Quantile Regression in Risk Calibration
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.
KeywordsCoVaR Value-at-Risk Quantile regression Locally linear quantile regression Partial linear model Semiparametric model
- 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
- 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.
- Adrian, T., & Brunnermeier, M. K. (2011). CoVaR (Staff reports 348). Federal Reserve Bank of New York.Google Scholar
- Berkowitz, J., Christoffersen, P., & Pelletier, D. (2011). Evaluating value-at-risk models with desk-level data management science, 57, 2213–2227.Google Scholar
- Brunnermeier, M., & Pedersen, L. H. (2008). Market liquidity and funding liquidity. Review of Financial Studies, 22(2201), 2238.Google Scholar
- Cai, Z., & Wang, X. (2008). Nonparametric estimation of conditional VaR and expected shortfall. Journal of Econometrics, 147(120), 130.Google Scholar
- Carroll, R., & Hardle, W. (1989). Symmetrized nearest neighbor regression estimates. Statistics & Probability Letters, 7(315), 318.Google Scholar
- Chernozhukov, V., & Umantsev, L. (2001). Conditional value-at-risk: Aspects of modeling and estimation. Empirical Economics, 26(271), 292.Google Scholar
- Engle, R., & Manganelli, S. (2004). CAViaR: Conditional autoregressive value at risk by regression quantiles. Journal of Business & Economic Statistics, 22(367), 381.Google Scholar
- Fan, J., Hu, T.-C., & Truong, Y. K. (1994). Robust nonparametric function estimation. Scandinavian Journal of Statistics, 21, 433–446.Google Scholar
- Hautsch, N., Schaumburg, J., & Schienle, M. (2011). Financial network systemic risk contributions (Discussion paper 2011-072). CRC 649, Humboldt-Universitat zu Berlin.Google Scholar
- Huang, X., Zhou, H., & Zhu, H. (2012). Systemic risk contributions Journal of Financial Services Research, 42, 55–83. Springer US.Google Scholar
- Ruppert, D., & Wand, M. P. (1995). Multivariate locally weighted least, squares regression. Annals of Statistics, 23(1346), 1370.Google Scholar
- 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
- 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
- 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
- Yu, K., & Jones, M. C. (1998). Local linear quantile regression. Journal of the American Statistical Association, 98(228), 237.Google Scholar