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Limit Theory for Moderate Deviations from a Unit Root under Weak Dependence

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Abstract

An asymptotic theory is given for autoregressive time series with weakly dependent innovations and a root of the form rho_{n} = 1+c/n^{alpha}, involving moderate deviations from unity when alpha in (0,1) and c in R are constant parameters. The limit theory combines a functional law to a diffusion on D[0,infinity) and a central limit theorem. For c > 0, the limit theory of the first order serial correlation coefficient is Cauchy and is invariant to both the distribution and the dependence structure of the innovations. To our knowledge, this is the first invariance principle of its kind for explosive processes. The rate of convergence is found to be n^{alpha}rho_{n}^{n}, which bridges asymptotic rate results for conventional local to unity cases (n) and explosive autoregressions ((1 + c)^{n}). For c

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File URL: http://cowles.econ.yale.edu/P/cd/d15a/d1517.pdf
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Bibliographic Info

Paper provided by Cowles Foundation for Research in Economics, Yale University in its series Cowles Foundation Discussion Papers with number 1517.

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Length: 46 pages
Date of creation: Jun 2005
Date of revision:
Publication status: Published in G. D. A. Phillips and E. Tzavalis, eds., The Refinement of Econometric Estimation and Test Procedures: Finite Sample and Asymptotic Analysis. Cambridge University, 2007, pp.123-162
Handle: RePEc:cwl:cwldpp:1517

Note: CFP 1202.
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Postal: Cowles Foundation, Yale University, Box 208281, New Haven, CT 06520-8281 USA

Related research

Keywords: Central limit theory; Diffusion; Explosive autoregression; Local to unity; Moderate deviations; Unit root distribution; Weak dependence;

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References

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  1. Peter C.B. Phillips & Victor Solo, 1989. "Asymptotics for Linear Processes," Cowles Foundation Discussion Papers 932, Cowles Foundation for Research in Economics, Yale University.
  2. Peter C.B. Phillips, 1986. "Regression Theory for Near-Integrated Time Series," Cowles Foundation Discussion Papers 781R, Cowles Foundation for Research in Economics, Yale University, revised Jan 1987.
  3. Liudas Giraitis & Peter C. B. Phillips, 2006. "Uniform Limit Theory for Stationary Autoregression," Journal of Time Series Analysis, Wiley Blackwell, vol. 27(1), pages 51-60, 01.
  4. Phillips, Peter C.B. & Magdalinos, Tassos, 2007. "Limit theory for moderate deviations from a unit root," Journal of Econometrics, Elsevier, vol. 136(1), pages 115-130, January.
  5. Phillips, P C B, 1987. "Time Series Regression with a Unit Root," Econometrica, Econometric Society, vol. 55(2), pages 277-301, March.
  6. Peter C.B. Phillips, 1999. "Unit Root Log Periodogram Regression," Cowles Foundation Discussion Papers 1244, Cowles Foundation for Research in Economics, Yale University.
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Cited by:
  1. Eiji Kurozumi & Kazuhiko Hayakawa, 2006. "Asymptotic Properties of the Efficient Estimators for Cointegrating Regression Models with Serially Dependent Errors," Hi-Stat Discussion Paper Series d06-197, Institute of Economic Research, Hitotsubashi University.
  2. Phillips, Peter C.B. & Magdalinos, Tassos, 2007. "Limit theory for moderate deviations from a unit root," Journal of Econometrics, Elsevier, vol. 136(1), pages 115-130, January.

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