Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-27T18:07:37.808Z Has data issue: false hasContentIssue false
  • English
  • Français

ESTIMATION-ADJUSTED VAR

Published online by Cambridge University Press:  08 January 2013

Christian Gourieroux  and
Jean-Michel Zakoïan
Christian Gourieroux
Affiliation:
CREST and University of Toronto
Jean-Michel Zakoïan*
Affiliation:
CREST and University Lille 3 (EQUIPPE)
*
*Address correspondence to Jean-Michel Zakoan, CREST, 15 Boulevard G. Péri, 92245 Malakoff Cedex, France; e-mail: [email protected].
Get access Rights & Permissions [Opens in a new window]

Abstract

Standard risk measures, such as the value-at-risk (VaR), or the expected shortfall, have to be estimated, and their estimated counterparts are subject to estimation uncertainty. Replacing, in the theoretical formulas, the true parameter value by an estimator based on n observations of the profit and loss variable induces an asymptotic bias of order 1/n in the coverage probabilities. This paper shows how to correct for this bias by introducing a new estimator of the VaR, called estimation-adjusted VaR (EVaR). This adjustment allows for a joint treatment of theoretical and estimation risks, taking into account their possible dependence. The estimator is derived for a general parametric dynamic model and is particularized to stochastic drift and volatility models. The finite sample properties of the EVaR estimator are studied by simulation and an empirical study of the S&P index is proposed.

Type
Research Articles
Information
Econometric Theory , Volume 29 , Issue 4 , August 2013 , pp. 735 - 770
Copyright
Copyright © Cambridge University Press 2012 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Bao, Y. & Ullah, A. (2004) Bias of a value-at-risk estimator. Finance Research Letters 1, 241–249.Google Scholar
Berkes, I. & Horváth, L. (2004) The efficiency of the estimators of the parameters in GARCH processes. The Annals of Statistics 32, 633–655.Google Scholar
Berkes, I., Horváth, L., & Kokoszka, P. (2003) GARCH processes: Structure and estimation. Bernoulli 9, 201–227.Google Scholar
Berkowitz, J., Christoffersen, P., & Pelletier, D. (2011) Evaluating value-at-risk models with desk-level data. Management Science 57, 2213–2227.CrossRefGoogle Scholar
Berkowitz, J. & O’Brien, J. (2002) How accurate are value-at-risk models at commercial banks? Journal of Finance 57, 1093–1111.Google Scholar
Chambers, M.J. (2012) Jacknife Estimation of Stationary Autoregressive Models. Discussion paper, University of Essex.Google Scholar
Chan, N.H., Deng, S.-J., Peng, L., & Xia, Z. (2007) Interval estimation of value-at-risk based on GARCH models with heavy-tailed innovations. Journal of Econometrics 137, 556–576.Google Scholar
Christoffersen, P. & Gonçalves, S. (2005) Estimation risk in financial risk management. Journal of Risk 7, 1–28.CrossRefGoogle Scholar
Dufour, J.-M. & Kiviet, J. (1997) Exact tests in single equation autoregressive distributed lag models. Journal of Econometrics 80, 325–353.Google Scholar
Engle, R.F., Lillien, D.M., & Robins, R.P. (1987) Estimating time varying risk premia in the term structure: The ARCH-M model. Econometrica 55, 391–407.Google Scholar
Escanciano, J.C. & Olmo, J. (2010) Backtesting parametric value-at-risk with estimation risk. Journal of Business and Economics Statistics 28, 36–51.CrossRefGoogle Scholar
Escanciano, J.C. & Olmo, J. (2011) Robust backtesting tests for value-at-risk models. Journal of Financial Econometrics 9, 132–161.CrossRefGoogle Scholar
Figlewski, S. (2004) Estimation Error in the Assessment of Financial Risk Exposure. Working paper, New York University.Google Scholar
Francq, C. & Zakoïan, J.-M. (2004) Maximum likelihood estimation of pure GARCH and ARMA-GARCH processes. Bernoulli 10, 605–637.CrossRefGoogle Scholar
Francq, C. & Zakoïan, J.-M. (2010) GARCH Models: Structure, Statistical Inference and Financial Applications. John Wiley.Google Scholar
Gouriéroux, C. & Jasiak, J. (2005) Nonlinear impulse response function. Annales d’Economie et de Statistique 78, 1–33.Google Scholar
Gouriéroux, C., J- Laurent, P., & Scaillet, O. (2000) Sensitivity analysis of value-at-risk. Journal of Empirical Finance 7, 225–246.Google Scholar
Hansen, B.E. (2006) Interval forecasts and parameter uncertainty. Journal of Econometrics 135, 377–398.Google Scholar
Hartz, C., Mittnik, S., & Paolella, M. (2006) Accurate value-at-risk forecasting based on the normal-GARCH model. Computational Statistics & Data Analysis 51, 2295–2312.Google Scholar
Inui, K., Kijima, M., & Kitano, A. (2005) VaR is subject to a significant positive bias. Statistics & Probability Letters 72, 299–311.Google Scholar
Koenker, R. & Zhao, Q. (1986) Conditional quantile estimation and inference for ARCH models. Econometric Theory 12, 793–813.Google Scholar
Lönnbark, C. (2010) A corrected value-at-risk predictor. Applied Economics Letters 17, 1193–1196.CrossRefGoogle Scholar
Martin, R. & Wilde, T. (2002) Unsystematic credit risk. Risk Magazine 15, 123–128.Google Scholar
Phillips, P.C.B. & Yu, J. (2005) Jacknifing bond option prices. Review of Financial Studies 18, 707–742.Google Scholar
Quenouille, M.H. (1956) Notes on bias in estimation. Biometrika 43, 353–360.Google Scholar