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

  • Carlos Martins-Filho (a1) (a2), Feng Yao (a3) (a4) and Maximo Torero (a2)

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.

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Corresponding author

*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.

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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.

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References

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Azzalini, A. (1981) A note on the estimation of a distribution function and quantiles by a kernel method. Biometrika 68, 326328.
Bowman, A., Hall, P., & Prvan, T. (1998) Bandwidth selection for the smoothing of distribution functions. Biometrika 85, 799808.
Cai, Z. (2002) Regression quantiles for time series. Econometric Theory 18, 169192.
Cai, Z. & Wang, X. (2008) Nonparametric estimation of conditional VaR and expected shortfall. Journal of Econometrics 147, 120130.
Chen, S.X. (2008) Nonparametric estimation of expected shortfall. Journal of Financial Econometrics 6, 87107.
Chen, S.X. & Tang, C. (2005) Nonparametric inference of value at risk for dependent financial returns. Journal of Financial Econometrics 3, 227255.
Chernozhukov, V. (2005) Extremal quantile regression. The Annals of Statistics 33, 806839.
Chernozhukov, V. & Umantsev, L. (2001) Conditional value-at-risk: Aspects of modelling and estimation. Empirical Economics 26, 271292.
Christoffersen, P. (1998) Evaluating internal forecasts. International Economic Review 39, 841862.
Christoffersen, P., Berkowitz, J., & Pelletier, D. (2009) Evaluating Value-At-Risk Models with Desk Level Data. Tech. rep. 2009–35, CREATES.
Christoffersen, P. & Pelletier, D. (2004) Backtesting value-at-risk: A duration-based approach. Journal of Financial Econometrics 2, 84108.
Cosma, A., Scaillet, O., & von Sachs, R. (2007) Multivariate wavelet-based shape preserving estimation for dependent observations. Bernoulli 13, 301329.
Danielsson, J. (2011) Financial Risk Forecasting. John Wiley and Sons.
Davidson, J. (1994) Stochastic Limit Theory. Oxford University Press.
Doukhan, P. (1994) Mixing: Properties and Examples. Springer-Verlag.
Drost, F.C. & Nijman, T.E. (1993) Temporal aggregation of GARCH processes. Econometrica 61, 909927.
Duffie, D. & Singleton, K. (2003) Credit Risk: Pricing, Measurement and Management. Princeton University Press.
Embrechts, P., Kluppelberg, C., & Mikosh, T. (1997) Modelling Extremal Events for Insurance and Finance. Springer Verlag.
Escanciano, J.C. (2009) Quasi-maximum likelihood estimation of semi-strong GARCH models. Econometric Theory 25, 561570.
Falk, M. (1985) Asymptotic normality of the kernel quantile estimator. Annals of Statistics 13, 428433.
Fan, J. & Yao, Q. (1998) Efficient estimation of conditional variance functions in stochastic regression. Biometrika 85, 645660.
Gao, J. (2007) Nonlinear Time Series: Nonparametric and Parametric Methods. Chapman and Hall.
Gnedenko, B.V. (1943) Sur la distribution limite du terme d’une série aléatoire. Annals of Mathematics 44, 423453.
Goldie, C.M. & Smith, R.L. (1987) Slow variation with remainder: A survey of the theory and its applications. Quarterly Journal of Mathematics 38, 4571.
Hall, P. (1982) On some simple estimates of an exponent of regular variation. Journal of Royal Satistical Society Series B 44, 3742.
Härdle, W. & Tsybakov, A.B. (1997) Local polynomial estimators of the volatility function in nonparametric autoregression. Journal of Econometrics 81, 233242.
Hill, B.M. (1975) A simple general approach to inference about the tail of a distribution. Annals of Statistics 3, 11631174.
Hill, J.B. (2015) Expected shortfall estimation and Gaussian inference for infinite variance time series. Journal of Financial Econometrics 13, 144.
Kato, K. (2012) Weighted Nadaraya-Watson estimation of conditional expected shortfall. Journal of Financial Econometrics 10, 265291.
Leadbetter, M., Lindgren, G., & Rootzen, H. (1983) Extremes and Related Properties of Random Sequences and Processes. Springer Verlag.
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.
Linton, O.B., Pan, J., & Wang, H. (2010) Estimation on nonstationary semi-strong GARCH(1,1) model with heavy-tailed errors. Econometric Theory 26, 128.
Linton, O. B. & Xiao, Z. (2013) Estimation of and inference about the expected shortfall for time series with infinite variance. Econometric Theory 29, 771807.
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, 141, Article 4.
Martins-Filho, C. & Yao, F. (2008) A smoothed conditional quantile frontier estimator. Journal of Econometrics 143, 317333.
Martins-Filho, C., Yao, F., & Torero, M. (2015) High order conditional quantile estimation based on nonparametric models of regression. Econometric Reviews 34, 906957.
Masry, E. & Fan, J. (1997) Local polynomial estimation of regression functions for mixing processes. Scandinavian Journal of Statistics 24, 19651979.
Masry, E. & Tjøstheim, D. (1995) Nonparametric estimation and identification of nonlinear ARCH time series: Strong convergence and asymptotic normality. Econometric Theory 11, 258289.
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, 271300.
McNeil, A.J., Frey, B., & Embrechts, P. (2005) Quantitative Risk Management: Concepts, Techniques and Tools. Princeton University Press.
Pagan, A. & Ullah, A. (1999) Nonparametric Econometrics. Cambridge University Press.
Peng, L. (1998) Asymptotically unbiased estimators for the extreme-value index. Statistics and Probability Letters 38, 107115.
Pickands, J. (1975) Statistical inference using extreme order statistics. Annals of Statistics 3, 119131.
Resnick, S.I. (1987) Extreme Values, Regular Variation and Point Processes. Springer Verlag.
Ruppert, D., Sheather, S., & Wand, M.P. (1995) An effective bandwidth selector for local least squares regression. Journal of the American Statistical Association 90, 12571270.
Scaillet, O. (2004) Nonparametric estimation and sensitivity analysis of expected shortfall. Mathematical Finance 14, 115129.
Scaillet, O. (2005) Nonparametric estimation of conditional expected shortfall. Revue Assurances et Gestion des Risques/Insurance and Risk Management Journal 72, 639660.
Smith, R.L. (1985) Maximum likelihood estimation in a class of nonregular cases. Biometrika 72, 6790.
Smith, R.L. (1987) Estimating tails of probability distributions. Annals of Statistics 15, 11741207.
Tsay, R. (2010) Analysis of Financial Time Series, 3rd ed. Wiley.
Yang, S.-S. (1985) A Smooth nonparametric estimator of a quantile function. Journal of the American Satistical Association 80, 10041011.
Yu, K. & Jones, M.C. (1998) Local linear quantile regression. Journal of the American Statistical Association 93, 228237.
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NONPARAMETRIC ESTIMATION OF CONDITIONAL VALUE-AT-RISK AND EXPECTED SHORTFALL BASED ON EXTREME VALUE THEORY

  • Carlos Martins-Filho (a1) (a2), Feng Yao (a3) (a4) and Maximo Torero (a2)

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