Hostname: page-component-76c49bb84f-h2dfl Total loading time: 0 Render date: 2025-07-13T07:39:18.176Z Has data issue: false hasContentIssue false
  • English
  • Français

NONPARAMETRIC ESTIMATION OF CONDITIONAL VALUE-AT-RISK AND EXPECTED SHORTFALL BASED ON EXTREME VALUE THEORY

Published online by Cambridge University Press:  19 December 2016

Carlos Martins-Filho ,
Feng Yao  and
Maximo Torero
Carlos Martins-Filho*
Affiliation:
University of Colorado IFPRI
Feng Yao
Affiliation:
West Virginia University Guangdong University of Foreign Studies
Maximo Torero
Affiliation:
IFPRI
*
*Address correspondence to Carlos Martins-Filho, Department of Economics, University of Colorado, Boulder, CO 80309-0256, USA; and IFPRI, 2033 K Street NW, Washington, DC 20006-1002, USA; e-mail: carlos.martins@colorado.edu, c.martins-filho@cgiar.org.
Get access Rights & Permissions [Opens in a new window]

Abstract

We propose nonparametric estimators for conditional value-at-risk (CVaR) and conditional expected shortfall (CES) associated with conditional distributions of a series of returns on a financial asset. The return series and the conditioning covariates, which may include lagged returns and other exogenous variables, are assumed to be strong mixing and follow a nonparametric conditional location-scale model. First stage nonparametric estimators for location and scale are combined with a generalized Pareto approximation for distribution tails proposed by Pickands (1975, Annals of Statistics 3, 119–131) to give final estimators for CVaR and CES. We provide consistency and asymptotic normality of the proposed estimators under suitable normalization. We also present the results of a Monte Carlo study that sheds light on their finite sample performance. Empirical viability of the model and estimators is investigated through a backtesting exercise using returns on future contracts for five agricultural commodities.

Information

Type
ARTICLES
Information
Econometric Theory , Volume 34 , Issue 1 , February 2018 , pp. 23 - 67
Copyright
Copyright © Cambridge University Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

Footnotes

We thank Peter C. B. Phillips, Eric Renault, an Associate Editor and an anonymous referee for comments that improved the paper substantially. Any remaining errors are the authors’ responsibility.

References

REFERENCES

Azzalini, A. (1981) A note on the estimation of a distribution function and quantiles by a kernel method. Biometrika 68, 326–328.Google Scholar
Bowman, A., Hall, P., & Prvan, T. (1998) Bandwidth selection for the smoothing of distribution functions. Biometrika 85, 799–808.CrossRefGoogle Scholar
Cai, Z. (2002) Regression quantiles for time series. Econometric Theory 18, 169–192.CrossRefGoogle Scholar
Cai, Z. & Wang, X. (2008) Nonparametric estimation of conditional VaR and expected shortfall. Journal of Econometrics 147, 120–130.CrossRefGoogle Scholar
Chen, S.X. (2008) Nonparametric estimation of expected shortfall. Journal of Financial Econometrics 6, 87–107.CrossRefGoogle Scholar
Chen, S.X. & Tang, C. (2005) Nonparametric inference of value at risk for dependent financial returns. Journal of Financial Econometrics 3, 227–255.CrossRefGoogle Scholar
Chernozhukov, V. (2005) Extremal quantile regression. The Annals of Statistics 33, 806–839.CrossRefGoogle Scholar
Chernozhukov, V. & Umantsev, L. (2001) Conditional value-at-risk: Aspects of modelling and estimation. Empirical Economics 26, 271–292.CrossRefGoogle Scholar
Christoffersen, P. (1998) Evaluating internal forecasts. International Economic Review 39, 841–862.CrossRefGoogle Scholar
Christoffersen, P., Berkowitz, J., & Pelletier, D. (2009) Evaluating Value-At-Risk Models with Desk Level Data. Tech. rep. 2009–35, CREATES.Google Scholar
Christoffersen, P. & Pelletier, D. (2004) Backtesting value-at-risk: A duration-based approach. Journal of Financial Econometrics 2, 84–108.Google Scholar
Cosma, A., Scaillet, O., & von Sachs, R. (2007) Multivariate wavelet-based shape preserving estimation for dependent observations. Bernoulli 13, 301–329.CrossRefGoogle Scholar
Danielsson, J. (2011) Financial Risk Forecasting. John Wiley and Sons.Google Scholar
Davidson, J. (1994) Stochastic Limit Theory. Oxford University Press.CrossRefGoogle Scholar
Doukhan, P. (1994) Mixing: Properties and Examples. Springer-Verlag.CrossRefGoogle Scholar
Drost, F.C. & Nijman, T.E. (1993) Temporal aggregation of GARCH processes. Econometrica 61, 909–927.CrossRefGoogle Scholar
Duffie, D. & Singleton, K. (2003) Credit Risk: Pricing, Measurement and Management. Princeton University Press.Google Scholar
Embrechts, P., Kluppelberg, C., & Mikosh, T. (1997) Modelling Extremal Events for Insurance and Finance. Springer Verlag.CrossRefGoogle Scholar
Escanciano, J.C. (2009) Quasi-maximum likelihood estimation of semi-strong GARCH models. Econometric Theory 25, 561–570.Google Scholar
Falk, M. (1985) Asymptotic normality of the kernel quantile estimator. Annals of Statistics 13, 428–433.Google Scholar
Fan, J. & Yao, Q. (1998) Efficient estimation of conditional variance functions in stochastic regression. Biometrika 85, 645–660.CrossRefGoogle Scholar
Gao, J. (2007) Nonlinear Time Series: Nonparametric and Parametric Methods. Chapman and Hall.CrossRefGoogle Scholar
Gnedenko, B.V. (1943) Sur la distribution limite du terme d’une série aléatoire. Annals of Mathematics 44, 423–453.CrossRefGoogle Scholar
Goldie, C.M. & Smith, R.L. (1987) Slow variation with remainder: A survey of the theory and its applications. Quarterly Journal of Mathematics 38, 45–71.CrossRefGoogle Scholar
Hall, P. (1982) On some simple estimates of an exponent of regular variation. Journal of Royal Satistical Society Series B 44, 37–42.Google Scholar
Härdle, W. & Tsybakov, A.B. (1997) Local polynomial estimators of the volatility function in nonparametric autoregression. Journal of Econometrics 81, 233–242.CrossRefGoogle Scholar
Hill, B.M. (1975) A simple general approach to inference about the tail of a distribution. Annals of Statistics 3, 1163–1174.CrossRefGoogle Scholar
Hill, J.B. (2015) Expected shortfall estimation and Gaussian inference for infinite variance time series. Journal of Financial Econometrics 13, 1–44.CrossRefGoogle Scholar
Kato, K. (2012) Weighted Nadaraya-Watson estimation of conditional expected shortfall. Journal of Financial Econometrics 10, 265–291.CrossRefGoogle Scholar
Leadbetter, M., Lindgren, G., & Rootzen, H. (1983) Extremes and Related Properties of Random Sequences and Processes. Springer Verlag.CrossRefGoogle Scholar
Ling, C. & Peng, Z. (2015) Approximations of Weyl fractional-order integrals with insurance applications. Available at http://arxiv.org/find/grp_physics/1/au:+Ling_Chengxiu/0/1/0/all/0/1, ArXiv.Google Scholar
Linton, O.B., Pan, J., & Wang, H. (2010) Estimation on nonstationary semi-strong GARCH(1,1) model with heavy-tailed errors. Econometric Theory 26, 1–28.CrossRefGoogle Scholar
Linton, O. B. & Xiao, Z. (2013) Estimation of and inference about the expected shortfall for time series with infinite variance. Econometric Theory 29, 771–807.CrossRefGoogle Scholar
Martins-Filho, C. & Yao, F. (2006) Estimation of value-at-risk and expected shortfall based on nonlinear models of return dynamics and extreme value theory. Studies in Nonlinear Dynamics & Econometrics 10, 1–41, Article 4.Google Scholar
Martins-Filho, C. & Yao, F. (2008) A smoothed conditional quantile frontier estimator. Journal of Econometrics 143, 317–333.CrossRefGoogle Scholar
Martins-Filho, C., Yao, F., & Torero, M. (2015) High order conditional quantile estimation based on nonparametric models of regression. Econometric Reviews 34, 906–957.CrossRefGoogle Scholar
Masry, E. & Fan, J. (1997) Local polynomial estimation of regression functions for mixing processes. Scandinavian Journal of Statistics 24, 1965–1979.CrossRefGoogle Scholar
Masry, E. & Tjøstheim, D. (1995) Nonparametric estimation and identification of nonlinear ARCH time series: Strong convergence and asymptotic normality. Econometric Theory 11, 258–289.CrossRefGoogle Scholar
McNeil, A. & Frey, B. (2000) Estimation of tail-related risk measures for heteroscedastic financial time series: An extreme value approach. Journal of Empirical Finance 7, 271–300.CrossRefGoogle Scholar
McNeil, A.J., Frey, B., & Embrechts, P. (2005) Quantitative Risk Management: Concepts, Techniques and Tools. Princeton University Press.Google Scholar
Pagan, A. & Ullah, A. (1999) Nonparametric Econometrics. Cambridge University Press.Google Scholar
Peng, L. (1998) Asymptotically unbiased estimators for the extreme-value index. Statistics and Probability Letters 38, 107–115.CrossRefGoogle Scholar
Pickands, J. (1975) Statistical inference using extreme order statistics. Annals of Statistics 3, 119–131.Google Scholar
Resnick, S.I. (1987) Extreme Values, Regular Variation and Point Processes. Springer Verlag.CrossRefGoogle Scholar
Ruppert, D., Sheather, S., & Wand, M.P. (1995) An effective bandwidth selector for local least squares regression. Journal of the American Statistical Association 90, 1257–1270.CrossRefGoogle Scholar
Scaillet, O. (2004) Nonparametric estimation and sensitivity analysis of expected shortfall. Mathematical Finance 14, 115–129.CrossRefGoogle Scholar
Scaillet, O. (2005) Nonparametric estimation of conditional expected shortfall. Revue Assurances et Gestion des Risques/Insurance and Risk Management Journal 72, 639–660.Google Scholar
Smith, R.L. (1985) Maximum likelihood estimation in a class of nonregular cases. Biometrika 72, 67–90.CrossRefGoogle Scholar
Smith, R.L. (1987) Estimating tails of probability distributions. Annals of Statistics 15, 1174–1207.CrossRefGoogle Scholar
Tsay, R. (2010) Analysis of Financial Time Series, 3rd ed. Wiley.CrossRefGoogle Scholar
Yang, S.-S. (1985) A Smooth nonparametric estimator of a quantile function. Journal of the American Satistical Association 80, 1004–1011.CrossRefGoogle Scholar
Yu, K. & Jones, M.C. (1998) Local linear quantile regression. Journal of the American Statistical Association 93, 228–237.CrossRefGoogle Scholar
Supplementary material: PDF

Martins-Filho supplementary material

Martins-Filho supplementary material

Download Martins-Filho supplementary material(PDF)
PDF 448.3 KB