Journal of the Operations Research Society of China >

2026 , Vol. 14 >Issue 2: 413 - 451

DOI: https://doi.org/10.1007/s40305-024-00564-x

Forecasting VaR and Returns Distribution Using the Real-Time GARCH Models with Standardized Two-Sided Lindley Distribution

Expand
  • Department of Statistics, Hangzhou City University, Hangzhou 310015, Zhejiang, China

Received date: 2023-11-21

  Revised date: 2024-08-27

  Online published: 2026-07-06

Supported by

The second author (Guang-Hui Cai) was supported by the National Social Science Foundation of China (No. 19BTJ013).

Fold

Abstract

The Real-time GARCH models which see the conditional volatility of financial returns as a mixture of past and current information have been confirmed as more effective models on modeling and forecasting financial volatility and risks, thus attracting abundant attention. However, the existing Real-time GARCH models only assume innovations follow the standard Normal distribution and fail to take into account the asymmetry of its distribution. In this paper, we consider the standardized two-sided Lindley (STSL) distribution as the distribution of innovations and then propose the Real-time GARCH-STSL models to estimate and forecast conditional density and VaR of financial returns. Besides, some important properties of the new models are discussed including the conditional density function and VaR of returns, the conditions of weak stationarity, and the maximum likelihood estimation method. Furthermore, Monte Carlo simulations are also conducted to observe the asymptotic performance of these proposed models. Finally, empirical results show that the new models are superior to competitors considering the Normal distribution for innovations in terms of the in-sample empirical fitting, out-of-sample multi-step-ahead returns density and extreme VaR predictions.

Key words： Real-time GARCH; Standardized two-sided Lindley distribution; VaR forecasts; Return density forecasts

Cite this article

Zhi-Min Wu, Guang-Hui Cai . Forecasting VaR and Returns Distribution Using the Real-Time GARCH Models with Standardized Two-Sided Lindley Distribution[J]. Journal of the Operations Research Society of China, 2026 , 14(2) : 413 -451 . DOI: 10.1007/s40305-024-00564-x

References

[1] Altun, E.: Two-sided Lindley distribution with inference and applications. J. Indian. Soc. Prob. Stat. 20, 255–279(2019)
[2] Asai, M., Chang, C.L., McAleer, M.: Realized stochastic volatility with general asymmetry and long memory. J. Econom. 199, 202–212(2017)
[3] Asai, M., Chang, C.L., McAleer, M.: Realized matrix-exponential stochastic volatility with asymmetry, long memory and higher-moment spillovers. J. Econom. 227, 285–304(2022)
[4] Bekaert, G., Engstrom, E., Ermolov, A.: Bad environments, good environments: a non-Gaussian asymmetric volatility model. J. Econom. 186, 258–275(2015)
[5] Berkowitz, J.: Testing density forecasts, with applications to risk management. J. Bus. Econ. Stat. 19, 465–474(2001)
[6] Bollerslev, T.: Generalized autoregressive conditional heteroskedasticity. J. Econom. 31, 307–327(1986)
[7] Bollerslev, T., Patton, A.J., Quaedvlieg, R.: Multivariate leverage effects and realized semicovariance GARCH models. J. Econom. 217, 411–430(2020)
[8] Cai, G.H., Wu, Z.M., Peng, L.: Forecasting volatility with outliers in realized GARCH models. J. Forecast. 40, 667–685(2021)
[9] Christoffersen, P.F.: Evaluating interval forecasts. Int. Econ. Rev. 39, 841–862(1998)
[10] Diebold, F.X., Lunde, A.: Comparing predictive accuracy. J. Bus. Econ. Stat. 13, 253–263(1995)
[11] Ding, Y.D.: A simple joint model for returns, volatility and volatility of volatility. J. Econom. 232, 521–541(2023)
[12] Ding, Z., Granger, C.W.J., Engle, R.F.: A long memory property of stock market returns and a new model. J. Empir. Finance 1, 83–106(1993)
[13] Fiszeder, P., Faldzinski, M., Molnár, P.: Range-based DCC models for covariance and value-at-risk forecasting. J. Empir. Finance 54, 58–76(2019)
[14] Ghysels, E., Harvey, A.C., Renault, E.: 5 Stochastic volatility. Handb. Stat. 14, 119–191(1996)
[15] Glosten, L.R., Jagannathan, R., Runkle, D.E.: On the relation between the expected value and the volatility of the nominal excess return on stocks. J. Finance 48, 1779–1801(1993)
[16] Hansen, P.R., Huang, Z.: Exponential GARCH modeling with realized measures of volatility. J. Bus. Econ. Stat. 34, 267–287(2016)
[17] Hansen, P.R., Huang, Z., Shek, H.H.: Realized GARCH: a joint model for returns and realized measures of volatility. J. Appl. Econom. 27, 877–906(2012)
[18] Hansen, P.R., Lunde, A.: A forecast comparison of volatility models: Does anything beat a GARCH (1,1)? J. Appl. Econom. 20, 873–889(2005)
[19] Hansen, P.R., Lunde, A., Nason, J.M.: The model confidence set. Econometrica 79, 453–497(2011)
[20] Kupiec, P.H.: Techniques for verifying the accuracy of risk measurement models. J. Deriv. 3, 73–84(1995)
[21] Lee, H.T., Lee, C.C.: A regime-switching real-time copula GARCH model for optimal futures hedging. Int. Rev. Financ. Anal. 84, 102395(2022)
[22] Li, C.W., Li, W.K.: On a double-threshold autoregressive heteroscedastic time series model. J. Appl. Econom. 11, 253–274(1996)
[23] Louzis, D.P., Xanthopoulos-Sisinis, S., Refenes, A.P.: Realized volatility models and alternative valueat-risk prediction strategies. Econ. Model. 40, 101–116(2014)
[24] Matheson, J.E., Winkler, R.L.: Scoring rules for continuous probability distributions. Manag. Sci. 22, 1087–1096(1976)
[25] Nelson, D.B.: Conditional heteroskedasticity in asset returns: a new approach. Econometrica 59, 347–370(1991)
[26] Pan, J., Wang, H., Tong, H.: Estimation and tests for power-transformed and threshold GARCH models. J. Econom. 142, 352–378(2008)
[27] Sarma, M., Thomas, S., Shah, A.: Selection of value-at-risk models. J. Forecast. 22, 337–358(2003)
[28] Shephard, N., Sheppard, K.: Realising the future: forecasting with high-frequency-based volatility (HEAVY) models. J. Appl. Econom. 25, 197–231(2010)
[29] So, M.K.P., Li, W.K., Lam, K.: A threshold stochastic volatility model. J. Forecast. 21, 473–500(2002)
[30] Shirota, S., Hizu, T., Omori, Y.: Realized stochastic volatility with leverage and long memory. Comput. Stat. Data Anal. 76, 618–641(2014)
[31] Smetanina, E.: Real-time GARCH. J. Financ. Econom. 15, 561–601(2017)
[32] Smetanina, E., Wu, W.B.: Asymptotic theory for QMLE for the real-time GARCH (1,1) model. J. Time. Ser. Anal. 42, 752–776(2021)
[33] Takahashi, M., Watanabe, T., Omori, Y.: Volatility and quantile forecasts by realized stochastic volatility models with generalized hyperbolic distribution. Int. J. Forecast. 32, 437–457(2016)
[34] Wu, X., Zhao, A., Cheng, T.: A real-time GARCH-MIDAS model. Finance Res. Lett. 56, 104103(2023)
[35] Yu, J.: On leverage in a stochastic volatility model. J. Econom. 127, 165–178(2005)
Options
Outlines

/

〈 〉

Copyright © Operations Research Society of China, The Periodicals Agency of Shanghai University , and Springer-Verlag Berlin Heidelberg 2016