Nonparametric Estimation of Dynamic Value-at-Risk: Multifunctional GARCH Model Case
Abstract
1. Introduction
2. Statistical Framework Models and Estimators
- PO1:
- The top Lyapunov exponent of the sequence is such that
3. Notations, Assumptions and Comments
- PO2:
- The processes satisfies
- PO3:
- There exists such that, for all and , we have
- PO4:
- The kernel is a positive and differentiable function, which is supported within such that
- PO5:
Comments on the Hypotheses
4. Results
Proof of the Main Result
- The vectorial sequence of functions
- The vectorial sequence
- The sequence .
5. Empirical Data Analysis
6. Real Data Application
7. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Appendix A
References
- Engle, R.F. Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econom. J. Econom. Soc. 1982, 50, 987–1007. [Google Scholar] [CrossRef]
- Bollerslev, T. A conditionally heteroskedastic time series model for speculative prices and rates of return. Rev. Econ. Stat. 1987, 69, 542–547. [Google Scholar] [CrossRef]
- Hormann, S.; Horvath, L.; Reeder, R. A functional version of the ARCH model. Econom. Theory 2013, 29, 267–288. [Google Scholar] [CrossRef]
- Yoon, J.E.; Kim, J.M.; Hwang, S.Y. Functional ARCH (fARCH) for high-frequency time series: Illustration. Korean J. Appl. Stat. 2017, 30, 983–991. [Google Scholar]
- Cerovecki, C.; Francq, C.; Hörmann, S.; Zakoïan, J.M. Functional GARCH models: The quasi-likelihood approach and its applications. J. Econom. 2019, 209, 353–375. [Google Scholar] [CrossRef]
- Sun, H.; Yu, B. Volatility asymmetry in functional threshold GARCH model. J. Time Ser. Anal. 2020, 41, 95–109. [Google Scholar] [CrossRef]
- Li, Z.; Sun, H.; Liu, J. A Functional Garch Model with Multiple Constant Parameters. Comput. Econ. 2025, 1–25. [Google Scholar] [CrossRef]
- Engle, R.F.; Manganelli, S. CAViaR: Conditional autoregressive value at risk by regression quantiles. J. Bus. Econ. Stat. 2004, 22, 367–381. [Google Scholar] [CrossRef]
- Kuester, K.; Mittnik, S.; Paolella, M.S. Value-at-risk prediction: A comparison of alternative strategies. J. Financ. Econom. 2006, 4, 53–89. [Google Scholar] [CrossRef]
- Sun, P.; Lin, F.; Xu, H.; Yu, K. Estimation of value-at-risk by L p quantile regression. Ann. Inst. Stat. Math. 2025, 77, 25–59. [Google Scholar] [CrossRef]
- Lux, M.; Härdle, W.K.; Lessmann, S. Data driven value-at-risk forecasting using a SVR-GARCH-KDE hybrid. Comput. Stat. 2020, 35, 947–981. [Google Scholar] [CrossRef]
- Herrera, R.; Schipp, B. Value at risk forecasts by extreme value models in a conditional duration framework. J. Empir. Financ. 2013, 23, 33–47. [Google Scholar] [CrossRef]
- Huang, J.J.; Lee, K.J.; Liang, H.; Lin, W.F. Estimating value at risk of portfolio by conditional copula-GARCH method. Insur. Math. Econ. 2009, 45, 315–324. [Google Scholar] [CrossRef]
- Brooks, C.; Chris, A.D.; Clare, J.W.; Dalle Molle, W.; Gita, P. A comparison of extreme value theory approaches for determining value at risk. J. Empir. Financ. 2005, 12, 339–352. [Google Scholar] [CrossRef]
- Kim, T.H.; White, H. On more robust estimation of skewness and kurtosis. Financ. Res. Lett. 2004, 1, 56–73. [Google Scholar] [CrossRef]
- Cardot, H.; Crambes, C.; Sarda, P. Quantile regression when the covariates are functions. Nonparametric Stat. 2005, 17, 841–856. [Google Scholar] [CrossRef]
- Ferraty, F.; Laksaci, A.; Vieu, P. Estimating some characteristics of the conditional distribution in nonparametric functional models. Stat. Inference Stoch. Process. 2006, 9, 47–76. [Google Scholar] [CrossRef]
- Dabo-Niang, S.; Laksaci, A. Nonparametric quantile regression estimation for functional dependent data. Commun. Stat.-Theory Methods 2012, 41, 1254–1268. [Google Scholar] [CrossRef]
- Messaci, F.; Nemouchi, N.; Ouassou, I.; Rachdi, M. Local polynomial modelling of the conditional quantile for functional data. Stat. Methods Appl. 2015, 24, 597–622. [Google Scholar] [CrossRef]
- Ling, N.; Yang, Y.; Peng, Q. Partial linear quantile regression model with incompletely observed functional covariates. J. Nonparametric Stat. 2025, 1–27. [Google Scholar] [CrossRef]
- Mutis, M.; Beyaztas, U.; Karaman, F.; Lin Shang, H. On function-on-function linear quantile regression. J. Appl. Stat. 2025, 52, 814–840. [Google Scholar] [CrossRef] [PubMed]
- Beyaztas, U.; Tez, M.; Lin Shang, H. Robust scalar-on-function partial quantile regression. J. Appl. Stat. 2024, 51, 1359–1377. [Google Scholar] [CrossRef]
- Kong, D.; Xue, K.; Yao, F.; Zhang, H.H. Partially functional linear regression in high dimensions. Biometrika 2016, 103, 147–159. [Google Scholar] [CrossRef]
- Kim, J.M.; Ha, I.D. Vine copula MFPCA residual control chart for sparse multivariate functional data. Commun. Stat.-Simul. Comput. 2025, 1–21. [Google Scholar] [CrossRef]
- Agarwal, G.; Sun, Y. Bivariate Functional Quantile Envelopes with Application to Radiosonde Wind Data. Technometrics 2020, 63, 199–211. [Google Scholar] [CrossRef]
- Bouanani, O.; Bouzebda, S. Limit theorems for local polynomial estimation of regression for functional dependent data. AIMS Math. 2024, 9, 23651–23691. [Google Scholar] [CrossRef]
- Berrahou, N.E.; Bouzebda, S.; Douge, L. The Bahadur Representation for Empirical and Smooth Quantile Estimators Under Association. Methodol. Comput. Appl. Probab. 2024, 26, 17. [Google Scholar] [CrossRef]
- Ferraty, F.; Quintela-Del-Río, A. Conditional VAR and expected shortfall: A new functional approach. Econom. Rev. 2016, 35, 263–292. [Google Scholar] [CrossRef]
- Cai, T. Financial risk management based on functional data analysis. J. Discret. Math. Sci. Cryptogr. 2018, 21, 1397–1400. [Google Scholar] [CrossRef]
- Rice, G.; Wirjanto, T.; Zhao, Y. Forecasting value at risk with intra-day return curves. Int. J. Forecast. 2020, 36, 1023–1038. [Google Scholar] [CrossRef]
- Ait-Hennani, L.; Kaid, Z.; Laksaci, A.; Rachdi, M. Nonparametric estimation of the expected shortfall regression for quasi-associated functional data. Mathematics 2022, 10, 4508. [Google Scholar] [CrossRef]
- Almanjahie, I.M.; Abood, H.; Bouzebda, S.; Alshahrani, F.; Laksaci, A. Nonparametric expectile shortfall regression for functional data. Demonstr. Math. 2025, 58, 20250125. [Google Scholar] [CrossRef]
- D’Ambra, L.; Castellano, R.; D’Urso, P.; De Iaco, S. Statistical methods for decision making in public sector: From the quality assessment to the citizen satisfaction. Ann. Oper. Res. 2024, 342, 1369–1377. [Google Scholar] [CrossRef]
- Litimein, O.; Laksaci, A.; Ait-Hennani, L.; Mechab, B.; Rachdi, M. Asymptotic normality of the local linear estimator of the functional expectile regression. J. Multivar. Anal. 2024, 202, 105281. [Google Scholar] [CrossRef]
- Engle, R. Dynamic conditional correlation: A simple class of multivariate generalized autoregressive conditional heteroskedasticity models. J. Bus. Econ. Stat. 2002, 20, 339–350. [Google Scholar] [CrossRef]
- Küchnert, S. Functional arch and GARCH models: A Yule–Walker approach. Electron. J. Stat. 2020, 14, 4321–4360. [Google Scholar]
- Francq, C.; Zakoian, J.M. GARCH Models: Structure, Statistical Inference and Financial Applications; John Wiley & Sons: Hoboken, NJ, USA, 2019. [Google Scholar]
- Laib, N.; Louani, D. Rates of strong consistencies of the regression function estimator for functional stationary ergodic data. J. Stat. Plan. Inference 2011, 141, 359–372. [Google Scholar] [CrossRef]
- Bogachev, V.I. Gaussian Measures; American Mathematical Society: Rhode Island, USA, 1998; Volume 62. [Google Scholar]
- Ferraty, F. Nonparametric Functional Data Analysis; Springer: Berlin/Heidelberg, Germany, 2006. [Google Scholar]
- Al-Awadhi, F.A.; Kaid, Z.; Laksaci, A.; Ouassou, I.; Rachdi, M. Functional data analysis: Local linear estimation of the L 1-conditional quantiles. Stat. Methods Appl. 2019, 28, 217–240. [Google Scholar] [CrossRef]
- Iglesias-Pérez, M.C. Strong representation of a conditional quantile function estimator with truncated and censored data. Stat. Probab. Lett. 2003, 65, 79–91. [Google Scholar] [CrossRef]
- Yuan, M. GACV for quantile smoothing splines. Comput. Stat. Data Anal. 2006, 50, 813–829. [Google Scholar] [CrossRef]
- Benhenni, K.; Ferraty, F.; Rachdi, M.; Vieu, P. Local smoothing regression with functional data. Comput. Stat. 2007, 22, 353–369. [Google Scholar] [CrossRef]
- Rachdi, M.; Laksaci, A.; Demongeot, J.; Abdali, A.; Madani, F. Theoretical and practical aspects of the quadratic error in the local linear estimation of the conditional density for functional data. Comput. Stat. Data Anal. 2014, 73, 53–68. [Google Scholar] [CrossRef]
Selectors | N | = 0.1 | = 0.05 | = 0.01 |
---|---|---|---|---|
50 | 0.83 | 0.87 | 0.75 | |
150 | 0.42 | 0.39 | 0.32 | |
250 | 0.24 | 0.22 | 0.23 | |
50 | 0.62 | 0.73 | 0.58 | |
150 | 0.29 | 0.26 | 0.24 | |
250 | 0.09 | 0.13 | 0.12 | |
50 | 0.66 | 0.78 | 0.69 | |
150 | 0.38 | 0.31 | 0.33 | |
250 | 0.29 | 0.25 | 0.18 |
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Chikr-Elmezouar, Z.; Laksaci, A.; Almanjahie, I.M.; Alshahrani, F. Nonparametric Estimation of Dynamic Value-at-Risk: Multifunctional GARCH Model Case. Mathematics 2025, 13, 1961. https://doi.org/10.3390/math13121961
Chikr-Elmezouar Z, Laksaci A, Almanjahie IM, Alshahrani F. Nonparametric Estimation of Dynamic Value-at-Risk: Multifunctional GARCH Model Case. Mathematics. 2025; 13(12):1961. https://doi.org/10.3390/math13121961
Chicago/Turabian StyleChikr-Elmezouar, Zouaoui, Ali Laksaci, Ibrahim M. Almanjahie, and Fatimah Alshahrani. 2025. "Nonparametric Estimation of Dynamic Value-at-Risk: Multifunctional GARCH Model Case" Mathematics 13, no. 12: 1961. https://doi.org/10.3390/math13121961
APA StyleChikr-Elmezouar, Z., Laksaci, A., Almanjahie, I. M., & Alshahrani, F. (2025). Nonparametric Estimation of Dynamic Value-at-Risk: Multifunctional GARCH Model Case. Mathematics, 13(12), 1961. https://doi.org/10.3390/math13121961