## The Annals of Statistics

### The efficiency of the estimators of the parameters in GARCH processes

#### Abstract

We propose a class of estimators for the parameters of a GARCH(p,q) sequence. We show that our estimators are consistent and asymptotically normal under mild conditions. The quasi-maximum likelihood and the likelihood estimators are discussed in detail. We show that the maximum likelihood estimator is optimal. If the tail of the distribution of the innovations is polynomial, even a quasi-maximum likelihood estimator based on exponential density performs better than the standard normal density-based quasi-likelihood estimator of Lee and Hansen and Lumsdaine.

#### Article information

Source
Ann. Statist. Volume 32, Number 2 (2004), 633-655.

Dates
First available in Project Euclid: 28 April 2004

http://projecteuclid.org/euclid.aos/1083178941

Digital Object Identifier
doi:10.1214/009053604000000120

Mathematical Reviews number (MathSciNet)
MR2060172

Zentralblatt MATH identifier
1048.62082

#### Citation

Berkes, István; Horváth, Lajos. The efficiency of the estimators of the parameters in GARCH processes. Ann. Statist. 32 (2004), no. 2, 633--655. doi:10.1214/009053604000000120. http://projecteuclid.org/euclid.aos/1083178941.

#### References

• Berkes, I. and Horváth, L. (2003). The rate of consistency of the quasi-maximum likelihood estimator. Statist. Probab. Lett. 61 133--143.
• Berkes, I., Horváth, L. and Kokoszka, P. (2003). GARCH processes: Structure and estimation. Bernoulli 9 201--227.
• Billingsley, P. (1968.) Convergence of Probability Measures. Wiley, New York.
• Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. J. Econometrics 31 307--327.
• Bougerol, P. and Picard, N. (1992a). Strict stationarity of generalized autoregressive processes. Ann. Probab. 20 1714--1730.
• Bougerol, P. and Picard, N. (1992b). Stationarity of GARCH processes and of some nonnegative time series. J. Econometrics 52 115--127.
• Drost, F. C. and Klaassen, C. A. (1997). Efficient estimation in semiparametric GARCH models. J. Econometrics 81 193--221.
• Engle, R. F. and Ng, V. (1993). Measuring and testing the impact of news on volatility. J. Finance 48 1749--1778.
• Geweke, J. (1986). Modeling the persistence of conditional variances: Comment. Econometric Rev. 5 57--61.
• Lee, S.-W. and Hansen, B. E. (1994). Asymptotic theory for the GARCH$(1, 1)$ quasi-maximum likelihood estimator. Econometric Theory 10 29--52.
• Lehmann, E. L. (1991). Theory of Point Estimation. Wadsworth and Brooks/Cole, Pacific Grove, CA.
• Lumsdaine, R. L. (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.
• Nelson, D. (1991). Conditional heteroskedasticity in asset returns: A new approach. Econometrica 59 347--370.
• Newey, W. and Steigerwald, D. G. (1997). Asymptotic bias for quasi-maximum likelihood estimators in conditional heteroskedasticity models. Econometrica 65 587--599.
• Stout, W. F. (1974). Almost Sure Convergence. Academic Press, New York.