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ASYMPTOTIC INFERENCE FOR AR MODELS WITH HEAVY-TAILED G-GARCH NOISES

Published online by Cambridge University Press:  03 November 2014

Rongmao Zhang  and
Shiqing Ling
Rongmao Zhang*
Affiliation:
Zhejiang University
Shiqing Ling
Affiliation:
Hong Kong University of Science and Technology
*
*Address correspondence to Rongmao Zhang, Zhejiang University, Hangzhou, 310027, China; e-mail: rmzhang@zju.edu.cn
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Abstract

It is well known that the least squares estimator (LSE) of an AR(p) model with i.i.d. (independent and identically distributed) noises is n1/αL(n)-consistent when the tail index α of the noise is within (0,2) and is n1/2-consistent when α ≥ 2, where L(n) is a slowly varying function. When the noises are not i.i.d., however, the case is far from clear. This paper studies the LSE of AR(p) models with heavy-tailed G-GARCH(1,1) noises. When the tail index α of G-GARCH is within (0,2), it is shown that the LSE is not a consistent estimator of the parameters, but converges to a ratio of stable vectors. When α ε [2,4], it is shown that the LSE is n1–2/α-consistent if α ε (2,4), logn-consistent if α = 2, and n1/2 / logn-consistent if α = 4, and its limiting distribution is a functional of stable processes. Our results are significantly different from those with i.i.d. noises and should warn practitioners in economics and finance of the implications, including inconsistency, of heavy-tailed errors in the presence of conditional heterogeneity.

Information

Type
MISCELLANEA
Information
Econometric Theory , Volume 31 , Issue 4 , August 2015 , pp. 880 - 890
Copyright
Copyright © Cambridge University Press 2014 

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Footnotes

We thank Ms. Alice Cheng for her editing comments and three referees, the co-editor Giuseppe Cavaliere, and the editor Peter C.B. Phillips for their very helpful and professional comments. Zhang’s research was supported by NSFC grants 11371318 and 11171074, the Fundamental Research Funds for the Central Universities, and Scientific Research Fund of Zhejiang Provincial Education Department (Y201009944). Ling’s research was supported by the Hong Kong Research Grants Council (Grants HKUST641912, 603413 and FSGRF12SC12).

References

REFERENCES

An, H.Z. & Chen, Z.G. (1982) On convergence of LAD estimates in autoregression with infinite variance. Journal of Multivariate Analysis 12, 335–345.CrossRefGoogle Scholar
Basrak, B., Davis, R.A., & Mikosch, T. (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
Breiman, L. (1965) On some limit theorems similar to the arc-sin law. Theory of Probability and Its Applications 10, 323–331.CrossRefGoogle Scholar
Cavaliere, G. & Georgiev, I. (2013) Exploiting infinite variance through dummy variables in non-stationary autoregressions. Econometric Theory 29, 1162–1195.CrossRefGoogle Scholar
Chan, K.S., Li, D., Ling, S., & Tong, H. (2014) On Conditionally Heteroscedastic AR Models with Thresholds. Statistica Sinica 24, 625–652.Google Scholar
Chan, N.H. & Zhang, R.M. (2010) Inference for unit-root models with infinite variance GARCH errors. Statistica Sinica 20, 1363–1393.Google Scholar
Davis, R.A., Knight, K., & Liu, J. (1992) M-estimation for autoregressions with infinite variance. Stochastic Processes and Their Applications 40, 145–180.Google Scholar
Davis, R.A. & Mikosch, T. (1998) The sample autocorrelations of heavy-tailed processes with applications to ARCH. Annals of Statistics 26, 2049–2080.Google Scholar
Davis, R.A. & Resnick, S. (1986) Limit theory for the sample covariance and correlation functions of moving averages. Annals of Statistics 14, 533–558.CrossRefGoogle Scholar
Engle, R.F. (1982) Autoregressive conditional heteroskedasticity with estimates of variance of U.K. inflation. Econometrica 50, 987–1008.CrossRefGoogle Scholar
Engle, R.F. (1990) Discussion: Stock market volatility and the crash of 1987. Review of Financial Studies 3, 103–106.CrossRefGoogle Scholar
Fornari, F. & Mele, A. (1997) Sign- and volatility-switching ARCH models: Theory and applications to international stock markets. Journal of Applied Econometrics 12, 1779–1801.Google Scholar
Francq, C. & Zakoïan, J.M. (2006) Mixing properties of a general class of GARCH(1,1) models without moment assumptions on the observed process. Econometric Theory 22, 815–834.CrossRefGoogle Scholar
Hannan, E.J. & Kanter, M. (1977) Autoregressive processes with infinite variance. Journal of Applied Probability 14, 411–415.CrossRefGoogle Scholar
He, C. & Terasvirta, T. (1999) Properties of moments of a family of GARCH processes. Journal of Econometrics 92, 173–192.CrossRefGoogle Scholar
Hill, J.B. & Renault, E. (2010) Generalized Method of Moments with Tail Trimming. Working paper, University of North Carolina.Google Scholar
Knight, K. (1987) Rate of convergence of centred estimates of autoregressive parameters for infinite variance autoregressions. Journal of Time Series Analysis 8, 51–60.Google Scholar
Knight, K. (1991) Limit theory for M-estimates in an integrated infinite variance. Econometric Theory 7, 200–212.CrossRefGoogle Scholar
Kokoszka, P.S. & Taqqu, M.S. (1996) Parameter estimation for infinite variance fractional ARIMA. Annals of Statistics 24, 1880–1913.Google Scholar
Lange, T. (2011) Tail behavior and OLS estimation in AR-GARCH models. Statistica Sinica 21, 1191–1200.CrossRefGoogle Scholar
Lange, T., Rahbek, A., & Jensen, S.T. (2011) Estimation and asymptotic inference in the AR-ARCH model. Econometric Reviews 30, 129–153.Google Scholar
Ling, S. (2007) Self-weighted and local quasi-maximum likelihood estimators for ARMA-GARCH/IGARCH models. Journal of Econometrics 140, 849–873.CrossRefGoogle Scholar
Mikosch, T., Gadrich, T., Klüppelberg, C., & Adler, R.J. (1995) Parameter estimation for ARMA models with infinite variance innovations. Annals of Statistics 23, 305–326.CrossRefGoogle Scholar
Mikosch, T. & Stărică, C. (2000) Limit theory for the sample autocorrelations and extremes of a GARCH (1,1) process. Annals of Statistics 28, 1427–1451.Google Scholar
Nelson, D.B. (1990) Stationary and persistence in GARCH (1,1) models. Econometric Theory 6, 318–334.CrossRefGoogle Scholar
Phillips, P.C.B. (1991) A shortcut to LAD asymptotics. Econometric Theory 7, 450–463.CrossRefGoogle Scholar
Schwert, G.W. (1989) Why does stock market volatility change over time? Journal of Finance 45, 1129–1155.Google Scholar
Sentana, E. (1995) Quadratic ARCH models. Review of Economic Studies 62, 639–661.CrossRefGoogle Scholar
Taylor, S. (1986) Modelling Financial Time Series. Wiley.Google Scholar
Zakoian, J.M. (1994) Threshold heteroskedastic models. Journal of Economic Dynamics and Control 18, 931–955.CrossRefGoogle Scholar
Zhang, R.M., Sin, C.Y., & Ling, S. (2014) On functional limits of short- and long-memory linear processes with GARCH(1,1) noises. Stochastic Processes and Their Applications, to appear.Google Scholar
Zhu, K. & Ling, S. (2011) Global self-weighted and local quasi-maximum exponential likelihood estimators for ARMA–GARCH/IGARCH models. Annals of Statistics 39, 2131–2163.CrossRefGoogle Scholar