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EMPIRICAL LIKELIHOOD FOR GARCH MODELS

Published online by Cambridge University Press:  15 March 2006

Ngai Hang Chan
Affiliation:
Chinese University of Hong Kong
Shiqing Ling
Affiliation:
Hong Kong University of Science and Technology

Abstract

This paper develops an empirical likelihood approach for regular generalized autoregressive conditional heteroskedasticity (GARCH) models and GARCH models with unit roots. For regular GARCH models, it is shown that the log empirical likelihood ratio statistic asymptotically follows a χ2 distribution. For GARCH models with unit roots, two versions of the empirical likelihood methods, the least squares score and the maximum likelihood score functions, are considered. For both cases, the limiting distributions of the log empirical likelihood ratio statistics are established. These two statistics can be used to test for unit roots under the GARCH framework. Finite-sample performances are assessed through simulations for GARCH models with unit roots.This research was supported in part by Hong Kong Research grants Council Grants CUHK4043/02P and HKUST6273/03H. The authors thank two referees and the Co-Editor, Bruce Hansen, for insightful and helpful comments about the relationship between QMLE and MELE, which led to substantial improvement of the presentation. Computational assistance from Jerry Wong and Chun-Yip Yau is also gratefully acknowledged.

Type
Research Article
Copyright
© 2006 Cambridge University Press

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References

REFERENCES

Chan, N.H. & C.Z. Wei (1987) Asymptotic inference for nearly nonstationary AR(1) processes. Annals of Statistics 15, 10501062.CrossRefGoogle Scholar
Chan, N.H. & C.Z. Wei (1988) Limiting distributions of least squares estimates of unstable autoregressive processes. Annals of Statistics 16, 367401.CrossRefGoogle Scholar
Chuang, C.S. & N.H. Chan (2002) Empirical likelihood for autoregressive models with applications to unstable time series. Statistica Sinica 12, 387407.Google Scholar
Chung, K.L. (1968) A Course in Probability Theory. Academic Press.
Francq, C. & J.M. Zakoïan (2004) Maximum likelihood estimation of pure GARCH and ARMA-GARCH processes. Bernoulli 10, 605637.CrossRefGoogle Scholar
Kitamura, Y. (1997) Empirical likelihood methods with weakly dependent processes. Annals of Statistics 25, 20842102.CrossRefGoogle Scholar
Kitamura, Y. (2001) Asymptotic optimality of empirical likelihood for testing moment restrictions. Econometrica 69, 16611672.CrossRefGoogle Scholar
Kitamura, Y., G. Tripathi, & H. Ahn (2004) Empirical likelihood-based inference in conditional moment restriction models. Econometrica 72, 16671714.CrossRefGoogle Scholar
Kolaczyk, E.D. (1994) Empirical likelihood for generalized linear models. Statistica Sinica 4, 199218.Google Scholar
Lee, S.-W. & B.E. Hansen (1994) Asymptotic theory for GARCH(1,1) quasi-maximum likelihood estimator. Econometric Theory 10, 2952.CrossRefGoogle Scholar
Ling, S. (2006) Self-weighted and local quasi-maximum likelihood estimators for ARMA-GARCH /IGARCH models. Journal of Econometrics, submitted.Google Scholar
Ling, S. & W.K. Li (1998) Limiting distributions of maximum likelihood estimators for unstable autoregressive moving-average time series with general autoregressive heteroskedastic errors. Annals of Statistics 26, 84125.Google Scholar
Ling, S. & W.K. Li (2003) Asymptotic inference for unit root with GARCH(1,1) errors. Econometric Theory 19, 541564.CrossRefGoogle Scholar
Ling, S., W.K. Li, & M. McAleer (2003) Estimation and testing for unit root processes with GARCH (1,1) errors: Theory and Monte Carlo study. Econometric Reviews 22, 179202.CrossRefGoogle Scholar
Owen, A. (1988) Empirical likelihood ratio confidence intervals for a single functional. Biometrika 75, 237249.CrossRefGoogle Scholar
Owen, A. (1990) Empirical likelihood ratio confidence regions. Annals of Statistics 18, 90120.CrossRefGoogle Scholar
Owen, A. (1991) Empirical likelihood for linear models. Annals of Statistics 19, 17251747.CrossRefGoogle Scholar
Qin, J. & J. Lawless (1994) Empirical likelihood and general estimating equations. Annals of Statistics 22, 300325.CrossRefGoogle Scholar
Pantula, S.G. (1989) Estimation of autoregressive models with ARCH errors. Sankhya B 50, 119138.Google Scholar
Phillips, P.C.B. (1989) Partially identified econometric models. Econometric Theory 5, 181240.CrossRefGoogle Scholar
Seo, B. (1999) Distribution theory for unit root tests with conditional heteroskedasticity. Journal of Econometrics 91, 113144.CrossRefGoogle Scholar
Wright, J. (1999) An empirical likelihood ratio test for a unit root, problem 99.2.1. Econometric Theory 15, 257.Google Scholar