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Asymptotic Theory for the Garch(1,1) Quasi-Maximum Likelihood Estimator

Published online by Cambridge University Press:  11 February 2009

Sang-Won Lee  and
Bruce E. Hansen
Sang-Won Lee
Affiliation:
Indiana University
Bruce E. Hansen
Affiliation:
University of Rochester
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Abstract

This paper investigates the sampling behavior of the quasi-maximum likelihood estimator of the Gaussian GARCH(1,1) model. The rescaled variable (the ratio of the disturbance to the conditional standard deviation) is not required to be Gaussian nor independent over time, in contrast to the current literature. The GARCH process may be integrated (α + β = 1), or even mildly explosive (α + β > 1). A bounded conditional fourth moment of the rescaled variable is sufficient for the results. Consistent estimation and asymptotic normality are demonstrated, as well as consistent estimation of the asymptotic covariance matrix.

Type
Articles
Information
Econometric Theory , Volume 10 , Issue 1 , March 1994 , pp. 29 - 52
Copyright
Copyright © Cambridge University Press 1994

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References

1.Amemiya, T.Advanced Econometrics. Cambridge: Harvard University Press, 1985.Google Scholar
2.Andrews, D.W.K.Generic uniform convergence. Econometric Theory 8 (1992): 241–257.CrossRefGoogle Scholar
3.Billingsley, P.Convergence of Probability Measures. New York: Wiley, 1968.Google Scholar
4.Billingsley, P.IProbability and Measure. New York: Wiley, 1979.Google Scholar
5.Bollerslev, T.Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics 31 (1986): 307–327.CrossRefGoogle Scholar
6.Bollerslev, T. & Engle, R.F.. Common persistence in conditional variances. Econometrica 61 (1993): 167–186.CrossRefGoogle Scholar
7.Bollerslev, T., Chou, R.Y. & Kroner, K.F.. ARCH modelling in finance: A review of the theory and empirical evidence. Journal of Econometrics 52 (1992): 5–59.CrossRefGoogle Scholar
8.Bollerslev, T. & Wooldridge, J.M.. Quasi-maximum likelihood estimation and inference in dynamic models with time-varying covariances. Econometric Reviews 11 (1992): 143–172.CrossRefGoogle Scholar
9.Engle, R.F.Autoregressive conditional heteroskedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50 (1982): 987–1007.CrossRefGoogle Scholar
10.Engle, R.F. & Bollerslev, T.. Modelling the persistence of conditional variances. Econometric Reviews 5 (1986): 1–50.CrossRefGoogle Scholar
11.Hardy, G.H., Littlewood, J.E. & Polya, G.. Inequalities. Cambridge: Cambridge University Press, 1952.Google Scholar
12.Lumsdaine, R.L.Asymptotic Properties of the Quasi-Maximum Likelihood Estimator in GARCH(1,1) and IGARCH(l.l) Models. Mimeo, Princeton University, 1991.Google Scholar
13.Nelson, D.B.Stationarity and persistence in the GARCH(l.l) model. Econometric Theory 6 (1990): 318–334.CrossRefGoogle Scholar
14.Pantula, S.G.Autoregressive Conditionally Heteroscedastic Models. Mimeo, North Carolina State University, 1984.Google Scholar
15.Stout, W.F.Almost Sure Convergence. New York: Academic Press, 1974.Google Scholar
16.Weiss, A.A.Asymptotic theory for ARCH models: Estimation and testing. Econometric Theory 2 (1986): 107–131.CrossRefGoogle Scholar