Abstract
Conditional Autoregressive Value-at-Risk and Conditional Autoregressive Expectile have become two popular approaches for direct measurement of market risk. Since their introduction several improvements both in the Bayesian and in the classical framework have been proposed to better account for asymmetry and local non-linearity. Here we propose a unified Bayesian Conditional Autoregressive Risk Measures approach by using the Skew Exponential Power distribution. Further, we extend the proposed models using a semiparametric P-Spline approximation answering for a flexible way to consider the presence of non-linearity. To make the statistical inference we adapt the MCMC algorithm proposed in Bernardi et al. (2018) to our case. The effectiveness of the whole approach is demonstrated using real data on daily return of five stock market indices.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
For this reason, while for the \({\mathrm{LR}}_{uc}\) and the \({\mathrm{LR}}_{cc}\) tests we consider the null hypothesis rejected at the 5% level we evaluate the DQ test on a 1% significance level, as in Engle and Manganelli (2004)
References
Andrieu C, Moulines R (2006) On the ergodicity properties of some adaptive MCMC algorithms. Ann Appl Probab 16(3):1462–1505. https://doi.org/10.1214/105051606000000286
Andrieu C, Thoms J (2008) A tutorial on adaptive MCMC. Stat Comput 18(4):343–373
Artzner P, Delbaen F, Eber JM, Heath D (1999) Coherent measures of risk. Math Finance 9(3):203–228. https://doi.org/10.1111/1467-9965.00068
Atchad’e Y, Fort G, Moulines E, Priouret P (2011) Adaptive Markov chain Monte Carlo: theory and methods. In: Barber D, Cemgil AT, Chiappa S (eds) Bayesian time series models. Cambridge University Press, Cambridge, pp 32–51. https://doi.org/10.1017/CBO9780511984679.003
Atchad’e YF, Rosenthal JS (2005) On adaptive Markov chain Monte Carlo algorithms. Bernoulli 11(5):815–828. https://doi.org/10.3150/bj/1130077595
Bernardi M, Gayraud G, Petrella L (2015) Bayesian tail risk interdependence using quantile regression. Bayesian Anal 10(3):553–603
Bernardi M, Bottone M, Petrella L (2018) Bayesian quantile regression using the skew exponential power distribution. Comput Stat Data Anal 126:92–111
Boor CD (2001) A pratical guide to splines. Springer, New York
Brezger A, Lang S (2006) Generalized structured additive regression based on bayesian p-splines. Comput Stat Data Anal 50(4):967–991. https://doi.org/10.1016/j.csda.2004.10.011
Brezger A, Steiner WJ (2008) Monotonic regression based on Bayesian p-splines: an application to estimating price response functions from store-level scanner data. J Bus Econ Stat 26(1):90–104
Chen C, Lin S, Yu P (2012) Smooth transition quantile capital asset pricing models with heteroscedasticity. Comput Econ 40:19–48
Chen CWS, So MKP, Lin EMH (2009) Volatility forecasting with double Markov switching garch models. J Forecast 28(8):681–697. https://doi.org/10.1002/for.1119
Christoffersen P (1998) Evaluating interval forecasts. J Empir Finance 39:841–862
Efron B (1991) Regression percentiles using asymmetric squared error loss. Stat Sin A:93–125
Efron B, Tibshirani J (1993) An introduction to the bootstrap. Chapman and Hall, New York
Eilers PHC, Rijnmond DM, Marx BD (1996) Flexible smoothing with b-splines and penalties. Stat Sci 11:89–121
Engle R, Manganelli S (2004) Caviar: conditional autoregressive value at risk by regression quantiles. J Bus Econ Stat 22:367–381
Engle RF, Ng VK (1993) Measuring and testing the impact of news on volatility. J Finance 48(5):1749–1778. https://doi.org/10.1111/j.1540-6261.1993.tb05127.x
Gerlach R, Chen CWS (2014) Bayesian expected shortfall forecasting incorporating the intraday range. J Financ Econom. https://doi.org/10.1093/jjfinec/nbu022
Gerlach R, Chen CWS (2016) Semi-parametric expected shortfall forecasting in financial markets. J Stat Comput Simul 87:1084–1106. https://doi.org/10.1080/00949655.2016.1246549
Gerlach R, Wang C (2015) Bayesian semi-parametric realized-care models for tail risk forecasting incorporating range and realized measures. In: BA working paper no: BAWP-2015-07
Gerlach R, Chen C, Chan N (2011) Bayesian time-varying quantile forecasting for value-at-risk in financial markets. JBES 29:481–492
Gerlach R, Chen CWS, Lin LY (2012) Bayesian semi-parametric expected shortfall forecasting in financial markets. In: University of Sydney working paper
Gerlach R, Walpole D, Wang C (2016) Semi-parametric Bayesian tail risk forecasting incorporating realized measures of volatility. Quant Finance 17(2):199–215. https://doi.org/10.1080/14697688.2016.1192295
Jorion P (2007) Value-at-risk: the new benchmark for managing financial risk, 3rd edn. McGraw-Hill, Chicago
Kobayashi G (2015) Skew exponential power stochastic volatility model for analysis of skewness, non-normal tails, quantiles and expectiles. Comput Stat 31(1):49–88. https://doi.org/10.1007/s00180-015-0596-4
Koenker B (2005) Quantile regression. Cambridge University Press, Cambridge
Koenker B, Basset G (1978) Regression quantiles. Econometrica 46:33–50
Kottas A, Gelfand A (2001) Bayesian semiparametric median regression modeling. J Am Stat Assoc 96:1458–1468
Kottas A, Krnjajic M (2009) Bayesian semiparametric modelling in quantile regression. Scand J Stat 36:297–319
Kupiec P (1995) Techniques for verifying the accuracy of risk measurement models. J Deriv 3:73–84
Lang S, Brezger A (2004) Bayesian p-splines. J Comput Graph Stat 13(1):183–212
Liang F, Liu C, Carroll R (2010) Advanced Markov chain Monte Carlo methods: learning from past samples. Wiley, Hoboken
McNeil A, Frey R, Embrechts P (2005) Quantitative risk management: concepts, techniques, tools. Princeton Series in Finance, Princeton
Newey W, Powell J (1987) Asymmetric least squares estimation and testing. Econometrica 55:819–847
O’Sullivan F (1986) A statistical perspective on ill-posedinverse problems (with discussion). Stat Sci 1:505–527
O’Sullivan F (1988) Fast computation of fully automated log-density and log-hazard estimators. SIAM J Sci Stat Comput 9(2):363–379. https://doi.org/10.1137/0909024
Roberts GO, Rosenthal JS (2007) Coupling and ergodicity of adaptive Markov chain Monte Carlo algorithms. J Appl Probab 44(2):458–475
Sriram K, Ramamoorthi R, Ghosh P (2013) Posterior consistency of bayesian quantile regression based on the misspecified asymmetric laplace density. Bayesian Anal 8:479–504
Taylor JW (2008) Estimating value at risk and expected shortfall using expectiles. J Financ Econom 6(2):231–252. https://doi.org/10.1093/jjfinec/nbn001
Wichitaksorn N, Choy STB, Gerlach R (2014) A generalized class of skew distributions and associated robust quantile regression models. Can J Stat 42(4):579–596. https://doi.org/10.1002/cjs.11228
Yu K, Moyeed R (2001) Bayesian quantile regression. Stat Probab Lett 54:437–447
Acknowledgements
The views expressed in this paper are those of the authors and do not involve the responsibility of the Bank of Italy. The authors would like to thank the anonymous reviewers for their helpful and constructive comments that greatly contributed to improving the final version of the paper. They would also like to thank the Editors for their generous support during the review process.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Bottone, M., Petrella, L. & Bernardi, M. Unified Bayesian conditional autoregressive risk measures using the skew exponential power distribution. Stat Methods Appl 30, 1079–1107 (2021). https://doi.org/10.1007/s10260-020-00550-6
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10260-020-00550-6