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Estimation and Testing for Unit Root Processes with GARCH (1, 1) Errors: Theory and Monte Carlo Evidence

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  • Shiqing Ling
  • W. K. Li
  • Michael McAleer

Abstract

Least squares (LS) and maximum likelihood (ML) estimation are considered for unit root processes with GARCH (1, 1) errors. The asymptotic distributions of LS and ML estimators are derived under the condition α + β < 1. The former has the usual unit root distribution and the latter is a functional of a bivariate Brownian motion, as in Ling and Li [Ling, S., Li, W. K. (1998). Limiting distributions of maximum likelihood estimators for unstable autoregressive moving-average time series with GARCH errors. Ann. Statist.26:84-125]. Several unit root tests based on LS estimators, ML estimators, and mixing LS and ML estimators, are constructed. Simulation results show that tests based on mixing LS and ML estimators perform better than Dickey-Fuller tests which are based on LS estimators, and that tests based on the ML estimators perform better than the mixed estimators.

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Bibliographic Info

Article provided by Taylor and Francis Journals in its journal Econometric Reviews.

Volume (Year): 22 (2003)
Issue (Month): 2 ()
Pages: 179-202

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Handle: RePEc:taf:emetrv:v:22:y:2003:i:2:p:179-202

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Related research

Keywords: Asymptotic distribution; Brownian motion; GARCH model; Least squares estimator; Maximum likelihood estimator; Unit root;

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References

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  1. Phillips, P C B & Durlauf, S N, 1986. "Multiple Time Series Regression with Integrated Processes," Review of Economic Studies, Wiley Blackwell, vol. 53(4), pages 473-95, August.
  2. Peter C.B. Phillips, 1987. "Partially Identified Econometric Models," Cowles Foundation Discussion Papers 845R, Cowles Foundation for Research in Economics, Yale University, revised Aug 1988.
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  4. W. K. Li & Shiqing Ling & Michael McAleer, 2001. "A Survey of Recent Theoretical Results for Time Series Models with GARCH Errors," ISER Discussion Paper 0545, Institute of Social and Economic Research, Osaka University.
  5. Li, W K & Ling, Shiqing & McAleer, Michael, 2002. " Recent Theoretical Results for Time Series Models with GARCH Errors," Journal of Economic Surveys, Wiley Blackwell, vol. 16(3), pages 245-69, July.
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  9. Seo, Byeongseon, 1999. "Distribution theory for unit root tests with conditional heteroskedasticity1," Journal of Econometrics, Elsevier, vol. 91(1), pages 113-144, July.
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  11. Hansen, Bruce E., 1995. "Rethinking the Univariate Approach to Unit Root Testing: Using Covariates to Increase Power," Econometric Theory, Cambridge University Press, vol. 11(05), pages 1148-1171, October.
  12. Herce, Miguel A., 1996. "Asymptotic Theory of LAD Estimation in a Unit Root Process with Finite Variance Errors," Econometric Theory, Cambridge University Press, vol. 12(01), pages 129-153, March.
  13. repec:cup:etheor:v:12:y:1996:i:1:p:129-53 is not listed on IDEAS
  14. Lee, Sang-Won & Hansen, Bruce E., 1994. "Asymptotic Theory for the Garch(1,1) Quasi-Maximum Likelihood Estimator," Econometric Theory, Cambridge University Press, vol. 10(01), pages 29-52, March.
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Citations

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Cited by:
  1. Matei Demetrescu, 2010. "On the Dickey–Fuller test with White standard errors," Statistical Papers, Springer, vol. 51(1), pages 11-25, January.
  2. Werner Ploberger & Peter C.B. Phillips, 2010. "Optimal Estimation under Nonstandard Conditions," Cowles Foundation Discussion Papers 1748, Cowles Foundation for Research in Economics, Yale University.
  3. Riccardo LUCCHETTI & Giulio PALOMBA, 2006. "Forecasting US bond yields at weekly frequency," Working Papers 261, Universita' Politecnica delle Marche (I), Dipartimento di Scienze Economiche e Sociali.
  4. Paulo M. M. Rodrigues & Antonio Rubia, 2004. "On the Small Sample Properties of Dickey Fuller and Maximum Likelihood Unit Root Tests on Discrete-Sampled Short-Term Interest Rates," Econometrics 0405004, EconWPA.
  5. Chor-yiu SIN, 2004. "Estimation and Testing for Partially Nonstationary Vector Autoregressive Models with GARCH: WLS versus QMLE," Econometric Society 2004 North American Summer Meetings 476, Econometric Society.
  6. Li, Yushu & Shukur, Ghazi, 2009. "Testing for Unit Root against LSTAR Model: Wavelet Improvement under GARCH Distortion," CAFO Working Papers 2009:6, Centre for Labour Market Policy Research (CAFO), School of Business and Economics, Linnaeus University.
  7. Cavaliere, Giuseppe & Taylor, A.M. Robert, 2007. "Testing for unit roots in time series models with non-stationary volatility," Journal of Econometrics, Elsevier, vol. 140(2), pages 919-947, October.
  8. Chor-yiu SIN, 2004. "Estimation and Testing for Partially Nonstationary Vector Autoregressive Models with GARCH: WLS versus QMLE," Econometric Society 2004 Australasian Meetings 92, Econometric Society.
  9. Wang, Gaowen, 2006. "A note on unit root tests with heavy-tailed GARCH errors," Statistics & Probability Letters, Elsevier, vol. 76(10), pages 1075-1079, May.
  10. Gospodinov, Nikolay, 2008. "Asymptotic and bootstrap tests for linearity in a TAR-GARCH(1,1) model with a unit root," Journal of Econometrics, Elsevier, vol. 146(1), pages 146-161, September.
  11. repec:ebl:ecbull:v:3:y:2005:i:55:p:1-12 is not listed on IDEAS
  12. W. K. Li & Shiqing Ling & Michael McAleer, 2001. "A Survey of Recent Theoretical Results for Time Series Models with GARCH Errors," ISER Discussion Paper 0545, Institute of Social and Economic Research, Osaka University.
  13. Li, Yushu & Shukur, Ghazi, 2009. "Testing for Unit Root against LSTAR model – wavelet improvements under GARCH distortion," Working Paper Series in Economics and Institutions of Innovation 184, Royal Institute of Technology, CESIS - Centre of Excellence for Science and Innovation Studies.
  14. Berkes, István & Hörmann, Siegfried & Horváth, Lajos, 2008. "The functional central limit theorem for a family of GARCH observations with applications," Statistics & Probability Letters, Elsevier, vol. 78(16), pages 2725-2730, November.
  15. Alvaro Escribano & J. Ignacio Peña & Pablo Villaplana, 2011. "Modelling Electricity Prices: International Evidence," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 73(5), pages 622-650, October.
  16. Lucchetti, Riccardo & Palomba, Giulio, 2008. "Nonlinear Adjustment in US Bond Yields: an Empirical Analysis with Conditional Heteroskedasticity," MPRA Paper 11571, University Library of Munich, Germany.
  17. Lucchetti, Riccardo & Palomba, Giulio, 2009. "Nonlinear adjustment in US bond yields: An empirical model with conditional heteroskedasticity," Economic Modelling, Elsevier, vol. 26(3), pages 659-667, May.

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