Looking for efficient qml estimation of conditional value-at-risk at multiple risk levels

Francq, Christian and Zakoian, Jean-Michel (2015): Looking for efficient qml estimation of conditional value-at-risk at multiple risk levels.

Abstract

We consider joint estimation of conditional Value-at-Risk (VaR) at several levels, in the framework of general GARCH-type models. The conditional VaR at level $\alpha$ is expressed as the product of the volatility and the opposite of the $\alpha$-quantile of the innovation. A standard method is to estimate the volatility parameter by Gaussian Quasi-Maximum Likelihood (QML) in a first step, and to use the residuals for estimating the innovations quantiles in a second step. We argue that the Gaussian QML may be inefficient with respect to more general QML and can even be in failure for heavy tailed conditional distributions. We therefore study, for a vector of risk levels, a two-step procedure based on a generalized QML. For a portfolio of VaR's at different levels, confidence intervals accounting for both market and estimation risks are deduced. An empirical study based on stock indices illustrates the theoretical results.

• Item Type: MPRA Paper
• Language: English
• Keywords: Asymmetric Power GARCH; Distortion Risk Measures; Estimation risk; Non-Gaussian Quasi-Maximum Likelihood; Value-at-Risk
• Subjects: C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C13 - Estimation: General; C - Mathematical and Quantitative Methods > C2 - Single Equation Models ; Single Variables > C22 - Time-Series Models ; Dynamic Quantile Regressions ; Dynamic Treatment Effect Models ; Diffusion Processes; C - Mathematical and Quantitative Methods > C5 - Econometric Modeling > C58 - Financial Econometrics

References

Berkes, I. and L. Horv\'ath (2004) The efficiency of the estimators of the parameters in GARCH processes. The Annals of Statistics 32, 633--655.

Chan, N.H., Deng, S.J., Peng, L. and Z. Xia (2007) Interval estimation of value-at-risk based on GARCH models with heavy-tailed innovations. Journal of Econometrics 137, 556--576.

Ding, Z., Granger C. et R.F. Engle (1993) A long memory property of stock market returns and a new model. Journal of Empirical Finance 1, 83--106.

El Ghourabi, M., Francq, C. and F. Telmoudi (2015) Consistent estimation of the Value-at-Risk when the error distribution of the volatility model is misspecified. Journal of Time Series Analysis, forthcoming.

Engle, R.F. and S. Manganelli (2004) CAViaR: Conditional Value-at-Risk by quantile regression. Journal of Business and Economic Statistics 22, 367--381.

Escancianio, J.C. and J. Olmo (2010) Backtesting parametric Value-at-Risk with estimation risk. Journal of Business and Economic Statistics 28, 36--51.

Fan, J., Qi, L. and D. Xiu (2014) Quasi-maximum likelihood estimation of GARCH Models with heavy-tailed likelihoods. Journal of Business and Economic Statistics 32, 178--191.

Francq, C., Lepage, G. and J-M. Zakoïan (2011) Two-stage non Gaussian QML estimation of GARCH Models and testing the efficiency of the Gaussian QMLE. Journal of Econometrics 165, 246--257.

Francq, C. and J.M. Zakoïan (2013) Optimal predictions of powers of conditionally heteroskedastic processes. Journal of the Royal Statistical Society - Series B 75, 345--367.

Francq, C. and J-M. Zakoïan (2014) Multi-level conditional VaR estimation in dynamic models. In Modeling Dependence in Econometrics. Advances in Intelligent Systems and Computing Volume 251. Edts: V-N. Huynh et al., Springer.

Francq, C. and J-M. Zakoïan (2015) Risk-parameter estimation in volatility models. Journal of Econometrics 184, 158--173.

Glosten, L.R., Jaganathan, R. and D. Runkle (1993) On the relation between the expected values and the volatility of the nominal excess return on stocks. Journal of Finance 48, 1779--1801.

Gouriéroux, C. and J. Jasiak (2008) Dynamic Quantile Models. Journal of Econometrics 147, 198--205.

Gouriéroux, C. and J.M. Zakoïan (2013) Estimation adjusted VaR. Econometric Theory 29, 735--770.

Hall, P. and Q. Yao (2003) Inference in ARCH and GARCH models with heavy-tailed errors. Econometrica 71, 285--317.

Hamadeh, T. and J-M. Zakoïan (2011) Asymptotic properties of LS and QML estimators for a class of nonlinear GARCH Processes. Journal of Statistical Planning and Inference 141, 488-507.

Hurlin, C., Laurent, S., Quaedvlieg, R., and S. Smeekes (2013) Risk measure inference. Available at SSRN 2345299.

Koenker, R. (2005) Quantile Regression. Cambridge: Cambridge University Press.

Koenker, R. and Z. Xiao (2006) Quantile autoregression. Journal of the American Statistical Association 101, 980--990.

Kuester, K., Mittnik, S. and M.S. Paolella (2006) Value-at-Risk predictions: A comparison of alternative strategies. Journal of Financial Econometrics 4, 53--89.

McNeil, A. J., Frey, R. and P. Embrechts (2005) Quantitative Risk Management. Princeton University Press.

Spierdijk, L. (2013) Confidence intervals for ARMA-GARCH Value-at-Risk. Working paper, University of Groningen.

Wang, S. (2000) A class of distortion operators for pricing financial and insurance risks. Journal of Risk and Insurance 67, 15--36.

Zakoïan J.M. (1994) Threshold Heteroskedastic Models. Journal of Economic Dynamics and Control, 18, 931--955.