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ASYMPTOTIC INFERENCE FOR NONSTATIONARY GARCH

Published online by Cambridge University Press:  01 December 2004

Søren Tolver Jensen  and
Anders Rahbek
Søren Tolver Jensen
Affiliation:
University of Copenhagen
Anders Rahbek
Affiliation:
University of Copenhagen
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Abstract

Consistency and asymptotic normality are established for the highly applied quasi-maximum likelihood estimator in the GARCH(1,1) model. Contrary to existing literature we allow the parameters to be in the region where no stationary version of the process exists. This has the important implication that the likelihood-based estimator for the GARCH parameters is consistent and asymptotically normal in the entire parameter region including both stationary and explosive behavior. In particular, there is no “knife edge result like the unit root case” as hypothesized in Lumsdaine (1996, Econometrica 64, 575–596).Anders Rahbek is grateful for support from the Danish Social Sciences Research Council, the Centre for Analytical Finance (CAF), and the EU network DYNSTOCH. Both authors thank the two anonymous referees and the editor for highly valuable and detailed comments that have, we believe, led to a much improved version of the paper, both in terms of the econometric theory and of the presentation.

Type
Research Article
Information
Econometric Theory , Volume 20 , Issue 6 , December 2004 , pp. 1203 - 1226
Copyright
© 2004 Cambridge University Press

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References

REFERENCES

Basawa, I.V., P.D. Feigin, & C.C. Heyde (1976) Asymptotic properties of maximum likelihood estimators for stochastic processes. Sankya Series A 38, 259–270.Google Scholar
Bollerslev, T. (1986) Generalized autoregressive conditional heteroscedasticity. Journal of Econometrics 31, 307–327.Google Scholar
Bougerol, P. & N. Picard (1992) Stationarity of GARCH processes and of some nonnegative time series. Journal of Econometrics 52, 115–127.Google Scholar
Brown, B.M. (1971) Martingale central limit theorems. Annals of Mathematical Statistics 42, 59–66.Google Scholar
Engle, R.F. (1982) Autoregressive conditional heteroscedasticity with estimates of United Kingdom inflation. Econometrica 50, 987–1007.Google Scholar
Jensen, S.T. & A. Rahbek (2004) Non-stationary and no-moments asymptotics for the ARCH(1) model. Econometrica 72, 641–646.Google Scholar
Kristensen, D. & A. Rahbek (2002) Asymptotics of the QMLE for a Class of ARCH(q) models. Preprint 10, Department of Applied Mathematics and Statistics, University of Copenhagen.
Lee, S.-W. & B. Hansen (1994) Asymptotic theory for the GARCH(1,1) quasi-maximum likelihood estimator. Econometric Theory 10, 29–53.Google Scholar
Lehmann, E.L. (1999) Elements of Large-Sample Theory. Springer-Verlag.
Lumsdaine, R. (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.Google Scholar
Nelson, D.B. (1990) Stationarity and persistence in the GARCH(1,1) model. Econometric Theory 6, 318–334.Google Scholar
Weiss, A. (1986) Asymptotic theory for ARCH models. Econometric Theory 2, 107–131.Google Scholar