Hostname: page-component-74656f9884-54qdv Total loading time: 0 Render date: 2026-11-09T19:59:08.806Z Has data issue: false hasContentIssue false
  • English
  • Français

M-ESTIMATION IN GARCH MODELS

Published online by Cambridge University Press:  09 July 2008

Kanchan Mukherjee
Kanchan Mukherjee*
Affiliation:
Lancaster University
*
Address correspondence to Kanchan Mukherjee, Department of Mathematics and Statistics, Lancaster University, Lancaster LA1 4YF, United Kingdom; e-mail: k.mukherjee@lancaster.ac.uk
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

This paper derives asymptotic normality of a class of M-estimators in the generalized autoregressive conditional heteroskedastic (GARCH) model. The class of estimators includes least absolute deviation and Huber's estimator in addition to the well-known quasi maximum likelihood estimator. For some estimators, the asymptotic normality results are obtained only under the existence of fractional unconditional moment assumption on the error distribution and some mild smoothness and moment assumptions on the score function.

Information

Type
Research Article
Information
Econometric Theory , Volume 24 , Issue 6 , December 2008 , pp. 1530 - 1553
Copyright
Copyright © Cambridge University Press 2008

References

REFERENCES

Berkes, I. & Horvath, L. (2004) The efficiency of the estimators of the parameters in GARCH processes. Annals of Statistics 32, 633–655.CrossRefGoogle Scholar
Berkes, I., Horvath, L., & Kokoszka, P. (2003) GARCH processes: Structure and estimation. Bernoulli 9, 201–228.CrossRefGoogle Scholar
Bollerslev, T. (1986) Generalized autoregressive conditional heteroscedasticity. Journal of Econometrics 31, 307–327.CrossRefGoogle Scholar
Bougerol, P. & Picard, N. (1992) Stationarity of GARCH processes and of some nonnegative time series. Journal of Econometrics 52, 115–127.CrossRefGoogle Scholar
Comte, F. & Lieberman, O. (2003) Asymptotic theory for multivariate GARCH processes. Journal of Multivariate Analysis 84, 61–84.CrossRefGoogle Scholar
Engle, R.F. (1982) Autoregressive conditional heteroscedasticity and estimates of the variance of UK inflation. Econometrica 50, 987–1008.CrossRefGoogle Scholar
Engle, R.F. & Gonzalez-Rivera, G. (1991) Semiparametric ARCH models. Journal of Business & Economic Statistics 9, 345–349.Google Scholar
Giraitis, L., Kokoszka, P., & Leipus, R. (2000) Stationary ARCH models: Dependence structure and central limit theorem. Econometric Theory 16, 3–22.CrossRefGoogle Scholar
Hall, P. & Heyde, C.C. (1980) Martingale Limit Theory and Its Applications. Academic Press.Google Scholar
Hall, P. & Yao, Q. (2003) Inference in ARCH and GARCH models with heavy-tailed errors. Econometrica 71, 285–317.CrossRefGoogle Scholar
Iqbal, F. & Mukherjee, K. (2007) Computation of M-Estimates in GARCH Models. Technical Report, Department of Mathematical Sciences, University of Liverpool.Google Scholar
Jeantheau, T. (1998) Strong consistency of estimators for multivariate ARCH models. Econometric Theory 14, 70–86.CrossRefGoogle Scholar
Klimko, L.A. & Nelson, P.I. (1978) On conditional least squares estimation for stochastic processes. Annals of Statistics 6, 629–642.CrossRefGoogle Scholar
Lee, S.W. & Hansen, B.E. (1994) Asymptotic theory for the GARCH(1,1) quasi-maximum likelihood estimator. Econometric Theory 10, 29–52.CrossRefGoogle Scholar
Li, D. & Turtle, H. (2000) Semiparametric ARCH models: An estimating function approach. Journal of Business & Economic Statistics 18, 174–186.Google Scholar
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.CrossRefGoogle Scholar
Mak, T., Wong, H., & Li, W. (1997) Estimation of nonlinear time series with conditional heteroscedastic variances by iteratively weighted least squares. Computational Statistics & Data Analysis 24, 169–178.CrossRefGoogle Scholar
Ndebu, J. (2006) Some estimation methods in ARCH and GARCH models. M.Sc. Dissertation, Department of Mathematical Sciences, University of Liverpool.Google Scholar
Nelson, D. (1991) Conditional heteroscedasticity in asset returns: A new approach. Econometrica 59, 347–370.CrossRefGoogle Scholar
Newey, W. & Steigerwald, D. (1997) Asymptotic bias for quasi-maximum-likelihood estimators in conditional heteroscedasticity models. Econometrica 65, 587–599.CrossRefGoogle Scholar
Peng, L. & Yao, Q. (2003) Least absolute deviation estimation for ARCH and GARCH models. Biometrika 90, 967–975.CrossRefGoogle Scholar
Robinson, P. & Zaffaroni, P. (2006) Pseudo-maximum-likelihood estimation of ARCH(∞) models. Annals of Statistics 34, 1049–1074.CrossRefGoogle Scholar
Straumann, D. & Mikosch, T. (2006) Quasi-maximum-likelihood estimation in conditionally heteroscedastic time series: A stochastic recurrence equations approach. Annals of Statistics 34, 2449–2495.CrossRefGoogle Scholar
Tsay, R.S. (2005) Analysis of Financial Time Series, 2nd ed.Wiley.CrossRefGoogle Scholar
Weiss, A.A. (1986) Asymptotic Theory for ARCH models: Estimation and testing. Econometric Theory 2, 107–131.CrossRefGoogle Scholar