Hostname: page-component-74656f9884-rdgjk Total loading time: 0 Render date: 2026-11-09T03:59:07.353Z Has data issue: false hasContentIssue false
  • English
  • Français

MODEL-FREE INFERENCE FOR TAIL RISK MEASURES

Published online by Cambridge University Press:  10 November 2014

Ke-Li Xu
Ke-Li Xu*
Affiliation:
Texas A&M University
*
*Address correspondence to Ke-Li Xu, Department of Economics, Texas A&M University, 3063 Allen Building, 4228 TAMU, College Station, Texas 77843-4228, USA; e-mail: keli.xu@tamu.edu.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

Understanding uncertainty in estimating risk measures is important in modern financial risk management. In this paper we consider a nonparametric framework that incorporates auxiliary information available in covariates and propose a family of inferential methods for the value at risk, expected shortfall, and related risk measures. A two-step generalized empirical likelihood test statistic is constructed and is shown to be asymptotically pivotal without requiring variance estimation. We also show its validity when applied to a semiparametric index model. Asymptotic theories are established allowing for serially dependent data. Simulations and an empirical application to Canadian stock return index illustrate the finite sample behavior of the methodologies proposed.

Information

Type
ARTICLES
Information
Econometric Theory , Volume 32 , Issue 1 , February 2016 , pp. 122 - 153
Copyright
Copyright © Cambridge University Press 2014 

References

REFERENCES

Acerbi, C. & Tasche, D. (2002) On the coherence of expected shortfall. Journal of Banking and Finance 26, 1487–1503.CrossRefGoogle Scholar
Andrews, D.W.K. (1987) Consistency in nonlinear econometric models: A generic uniform law of large numbers. Econometrica 55, 1465–1472.CrossRefGoogle Scholar
Artzner, P., Delbaen, F., Eber, J.-M., & Heath, D. (1999) Coherent measures of risk. Mathematical Finance 9, 203–228.CrossRefGoogle Scholar
Baggerly, K.A. (1998) Empirical likelihood as a goodness-of-fit measure. Biometrika 85, 535–547.Google Scholar
Basak, S. & Shapiro, A. (2001) Value-at-risk based risk management: Optimal policies and asset prices. Review of Financial Studies 14, 371–405.CrossRefGoogle Scholar
Bassett, G., Koenker, R., & Kordas, G. (2004) Pessimistic portfolio allocation and Choquet expected utility. Journal of Financial Econometrics 2, 477–492.CrossRefGoogle Scholar
Cai, Z. & Wang, X. (2008) Nonparametric estimation of conditional VaR and expected shortfall. Journal of Econometrics 147, 120–130.Google Scholar
Campbell, J.Y. & Yogo, M. (2006) Efficient tests of stock return predictability. Journal of Financial Economics 81, 27–60.Google Scholar
Chan, N.H., Deng, S.J., Peng, L., & Xia, Z. (2007) Interval estimation for the conditional value-at-risk based on GARCH models with heavy tailed innovations. Journal of Econometrics 137, 556–576.Google Scholar
Chan, N.H., Peng, L., & Zhang, D. (2011) Empirical likelihood based confidence intervals for conditional variance in heteroskedastic regression models. Econometric Theory 27, 154–177.Google Scholar
Chaudhuri, P., Doksum, K., & Samarov, A. (1997) On average derivative quantile regression. Annals of Statistics 25, 715–744.Google Scholar
Chen, S.X. (2008) Nonparametric estimation of expected shortfall. Journal of Financial Econometrics 6, 87–107.Google Scholar
Chen, S.X. & Qin, Y.S. (2000) Empirical likelihood confidence intervals for local linear smoothers. Biometrika 87, 946–953.CrossRefGoogle Scholar
Christoffersen, P. & Gonçalves, S. (2005) Estimation risk in financial risk management. Journal of Risk 7, 1–28.Google Scholar
Corcoran, S.A. (1998) Bartlett adjustment of empirical discrepancy statistics. Biometrika 85, 967–972.CrossRefGoogle Scholar
Cressie, N. & Read, T. (1984) Multinomial goodness-of-fit tests. Journal of the Royal Statistical Society (Series B) 46, 440–464.Google Scholar
Elliott, G., Rothenberg, T.J., & Stock, J.H. (1996) Efficient tests for an autoregressive unit root. Econometrica 64, 813–836.Google Scholar
Engle, R.F. & Manganelli, S. (2004) CAViaR: Conditional autoregressive value at risk by regression quantiles. Journal of Business and Economic Statistics 22, 367–381.Google Scholar
Fan, J. & Gijbels, I. (1996) Local Polynomial Modelling and Its Applications. Chapman & Hall/CRC.Google Scholar
Fan, J. & Yao, Q. (2003) Nonlinear Time Series: Nonparametric and Parametric Methods. Springer.CrossRefGoogle Scholar
Gastwirth, J.L. (1971) A general definition of the Lorenz curve. Econometrica 39, 1037–1039.Google Scholar
Glosten, L.R., Jagannathan, R., & Runkle, D.E. (1993) On the relation between the expected value and the volatility of the nominal excess returns on stocks. Journal of Finance 48, 1779–1801.CrossRefGoogle Scholar
Gourieroux, C., Scaillet, O., & Laurent, J.P. (2000) Sensitivity analysis of values at risk. Journal of Empirical Finance 7, 225–245.CrossRefGoogle Scholar
Hall, P. & Yao, Q. (2005) Approximating conditional distribution functions using dimension reduction. Annals of Statistics 33, 1404–1421.Google Scholar
Hansen, B.E. (1994) Autoregressive conditional density estimation. International Economic Review 35, 705–730.CrossRefGoogle Scholar
Hart, J.D. (1996) Some automated methods of smoothing time-dependent data. Journal of Nonparametric Statistics 6, 115–142.Google Scholar
Hill, J. (2014) Expected shortfall estimation and gaussian inference for infinite variance time series. Journal of Financial Econometrics, forthcoming.Google Scholar
Kitamura, Y. (2006) Empirical likelihood methods in econometrics: Theory and practice. In Blundell, R., Torsten, P., & Newey, W.K. (eds.), Advances in Economics and Econometrics, Theory and Applications, Ninth World Congress. Cambridge University Press.Google Scholar
Kleiber, C. (2008) The Lorenz curve in economics and econometrics. In Betti, G. & Lemmi, A. (eds.), Advances on Income Inequality and Concentration Measures: Collected Papers in Memory of C. Gini and M.O. Lorenz, pp. 225–242. Routledge.Google Scholar
Komunjer, I. (2007) Asymmetric power distribution: Theory and applications to risk measurement. Journal of Applied Econometrics 22, 891–921.CrossRefGoogle Scholar
Kuan, C.-M., Yeh, J.-H., & Hsu, Y.-C. (2009) Assessing value at risk with CARE: The conditional autoregressive expectile models. Journal of Econometrics 150, 261–270.Google Scholar
Li, Q. & Racine, J.S. (2008) Nonparametric estimation of conditional CDF and quantile functions with mixed categorical and continuous data. Journal of Business and Economic Statistics 26, 423–434.Google Scholar
Linton, O. & Xiao, Z. (2013) Estimation of and inference about the expected shortfall for time series with infinite variance. Econometric Theory 29, 771–807.CrossRefGoogle Scholar
Masry, E. & Fan, J. (1997) Local polynomial estimation of regression functions for mixing processes. Scandinavian Journal of Statistics 24, 165–179.CrossRefGoogle Scholar
McNeil, A.J., Frey, R., & Embrechts, P. (2005) Quantitative Risk Management. Princeton University Press.Google Scholar
Mittnik, S. & Rachev, S.T. (2000) Stable Paretian Models in Finance. Wiley.Google Scholar
Newey, W.K. & Smith, R.J. (2004) Higher order properties of GMM and generalized empirical likelihood estimators. Econometrica 72, 219–255.Google Scholar
Owen, A.B. (1988) Empirical likelihood ratio confidence intervals for a single functional. Biometrika 75, 237–249.Google Scholar
Owen, A.B. (1990) Empirical likelihood ratio confidence regions. Annals of Statistics 18, 90–120.Google Scholar
Owen, A.B. (2001) Empirical Likelihood. Chapman and Hall/CRC.Google Scholar
Peracchi, F. & Tanase, A.V. (2008) On estimating the conditional expected shortfall. Applied Stochastic Models in Business and Industry 24, 471–493.Google Scholar
Qin, J. & Lawless, J. (1994) Empirical likelihood and general estimating equations. Annals of Statistics 22, 300–325.Google Scholar
Ruszczyski, A. & Vanderbei, R. (2003) Frontiers of stochastically nondominated portfolios. Econometrica 71, 1287–1298.Google Scholar
Scaillet, O. (2004) Nonparametric estimation and sensitivity analysis of expected shortfall. Mathematical Finance 14, 115–129.Google Scholar
Scaillet, O. (2005) Nonparametric estimation of conditional expected shortfall. Insurance and Risk Management Journal 72(4), 639–660.Google Scholar
Shorrocks, A.F. (1983) Ranking income distributions. Economica 50, 3–17.Google Scholar
Smith, R.J. (1997) Alternative semi-parametric likelihood approaches to generalized method of moments estimation. Economic Journal 107, 503–519.Google Scholar
Taylor, J.W. (2008) Using exponentially weighted quantile regression to estimate value at risk and expected shortfall. Journal of Financial Econometrics 6, 382–406.Google Scholar
Tsay, R.S. (2005) Analysis of Financial Time Series, 2nd ed. Wiley-Interscience.CrossRefGoogle Scholar
Wu, T.Z., Yu, K., & Yu, Y. (2010) Single-index quantile regression. Journal of Multivariate Analysis 101, 1607–1621.Google Scholar
Xu, K.-L. (2009) Empirical likelihood based inference for recurrent nonparametric diffusions. Journal of Econometrics 153, 65–82.CrossRefGoogle Scholar
Xu, K.-L. (2013) Nonparametric inference of conditional quantiles of time series. Econometric Theory 29(4), 673–698.Google Scholar
Yamai, Y. & Yoshiba, T. (2005) Value-at-risk versus expected shortfall: A practical perspective. Journal of Banking and Finance 29, 997–1015.CrossRefGoogle Scholar