Analyzing the Information Contained in the Skewness and Kurtosis of TEPIX Returns for Forecasting Risk: GARCH Model with Gram-Charlier Expansions for Innovations

Document Type : Original Article

Author

Assistant Professor, Nahavand Higher Education Complex, Bu-Ali Sina University, Hamedan, Iran

Abstract

Purpose: One of the well-known stylized facts of the distribution of asset returns is the pattern of skewness and kurtosis. Previous research has shown that financial crises and turbulences induce shocks that significantly affect the return distributions, which in addition to creating fat tails, also leads to an asymmetric reaction of the tails. Although skewness and kurtosis have a common impact on tail risk, their significance for risk forecasting has not been considered in empirical financial studies.Developing models for accurate risk forecasting is an important consideration that has always received considerable attention from policymakers, economists, financial market participants, and researchers.For this purpose, in this study, following Jimenez et al. (2022), to estimate the return density, a semi-nonparametric approach is adopted which is based on the asymptotic properties of Gram-Charlier extensions. This approach allows examining the significance of the inclusion of Hermit polynomials and their crossed products in the Gram-Charlier densities for risk forecasting. In fact, in the framework. Evaluating the risk measures in a semi-nonparametric framework allows for capturing all stylized facts of the return series for assessing skewness and kurtosis and their interactions by adding new parameters to the density function as a relevant source of information.
Method: This research, for the first time, employs the modified Gram-Charlier density function (mGC), including the second and third moments (skewness and kurtosis) and their interactions for modeling the risk of distribution of the daily losses of the TEPIX. Moreover, the performance of alternative models based on different specifications of Gram-Charlier is evaluated in terms of the accuracy of risk forecasting measures using modern backtesting tests. For this purpose, the Value-at-Risk criteria and Median Shortfall measure, implied for the first time in the present study, are used. The sample includes the daily series of the TEPIX index covering the period from May 20, 2008, to August 22, 2023. Focusing on the right-tail of the TEPIX distribution, the loss series is calculated as a negative of log differences of prices. The models are estimated by R and MATLAB software. Modeling the losses is done through a two-step estimation process according to the following steps. Step 1: the ARMA(1,1)-GARCH(1,1) model is estimated  using the quasi-maximum likelihood (QML) approach by assuming the Gaussian distribution for error terms. Step 2: the modified Gram-Charlier expansion and alternative specifications are estimated using standardized residuals extracted from the previous step. Different specifications of the Gram-Charlier density density fit using the maximum likelihood method. For in-sample fitting of the model, the estimation window size is chosen to be W=2656 observations and the step is chosen to be one new observation. The remaining 1000 observations are used for out-of-sample forecasts.
Findings: The empirical findings from the in-sample fitting of the ARMA(1,1)-GARCH(1,1) model, under the Gram-Charlier densities for the innovations, indicate that skewness and kurtosis and their interactions are economically and financially significant. The results of the Backtesting for both 99%-VaR and 99%-MS confirm the out-of-sample forecasting performance of the Gram-Charlier density specification incorporating the skewness parameter, especially for the tails, compared to other specifications that have been taken into account in this research.
Conclusion: Overall, the results show that the parameter related to the asymmetry of distribution alone can be a valuable source of information to the market participants by providing accurate risk measures. The empirical findings have practical implications for designing strategies for managing risk and decision-making in times of market instability. The novelity of this study is the application of a semi non-parametric approach to evaluate the risk forecasting of the TEPIX index. The previous studies have mainly modeled the return series based on parametric and non-parametric distributions. Therefore, the empirical findings of the present study provide a novel application for risk management in the Tehran Stock Exchange. The empirical findings can have useful implications for stabilizing the financial markets.

Keywords

References

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Financial Management Perspective
Volume 14, Issue 45
2024
Pages 149-174

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