Elliptical and Skew-Elliptical Regression Models and Their Applications to Financial Data Analytics †
Abstract
:1. Introduction
2. The Big Picture
2.1. GARCH Formulations
2.2. Copula Formulations
2.3. GBM and FATGBM Models
3. Elliptical and Skew-Elliptical Distributions
4. Elliptical and Skew-Elliptical Regression
4.1. Elliptical Regression
4.2. Skew-Elliptical Regression
Median and Quantile Association Regression
5. Elliptical and Skew-Elliptical Regression Diagnostics
6. Copula Regression
7. Determining Elliptical or Skew-Elliptical Distributions—Illustrative examples
8. Concluding Remarks
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
- Adcock, Chris, and Adelchi Azzalini. 2020. A Selective Overview of Skew–Elliptical and Related Distributions and of their Applications. Symmetry 12: 118. [Google Scholar] [CrossRef] [Green Version]
- Arellano-Valle, Reinaldo B., and Adelchi Azzalini. 2013. The Centred Parameterization and Related Quantities of the Skew–t Distribution. Journal of Multivariate Analysis 113: 73–90. [Google Scholar] [CrossRef]
- Arellano-Vale, Reinaldo B., and Marc G. Genton. 2010. Multivariate Unified Skew–Elliptical Distributions. Chilean Journal of Statistics 1: 17–33. [Google Scholar]
- Azzalini, Adelchi. 2022. An Overview on the Progeny of the Skew–Normal Family—A Personal Perspective. Journal of Multivariate Analysis 188: 104851. [Google Scholar] [CrossRef]
- Azzalini, Adelchi, and Alessandra Dalla Valle. 1996. The Multivariate Skew–Normal Distribution. Biometrika 83: 715–26. [Google Scholar] [CrossRef]
- Azzalini, Adelchi, and Antonella Capitanio. 2013. The Skew–Normal and Related Families. Cambridge: Cambridge University Press, vol. 3. [Google Scholar]
- Bekaert, Geert, and Guojun Wu. 2000. Asymmetric Volatility and Risk in Equity Markets. The Review of Financial Studies 13: 1–42. [Google Scholar] [CrossRef] [Green Version]
- Bhatti, M. Ishaq, and Hung Quang Do. 2019. Recent Development In Copula And Its Applications To The Energy, Forestry And Environmental Sciences. International Journal of Hydrogen Energy 44: 19453–73. [Google Scholar] [CrossRef]
- Black, F. 1976. Studies of Stock Market Volatility Changes. In Proceedings of the American Statistical Association Business and Economic Statistics Section. Washington, DC: American Statistical Association. [Google Scholar]
- Branco, Marcia D., and Dipak K. Dey. 2002. Regression Model Under Skew Elliptical Error Distribution. Journal of Mathematical Sciences 1: 151–69. [Google Scholar]
- Campbell, Rachel A. J., Catherine S. Forbes, Kees G. Koedijk, and Paul Kofman. 2008. Increasing Correlations or Just Fat Tails? Journal of Empirical Finance 15: 287–309. [Google Scholar] [CrossRef]
- Campbell, Rachel, Kees Koedijk, and Paul Kofman. 2002. Increased Correlation in Bear Markets. Financial Analysts Journal 58: 87–94. [Google Scholar] [CrossRef]
- Cao, Linyu, Ruili Sun, Tiefeng Ma, and Conan Liu. 2023. On Asymmetric Correlations and Their Applications in Financial Markets. Journal of Risk and Financial Management 16: 187. [Google Scholar] [CrossRef]
- Cherubini, Umberto, Elisa Luciano, and Walter Vecchiato. 2004. Copula Methods in Finance. Hoboken: John Wiley & Sons. [Google Scholar]
- Christoffersen, Peter, Vihang Errunza, Kris Jacobs, and Hugues Langlois. 2012. Is the Potential for International Diversification Disappearing? A Dynamic Copula Approach. The Review of Financial Studies 25: 3711–51. [Google Scholar] [CrossRef] [Green Version]
- Cook, R. Dennis. 1986. Assessment of Local Influence. Journal of the Royal Statistical Society: Series B (Methodological) 48: 133–55. [Google Scholar]
- Dette, Holger, Ria Van Hecke, and Stanislav Volgushev. 2014. Some Comments on Copula–Based Regression. Journal of the American Statistical Association 109: 1319–24. [Google Scholar] [CrossRef]
- Dewick, Paul R. 2022. On Financial Distributions Modelling Methods: Application on Regression Models for Time Series. Journal of Risk and Financial Management 15: 461. [Google Scholar] [CrossRef]
- Dewick, Paul R., and Shuangzhe Liu. 2022. Copula Modelling to Analyse Financial Data. Journal of Risk and Financial Management 15: 104. [Google Scholar] [CrossRef]
- Dong, Hua, and Chuancun Yin. 2021. A Unified Treatment of Characteristic Functions of Symmetric Multivariate and Related Distributions. arXiv arXiv:2112.06472. [Google Scholar]
- Embrechts Paul, Filip Lindskog, and Alexander McNeil. 2001. Modelling dependence with copulas. Rapport technique, Département de mathématiques, Institut Fédéral de Technologie de Zurich, Zurich 14: 1–50. [Google Scholar]
- Fang, Kai-Tang, and Theodore W. Anderson. 1990. Statistical Inference in Elliptically Contoured and Related Distributions. New York: Allerton Press. [Google Scholar]
- Fang, Kai-Tang, and Yao-Ting Zhang. 1990. Generalized Multivariate Analysis. Beijing and Berlin: Science Press Beijing and Springer. [Google Scholar]
- Fernández, Carmen, and Mark F. J. Steel. 1999. Multivariate Student–t Regression Models: Pitfalls and Inference. Biometrika 86: 153–67. [Google Scholar] [CrossRef]
- Friendly, Michael, Georges Monette, and John Fox. 2013. Elliptical Insights: Understanding Statistical Methods Through Elliptical Geometry. Statistical Science 28: 1–39. [Google Scholar] [CrossRef]
- Fung, Thomas, and Eugene Seneta. 2021. Tail Asymptotics for the Bivariate Equi–skew Generalized Hyperbolic Distribution and its Variance–Gamma Special Case. Statistics & Probability Letters 178: 109182. [Google Scholar]
- Galea, Manuel, and Patricia Giménez. 2019. Local Influence Diagnostics for the Test of Mean–Variance Efficiency and Systematic Risks in the Capital Asset Pricing Model. Statistical Papers 60: 293–312. [Google Scholar] [CrossRef]
- Galea, Manuel, David Cademartori, Roberto Curci, and Alonso Molina. 2020. Robust inference in the capital asset pricing model using the multivariate t-distribution. Journal of Risk and Financial Management 6: 123. [Google Scholar] [CrossRef]
- Galea, Manuel P., Gilberto A. Paula, and Heleno Bolfarine. 1997. Local Influence in Elliptical Linear Regression Models. Journal of the Royal Statistical Society: Series D (The Statistician) 46: 71–79. [Google Scholar] [CrossRef]
- Galea, Manuel, Gilberto A. Paula, and Miguel Uribe-Opazo. 2003. On Influence Diagnostic in Univariate Elliptical Linear Regression Models. Statistical Papers 44: 23–45. [Google Scholar] [CrossRef]
- Galea, Manuel, Marco Riquelme, and Gilberto A. Paula. 2000. Diagnostic Methods in Elliptical Linear Regression Models. Brazilian Journal of Probability and Statistics 14: 167–84. [Google Scholar]
- Gani, Joseph M., and Eugene Seneta. 2004. Christopher Charles Heyde, AM, DSc, FAA, FAASA. Journal of Applied Probability, 41A (Special issue) 41: vii–x. [Google Scholar]
- Genton, Marc G. 2004. Skew-Elliptical Distributions and Their Applications: A Journey Beyond Normality. Boca Raton: CRC Press. [Google Scholar]
- Ghani, Intan M. Md., and Hanafi A. Rahim. 2019. Modeling and Forecasting of Volatility using ARMA–GARCH: Case Study on Malaysia Natural Rubber Prices. IOP Conference Series: Materials Science and Engineering 548: 012023. [Google Scholar] [CrossRef]
- Glasserman, Paul, and Steven Kou. 2006. A Conversation With Chris Heyde. Statistical Science 21: 286–98. [Google Scholar] [CrossRef] [Green Version]
- Gródek-Szostak, Zofia, Gabriela Malik, Danuta Kajrunajtys, Anna Szeląg-Sikora, Jakub Sikora, Maciej Kuboń, Marcin Niemiec, and Joanna Kapusta-Duch. 2019. Modeling The Dependency Between Extreme Prices Of Selected Agricultural Products On The Derivatives Market Using The Linkage Function. Sustainability 11: 4144. [Google Scholar] [CrossRef] [Green Version]
- Guan, Jing, Daoji Shi, and Yuanyuan He. 2008. Copula Quantile Regression and Measurement of Risk in Finance. Paper presented at 2008 4th International Conference on Wireless Communications, Networking and Mobile Computing, Dalian, China, October 12–14; pp. 1–4. [Google Scholar]
- Gupta, Arjun K., Tamas Varga, and Taras Bodnar. 2013. Elliptically Contoured Models in Statistics and Portfolio Theory. New York: Springer. [Google Scholar]
- Gupta, Neha, Arun Kumar, and Nikolai Leonenko. 2021. Stochastic Models with Mixtures of Tempered Stable Subordinators. Mathematical Communications 26: 77–99. [Google Scholar]
- Heyde, Christopher C. 1967. On the Influence of Moments on the Rate of Convergence to the Normal Distribution. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 8: 12–18. [Google Scholar] [CrossRef]
- Heyde, Christopher C. 1999. A Risky Asset Model with Strong Dependence through Fractal Activity Time. Journal of Applied Probability 36: 1234–39. [Google Scholar] [CrossRef]
- Heyde, Christopher C., and Nikolai N. Leonenko. 2005. Student Processes. Advances in Applied Probability 37: 342–65. [Google Scholar] [CrossRef] [Green Version]
- Heyde, Christopher C., and Shuangzhe Liu. 2001. Empirical Realities for a Minimal Description Risky Asset Model. The Need for Fractal Features. Journal of the Korean Mathematical Society 38: 1047–59. [Google Scholar]
- Heyde, Christopher C., Shuangzhe Liu, and Roger Gay. 2001. Fractal scaling and Black-Scholes: The Full Story. A New View of Long–Range Dependence in Stock Prices. JASSA 1: 29–32. [Google Scholar]
- Hofert, Marius, Ivan Kojadinovic, Martin Mächler, and Jun Yan. 2018. Elements of Copula Modeling with R. Cham: Springer Nature Switzerland. [Google Scholar]
- Joe, Harry. 2014. Dependence Modeling with Copulas. Boca Raton: CRC Press. [Google Scholar]
- Kariya, Takeaki, and Bimal K. Sinha. 1989. Robustness of Statistical Tests. Boca Raton: Academic Press. [Google Scholar]
- Kato, Shogo, Toshinao Yoshiba, and Shinto Eguchi. 2022. Copula-Based Measures of Asymmetry Between the Lower and Upper Tail Probabilities. Statistical Papers 63: 1907–29. [Google Scholar] [CrossRef]
- Kayalar, Derya E., C. Coşkun Küçüközmen, and A. Sevtap Selcuk-Kestel. 2017. The Impact of Crude Oil Prices on Financial Market Indicators: Copula Approach. Energy Economics 61: 162–73. [Google Scholar] [CrossRef]
- Kerss, Alexander D. J., Nikolai Leonenko, and Alla Sikorskii. 2014. Risky Asset Models with Tempered Stable Fractal Activity Time. Stochastic Analysis and Applications 32: 642–63. [Google Scholar] [CrossRef]
- Kwong, Hok S., and Saralees Nadarajah. 2022. A New Robust Class of Skew Elliptical Distributions. Methodology and Computing in Applied Probability 24: 1669–91. [Google Scholar] [CrossRef]
- Landsman, Zinoviy M., and Emiliano A. Valdez. 2003. Tail Conditional Expectations for Elliptical Distributions. North American Actuarial Journal 7: 55–71. [Google Scholar] [CrossRef] [Green Version]
- Lange, Kenneth L., Roderick J. A. Little, and Jeremy M. G. Taylor. 1989. Robust Statistical Modeling Using the t Distribution. Journal of the American Statistical Association 84: 881–96. [Google Scholar] [CrossRef] [Green Version]
- Lee, Sharon X., and Geoffrey J. McLachlan. 2013. On Mixtures of Skew Normal and Skew t–Distributions. Advances in Data Analysis and Classification 7: 241–66. [Google Scholar] [CrossRef] [Green Version]
- Lemonte, Artur J., and Alexandre G. Patriota. 2011. Multivariate Elliptical Models with General Parameterization. Statistical Methodology 8: 389–400. [Google Scholar] [CrossRef]
- Leonenko, Nikolai N., Stuart Petherick, and Alla Sikorskii. 2012. Fractal Activity Time Models for Risky Asset with Dependence and Generalized Hyperbolic Distributions. Stochastic Analysis and Applications 30: 476–92. [Google Scholar] [CrossRef]
- Li, Zhengxiao, Jan Beirlant, and Liang Yang. 2022. A New class of Copula Regression Models for Modelling Multivariate Heavy–Tailed Data. Insurance: Mathematics and Economics 104: 243–61. [Google Scholar] [CrossRef]
- Liu, Shuangzhe. 2000. On Local Influence for Elliptical Linear Models. Statistical Papers 41: 211–24. [Google Scholar] [CrossRef] [Green Version]
- Liu, Shuangzhe. 2002. Local Influence in Multivariate Elliptical Linear Regression Models. Linear Algebra and Its Applications 354: 159–74. [Google Scholar] [CrossRef] [Green Version]
- Liu, Shuangzhe. 2004. On Diagnostics in Conditionally Heteroskedastic Time Series Models under Elliptical Distributions. Journal of Applied Probability 41: 393–405. [Google Scholar] [CrossRef]
- Liu, Shuangzhe, and Christopher C. Heyde. 2008. On Estimation in Conditional Heteroskedastic Time Series Models under Non–Normal Distributions. Statistical Papers 49: 455–69. [Google Scholar] [CrossRef]
- Liu, Shuangzhe, and Milind Sathye, eds. 2021. Financial Statistics and Data Analytics. (A reprint of the Special Issue published in Journal of Risk and Financial Management). Basel: MDPI. [Google Scholar]
- Liu, Shuangzhe, Christopher C. Heyde, and Wing-Keung Wong. 2011. Moment Matrices in Conditional Heteroskedastic Models under Elliptical Distributions with Applications in AR–ARCH Models. Statistical Papers 52: 621–32. [Google Scholar] [CrossRef] [Green Version]
- Liu, Yonghui, Guocheng Ji, and Shuangzhe Liu. 2015. Influence Diagnostics in a Vector Autoregressive Model. Journal of Statistical Computation and Simulation 85: 2632–55. [Google Scholar] [CrossRef]
- Liu, Yonghui, Chaoxuan Mao, Victor Leiva, Shuangzhe Liu, and Waldemiro A. Silva Neto. 2022a. Asymmetric Autoregressive Models: Statistical Aspects and a Financial Application under COVID-19 Pandemic. Journal of Applied Statistics 49: 1323–347. [Google Scholar] [CrossRef] [PubMed]
- Liu, Yonghui, Guonhua Mao, Victor Leiva, Shuangzhe Liu, and Alejandra Tapia. 2020. Diagnostic Analytics for an Autoregressive Model Under the Skew–Normal Distribution. Mathematics 8: 693. [Google Scholar] [CrossRef]
- Liu, Yonghui, Jing Wang, Victor Leiva, Alejandra Tapia, Wei Tan, and Shuangzhe Liu. 2023. Robust Autoregressive Modeling and its Diagnostic Analytics with a COVID-19 Related Application. Journal of Applied Statistics, 1–26. [Google Scholar] [CrossRef]
- Liu, Yonghui, Jing Wang, Zhao Yao, Conan Liu, and Shuangzhe Liu. 2022b. Diagnostic Analytics for a GARCH Model Under Skew-Normal Distributions. Communications in Statistics-Simulation and Computation, 1–25. [Google Scholar] [CrossRef]
- Ma, Yanyuan, Marc G. Genton, and Anastasios A. Tsiatis. 2005. Locally Efficient Semiparametric Estimators for Generalized Skew-Elliptical Distributions. Journal of the American Statistical Association 100: 980–89. [Google Scholar] [CrossRef]
- Madan, Dilip B., and Wim Schoutens. 2020. Self-Similarity in Long-Horizon Returns. Mathematical Finance 30: 1368–391. [Google Scholar] [CrossRef]
- Maller, Ross, Ishwar Basawa, Peter Hall, and Eugene Seneta. 2010. Selected Works of C. C. Heyde. New York: Springer. [Google Scholar]
- Meerschaert, Mark M., and Alla Sikorskii. 2019. Stochastic Models for Fractional Calculus, 2nd ed. Berlin: De Gruyter. [Google Scholar]
- Najafi, Zeinolabedin, Karim Zare, Mohammad R. Mahmoudi, Soheil Shokri, and Amir Mosavi. 2022. Inference and Local Influence Assessment in a Multifactor Skew–Normal Linear Mixed Model. Mathematics 10: 2820. [Google Scholar] [CrossRef]
- Nelsen, Roger B. 2007. An Introduction to Copulas. New York: Springer. [Google Scholar]
- Noh, Hohsuk, Anouar El Ghouch, and Taoufik Bouezmarni. 2013. Copula-Based Regression Estimation and Inference. Journal of the American Statistical Association 108: 676–88. [Google Scholar] [CrossRef] [Green Version]
- Osorio, Felipe, Gilberto A. Paula, and Manuel Galea. 2007. Assessment of Local Influence in Elliptical Linear Models with Longitudinal Structure. Computational Statistics & Data Analysis 51: 4354–368. [Google Scholar]
- Parsa, Rahul A., and Stuart A. Klugman. 2011. Copula Regression. Variance Advancing and Science of Risk 5: 45–54. [Google Scholar]
- Patton, Andrew J. 2004. On the Out-of-Sample Importance of Skewness and Asymmetric Dependence for Asset Allocation. Journal of Financial Econometrics 1: 130–68. [Google Scholar] [CrossRef] [Green Version]
- Patton, Andrew J. 2006. Modelling Asymmetric Exchange Rate Dependence. International Economic Review 47: 527–56. [Google Scholar] [CrossRef] [Green Version]
- Schmidt, Thorsten. 2007. Coping with Copulas. Copulas—From Theory to Application in Finance 3: 1–34. [Google Scholar]
- Seneta, Eugene, and Joseph M. Gani. 2009. Christopher Charles Heyde 1939–2008. Historical Records of Australian Science 20: 67–90. [Google Scholar] [CrossRef]
- SenGupta, Ashis, and Barry C. Arnold. 2022. Directional Statistics for Innovative Applications: A Bicentennial Tribute to Florence Nightingale. Singapore: Springer Nature Singapore. [Google Scholar]
- Sklar, M. 1959. Fonctions de Repartition an Dimensions et Leurs Marges. Publications de l’Institut Statistique de l’Université de Paris 8: 229–31. [Google Scholar]
- Stamatatou, Nikoletta, Lampros Vasiliades, and Athanasios Loukas. 2018. Bivariate Flood Frequency Analysis using Copulas. Proceedings 2: 635. [Google Scholar]
- Sungur, Engin A. 2006. Some Observations on Copula Regression Functions. Communications in Statistics—Theory and Methods 34: 9–10. [Google Scholar] [CrossRef]
- Vuolo, Mike. 2017. Copula Models For Sociology: Measures of Dependence and Probabilities for Joint Distributions. Sociological Methods and Research 46: 604–48. [Google Scholar] [CrossRef]
- Welsh, Alan H., and Alice M. Richardson. 1997. 13 Approaches to the Robust Estimation of Mixed Models. Handbook of Statistics 15: 343–84. [Google Scholar]
- Wichitaksorn, Nuttanan, S. T. Boris Choy, and Richard Gerlach. 2014. A Generalized Class of Skew Distributions and Associated Robust Quantile Regression Models. Canadian Journal of Statistics 42: 579–96. [Google Scholar] [CrossRef]
- Zeller, Camila B., Victor H. Lachos, and Filidor E. Vilca-Labra. 2011. Local Influence Analysis for Regression Models with Scale Mixtures of Skew-Normal Distributions. Journal of Applied Statistics 38: 343–68. [Google Scholar] [CrossRef]
Author (Year) | Paper/Book/Thesis |
---|---|
Sklar (1959) | Fonctions Derépartitionàn Dimensions et Leurs Marges. |
Embrechts et al. (2001) | Modelling Dependence with Copulas. |
Cherubini et al. (2004) | Copula Methods in Finance |
Patton (2006) | Modeling Asymmetric Exchange Rate Dependence. |
Nelsen (2007) | An Introduction to Copulas. |
Christoffersen et al. (2012) | Is the Potential for International Diversification Disappearing? A Dynamic Copula Approach. |
Joe (2014) | Dependence Modeling with Copulas. |
Hofert et al. (2018) | Elements of Copula Modeling with R. |
Distribution | Notation | Generating Function— |
---|---|---|
Normal | , | |
Student’s t | ||
Contaminated Normal | ||
Cauchy | ||
Logistic | ||
Exponential Power |
Author (Year) | Paper/Book/Thesis |
---|---|
Azzalini and Dalla Valle (1996) | The Multivariate Skew–Normal Distribution. |
Arellano-Vale and Genton (2010) | Multivariate Unified Skew-Elliptical Distributions. |
Adcock and Azzalini (2020) | A Selective Overview of Skew-Elliptical and Related Distributions and of their Applications. |
Azzalini (2022) | An Overview on the Progeny of the Skew-Normal Family—A Personal Perspective. |
Kwong and Nadarajah (2022) | A New Class of Skew Elliptical Distributions. |
Liu et al. (2023) | Robust Autoregressive Modeling and its Diagnostic Analytics with a COVID-19 |
Related Application. |
Author Year | Paper/Book/Thesis |
---|---|
Galea et al. (1997) | Local Influence in Elliptical Linear Regression Models. |
Ma et al. (2005) | Locally Efficient Semiparametric Estimators for Generalized Skew-Elliptical Distributions. |
Galea at el. (2020) | Robust Inference in the Capital Asset Pricing Model using the Multivariate t-Distribution. |
Liu et al. (2022b) | Diagnostic Analytics for a GARCH Model under Skew–Normal Distribution. |
Najafi et al. (2022) | Inference and Local Influence Assessment in a Multifactor Skew–Normal Linear Mixed Model. |
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Dewick, P.R.; Liu, S.; Liu, Y.; Ma, T. Elliptical and Skew-Elliptical Regression Models and Their Applications to Financial Data Analytics. J. Risk Financial Manag. 2023, 16, 310. https://doi.org/10.3390/jrfm16070310
Dewick PR, Liu S, Liu Y, Ma T. Elliptical and Skew-Elliptical Regression Models and Their Applications to Financial Data Analytics. Journal of Risk and Financial Management. 2023; 16(7):310. https://doi.org/10.3390/jrfm16070310
Chicago/Turabian StyleDewick, Paul R., Shuangzhe Liu, Yonghui Liu, and Tiefeng Ma. 2023. "Elliptical and Skew-Elliptical Regression Models and Their Applications to Financial Data Analytics" Journal of Risk and Financial Management 16, no. 7: 310. https://doi.org/10.3390/jrfm16070310