Home Efficient estimation of financial risk by regressing the quantiles of parametric distributions: An application to CARR models
Article
Licensed
Unlicensed Requires Authentication

Efficient estimation of financial risk by regressing the quantiles of parametric distributions: An application to CARR models

  • Jennifer So Kuen Chan EMAIL logo , Kok-Haur Ng , Thanakorn Nitithumbundit and Shelton Peiris ORCID logo
Published/Copyright: September 6, 2018
Become an author with De Gruyter Brill
Submit Manuscript Author Information
Studies in Nonlinear Dynamics & Econometrics
From the journal Studies in Nonlinear Dynamics & Econometrics Volume 23 Issue 2

Abstract

Risk measures such as value-at-risk (VaR) and expected shortfall (ES) may require the calculation of quantile functions from quantile regression models. In a parametric set-up, we propose to regress directly on the quantiles of a distribution and demonstrate a method through the conditional autoregressive range model which has increasing popularity in recent years. Two flexible distribution families: the generalised beta type two on positive support and the generalised-t on real support (which requires log transformation) are adopted for the range data. Then the models are extended to allow the volatility dynamic and compared in terms of goodness-of-fit. The models are implemented using the module fmincon in Matlab under the classical likelihood approach and applied to analyse the intra-day high-low price ranges from the All Ordinaries index for the Australian stock market. Quantiles and upper-tail conditional expectations evaluated via VaR and ES respectively are forecast using the proposed models.

Keywords: conditional autoregressive range model; generalised beta type two distribution; generalised-t distribution; parametric quantile regression; tail conditional expectation; volatility model

Acknowledgement

This work is partially supported by the UMRG grant no: RG260-13AFR of the University of Malaya. The second author acknowledged the support from the School of Mathematics and Statistics, The University of Sydney during his visit in 2017.

A Appendix

For the quantile regression model at level u using GB2 distribution and (22), the upper TCE ξut,ρu for ρ∈(0.5,1) at time t is

ξut,ρu=11−ρ∫ρ∞FGB2−1(r)dr=11−ρ∫FGB2−1(ρ|at,bt,p,q)∞yfGB2(y)dy=at(1−ρ)B(p,q)∫FGB2−1(ρ|at,bt,p,q)∞(ybt)atp[1+(ybt)at]p+qdy=bt(1−ρ)B(p,q)∫FGB2−1(ρ|at,bt,p,q)∞(yb)atp−at+1[1+(ybt)at]p+q−2atbt(ybt)at−1[1+(ybt)at]2dy=btB(p+1at,q−1at)(1−ρ)B(p,q)∫FB−1(ρ|p,q)∞zp−1+1at(1−z)q−1−1atB(p+1at,q−1at)dz=btB(p+1at,q−1at)B(p,q)×1−FB(FB−1(ρ|p,q) |p+1at,q−1at)1−ρ=qut(1−FB−1(u|p,q)FB−1(u|p,q))1atB(p+1at,q−1at)B(p,q)×1−FB(FB−1(ρ|p,q) |p+1at,q−1at)1−ρ.

Similarly, the lower TCE ξut,ρl for ρ∈(0,0.5) at time t is

ξut,ρl=qut(1−FB−1(u|p,q)FB−1(u|p,q))1atB(p+1at,q−1at)B(p,q)×FB(FB−1(ρ|p,q) |p+1at,q−1at)ρ.

References

Andersen, T. G., and T. Bollerslev. 1998. “Answering the Skeptics: Yes, Standard Volatility Models Do Provide Accurate Forecasts.” International Economic Review 39: 885–905.10.2307/2527343Search in Google Scholar

Artzner, P., F. Delbaen, J. M. Eber, and D. Heath. 1999. “Coherent Measures of Risk.” Mathematical Finance 9: 203–228.10.1111/1467-9965.00068Search in Google Scholar

Bollerslev, T. 1986. “Generalized Autoregressive Conditional Heteroskedasticity.” Journal of Econometrics 31: 307–327.10.1016/0304-4076(86)90063-1Search in Google Scholar

Buchinsky, M. 1998. “Recent Advances in Quantile Regression Models: A Practical Guideline for Empirical Research.” Journal of Human Resources 33: 88–126.10.2307/146316Search in Google Scholar

Chan, J. S., S. B. Choy, and U. E. Makov. 2008. “Robust Bayesian Analysis of Loss Reserves Data Using the Generalized-t Distribution.” ASTIN Bulletin 38: 207–230.10.1017/S0515036100015142Search in Google Scholar

Chan, J. S. K., C. P. Y. Lam, P. L. H. Yu, S. T. B. Choy, and C. W. S. Chen. 2012. “A Bayesian Conditional Autoregressive Geometric Process Model for Range Data.” Computational Statistics & Data Analysis 56: 3006–3019.10.1016/j.csda.2011.01.006Search in Google Scholar

Chernozhukov, V., I. Fernández-Val, and A. Galichon. 2010. “Quantile and Probability Curves Without Crossing.” Econometrica 78: 1093–1125.10.3982/ECTA7880Search in Google Scholar

Chou, R. Y. 2005. “Forecasting Financial Volatilities with Extreme Values: The Conditional Autoregressive Range (CARR) Model.” Journal of Money, Credit and Banking 37: 561–582.10.1353/mcb.2005.0027Search in Google Scholar

Christoffersen, P. F. 1998. “Evaluating Interval Forecasts.” International Economic Review 39: 841–862.10.2307/2527341Search in Google Scholar

Cummins, J. D., G. Dionne, J. B. McDonald, and B. M. Pritchett. 1990. “Applications of the GB2 Family of Distributions in Modeling Insurance Loss Processes.” Insurance: Mathematics and Economics 9: 257–272.10.1016/0167-6687(90)90003-VSearch in Google Scholar

Dong, A. X., and J. Chan. 2013. “Bayesian Analysis of Loss Reserving Using Dynamic Models with Generalized Beta Distribution.” Insurance: Mathematics and Economics 53: 355–365.10.1016/j.insmatheco.2013.07.001Search in Google Scholar

Engle, R. F., and S. Manganelli. 2004. “CAViaR: Conditional Autoregressive Value at Risk by Regression Quantiles.” Journal of Business & Economic Statistics 22: 367–381.10.1198/073500104000000370Search in Google Scholar

Engle, R. F., and J. R. Russell. 1998. “Autoregressive Conditional Duration: A New Model for Irregularly Spaced Transaction Data.” Econometrica 66: 1127–1162.10.2307/2999632Search in Google Scholar

Gaglianone, W. P., L. R. Lima, O. Linton, and D. R. Smith. 2011. “Evaluating Value-at-Risk Models via Quantile Regression.” Journal of Business & Economic Statistics 29: 150–160.10.1198/jbes.2010.07318Search in Google Scholar

Gerlach, R., and C. Wang. 2016. “Bayesian Semi-Parametric Realized-CARE Models for Tail Risk Forecasting Incorporating Realized Measures.” arXiv:1612.08488.10.1080/14697688.2016.1192295Search in Google Scholar

Gilchrist, W. 2000. Statistical Modelling with Quantile Functions. Boca Raton, FL: CRC Press.10.1201/9781420035919Search in Google Scholar

Gneiting, T. 2011. “Making and Evaluating Point Forecasts.” Journal of the American Statistical Association 106: 746–762.10.1198/jasa.2011.r10138Search in Google Scholar

He, X. 1997. “Quantile Curves Without Crossing.” The American Statistician 51: 186–192.10.1080/00031305.1997.10473959Search in Google Scholar

Hull, J., and A. White. 1987. “The Pricing of Options on Assets with Stochastic Volatilities.” Journal of Finance 42: 281–300.10.1111/j.1540-6261.1987.tb02568.xSearch in Google Scholar

Jorion, P. 1996. “Risk2: Measuring the Risk in Value at Risk.” Financial Analysts Journal 52: 47–56.10.2469/faj.v52.n6.2039Search in Google Scholar

Koenker, R. 2005. Quantile Regression. New York: Cambridge University Press.10.1017/CBO9780511754098Search in Google Scholar

Koenker, R., and G. Bassett. 1978. “Regression Quantiles.” Econometrica 46: 33–50.10.2307/1913643Search in Google Scholar

Koenker, R., and K. F. Hallock. 2001. “Quantile Regression: An Introduction.” Journal of Economic Perspectives 15: 143–156.10.1257/jep.15.4.143Search in Google Scholar

Koenker, R., and J. A. Machado. 1999. “Goodness of Fit and Related Inference Processes for Quantile Regression.” Journal of the American Statistical Association 94: 1296–1310.10.1080/01621459.1999.10473882Search in Google Scholar

Koenker, R., and Q. Zhao. 1996. “Conditional Quantile Estimation and Inference for ARCH Models.” Econometric Theory 12: 793–813.10.1017/S0266466600007167Search in Google Scholar

Kuester, K., S. Mittnik, and M. S. Paolella. 2006. “Value-at-Risk Prediction: A Comparison of Alternative Strategies.” Journal of Financial Econometrics 4: 53–89.10.1093/jjfinec/nbj002Search in Google Scholar

Kupiec, P. H. 1995. “Techniques for Verifying the Accuracy of Risk Measurement Models.” Journal of Derivatives 3: 73–84.10.3905/jod.1995.407942Search in Google Scholar

Lawless, J. F. 1980. “Inference in the Generalized Gamma and Log Gamma Distributions.” Technometrics 22: 409–419.10.1080/00401706.1980.10486173Search in Google Scholar

McDonald, J. B. 1984. “Some Generalized Functions for the Size Distribution of Income.” Econometrica 52: 647–663.10.2307/1913469Search in Google Scholar

McDonald, J. B., and W. K. Newey. 1988. “Partially Adaptive Estimation of Regression Models via the Generalized t Distribution.” Econometric Theory 4: 428–457.10.1017/S0266466600013384Search in Google Scholar

McDonald, J. B., and D. O. Richards. 1988. “Hazard Rates and Generalized Beta Distributions.” In: 10th Advances in Reliability Technology Symposium, 378–386. Dordrecht: Springer.10.1007/978-94-009-1355-4_35Search in Google Scholar

McDonald, J. B., and Y. J. Xu. 1995. “A Generalization of the Beta Distribution with Applications.” Journal of Econometrics 66: 133–152.10.1016/0304-4076(94)01612-4Search in Google Scholar

Mielke, P. W., and E. S. Johnson. 1974. “Some Generalized Beta Distributions of the Second Kind Having Desirable Application Features in Hydrology and Meteorology.” Water Resources Research 10: 223–226.10.1029/WR010i002p00223Search in Google Scholar

Nieto, M. R., and E. Ruiz. 2016. “Frontiers in VaR Forecasting and Backtesting.” International Journal of Forecasting 32: 475–501.10.1016/j.ijforecast.2015.08.003Search in Google Scholar

Nitithumbundit, T., and J. S. Chan. 2015. “An ECM Algorithm for Skewed Multivariate Variance Gamma Distribution in Normal Mean-Variance Representation.” arXiv:1504.01239.Search in Google Scholar

Noufaily, A., and M. Jones. 2013. “Parametric Quantile Regression Based on the Generalized Gamma Distribution.” Journal of the Royal Statistical Society: Series C 62: 723–740.10.1111/rssc.12014Search in Google Scholar

Shao, X. D., Y. J. Lian, and L. Q. Yin. 2009. “Forecasting Value-at-Risk Using High Frequency Data: The Realized Range Model.” Global Finance Journal 20: 128–136.10.1016/j.gfj.2008.11.003Search in Google Scholar

Shaw, W. T., T. Luu, and N. Brickman. 2014. “Quantile Mechanics II: Changes of Variables in Monte Carlo Methods and GPU-Optimised Normal Quantiles.” European Journal of Applied Mathematics 25: 177–212.10.1017/S0956792513000417Search in Google Scholar

Wolters, M. H., and P. Tillmann. 2015. “The Changing Dynamics of US Inflation Persistence: A Quantile Regression Approach.” Studies in Nonlinear Dynamics and Econometrics 19: 161–182.10.1515/snde-2013-0080Search in Google Scholar

Supplementary Material

The online version of this article offers supplementary material (DOI: https://doi.org/10.1515/snde-2017-0012).

Published Online: 2018-09-06

©2019 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Research Articles
  2. A unified framework jointly explaining business conditions, stock returns, volatility and “volatility feedback news” effects
  3. Efficient estimation of financial risk by regressing the quantiles of parametric distributions: An application to CARR models
  4. A parametric stationarity test with smooth breaks
  5. Regression discontinuity designs with unknown state-dependent discontinuity points: estimation and testing
  6. Foster-Hart optimization for currency portfolios
  7. Asymmetric impact of uncertainty in recessions: are emerging countries more vulnerable?
https://doi.org/10.1515/snde-2017-0012
Keywords for this article
conditional autoregressive range model; generalised beta type two distribution; generalised-t distribution; parametric quantile regression; tail conditional expectation; volatility model

Articles in the same Issue

  1. Research Articles
  2. A unified framework jointly explaining business conditions, stock returns, volatility and “volatility feedback news” effects
  3. Efficient estimation of financial risk by regressing the quantiles of parametric distributions: An application to CARR models
  4. A parametric stationarity test with smooth breaks
  5. Regression discontinuity designs with unknown state-dependent discontinuity points: estimation and testing
  6. Foster-Hart optimization for currency portfolios
  7. Asymmetric impact of uncertainty in recessions: are emerging countries more vulnerable?
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 21.11.2025 from https://www.degruyterbrill.com/document/doi/10.1515/snde-2017-0012/html
Scroll to top button