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

doi:10.1007/s40305-024-00564-x

Journal of the Operations Research Society of China ›› 2026, Vol. 14 ›› Issue (2): 413-451.doi: 10.1007/s40305-024-00564-x

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

Zhi-Min Wu, Guang-Hui Cai

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

Received:2023-11-21 Revised:2024-08-27 Online:2026-06-30 Published:2026-07-06
Contact: Zhi-Min Wu E-mail:zhiminwu614@163.com
Supported by:
The second author (Guang-Hui Cai) was supported by the National Social Science Foundation of China (No. 19BTJ013).

Abstract

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

CLC Number:

62P05
91G70

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.

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

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