Variance targeting estimation of multivariate GARCH models

Francq, Christian and Horvath, Lajos and Zakoian, Jean-Michel (2014): Variance targeting estimation of multivariate GARCH models.

Abstract

We establish the strong consistency and the asymptotic normality of the variance-targeting estimator (VTE) of the parameters of the multivariate CCC-GARCH($p,q$) processes. This method alleviates the numerical difficulties encountered in the maximization of the quasi likelihood by using an estimator of the unconditional variance. It is shown that the distribution of the VTE can be consistently estimated by a simple residual bootstrap technique. We also use the VTE for testing the model adequacy. A test statistic in the spirit of the score test is constructed, and its asymptotic properties are derived under the null assumption that the model is well specified. An extension of the VT method to asymmetric CCC-GARCH models incorporating leverage effects is studied. Numerical illustrations are provided and an empirical application based on daily exchange rates is proposed.

• Item Type: MPRA Paper
• Language: English
• Keywords: Adequacy Test for CCC-GARCH models, Bootstrap, Leverage Effect, Quasi Maximum Likelihood Estimation, Variance Targeting Estimator
• 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

Aue, A., Hörmann, S., Horváth, L. and M. Reimherr (2009) Break detection in the covariance structure of multivariate time series models. The Annals of Statistics 37, 4046-4087.

Bauwens, L., Laurent, S. and J.V.K. Rombouts (2006) Multivariate GARCH models: a survey. Journal of Applied Econometrics 21, 79-109.

Billingsley, P.(1961) The Lindeberg-Levy theorem for martingales. Proceedings of the American Mathematical Society 12, 788-792.

Bollerslev, T. (1990) Modelling the coherence in short-run nominal exchange rates: a multivariate generalized ARCH model. Review of Economics and Statistics 72, 498-505.

Boudt, K., Danielsson J. and S. Laurent (2013) Robust forecasting of dynamic conditional correlation GARCH models. International Journal of Forecasting 29, 244-257.

Christoffersen, P.F. (2003) Elements of financial risk management. Academic Press, London.

Engle, R.F. and K. Kroner (1995) Multivariate simultaneous generalized ARCH. Econometric Theory 11, 122-150.

Engle, R.F. and J. Mezrich (1996) GARCH for groups. Risk 9, 36-40.

Francq, C., Horváth, L. and J-M. Zakoïan (2011) Merits and drawbacks of variance targeting in GARCH Models. Journal of Financial Econometrics 9, 619-656.

Francq C. and J-M. Zakoïan (2007) Quasi-maximum likelihood estimation in GARCH processes when some coefficients are equal to zero. Stochastic Processes and their Applications 117, 1265-1284.

Francq, C. and J-M. Zakoïan (2009) Testing the nullity of GARCH coefficients : correction of the standard tests and relative efficiency comparisons. Journal of the American Statistical Association 104, 313-324.

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

Francq, C. and J-M. Zakoïan (2012) QML estimation of a class of multivariate asymmetric GARCH models. Econometric Theory 28, 179-206.

Francq, C. and J-M. Zakoïan (2013) Optimal predictions of powers of conditionally heteroscedastic processes. J.R. Statist. Soc. B 75, 345-367.

Giraitis, L., Leipus, R. and D. Surgailis Recent advances in ARCH modelling. In: Teyssière, G. and Kirman, A. (Eds.): Long Memory in Economics, 3-38. Springer, New-York.

He, C. and T. Teräsvirta (2004) An extended constant conditional correlation GARCH model and its fourth-moment structure. EconometricTheory 20, 904-926.

Hill, J. and E. Renault (2012) Variance Targeting for Heavy Tailed Time Series. Unpublished document.

Hull, J. 2003. Options, Futures, and Other Derivatives. Princeton Hall, New Jersey, USA.

Jeantheau, T. (1998) Strong consistency of estimators for multivariate ARCH models. Econometric Theory 14, 70-86.

Kreiss, J.P., Paparoditis, E. and D.N. Politis (2011) On the range of validity of the autoregressive sieve bootstrap. The Annals of Statistics 39, 2103-2130.

Kristensen, D. and O. Linton (2004) Consistent standard errors for target variance approach to GARCH estimation. Econometric Theory 20, 990-993.

Laurent, S., Rombouts, J.V.K. and F. Violante (2012) On the forecasting accuracy of multivariate GARCH models. Journal of Applied Econometrics 27, 934-955.

Ling, S. and M. McAleer (2002) Necessary and sufficient moment conditions for the GARCH($r,s$) and asymmetric GARCH$r,s$) models. Econometric Theory 18, 722-729.

Pedersen, R.S. and A. Rahbek (2013) Multivariate variance targeting in the BEKK-GARCH model. Forthcoming in The Econometrics Journal.

Silvennoinen, A. and T. Teräsvirta (2009) Multivariate GARCH models. Handbook of Financial Time Series T.G. Andersen, R.A. Davis, J-P. Kreiss and T. Mikosch, eds. New York: Springer.

Shimizu, K. (2013) The bootstrap does not always work for heteroscedastic models. Statistics & Risk Modeling 30, 189-204.

Vaynman, I. and B. K. Beare (2013) Stable limit theory for the variance targeting estimator. Unpublished document, University of California, San Diego.

Zhu, D. and V. Zinde-Walsh (2009) Properties and estimation of asymmetric exponential power distribution. Journal of Econometrics 148, 86-99.