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

* 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.

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. AlbergH. 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. AngelidisA. 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. BallieT. 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

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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

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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. ChoiJ. 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

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P. F. Christoffersen, Evaluating interval forecasts, International Economic Review, 39 (1998), 841-862.  doi: 10.2307/2527341.  Google Scholar

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Z. DingC. 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

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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

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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

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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

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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

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L. R. GlostenR. 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. KosapattarapimY. 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. KuesterS. 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. LyuP. WangY. 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. McMillanS. Alan and A. Owain, Forecasting UK stock market volatility, Applied Financial Economics, 10 (2000), 435-448.  doi: 10.1080/09603100050031561.  Google Scholar

[30]

S. NadarajahE. 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. SmolovićM. 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. AlbergH. 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. AngelidisA. 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. BallieT. 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. ChoiJ. 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. DingC. 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. GlostenR. 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. KosapattarapimY. 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. KuesterS. 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. LyuP. WangY. 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. McMillanS. Alan and A. Owain, Forecasting UK stock market volatility, Applied Financial Economics, 10 (2000), 435-448.  doi: 10.1080/09603100050031561.  Google Scholar

[30]

S. NadarajahE. 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. SmolovićM. 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.
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
$ \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
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
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
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
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.
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
$ \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.
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
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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