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A CLOSED-FORM ESTIMATOR FOR THE GARCH(1,1) MODEL

Published online by Cambridge University Press:  09 February 2006

Dennis Kristensen  and
Oliver Linton
Dennis Kristensen
Affiliation:
University of Wisconsin-Madison
Oliver Linton
Affiliation:
London School of Economics
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Abstract

We propose a closed-form estimator for the linear GARCH(1,1) model. The estimator has the advantage over the often used quasi-maximum likelihood estimator (QMLE) that it can be easily implemented and does not require the use of any numerical optimization procedures or the choice of initial values of the conditional variance process. We derive the asymptotic properties of the estimator, showing T(κ−1)/κ-consistency for some κ ∈ (1,2) when the fourth moment exists and -asymptotic normality when the eighth moment exists. We demonstrate that a finite number of Newton–Raphson iterations using our estimator as starting point will yield asymptotically the same distribution as the QMLE when the fourth moment exists. A simulation study confirms our theoretical results.The first author's research was supported by the Shoemaker Foundation. The second author's research was supported by the Economic and Social Science Research Council of the United Kingdom.

Type
NOTES AND PROBLEMS
Information
Econometric Theory , Volume 22 , Issue 2 , April 2006 , pp. 323 - 337
Copyright
© 2006 Cambridge University Press

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References

REFERENCES

Baillie, R.T. & H. Chung (2001) Estimation of GARCH models from the autocorrelations of the squares of a process. Journal of Time Series Analysis 22, 631–650.CrossRefGoogle Scholar
Basrak, B., R.A. Davis, & T. Mikosch (2002) Regular variation of GARCH processes. Stochastic Processes and Their Applications 99, 95–115.CrossRefGoogle Scholar
Bollerslev, T. (1986) Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics 31, 307–327.CrossRefGoogle Scholar
Bollerslev, T. (1988) On the correlation structure for the generalized autoregressive conditional heteroskedastic process. Journal of Time Series Analysis 9, 121–131.CrossRefGoogle Scholar
Brooks, C., S.P. Burke, & G. Persand (2001) Benchmarks and the accuracy of GARCH model estimation. International Journal of Forecasting 17, 45–56.CrossRefGoogle Scholar
Coleman, T.F. & Y. Li (1996) An interior, trust region approach for nonlinear minimization subject to bounds. SIAM Journal on Optimization 6, 418–445.CrossRefGoogle Scholar
Davis, R.A. & T. Mikosch (1998) The sample autocorrelations of heavy-tailed processes with applications to ARCH. Annals of Statistics 26, 2049–2080.Google Scholar
Engle, R.F. & K. Sheppard (2001) Theoretical and Empirical Properties of Dynamic Conditional Correlation Multivariate GARCH. NBER Working paper w8554.
Fiorentini, G., G. Calzolari, & L. Panattoni (1996) Analytical derivatives and the computation of GARCH estimates. Journal of Applied Econometrics 11, 399–417.3.0.CO;2-R>CrossRefGoogle Scholar
Francq, C. & J.-M. Zakoïan (2000) Estimating weak GARCH representations. Econometric Theory 16, 692–728.CrossRefGoogle Scholar
Galbraith, J.W. & V. Zinde-Walsh (1994) A simple noniterative estimator for moving average models. Biometrika 81, 143–155.CrossRefGoogle Scholar
Giraitis, L. & P.M. Robinson (2001) Whittle estimation of ARCH models. Econometric Theory 17, 608–631.CrossRefGoogle Scholar
Gouriéroux, C. & J. Jasiak (2001) Financial Econometrics: Problems, Models and Methods. Princeton University Press.
Hannan, E.J. & C.C. Heyde (1972) On limit theorems for quadratic functions of discrete time series. Annals of Mathematical Statistics 43, 2058–2066.CrossRefGoogle Scholar
Harvey, A.C. (1993) Time Series Models, 2nd ed. Harvester Wheatsheaf.
He, C. & T. Teräsvirta (1999) Fourth moment structure of the GARCH(p,q) process. Econometric Theory 15, 824–846.CrossRefGoogle Scholar
Jeantheau, T. (1998) Strong consistency of estimators for multivariate ARCH models. Econometric Theory 14, 70–86.Google Scholar
Lee, S.-W. & B. Hansen (1994) Asymptotic theory for the GARCH(1,1) quasi-maximum likelihood estimator. Econometric Theory 10, 29–53.CrossRefGoogle Scholar
Ling, S. & M. McAleer (2003) Asymptotic theory for a new vector ARMA-GARCH model. Econometric Theory 19, 280–310.Google Scholar
Lumsdaine, R. (1995) Finite-sample properties of the maximum likelihood estimator in GARCH(1,1) and IGARCH(1,1) models: A Monte Carlo investigation. Journal of Business & Economic Statistics 13, 1–10.Google Scholar
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.CrossRefGoogle Scholar
Mayoral, L. (2004) A New Minimum Distance Estimation Procedure of ARFIMA Models. Manuscript, Pompeu Fabra.
Meitz, M. & P. Saikkonen (2004) Ergodicity, Mixing and Existence of Moments of a Class of Markov Models with Applications to GARCH and ACD Models. Stockholm School of Economics, SSE/EFI Working Paper Series in Economics and Finance 573.
Meyn, S.P. & R.L. Tweedie (1993) Markov Chains and Stochastic Stability. Communications and Control Engineering Series. Springer-Verlag.
McCullough, B.D. & C.G. Renfro (1999) Benchmarks and software standards: A case study of GARCH procedures. Journal of Economic and Social Measurement 25, 59–71.Google Scholar
Mikosch, T. & C. Stărică (2000) Limit theory for the sample autocorrelations and extremes of a GARCH(1,1) process. Annals of Statistics 28, 1427–1451.Google Scholar
Resnick, S.I. (1987) Extreme Values, Regular Variation, and Point Processes. Springer-Verlag.
Robinson, P.M. (1988) The stochastic difference between econometric statistics. Econometrica 56, 531–548.CrossRefGoogle Scholar
Robinson, P.M. & C. Velasco (1997) Autocorrelation-robust inference. In G.S. Maddala & C.R. Rao (eds.), Robust Inference, pp. 267–298. North-Holland.
Samorodnitsky, G. & M.S. Taqqu (1994) Stable Non-Gaussian Processes: Stochastic Models with Infinite Variance. Chapman & Hall.
Tuan, P.-D. (1979) The estimation of parameters for autoregressive-moving average models from sample autocovariances. Biometrika 66, 555–560.CrossRefGoogle Scholar