Electronic Journal of Statistics

A joint quantile and expected shortfall regression framework

Timo Dimitriadis and Sebastian Bayer

Full-text: Open access

Abstract

We introduce a novel regression framework which simultaneously models the quantile and the Expected Shortfall (ES) of a response variable given a set of covariates. This regression is based on strictly consistent loss functions for the pair consisting of the quantile and the ES, which allow for M- and Z-estimation of the joint regression parameters. We show consistency and asymptotic normality for both estimators under weak regularity conditions. The underlying loss functions depend on two specification functions, whose choices affect the properties of the resulting estimators. We find that the Z-estimator is numerically unstable and thus, we rely on M-estimation of the model parameters. Extensive simulations verify the asymptotic properties and analyze the small sample behavior of the M-estimator for different specification functions. This joint regression framework allows for various applications including estimating, forecasting and backtesting ES, which is particularly relevant in light of the recent introduction of the ES into the Basel Accords. We illustrate this through two exemplary empirical applications in forecasting and forecast combination of the ES.

Article information

Source
Electron. J. Statist., Volume 13, Number 1 (2019), 1823-1871.

Dates
Received: May 2018
First available in Project Euclid: 7 June 2019

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1559872834

Digital Object Identifier
doi:10.1214/19-EJS1560

Mathematical Reviews number (MathSciNet)
MR3959874

Zentralblatt MATH identifier
07080063

Keywords
Expected shortfall joint elicitability joint regression M-estimation quantile regression

Rights
Creative Commons Attribution 4.0 International License.

Citation

Dimitriadis, Timo; Bayer, Sebastian. A joint quantile and expected shortfall regression framework. Electron. J. Statist. 13 (2019), no. 1, 1823--1871. doi:10.1214/19-EJS1560. https://projecteuclid.org/euclid.ejs/1559872834


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References

  • Andersen, T. G. and Bollerslev, T. (1998). Answering the skeptics: Yes, standard volatility models do provide accurate forecasts., International Economic Review, 39:885–905.
  • Andrews, D. (1994). Empirical process methods in econometrics. In Engle, R. and McFadden, D., editors, Handbook of Econometrics, volume 4, chapter 37, pages 2247–2294. Elsevier.
  • Artzner, P., Delbaen, F., Eber, J.-M., and Heath, D. (1999). Coherent measures of risk., Mathematical Finance, 9(3):203–228.
  • Barendse, S. (2017). Interquantile expectation regression. Available at, https://ssrn.com/abstract=2937665.
  • Basel Committee (2016). Minimum capital requirements for market risk. Technical report, Basel Committee on Banking Supervision. Available at, http://www.bis.org/bcbs/publ/d352.pdf.
  • Bayer, S. (2018). Combining value-at-risk forecasts using penalized quantile regressions., Econometrics and Statistics, 8:56–77.
  • Bayer, S. and Dimitriadis, T. (2019). Regression based expected shortfall backtesting., arXiv:1801.04112 [q-fin.RM].
  • Bayer, S. and Dimitriadis, T. (2019)., esreg: Joint Quantile and Expected Shortfall Regression. R package version 0.4.0, available at https://cran.r-project.org/package=esreg.
  • Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity., Journal of Econometrics, 31(3):307–327.
  • Brazauskas, V., Jones, B. L., Puri, M. L., and Zitikis, R. (2008). Estimating conditional tail expectation with actuarial applications in view., Journal of Statistical Planning and Inference, 138(11):3590–3604.
  • Chao, W., Gerlach, R., and Chen, Q. (2018). A semi-parametric realized joint value-at-risk and expected shortfall regression framework., arXiv:1807.02422 [q-fin.RM].
  • Chen, S. X. (2008). Nonparametric estimation of expected shortfall., Journal of Financial Econometrics, 6(1):87–107.
  • Chernozhukov, V. and Umantsev, L. (2001). Conditional value-at-risk: Aspects of modeling and estimation., Empirical Economics, 26(1):271–292.
  • Corsi, F. (2009). A simple approximate long-memory model of realized volatility., Journal of Financial Econometrics, 7(2):174–196.
  • Efron, B. (1979). Bootstrap methods: Another look at the jackknife., The Annals of Statistics, 7(1):1–26.
  • Efron, B. (1991). Regression percentiles using asymmetric squared error loss., Statistica Sinica, 1:93–125.
  • Ehm, W., Gneiting, T., Jordan, A., and Krüger, F. (2016). Of quantiles and expectiles: consistent scoring functions, choquet representations and forecast rankings., Journal of the Royal Statistical Society: Series B, 78(3):505–562.
  • Engle, R. and Manganelli, S. (2004). CAViaR: Conditional autoregressive value at risk by regression quantiles., Journal of Business and Economic Statistics, 22(4):367–381.
  • Fissler, T. (2017)., On Higher Order Elicitability and Some Limit Theorems on the Poisson and Wiener Space. PhD thesis, Universität Bern.
  • Fissler, T. and Ziegel, J. F. (2016). Higher order elicitability and Osband’s principle., Annals of Statistics, 44(4):1680–1707.
  • Fissler, T., Ziegel, J. F., and Gneiting, T. (2016). Expected shortfall is jointly elicitable with value at risk – implications for backtesting., Risk Magazine, Janaury 2016.
  • Gaglianone, W. P., Lima, L. R., Linton, O., and Smith, D. R. (2011). Evaluating value-at-risk models via quantile regression., Journal of Business and Economic Statistics, 29(1):150–160.
  • Giacomini, R. and Komunjer, I. (2005). Evaluation and combination of conditional quantile forecasts., Journal of Business and Economic Statistics, 23:416–431.
  • Gikhman, I. and Skorokhod, A. (2004)., The Theory of Stochastic Processes I, volume 210 of Classics in Mathematics. Springer Berlin Heidelberg.
  • Gneiting, T. (2011). Making and evaluating point forecasts., Journal of the American Statistical Association, 106(494):746–762.
  • Gourieroux, C. and Monfort, A. (1995)., Statistics and Econometric Models: Volume 1, General Concepts, Estimation, Prediction and Algorithms. Cambridge University Press.
  • Gourieroux, C., Monfort, A., and Trognon, A. (1984). Pseudo maximum likelihood methods: Theory., Econometrica, 52(3):681–700.
  • Halbleib, R. and Pohlmeier, W. (2012). Improving the value at risk forecasts: Theory and evidence from the financial crisis., Journal of Economic Dynamics and Control, 36(8):1212–1228.
  • Hall, P. and Sheather, S. J. (1988). On the distribution of a studentized quantile., Journal of the Royal Statistical Society: Series B, 50(3):381–391.
  • Hansen, P. R., Lunde, A., and Nason, J. M. (2011). The model confidence set., Econometrica, 79(2):453–497.
  • Heber, G., Lunde, A., Shephard, N., and Sheppard, K. (2009)., Oxford-Man Institute’s realized library, version 0.3. Oxford-Man Institute, University of Oxford.
  • Hendricks, W. and Koenker, R. (1992). Hierarchical spline models for conditional quantiles and the demand for electricity., Journal of the American Statistical Association, 87(417):58–68.
  • Huber, P. (1967). The behavior of maximum likelihood estimates under nonstandard conditions. In, Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, pages 221–233. Berkeley: University of California Press.
  • Koenker, R. (1994). Confidence intervals for regression quantiles. In Mandl, P. and Hušková, M., editors, Asymptotic Statistics: Proceedings of the Fifth Prague Symposium, held from September 4–9, 1993, pages 349–359. Physica-Verlag Heidelberg.
  • Koenker, R. (2005)., Quantile Regression. Econometric Society Monographs. Cambridge University Press.
  • Koenker, R. and Machado, J. A. F. (1999). Goodness of fit and related inference processes for quantile regression., Journal of the American Statistical Association, 94(448):1296–1310.
  • Koenker, R. and Xiao, Z. (2006). Quantile autoregression., Journal of the American Statistical Association, 101(475):980–990.
  • Komunjer, I. (2013). Quantile prediction. In, Handbook of Economic Forecasting, volume 2, chapter 17, pages 961–994. Elsevier.
  • Komunjer, I. and Vuong, Q. (2010). Semiparametric efficiency bound in time-series models for conditional quantiles., Econometric Theory, 26(02):383–405.
  • Lambert, N. S., Pennock, D. M., and Shoham, Y. (2008). Eliciting properties of probability distributions. In, Proceedings of the 9th ACM Conference on Electronic Commerce, pages 129–138. ACM.
  • Lourenço, H. R., Martin, O. C., and Stützle, T. (2003). Iterated local search. In Glover, F. and Kochenberger, G. A., editors, Handbook of Metaheuristics, pages 320–353. Springer US, Boston, MA.
  • Nadarajah, S., Zhang, B., and Chan, S. (2014). Estimation methods for expected shortfall., Quantitative Finance, 14(2):271–291.
  • Nelder, J. A. and Mead, R. (1965). A simplex method for function minimization., The Computer Journal, 7(4):308–313.
  • Newey, W. and McFadden, D. (1994). Large sample estimation and hypothesis testing. In Engle, R. and McFadden, D., editors, Handbook of Econometrics, volume 4, chapter 36, pages 2111–2245. Elsevier.
  • Nolde, N. and Ziegel, J. F. (2017). Elicitability and backtesting: Perspectives for banking regulation., The Annals of Applied Statistics, 11(4):1833–1874.
  • Patton, A. J., Ziegel, J. F., and Chen, R. (2019). Dynamic semiparametric models for expected shortfall (and value-at-risk)., Forthcoming in Journal of Econometrics.
  • Taylor, J. W. (2008a). Estimating value at risk and expected shortfall using expectiles., Journal of Financial Econometrics, 6(2):231–252.
  • Taylor, J. W. (2008b). Using exponentially weighted quantile regression to estimate value at risk and expected shortfall., Journal of Financial Econometrics, 6(3):382–406.
  • Taylor, J. W. (2019). Forecasting value at risk and expected shortfall using a semiparametric approach based on the asymmetric laplace distribution., Journal of Business and Economic Statistics, 37:1, 121–133.
  • Timmermann, A. (2006). Forecast combinations. In Elliott, G., Granger, C. W., and Timmermann, A., editors, Handbook of Economic Forecasting, volume 1, chapter 4, pages 135–196. Elsevier.
  • van der Vaart, A. W. (1998)., Asymptotic statistics. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press.
  • Žikeš, F. and Baruník, J. (2016). Semi-parametric conditional quantile models for financial returns and realized volatility., Journal of Financial Econometrics, 14(1):185–226.
  • Weber, S. (2006). Distribution invariant risk measures, information, and dynamic consistency., Mathematical Finance, 16(2):419–441.
  • Xiao, Z., Guo, H., and Lam, M. S. (2015). Quantile regression and value at risk. In Lee, C.-F. and Lee, J. C., editors, Handbook of Financial Econometrics and Statistics, pages 1143–1167. Springer.
  • Ziegel, J. F., Krüger, F., Jordan, A., and Fasciati, F. (2019). Robust Forecast Evaluation of Expected Shortfall., Journal of Financial Econometrics.
  • Zwingmann, T. and Holzmann, H. (2016). Asymptotics for the expected shortfall., arXiv:1611.07222 [math.ST].