Model selection based on value-at-risk backtesting approach for GARCH-Type models

Hao-Zhe Tay; Kok-Haur Ng; You-Beng Koh; Kooi-Huat Ng

doi:10.3934/jimo.2019021

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doi: 10.3934/jimo.2019021

Model selection based on value-at-risk backtesting approach for GARCH-Type models

Hao-Zhe Tay ^1, , Kok-Haur Ng ^1,, , You-Beng Koh ^1, and Kooi-Huat Ng ^2,

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

^* Corresponding author: Ng, Kok-Haur

Received August 2018 Revised October 2018 Published March 2019

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.

Keywords: Model selection, backtest, value-at-risk, GARCH model, skewed distribution.

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 & Management Optimization, doi: 10.3934/jimo.2019021

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. Google Scholar
[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. Google Scholar
[3]	A. Azzalini, A class of distributions which includes the normal ones, Scandinavian Journal of Statistics, 12 (1985), 171-178. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[7]	T. Bollerslev, Generalized autoregressive conditional heteroskedasticity, Journal of Econometrics, 31 (1986), 307-327. doi: 10.1016/0304-4076(86)90063-1. Google Scholar
[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. Google Scholar
[9]	T. Bollerslev and E. Ghysels, Periodic autoregressive conditional heteroscedasticity, Journal of Business and Economic Statisticss, 14 (1996), 139-151. Google Scholar
[10]	M. Braione and N. K. Scholtes, Forecasting value-at-risk under different distributional assumptions, Econometrics, 4 (2016), 1-27. doi: 10.3390/econometrics4010003. Google Scholar
[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. Google Scholar
[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. Google Scholar
[13]	P. F. Christoffersen, Evaluating interval forecasts, International Economic Review, 39 (1998), 841-862. doi: 10.2307/2527341. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[28]	B. Mandelbrot, The variation of certain speculative prices, Journal of Business, 36 (1963), 394-419. Google Scholar
[29]	D. McMillan, S. Alan and A. Owain, Forecasting UK stock market volatility, Applied Financial Economics, 10 (2000), 435-448. doi: 10.1080/09603100050031561. Google Scholar
[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. Google Scholar
[31]	D. B. Nelson, Conditional heteroskedasticity in asset returns: A new approach, Econometrica, 59 (1991), 347-370. doi: 10.2307/2938260. Google Scholar
[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. Google Scholar
[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. Google Scholar
[34]	E. Sentana, Quadratic ARCH models, Review of Economic Studies, 62 (1995), 639-661. doi: 10.2307/2298081. Google Scholar
[35]	O. Scaillet, Nonparametric estimation of conditional expected shortfall, Insurance and Risk Management Journal, 74 (2005), 639-660. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[39]	R. S. Tsay, Analysis of Financial Time Series, 3$ ^{rd} $ edition, John Wiley & Sons, Hoboken, New Jersey, 2010. doi: 10.1002/9780470644560. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

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. Google Scholar
[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. Google Scholar
[3]	A. Azzalini, A class of distributions which includes the normal ones, Scandinavian Journal of Statistics, 12 (1985), 171-178. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[7]	T. Bollerslev, Generalized autoregressive conditional heteroskedasticity, Journal of Econometrics, 31 (1986), 307-327. doi: 10.1016/0304-4076(86)90063-1. Google Scholar
[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. Google Scholar
[9]	T. Bollerslev and E. Ghysels, Periodic autoregressive conditional heteroscedasticity, Journal of Business and Economic Statisticss, 14 (1996), 139-151. Google Scholar
[10]	M. Braione and N. K. Scholtes, Forecasting value-at-risk under different distributional assumptions, Econometrics, 4 (2016), 1-27. doi: 10.3390/econometrics4010003. Google Scholar
[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. Google Scholar
[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. Google Scholar
[13]	P. F. Christoffersen, Evaluating interval forecasts, International Economic Review, 39 (1998), 841-862. doi: 10.2307/2527341. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[28]	B. Mandelbrot, The variation of certain speculative prices, Journal of Business, 36 (1963), 394-419. Google Scholar
[29]	D. McMillan, S. Alan and A. Owain, Forecasting UK stock market volatility, Applied Financial Economics, 10 (2000), 435-448. doi: 10.1080/09603100050031561. Google Scholar
[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. Google Scholar
[31]	D. B. Nelson, Conditional heteroskedasticity in asset returns: A new approach, Econometrica, 59 (1991), 347-370. doi: 10.2307/2938260. Google Scholar
[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. Google Scholar
[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. Google Scholar
[34]	E. Sentana, Quadratic ARCH models, Review of Economic Studies, 62 (1995), 639-661. doi: 10.2307/2298081. Google Scholar
[35]	O. Scaillet, Nonparametric estimation of conditional expected shortfall, Insurance and Risk Management Journal, 74 (2005), 639-660. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[39]	R. S. Tsay, Analysis of Financial Time Series, 3$ ^{rd} $ edition, John Wiley & Sons, Hoboken, New Jersey, 2010. doi: 10.1002/9780470644560. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

Figure 1. Time series plots for the prices of NASDAQ index and returns of NASDAQ index.

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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)

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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)

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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)

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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 $ \left(v=5\right)$	GED $ \left(v=1.5\right)$	SNORMD $ \left(\xi=2.5\right)$	SSTD $ \left(\xi=2.5\right)$	SGED $ \left(v, \xi=1.5, 2.5\right)$
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

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		Fitted Model with Related Innovation Distribution
True Dis-tribution	Measure	NORMD	STD $ \left(v=5\right)$	GED $ \left(v=1.5\right)$	SNORMD $ \left(\xi=2.5\right)$	SSTD $ \left(\xi=2.5\right)$	SGED $ \left(v, \xi=1.5, 2.5\right)$
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 $ \left(v=5\right)$	GED $ \left(v=1.5\right)$	SNORMD $ \left(\xi=2.5\right)$	SSTD $ \left(\xi=2.5\right)$	SGED $ \left(v, \xi=1.5, 2.5\right)$
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

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		Fitted Model with Related Innovation Distribution
True Distribution	Measure	NORMD	STD $ \left(v=5\right)$	GED $ \left(v=1.5\right)$	SNORMD $ \left(\xi=2.5\right)$	SSTD $ \left(\xi=2.5\right)$	SGED $ \left(v, \xi=1.5, 2.5\right)$
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 $ \left(v=5\right) $	GED $ \left(v=1.5\right) $	SNORMD $ \left(\xi=2.5\right) $	SSTD $ \left(\xi=2.5\right) $	SGED $ \left(v, \xi=1.5, 2.5\right) $
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

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		Fitted Model with Related Innovation Distribution
True Distribution	Measure	NORMD	STD $ \left(v=5\right) $	GED $ \left(v=1.5\right) $	SNORMD $ \left(\xi=2.5\right) $	SSTD $ \left(\xi=2.5\right) $	SGED $ \left(v, \xi=1.5, 2.5\right) $
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 $ \left(v=5\right)$	GED $ \left(v=1.5\right)$	SNORMD $ \left(\xi=2.5\right)$	SSTD $ \left(\xi=2.5\right)$	SGED $ \left(v, \xi=1.5, 2.5\right)$
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

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		Fitted Model with Related Innovation Distribution
True Distribution	Measure	NORMD	STD $ \left(v=5\right)$	GED $ \left(v=1.5\right)$	SNORMD $ \left(\xi=2.5\right)$	SSTD $ \left(\xi=2.5\right)$	SGED $ \left(v, \xi=1.5, 2.5\right)$
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.

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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

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	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

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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\|.

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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
$ \mu$	0.043^*** [0.008]	0.041^*** [0.011]
$ \phi$	0.147^*** [0.014]	0.153^*** [0.014]
$ \alpha_{0} $	0.011^*** [0.002]	0.012^*** [0.003]
$ \alpha_{1} $	-	0.111^*** [0.012]
$ \alpha_{1}^+ $	0.067^*** [0.009]	-
$ \alpha_{1}^- $	0.134^*** [0.013]	-
$ \beta_{1} $	0.913^*** [0.009]	0.887^*** [0.011]
$ \zeta_{1} $	-	- 0.079^*** [0.013]
$ v $	8.043^*** [0.787]	8.175^*** [0.813]
$ \xi $	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.

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	AIC	VaR
Model	TGARCH-SSTD	BLGARCH-SSTD
$ \mu$	0.043^*** [0.008]	0.041^*** [0.011]
$ \phi$	0.147^*** [0.014]	0.153^*** [0.014]
$ \alpha_{0} $	0.011^*** [0.002]	0.012^*** [0.003]
$ \alpha_{1} $	-	0.111^*** [0.012]
$ \alpha_{1}^+ $	0.067^*** [0.009]	-
$ \alpha_{1}^- $	0.134^*** [0.013]	-
$ \beta_{1} $	0.913^*** [0.009]	0.887^*** [0.011]
$ \zeta_{1} $	-	- 0.079^*** [0.013]
$ v $	8.043^*** [0.787]	8.175^*** [0.813]
$ \xi $	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

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	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

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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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American Institute of Mathematical Sciences

Model selection based on value-at-risk backtesting approach for GARCH-Type models

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Model selection based on value-at-risk backtesting approach for GARCH-Type models

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