff | 44 | 74 | 30 | 60 | 29 | 61 | 22 | 58 |
| STD | 346 | 207 | 334 | 220 | 335 | 214 | 327 | 223 |
| Diff | 68 | 71 | 56 | 58 | 57 | 64 | 49 | 55 |
| GED | 332 | 193 | 313 | 197 | 313 | 193 | 311 | 203 |
| Diff | 54 | 85 | 35 | 81 | 35 | 85 | 33 | 75 |
| SNORM | 287 | 255 | 270 | 266 | 269 | 268 | 262 | 273 |
| Diff | 9 | 23 | 8 | 12 | 9 | 10 | 16 | 5 |
| SSTD | 298 | 268 | 281 | 283 | 286 | 282 | 281 | 282 |
| Diff | 20 | 10 | 3 | 5 | 8 | 4 | 3 | 4 |
| SGED | 281 | 251 | 265 | 266 | 265 | 264 | 262 | 269 |
| Diff | 3 | 27 | 13 | 12 | 13 | 14 | 16 | 9 |
| Notes: Expected exceed = 5551 × 0:05 ≈ 278 and Diff = |actual exceed − expected exceed|. | ||||||||
| Model | GARCH (1, 1) |
GJRGARCH (1, 1) |
TGARCH (1, 1) |
BLGARCH (1, 1) |
||||
| α% | 5% Long |
5% Short |
5% Long |
5% Short |
5% Long |
5% Short |
5% Long |
5% Short |
| NORM | 322 | 204 | 308 | 218 | 307 | 217 | 300 | 220 |
| Diff | 44 | 74 | 30 | 60 | 29 | 61 | 22 | 58 |
| STD | 346 | 207 | 334 | 220 | 335 | 214 | 327 | 223 |
| Diff | 68 | 71 | 56 | 58 | 57 | 64 | 49 | 55 |
| GED | 332 | 193 | 313 | 197 | 313 | 193 | 311 | 203 |
| Diff | 54 | 85 | 35 | 81 | 35 | 85 | 33 | 75 |
| SNORM | 287 | 255 | 270 | 266 | 269 | 268 | 262 | 273 |
| Diff | 9 | 23 | 8 | 12 | 9 | 10 | 16 | 5 |
| SSTD | 298 | 268 | 281 | 283 | 286 | 282 | 281 | 282 |
| Diff | 20 | 10 | 3 | 5 | 8 | 4 | 3 | 4 |
| SGED | 281 | 251 | 265 | 266 | 265 | 264 | 262 | 269 |
| Diff | 3 | 27 | 13 | 12 | 13 | 14 | 16 | 9 |
| Notes: Expected exceed = 5551 × 0:05 ≈ 278 and Diff = |actual exceed − expected exceed|. | ||||||||
Table 9.
Parameter estimates based on AIC and VaR backtests criteria for the selected GARCH-Type models using the returns of NASDAQ index
| AIC | VaR | |
| Model | TGARCH-SSTD | BLGARCH-SSTD |
| 0.043*** [0.008] | 0.041*** [0.011] | |
| 0.147*** [0.014] | 0.153*** [0.014] | |
| 0.011*** [0.002] | 0.012*** [0.003] | |
| - | 0.111*** [0.012] | |
| 0.067*** [0.009] | - | |
| 0.134*** [0.013] | - | |
| 0.913*** [0.009] | 0.887*** [0.011] | |
| - | - 0.079*** [0.013] | |
| 8.043*** [0.787] | 8.175*** [0.813] | |
| 0.866*** [0.017] | 0.869*** [0.017] | |
| Notes: the numbers in square brackets are the standard error of the estimates. *Denote rejection of the null hypothesis at the 10% significance level. **Denote rejection of the null hypothesis at the 5% significance level. ***Denote rejection of the null hypothesis at the 1% significance level. |
||
| AIC | VaR | |
| Model | TGARCH-SSTD | BLGARCH-SSTD |
| 0.043*** [0.008] | 0.041*** [0.011] | |
| 0.147*** [0.014] | 0.153*** [0.014] | |
| 0.011*** [0.002] | 0.012*** [0.003] | |
| - | 0.111*** [0.012] | |
| 0.067*** [0.009] | - | |
| 0.134*** [0.013] | - | |
| 0.913*** [0.009] | 0.887*** [0.011] | |
| - | - 0.079*** [0.013] | |
| 8.043*** [0.787] | 8.175*** [0.813] | |
| 0.866*** [0.017] | 0.869*** [0.017] | |
| Notes: the numbers in square brackets are the standard error of the estimates. *Denote rejection of the null hypothesis at the 10% significance level. **Denote rejection of the null hypothesis at the 5% significance level. ***Denote rejection of the null hypothesis at the 1% significance level. |
||
Table 10.
Out-of-sample volatility forecasting evaluated under the six criteria of loss functions for returns of NASDAQ index
| MAE | MSE | MAPE | HMAE | HMSE | LL | |
| TGARCH (1, 1)-SSTD | 0.9109 | 1.9433 | 874.463 | 1.0313 | 2.5931 | 6.9049 |
| BLGARCH (1, 1)-SSTD | 0.9151 | 1.9831 | 513.787 | 1.0589 | 2.8460 | 6.4131 |
| MAE | MSE | MAPE | HMAE | HMSE | LL | |
| TGARCH (1, 1)-SSTD | 0.9109 | 1.9433 | 874.463 | 1.0313 | 2.5931 | 6.9049 |
| BLGARCH (1, 1)-SSTD | 0.9151 | 1.9831 | 513.787 | 1.0589 | 2.8460 | 6.4131 |
Table 11.
Out-of-sample VaR for various tests of the returns of NASDAQ index
| α% | 5% | 2.5% | 1% | |||
| Position | Long | Short | Long | Short | Long | Short |
| TGARCH-SSTD | ||||||
| UCK-Test | 0.9178 | 0.7625 | 0.9425 | 0.3224 | 0.8159 | 0.5351 |
| CCC-Test | 0.7330 | 0.6881 | 0.9259 | 0.5184 | 0.9653 | 0.8117 |
| DQ-Test | 0.8268 | 0.9681 | 0.9968 | 0.8134 | 0.9960 | 0.9913 |
| BLGARCH-SSTD | ||||||
| UCK-Test | 0.9176 | 0.9176 | 0.9425 | 0.3224 | 0.5352 | 0.5352 |
| CCC-Test | 0.7330 | 0.7721 | 0.9259 | 0.5184 | 0.7985 | 0.8117 |
| DQ-Test | 0.8843 | 0.9857 | 0.9935 | 0.8551 | 0.8960 | 0.9918 |
| α% | 5% | 2.5% | 1% | |||
| Position | Long | Short | Long | Short | Long | Short |
| TGARCH-SSTD | ||||||
| UCK-Test | 0.9178 | 0.7625 | 0.9425 | 0.3224 | 0.8159 | 0.5351 |
| CCC-Test | 0.7330 | 0.6881 | 0.9259 | 0.5184 | 0.9653 | 0.8117 |
| DQ-Test | 0.8268 | 0.9681 | 0.9968 | 0.8134 | 0.9960 | 0.9913 |
| BLGARCH-SSTD | ||||||
| UCK-Test | 0.9176 | 0.9176 | 0.9425 | 0.3224 | 0.5352 | 0.5352 |
| CCC-Test | 0.7330 | 0.7721 | 0.9259 | 0.5184 | 0.7985 | 0.8117 |
| DQ-Test | 0.8843 | 0.9857 | 0.9935 | 0.8551 | 0.8960 | 0.9918 |
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