Quantile Regression in Risk Calibration

  • Shih-Kang Chao
  • Wolfgang Karl Härdle
  • Weining Wang
Reference work entry

Abstract

Financial risk control has always been challenging and becomes now an even harder problem as joint extreme events occur more frequently. For decision makers and government regulators, it is therefore important to obtain accurate information on the interdependency of risk factors. Given a stressful situation for one market participant, one likes to measure how this stress affects other factors. The CoVaR (Conditional VaR) framework has been developed for this purpose. The basic technical elements of CoVaR estimation are two levels of quantile regression: one on market risk factors; another on individual risk factor.

Tests on the functional form of the two-level quantile regression reject the linearity. A flexible semiparametric modeling framework for CoVaR is proposed. A partial linear model (PLM) is analyzed. In applying the technology to stock data covering the crisis period, the PLM outperforms in the crisis time, with the justification of the backtesting procedures. Moreover, using the data on global stock markets indices, the analysis on marginal contribution of risk (MCR) defined as the local first order derivative of the quantile curve sheds some light on the source of the global market risk.

Keywords

CoVaR Value-at-Risk Quantile regression Locally linear quantile regression Partial linear model Semiparametric model 

References

  1. Acharya, V. V., Pedersen, L. H., Philippon, T., & Richardson, M. (2010). Measuring systemic risk (Working paper 10-02). Federal Reserve Bank of Cleveland.Google Scholar
  2. Adams, Z., Füss, R., & Gropp, R. (2012). Spillover effects among financial institutions: A state-dependent sensitivity value-at-risk approach, SAFE Working Paper Series, No. 20, http://dx.doi.org/10.2139/ssrn.2267853.
  3. Adrian, T., & Brunnermeier, M. K. (2011). CoVaR (Staff reports 348). Federal Reserve Bank of New York.Google Scholar
  4. Bae, K.-H., Karolyi, G. A., & Stulz, R. M. (2003). A new approach to measuring financial contagion. Review of Financial Studies, 16(3), 717–763.CrossRefGoogle Scholar
  5. Berkowitz, J., Christoffersen, P., & Pelletier, D. (2011). Evaluating value-at-risk models with desk-level data management science, 57, 2213–2227.Google Scholar
  6. Brownlees, C. T., & Engle, R. F. (2012). Volatility, correlation and tails for systemic risk measurement. Available at SSRN: http://ssrn.com/abstract=1611229 or http://dx.doi.org/10.2139/ssrn.1611229.
  7. Brunnermeier, M., & Pedersen, L. H. (2008). Market liquidity and funding liquidity. Review of Financial Studies, 22(2201), 2238.Google Scholar
  8. Cai, Z., & Wang, X. (2008). Nonparametric estimation of conditional VaR and expected shortfall. Journal of Econometrics, 147(120), 130.Google Scholar
  9. Carroll, R., & Hardle, W. (1989). Symmetrized nearest neighbor regression estimates. Statistics & Probability Letters, 7(315), 318.Google Scholar
  10. Chernozhukov, V., & Umantsev, L. (2001). Conditional value-at-risk: Aspects of modeling and estimation. Empirical Economics, 26(271), 292.Google Scholar
  11. Engle, R., & Manganelli, S. (2004). CAViaR: Conditional autoregressive value at risk by regression quantiles. Journal of Business & Economic Statistics, 22(367), 381.Google Scholar
  12. Fan, J., Hu, T.-C., & Truong, Y. K. (1994). Robust nonparametric function estimation. Scandinavian Journal of Statistics, 21, 433–446.Google Scholar
  13. Härdle, W., & Song, S. (2010). Confidence bands in quantile regression. Econometric Theory, 26, 1180–1200.CrossRefGoogle Scholar
  14. Härdle, W., Liang, H., & Gao, J. (2000). Partially linear models. Heidelberg: Physica-Verlag.CrossRefGoogle Scholar
  15. Härdle, W., Muller, M., Sperlicli, S., & Werwatz, A. (2004). Nonparametric and semiparametric models. Berlin: Springer.CrossRefGoogle Scholar
  16. Hautsch, N., Schaumburg, J., & Schienle, M. (2011). Financial network systemic risk contributions (Discussion paper 2011-072). CRC 649, Humboldt-Universitat zu Berlin.Google Scholar
  17. Huang, X., Zhou, H., & Zhu, H. (2012). Systemic risk contributions Journal of Financial Services Research, 42, 55–83. Springer US.Google Scholar
  18. Koenker, R., & Bassett, G. S. (1978). Regression quantiles. Econometrica, 46, 33.CrossRefGoogle Scholar
  19. 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.CrossRefGoogle Scholar
  20. Lobato, I., Nankervis, J. C., & Savin, N. (2001). Testing for autocorrelation using a modified Box-Pierce Q test. International Economic Review, 42(1), 187–205.CrossRefGoogle Scholar
  21. Ruppert, D., & Wand, M. P. (1995). Multivariate locally weighted least, squares regression. Annals of Statistics, 23(1346), 1370.Google Scholar
  22. Ruppert, D., Sheatlier, S. J., & Wand, M. P. (1995). An effective bandwidth selector for local least squares regression. Journal of the American Statistical Association, 90(1257), 1270.Google Scholar
  23. Schaumburg, J. (2011). Predicting extreme VaR: Nonparametric quantile regression with refinements from extreme value theory (Discussion Paper 2010-009). CRC 649, Humboldt-Universitat zu Berlin.Google Scholar
  24. Song, S., Ritov, Y., & Härdle, W. (2012). Partial linear quantile regression and bootstrap confidence bands. Journal of Multivariate Analysis, 107, 244–262.CrossRefGoogle Scholar
  25. Spokoiny, V., Wang, W., & Härdle, W. (2013). Local quantile regression. Journal of Statistical Planning and Inference, 143(7), 1109–1129. doi:10.1016/j.jspi.2013.03.008, with discussions.Google Scholar
  26. Taylor, J. W. (2008). Using exponentially weighted quantile regression to estimate value at risk and expected shortfall. Journal of Financial Econometrics, 6, 382–406.CrossRefGoogle Scholar
  27. Yu, K., & Jones, M. C. (1998). Local linear quantile regression. Journal of the American Statistical Association, 98(228), 237.Google Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Shih-Kang Chao
    • 1
  • Wolfgang Karl Härdle
    • 1
    • 2
  • Weining Wang
    • 1
  1. 1.Ladislaus von Bortkiewicz Chair of Statistics, C.A.S.E. – Center for Applied Statistics and EconomicsHumboldt–Universität zu BerlinBerlinGermany
  2. 2.Lee Kong Chian School of Business, Singapore Management UniversitySingaporeSingapore

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