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LEAST ABSOLUTE DEVIATION ESTIMATION FOR UNIT ROOT PROCESSES WITH GARCH ERRORS

Published online by Cambridge University Press:  01 October 2009

Guodong Li  and
Wai Keung Li
Guodong Li
Affiliation:
Nanyang Technological University and University of Hong Kong
Wai Keung Li*
Affiliation:
University of Hong Kong
*
*Address correspondence to Wai Keung Li, Department of Statistics and Actuarial Sciences, University of Hong Kong, Pokfulam Road, Hong Kong; e-mail: hrntlwk@hku.hk.
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Abstract

This paper considers a local least absolute deviation estimation for unit root processes with generalized autoregressive conditional heteroskedastic (GARCH) errors and derives its asymptotic properties under only finite second-order moment for both errors and innovations. When the innovations are symmetrically distributed, the asymptotic distribution of the estimated unit root is shown to be a functional of a bivariate Brownian motion, and then two unit root tests are derived. The simulation results demonstrate that the tests outperform those based on the Gaussian quasi maximum likelihood estimators with heavy-tailed innovations and those based on the simple least absolute deviation estimators.

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ARTICLES
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Econometric Theory , Volume 25 , Issue 5 , October 2009 , pp. 1208 - 1227
Copyright
Copyright © Cambridge University Press 2009

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References

REFERENCES

Bassett, G.J.R. & Koenker, R. (1978) Asymptotic theory of least absolute error regression. Journal of the American Statistical Association 73, 618–622.CrossRefGoogle Scholar
Billingsley, P. (1968) Convergence of Probability Measures. Wiley.Google Scholar
Bollerslev, T. (1986) Generalized autoregression conditional heteroscedasticity. Journal of Econometrics 31, 307–327.CrossRefGoogle Scholar
Chan, N.H. & Wei, C.Z. (1988) Limiting distributions of least squares estimates of unstable autoregressive processes with general autoregressive heteroscedastic errors. Annals of Statistics 16, 367–401.CrossRefGoogle Scholar
Davis, R.A. & Dunsmuir, W.T.M. (1997) Least absolute deviation estimation for regression with ARMA errors. Journal of Theoretical Probability 10, 481–497.CrossRefGoogle Scholar
Davis, R.A., Knight, K., & Liu, J. (1992) M-estimation for autoregressions with infinite variances. Stochastic Processes and Their Applications 40, 145–180.CrossRefGoogle Scholar
Fan, J. & Li, R. (2001) Variable selection via nonconcave penalized likelihood and its oracle properties. Journal of the American Statistical Association 96, 1348–1360.CrossRefGoogle Scholar
Fuller, W.A. (1996) Introduction to Statistical Time Series, 2nd ed.Wiley.Google Scholar
Hall, P. & Heyde, C.C. (1980) Martingale Limit Theory and Its Application. 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
Herce, M.A. (1996) Asymptotic theory of LAD estimation in a unit root process with finite variance errors. Econometric Theory 12, 129–153.CrossRefGoogle Scholar
Jones, M.C., Marron, J.S., & Sheather, S.J. (1996) A brief survey of bandwidth selection for density estimation. Journal of the American Statistical Association 91, 401–407.CrossRefGoogle Scholar
Kim, K. & Schmidt, P. (1993) Unit root tests with conditional heteroskedasticity. Journal of Econometrics 59, 287–300.CrossRefGoogle Scholar
Knight, K. (1998) Limiting distributions for l 1 regression estimators under general conditions. Annals of Statistics 26, 755–770.CrossRefGoogle Scholar
Kurtz, T.G. & Protter, P. (1991) Weak limit theorems to stochastic integrals and stochastic differential equations. Annals of Probability 19, 1035–1070.CrossRefGoogle Scholar
Li, G. & Li, W.K. (2005) Diagnostic checking for time series models with conditional heteroskedasticity estimated by the least absolute deviation approach. Biometrika 92, 691–701.CrossRefGoogle Scholar
Li, G. & Li, W.K. (2008) Testing for threshold moving average with conditional heteroscedasticity. Statistica Sinica 18, 647–665.Google Scholar
Ling, S. (2007) A double Ar(p) model: Structure and estimation. Statistica Sinica 17, 161–175.Google Scholar
Ling, S. & Li, W.K. (1998) Limiting distributions of maximum likelihood estimators for unstable autoregressive moving-average time series with general autoregressive heteroskedastic errors. Annals of Statistics 26, 84–125.CrossRefGoogle Scholar
Ling, S. & Li, W.K. (2003) Asymptotic inference for unit root processes with GARCH(1,1) errors. Econometric Theory 19, 541–564.CrossRefGoogle Scholar
Ling, S., Li, W.K., & McAleer, M. (2003) Estimation and testing for unit root processes with GARCH(1,1) errors: Theory and Monte Carlo evidence. Econometric Reviews 22, 179–202.CrossRefGoogle Scholar
Lucas, A. (1995) Unit root tests based on M estimators. Econometric Theory 11, 331–346.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.CrossRefGoogle Scholar
Mikosch, T. & Straumann, D. (2006) Stable limits of martingale transforms with application to the estimation of GARCH parameters. Annals of Statistics 34, 493–522.CrossRefGoogle Scholar
Mittnik, S., Rachev, S.T., & Paolella, M.S. (1998) Stable Paretian models in finance: Some empirical and theoretical aspects. In Adler, R.J., Feldman, R.E., & Taqqu, M.S. (eds.), A Practical Guide to Heavy Tails, pp. 79–110. Birkhauser.Google Scholar
Pantula, S.G. (1988) Estimation of autoregressive models with ARCH errors. Sankhya B 50, 119–138.Google Scholar
Phillips, P.C.B. (1987) Time series regression with a unit root. Econometrica 55, 277–301.CrossRefGoogle Scholar
Phillips, P.C.B. (1989) Partially identified econometric models. Econometric Theory 5, 181–240.CrossRefGoogle Scholar
Phillips, P.C.B. & Durlauf, S.N. (1986) Multiple time series regression with integrated processes. Review of Economic Studies 53, 473–495.CrossRefGoogle Scholar
Phillips, P.C.B. & Perron, P. (1988) Testing for a unit root in time series regression. Biometrika 75, 335–346.CrossRefGoogle Scholar
Politis, D.N. (2004) A heavy-tailed distribution for ARCH residuals with application to volatility prediction. Annals of Economics and Finance 5, 283–298.Google Scholar
Roussas, G.G. (1988) Nonparametric estimation in mixing sequences of random variable. Journal of Statistical Planning and Inference 18, 135–149.CrossRefGoogle Scholar
Seo, B. (1999) Distribution theory for unit root tests with conditional heteroskedasticity. Journal of Econometrics 91, 113–144.CrossRefGoogle Scholar
Silverman, B.W. (1986) Density Estimation for Statistics and Data Analysis. Chapman and Hall.Google Scholar
Zhang, Z., Li, W.K., & Yuen, K.C. (2006) On a mixture GARCH time-series model. Journal of Time Series Analysis 27, 577–597.CrossRefGoogle Scholar