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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, 1999. "Unit Root Log Periodogram Regression," Cowles Foundation Discussion Papers 1244, Cowles Foundation for Research in Economics, Yale 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.
  3. 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.
  4. Peter C.B. Phillips, 1985. "Time Series Regression with a Unit Root," Cowles Foundation Discussion Papers 740R, Cowles Foundation for Research in Economics, Yale University, revised Feb 1986.
  5. L Giraitis & P C B Phillips, . "Uniform limit theory for stationary autoregression," Discussion Papers 05/23, Department of Economics, University of York.
  6. Peter C.B. Phillips & Victor Solo, 1989. "Asymptotics for Linear Processes," Cowles Foundation Discussion Papers 932, Cowles Foundation for Research in Economics, Yale University.
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Cited by:
  1. Peter C.B. Phillips & Tassos Magdalinos, 2004. "Limit Theory for Moderate Deviations from a Unit Root," Cowles Foundation Discussion Papers 1471, Cowles Foundation for Research in Economics, Yale University.
  2. Kurozumi, Eiji & Hayakawa, Kazuhiko, 2009. "Asymptotic properties of the efficient estimators for cointegrating regression models with serially dependent errors," Journal of Econometrics, Elsevier, vol. 149(2), pages 118-135, April.

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