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  • Søren Tolver Jensen (a1) and Anders Rahbek (a1)

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.

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Address correspondence to Anders Rahbek, Department of Applied Mathematics and Statistics, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen Ø, Denmark; e-mail:
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Bollerslev, T. (1986) Generalized autoregressive conditional heteroscedasticity. Journal of Econometrics 31, 307327.

Bougerol, P. & N. Picard (1992) Stationarity of GARCH processes and of some nonnegative time series. Journal of Econometrics 52, 115127.

Brown, B.M. (1971) Martingale central limit theorems. Annals of Mathematical Statistics 42, 5966.

Engle, R.F. (1982) Autoregressive conditional heteroscedasticity with estimates of United Kingdom inflation. Econometrica 50, 9871007.

Jensen, S.T. & A. Rahbek (2004) Non-stationary and no-moments asymptotics for the ARCH(1) model. Econometrica 72, 641646.

Lee, S.-W. & B. Hansen (1994) Asymptotic theory for the GARCH(1,1) quasi-maximum likelihood estimator. Econometric Theory 10, 2953.

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, 575596.

Nelson, D.B. (1990) Stationarity and persistence in the GARCH(1,1) model. Econometric Theory 6, 318334.

Weiss, A. (1986) Asymptotic theory for ARCH models. Econometric Theory 2, 107131.

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Econometric Theory
  • ISSN: 0266-4666
  • EISSN: 1469-4360
  • URL: /core/journals/econometric-theory
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