Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-24T21:02:53.688Z Has data issue: false hasContentIssue false
  • English
  • Français

AN ALMOST CLOSED FORM ESTIMATOR FOR THE EGARCH MODEL

Published online by Cambridge University Press:  22 August 2016

Christian M. Hafner  and
Oliver Linton
Christian M. Hafner*
Affiliation:
Université catholique de Louvain
Oliver Linton
Affiliation:
University of Cambridge
*
*Address correspondence to Christian M. Hafner, CORE and Institut de statistique, biostatistique et sciences actuarielles, Université catholique de Louvain, Voie du Roman Pays 20, B-1348 Louvain-la-Neuve, Belgium; e-mail: christian.hafner@uclouvain.be.
Get access Rights & Permissions [Opens in a new window]

Abstract

The exponential GARCH (EGARCH) model introduced by Nelson (1991) is a popular model for discrete time volatility since it allows for asymmetric effects and naturally ensures positivity even when including exogenous variables. Estimation and inference are usually done via maximum likelihood. Although some progress has been made recently, a complete distribution theory of MLE for EGARCH models is still missing. Furthermore, the estimation procedure itself may be highly sensitive to starting values, the choice of numerical optimization algorithm, etc. We present an alternative estimator that is available in a simple closed form and which could be used, for example, as starting values for MLE. The estimator of the dynamic parameter is independent of the innovation distribution. For the other parameters we assume that the innovation distribution belongs to the class of Generalized Error Distributions (GED), profiling out its parameter in the estimation procedure. We discuss the properties of the proposed estimator and illustrate its performance in a simulation study and an empirical example.

Type
ARTICLES
Information
Econometric Theory , Volume 33 , Issue 4 , August 2017 , pp. 1013 - 1038
Copyright
Copyright © Cambridge University Press 2016 

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.)

Footnotes

We thank Eric Renault, Piotr Fryzlewicz, Dimitra Kyriakopoulou, Enno Mammen, Paolo Zaffaroni, Jean-Michel Zakoian, and three referees for helpful comments. Financial support of the Académie universitaire Louvain is gratefully acknowledged. Thanks to the ERC for financial support.

References

REFERENCES

Andrews, D.W.K. (1997) A conditional Kolmogorov test. Econometrica 65, 1097–1128.Google Scholar
Andrews, D.W.K. & Guggenberger, P. (2009) Hybrid and size-corrected subsampling methods. Econometrica 77, 721–762.Google Scholar
Bai, J. (2003) Testing parametric conditional distributions of dynamic models. The Review of Economics and Statistics 85, 531–554.Google Scholar
Bollerslev, T. (1986) Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics 31, 307–327.Google Scholar
Brooks, C., Burke, S.P., & Persand, G. (2001) Benchmarks and the accuracy of GARCH model estimation. International Journal of Forecasting 17, 45–56.Google Scholar
Carrasco, M. & Chen, X. (2002) Mixing and moment properties of various GARCH and stochastic volatility models. Econometric Theory 18, 17–39.Google Scholar
Davidson, J. (1994) Stochastic Limit Theory. Oxford University Press.Google Scholar
Davis, R.A. & Mikosch, T. (1998) The sample autocorrelations of heavy-tailed processes with applications to ARCH. Annals of Statistics 26, 2049–2080.Google Scholar
Doukhan, P. (1994) Mixing: Properties and Examples. Springer-Verlag.Google Scholar
Engle, R.F. (1982) Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50, 987–1007.CrossRefGoogle Scholar
Engle, R.F. & Mezrich, J. (1996) GARCH for groups. Risk 9, 36–40.Google Scholar
Francq, C., Horváth, L., & Zakoïan, J.-M. (2011) Merits and drawbacks of variance targeting in GARCH models. Journal of Financial Econometrics 9, 619–656.Google Scholar
Gradshteyn, I.S. & Ryzhik, I.M. (2007) Tables of Integrals, Series, and Products. 7th ed. Academic Press.Google Scholar
Kheifets, I (2015) Specification tests for nonlinear dynamic models. Econometrics Journal 18, 67–94.CrossRefGoogle Scholar
Kristensen, D. & Linton, O. (2006) A closed form estimator for the GARCH(1,1) model. Econometric Theory 22, 323–337.Google Scholar
Kyriakopoulou, D. (2015) Asymptotic Normality of the QML Estimator of the EGARCH(1,1) Model. SSRN Working paper. Available at SSRN: http://ssrn.com/abstract=2236055 or http://dx.doi.org/10.2139/ssrn.2236055.Google Scholar
McCullough, B.D. & Renfro, C.G. (1999) Benchmarks and software standards: A case study of GARCH procedures. Journal of Economic and Social Measurement 25, 59–71.Google Scholar
Mineo, A.M. & Ruggieri, M. (2005) A software tool for the exponential power distribution: The normalp package. The Journal of Statistical Software 12, 1–24.Google Scholar
Nelson, D. (1991) Conditional heteoskedasticity in asset returns: A new approach. Econometrica 59, 347–370.CrossRefGoogle Scholar
Politis, D.N. & Romano, J.P. (1994) Large sample confidence regions based on subsamples under minimal assumptions. Annals of Statistics 22, 2031–2050.Google Scholar
Renault, E. (2009) Moment-based estimation of stochastic volatility models. In Andersen, T., Davis, R., Kreiss, J.-P., & Mikosch, T. (eds.), Handbook of Financial Time Series, pp. 269–311, Springer Verlag.Google Scholar
Robinson, P.M. (1988) The stochastic difference between econometric statistics. Econometrica 56, 531–548.Google Scholar
Saumard, A. & Wellner, J.A. (2014) Log-concavity and strong log-concavity: A review. Statistics Surveys 8, 45–114.Google Scholar
Straumann, D. & Mikosch, T. (2006) Quasi-maximum likelihood estimation in conditionally heteroskedastic time series: A stochastic recurrence equations approach. Annals of Statistics 34, 2449–2495.Google Scholar
Subbotin, M.T. (1923) On the law of frequency of errors. Matematicheskii Sbornik 31, 296–301.Google Scholar
Wintenberger, O. (2013) Continuous invertibility and stable QML estimation of the EGARCH(1,1) model. Scandinavian Journal of Statistics 40, 846–867.Google Scholar
Zaffaroni, P. (2009) Whittle estimation of EGARCH and other exponential volatility models. Journal of Econometrics 151, 190–200.CrossRefGoogle Scholar