Risk-parameter estimation in volatility models

Francq, Christian and Zakoian, Jean-Michel (2012): Risk-parameter estimation in volatility models.

Abstract

This paper introduces the concept of risk parameter in conditional volatility models of the form $\epsilon_t=\sigma_t(\theta_0)\eta_t$ and develops statistical procedures to estimate this parameter. For a given risk measure $r$, the risk parameter is expressed as a function of the volatility coefficients $\theta_0$ and the risk, $r(\eta_t)$, of the innovation process. A two-step method is proposed to successively estimate these quantities. An alternative one-step approach, relying on a reparameterization of the model and the use of a non Gaussian QML, is proposed. Asymptotic results are established for smooth risk measures as well as for the Value-at-Risk (VaR). Asymptotic comparisons of the two approaches for VaR estimation suggest a superiority of the one-step method when the innovations are heavy-tailed. For standard GARCH models, the comparison only depends on characteristics of the innovations distribution, not on the volatility parameters. Monte-Carlo experiments and an empirical study illustrate these findings.

• Item Type: MPRA Paper
• Language: English
• Keywords: GARCH; Quantile Regression; Quasi-Maximum Likelihood; Risk measures; 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

References

Acerbi, C. and D. Tasche (2002) On the coherence of expected shortfall Journal of Banking \& Finance 26, 1487--1503.

Artzner, P., Delbaen, F., Eber, J-M. and D. Heath (1999) Coherent measures of risk. Mathematical Finance 9, 203--228.

Bardet, J-M. and O. Wintenberger (2009) Asymptotic normality of the Quasi-maximum likelihood estimator for multidimensional causal processes. The Annals of Statistics 37, 2730--2759.

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.

Berkes, I., Horv\'ath, L. and P. Kokoszka (2003) GARCH processes: structure and estimation. Bernoulli 9, 201--227.

Bollerslev, T. (1986) Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics 31, 307--327.

Breidt, F.J., Davis, R.A. and A. A. Trindade (2001) Least absolute deviation estimation for all-pass time series models. The Annals of Statistics 29, 919--946.

Chen, S.X. (2008) Nonparametric estimation of expected shortfall. Journal of Financial Econometrics 6, 87--107.

Davis, R.A., Knight, K. and J. Liu (1992) M-estimation for autoregressions with infinite variance. Stochastic Processes and their Applications 40, 145--180.

Davis, R.A. and W.T.M. Dunsmuir (1997) Least absolute deviation estimation for regression with ARMA errors. Journal of Theoretical Probability 10, 481--497.

Engle, R.F. (1982) Autoregressive conditional heteroskedasticity with estimates of the variance of the United Kingdom inflation. Econometrica 50, 987--1007.

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 (2004) Maximum likelihood estimation of pure GARCH and ARMA-GARCH processes. Bernoulli 10, 605--637.

Francq, C. and J-M. Zakoïan (2010) GARCH Models: Structure, Statistical Inference and Financial Applications. John Wiley.

Francq, C. and J-M. Zakoïan (2012) Optimal predictions of powers of conditionally heteroskedastic processes. Forthcoming in the Journal of the Royal Statistical Society - Series B.

Knight, K. (1998) Limiting distributions for $L_1$ regression estimators under general conditions. The Annals of Statistics 26, 755--770.

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.

Lee, S.W. and B.E. Hansen (1994) Asymptotic theory for the GARCH(1,1) quasi-maximum likelihood estimator, Econometric Theory 10, 29--52.

Ling, S. (2004) Estimation and testing stationarity for double-autoregressive models. Journal of the Royal Statistical Society B, 66, 63--78.

Ling, S. (2005). Self-weighted LAD estimation for infinite variance autoregressive models. Journal of the Royal Statistical Society B, 67, 381--393.

Lumsdaine, R.L. (1996) Consistency and asymptotic normality of the quasi-maximum likelihood estimator in IGARCH(1,1) and covariance stationary GARCH(1,1) models. Econometrica 64, 575--596.

Mikosch, T. and D. Straumann (2006) Stable limits of martingale transforms with application to the estimation of GARCH parameters. The Annals of Statistics 34, 493--522.

Pollard, D. (1991) Asymptotics for Least Absolute Deviation Regression Estimators Econometric Theory 7, 186--199.

Straumann, D. and T. Mikosch (2006) Quasi-maximum likelihood estimation in conditionally heteroscedastic Time Series: a stochastic recurrence equations approach. The Annals of Statistics 5, 2449--2495.

Xiao, Z. and R. Koenker (2009) Conditional quantile estimation for generalized autoregressive conditional heteroscedasticity models. Journal of the American Statistical Association 104, 1696--1712.

Xiao, Z. and C. Wan (2010) A robust estimator of conditional volatility. Unpublished document.

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

Zhu, K. and S. Ling (2011) Global self-weighted and local quasi-maximum exponential likelihood estimators for ARMA-GARCH/IGARCH models. The Annals of Statistics 32, 2131--2163.