×

Backtesting VaR and expectiles with realized scores. (English) Zbl 1427.62121

Summary: Several statistical functionals such as quantiles and expectiles arise naturally as the minimizers of the expected value of a scoring function, a property that is called elicitability (see [T. Gneiting, J. Am. Stat. Assoc. 106, No. 494, 746–762 (2011; Zbl 1232.62028)] and the references therein). The existence of such scoring functions gives a natural way to compare the accuracy of different forecasting models, and to test comparative hypotheses by means of the Diebold-Mariano test as suggested in a recent work. In this paper we suggest a procedure to test the accuracy of a quantile or expectile forecasting model in an absolute sense, as in the original Basel I backtesting procedure of value-at-risk. To this aim, we study the asymptotic and finite-sample distributions of empirical scores for normal and uniform i.i.d. samples. We compare on simulated data the empirical power of our procedure with alternative procedures based on empirical identification functions (i.e. in the case of VaR the number of violations) and we find an higher power in detecting at least misspecification in the mean. We conclude with a real data example where both backtesting procedures are applied to AR(1)-GARCH(1,1) models fitted to SP500 logreturns for VaR and expectiles’ forecasts.

MSC:

62P05 Applications of statistics to actuarial sciences and financial mathematics
62G08 Nonparametric regression and quantile regression
62M20 Inference from stochastic processes and prediction
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)

Citations:

Zbl 1232.62028
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Acerbi C, Szekely B (2014) Backtesting expected shortfall. Risk Mag 27:76-81
[2] Artzner P, Delbaen F, Eber JM, Heath D (1999) Coherent measures of risk. Math. Finance 9(3):203-228 · Zbl 0980.91042
[3] Bellini F (2012) Isotonicity results for generalized quantiles. Stat Prob Lett 82:2017-2024 · Zbl 1312.62055 · doi:10.1016/j.spl.2012.07.003
[4] Bellini F, Bignozzi V (2014) On elicitable risk measures. Quant Finance 15:725-733 · Zbl 1395.91506 · doi:10.1080/14697688.2014.946955
[5] Bellini F, Di Bernardino E (2017) Risk management with expectiles. Eur J Finance 23(6):487-506 · doi:10.1080/1351847X.2015.1052150
[6] Bellini F, Klar B, Muller A (2014) Generalized quantiles as risk measures. Insur Math Econ 54:41-48 · Zbl 1303.91089 · doi:10.1016/j.insmatheco.2013.10.015
[7] Berkowitz J (2001) Testing the accuracy of density forecasts, applications to risk management. J Bus Econ Stat 19:465-474 · doi:10.1198/07350010152596718
[8] Campbell SD (2005) A review of backtesting and backtesting procedures. Finance and Economics Discussion Series, Federal Reserve Board, 21
[9] Christoffersen PF (1998) Evaluating interval forecasts. Int Econ Rev 39:841-862 · doi:10.2307/2527341
[10] Christoffersen PF, Pelletier D (2004) Backtesting value-at-risk: a duration-based approach. J Financ Econom 2:84-108 · doi:10.1093/jjfinec/nbh004
[11] Corbetta J, Peri I (2016) A new approach to backtesting and risk model selection. Preprint, ssrn.com/abstract=2796253
[12] Delbaen F, Bellini F, Bignozzi V, Ziegel J (2016) Risk measures with the CxLS property. Finance Stoch 20(2):433-453 · Zbl 1376.91173 · doi:10.1007/s00780-015-0279-6
[13] Ehm W, Gneiting T, Jordan A, Krüger F (2016) Of quantiles and expectiles: consistent scoring functions, Choquet representations and forecast rankings. J R Stat Soc Ser B 78(3):505-562 · Zbl 1414.62038 · doi:10.1111/rssb.12154
[14] Fissler T, Ziegel JF (2016) Higher order elicitability and Osband’s principle. Ann Stat 44(4):1680-1707 · Zbl 1355.62006 · doi:10.1214/16-AOS1439
[15] Fissler T, Ziegel JF, Gneiting T (2016) Expected shortfall is jointly elicitable with value at risk—implications for backtesting. Risk Mag 2016:58-611
[16] Gneiting T (2011) Making and evaluating point forecasts. J Am Stat Assoc 106:746-762 · Zbl 1232.62028 · doi:10.1198/jasa.2011.r10138
[17] Holzmann H, Eulert M (2014) The role of the information set for forecasting—with applications to risk management. Ann Appl Stat 8(1):595-621 · Zbl 1454.62277 · doi:10.1214/13-AOAS709
[18] Kerkhof J, Melenberg B (2004) Backtesting for risk-based regulatory capital. J Bank Finance 28:1845-1865 · doi:10.1016/j.jbankfin.2003.06.007
[19] Kupiec P (1995) Techniques for verifying the accuracy of risk measurement models. J Deriv 3:73-84 · doi:10.3905/jod.1995.407942
[20] Lopez JA (1999a) Evaluation of value-at-risk models. J. Risk 1:37-64 · doi:10.21314/JOR.1999.005
[21] Lopez JA (1999b) Methods for evaluating value-at-risk models. Federal Reserve Bank San Franc Econ Rev 2:3-17
[22] Macneil AJ, Frey R (2000) Estimation of tail-related risk measures for heteroschedastic financial time series: an extreme value approach. J Empir Finance 7:271-300 · doi:10.1016/S0927-5398(00)00012-8
[23] Newey W, Powell J (1987) Asymmetric least squares estimation and testing. Econometrica 55(4):819-847 · Zbl 0625.62047 · doi:10.2307/1911031
[24] Nolde N, Ziegel J (2016) Elicitability and backtesting. Preprint. arXiv:1608.05498 · Zbl 1383.62247
[25] Saerens M (2000) Building cost functions minimizing to some summary statistics. IEEE Trans Neural Netw 11:1263-1271 · doi:10.1109/72.883416
[26] Thomson W (1979) Eliciting production possibilities from a well-informed manager. J Econ Theory 20(3):360-380 · Zbl 0414.90055 · doi:10.1016/0022-0531(79)90042-5
[27] Weber S (2006) Distribution-invariant risk measures, information and dynamic consistency. Math Finance 16:419-441 · Zbl 1145.91037 · doi:10.1111/j.1467-9965.2006.00277.x
[28] Wong WK (2010) Backtesting value-at-risk based on tail losses. J Empir Finance 17:526-538 · doi:10.1016/j.jempfin.2009.11.004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.