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TAIL INDEX OF AN AR(1) MODEL WITH ARCH(1) ERRORS

Published online by Cambridge University Press:  21 February 2013

Ngai Hang Chan ,
Deyuan Li ,
Liang Peng  and
Rongmao Zhang
Ngai Hang Chan
Affiliation:
Chinese University of Hong Kong
Deyuan Li
Affiliation:
Fudan University
Liang Peng
Affiliation:
Georgia Institute of Technology
Rongmao Zhang*
Affiliation:
Zhejiang University
*
*Address correspondence to Rongmao Zhang, Department of Mathematics, Zhejiang University, Hangzhou, 310027, China; e-mail: [email protected].
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Abstract

Relevant sample quantities such as the sample autocorrelation function and extremes contain useful information about autoregressive time series with heteroskedastic errors. As these quantities usually depend on the tail index of the underlying heteroskedastic time series, estimating the tail index becomes an important task. Since the tail index of such a model is determined by a moment equation, one can estimate the underlying tail index by solving the sample moment equation with the unknown parameters being replaced by their quasi-maximum likelihood estimates. To construct a confidence interval for the tail index, one needs to estimate the complicated asymptotic variance of the tail index estimator, however. In this paper the asymptotic normality of the tail index estimator is first derived, and a profile empirical likelihood method to construct a confidence interval for the tail index is then proposed. A simulation study shows that the proposed empirical likelihood method works better than the bootstrap method in terms of coverage accuracy, especially when the process is nearly nonstationary.

Type
ARTICLES
Information
Econometric Theory , Volume 29 , Issue 5 , October 2013 , pp. 920 - 940
Copyright
Copyright © Cambridge University Press 2013 

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Footnotes

We would like to thank the co-editor and two anonymous referees for their careful readings and helpful suggestions. Research was supported in part by grants from HKSAR-RGC-GRF: 400306, 400408, and 400410; HKSAR-RGC-CRF: CityU8/CRF/09; NSA: H98230-10-1-0170; NSF: DMS-1005336; NSFC: 11171074 and 10801118; and the Ph.D. Programs Foundation of the Ministry of Education of China (200803351094).

References

REFERENCES

Arcones, M.A. (1996) The Bahadur-Kiefer representation of Lp regression estimators. Econometric Theory 12, 257–283.CrossRefGoogle Scholar
Berkes, I. L Horvath, & Kokoszka, P. (2003) Estimation of the maximal moment exponent of a GARCH (1,1) sequence. Econometric Theory 19, 565–586.CrossRefGoogle Scholar
Borkovec, M. (2000) Extremal behavior of the autoregressive process with ARCH(1) errors. Stochastic Processes and Their Applications 85, 189–207.CrossRefGoogle Scholar
Borkovec, M. (2001) Asymptotic behavior of the sample autocovariance and autocorrelation function of the AR(1) process with ARCH errors. Bernoulli 7, 847–872.CrossRefGoogle Scholar
Borkovec, M. & Klüppelberg, C. (2001) The tail of the stationary distribution of an autoregressive process with ARCH(1) errors. Annals of Applied Probability 11, 1220–1241.Google Scholar
Chan, N.H. & Peng, L. (2005) Weighted least absolute deviations estimation for an AR(1) process with ARCH(1) errors. Biometrika 92, 477–484.CrossRefGoogle Scholar
Chen, S.X. & Van Keilegom, I. (2009) A review on empirical likelihood methods for regression. Test 18, 415–447.CrossRefGoogle Scholar
Cont, R. & Tankov, P. (2004) Financial Modeling with Jump Processes. Chapman & Hall.Google Scholar
Drees, H. (2000) Weighted approximations for tail processes for β-mixing random variables. Annals of Applied Probability 10, 1274–1301.Google Scholar
Guo, B. & Phillips, P.C.B. (1998) Efficient Estimation of Second Moment Parameters in ARCH Models. Unpublished manuscript.Google Scholar
Hall, P. & Heyde, C.C. (1980) Martingale Limit Theory and Its Application. Academic Press.Google Scholar
Klüppelberg, C. & Peng, L. (2006) Empirical likelihood method for an AR(1) process with ARCH(1) errors. International Journal of Statistics and Management Systems 1, 48–58.Google Scholar
Ling, S. (2004) Estimation and testing stationarity for double autoregressive models. Journal of the Royal Statistical Society B 66, 63–78.CrossRefGoogle 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
Owen, A.B. (1990) Empirical likelihood confidence regions. Annals of Statistics 18, 90–120.CrossRefGoogle Scholar
Owen, A.B. (2001) Empirical Likelihood. Chapman & Hall.CrossRefGoogle Scholar
Qin, J. & Lawless, J. (1994) Empirical likelihood and general estimating equations. Annals of Statistics 22, 300–325.CrossRefGoogle Scholar
Weiss, A.A. (1984) ARMA models with ARCH errors. Journal of Time Series Analysis 3, 129–143.CrossRefGoogle Scholar