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ESTIMATION OF HIGH CONDITIONAL TAIL RISK BASED ON EXPECTILE REGRESSION

Published online by Cambridge University Press:  15 February 2021

Jie Hu Open the ORCID record for Jie Hu [Opens in a new window] ,
Yu Chen Open the ORCID record for Yu Chen [Opens in a new window]  and
Keqi Tan
Jie Hu
Affiliation:
Department of Statistics and Finance, School of Management University of Science and Technology of China, Hefei, China
Yu Chen*
Affiliation:
Department of Statistics and Finance, School of Management University of Science and Technology of China, Hefei, China E-Mail: cyu@ustc.edu.cn
Keqi Tan
Affiliation:
Department of Statistics and Finance, School of Management University of Science and Technology of China, Hefei, China
*
E-Mail: cyu@ustc.edu.cn
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Abstract

Assessing conditional tail risk at very high or low levels is of great interest in numerous applications. Due to data sparsity in high tails, the widely used quantile regression method can suffer from high variability at the tails, especially for heavy-tailed distributions. As an alternative to quantile regression, expectile regression, which relies on the minimization of the asymmetric l2-norm and is more sensitive to the magnitudes of extreme losses than quantile regression, is considered. In this article, we develop a new estimation method for high conditional tail risk by first estimating the intermediate conditional expectiles in regression framework, and then estimating the underlying tail index via weighted combinations of the top order conditional expectiles. The resulting conditional tail index estimators are then used as the basis for extrapolating these intermediate conditional expectiles to high tails based on reasonable assumptions on tail behaviors. Finally, we use these high conditional tail expectiles to estimate alternative risk measures such as the Value at Risk (VaR) and Expected Shortfall (ES), both in high tails. The asymptotic properties of the proposed estimators are investigated. Simulation studies and real data analysis show that the proposed method outperforms alternative approaches.

Keywords

Quantile regressionexpectile regressionasymptoticheavy-tailed distributiontail index
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 51 , Issue 2 , May 2021 , pp. 539 - 570
Copyright
© 2021 by Astin Bulletin. All rights reserved

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References

Acerbi, C. (2002) Spectral measures of risk: a coherent representation of subjective risk aversion. Journal of Banking and Finance, 26, 1505–1518.CrossRefGoogle Scholar
Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D. (1999) Coherent measures of risk. Mathematical Finance, 9, 203–228.CrossRefGoogle Scholar
Bellini, F., Klar, B., Müller, A. and Rosazza Gianina, E. (2014) Generalized quantiles as risk measures. Insurance: Mathematics and Economics, 54, 41–48.Google Scholar
Bellini, F. and Di Bernardino, E. (2017). Risk management with expectiles. The European Journal of Finance, 23, 487–506CrossRefGoogle Scholar
Chan, N.H., Deng, S. J., Peng, L. and Xia, Z. (2007) Interval estimation of value-at-risk based on GARCH models with heavy-tailed innovations. Journal of Econometrics 137(2), 556–576.CrossRefGoogle Scholar
Cai, Z., Fang, Y. and Tian, D. (2018). Assessing tail risk using expectile regressions with partially varying coefficients. Journal of Management Science and Engineering, 3, 183–213.CrossRefGoogle Scholar
de Haan, L. and Ferreira, A. (2006) Extreme Value Theory: An Introduction. New York: Springer.CrossRefGoogle Scholar
Daouia, A., Girard, S. and Stupfler, G. (2018) Estimation of tail risk based on extreme expectiles. Journal of the Royal Statistical Society, Series B, 80, 263–292.CrossRefGoogle Scholar
Daouia, A., Girard, S. and Stupfler, G. (2020a) Tail expectile process and risk assessment. Bernoulli, 26(1), 531–556.CrossRefGoogle Scholar
Daouia, A., Girard, S. and Stupfler, G. (2020b) ExpectHill estimation, extreme risk and heavy tails. Journal of Econometrics, preprint.CrossRefGoogle Scholar
Fang, L.B., Sun, B.Y., Li, H.J. and Yu, H. (2018) Systemic risk network of chinese financial institutions. Emerging Markets Review, 35, 190–206.CrossRefGoogle Scholar
Gomes, M. and Pestana, D. (2007) A simple second-order reduced bias tail index estimator. Journal of Statistical Computation and Simulation, 77, 487–504.CrossRefGoogle Scholar
Hill, B.M. (1975) A simple general approach to inference about the tail of a distribution. The Annals of Statistics, 3, 1163–1174.CrossRefGoogle Scholar
Koenker, R. and Bassett, G. (1978) Regression quantiles. Econometrica, 46, 33–50.CrossRefGoogle Scholar
Kuan, C.-M., Yeh, J.-H. and Hsu, Y.-C. (2009) Assessing value at risk with CARE, the Conditional Autoregressive Expectile models. Journal of Econometrics, 2, 261–270.CrossRefGoogle Scholar
Li, D., Peng, L. and Yang, J. (2010) Bias reduction for high quantiles. Journal of Statistical Planning and Inference, 140, 2433–2441.CrossRefGoogle Scholar
Newey, W.K. and Powell, J.L. (1987) Asymmetric least squares estimation and testing. Econometrica, 55, 819–847.CrossRefGoogle Scholar
Rockafellar, R.T. and Uryasev, S. (2000) Optimization of conditional value-at-risk. Journal of Risk, 2, 21–42.CrossRefGoogle Scholar
Sobotka, F., Radice, R., Marra, G. and Kneib, T. (2013a) Estimating the relationship between women’s education and fertility in Bostwana by using and instumental variable approach to semiparametric expectile regression. Journal of the Royal Statistical Society, 62, 25–45.Google Scholar
Taylor, J.W. (2008) Estimating value at risk and expected shortfall using expectiles. Journal of Financial Econometrics, 6, 231–252CrossRefGoogle Scholar
Weissman, I. (1978) Estimation of parameters and large quantiles based on the k largest observations. Journal of the American Statistical Association, 73, 812–815.Google Scholar
Wirch, J.L. and Hardy, M.R. (1999) A synthesis of risk measures for capital adequacy. Insurance: Mathematics and Economics, 25, 337–347.Google Scholar
Wang, H. and Tsai, C.L. (2009) Tail index regression. Journal of the American Statistical Association, 104, 1233–1240.CrossRefGoogle Scholar
Wang, H.J., Li, D. and He, X. (2012) Estimation of high conditional quantiles for heavy-tailed distributions. Journal of the American Statistical Association, 107, 1453–1464.CrossRefGoogle Scholar
Yao, Q. and Tong, H. (1996) Asymmetric least squares regression estimation: a nonparametric approach. Journal of Nonparametric Statistics, 6, 273–292.CrossRefGoogle Scholar