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Limit theory for moderate deviations from a unit root

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  • Phillips, Peter C.B.
  • Magdalinos, Tassos

Abstract

An asymptotic theory is given for autoregressive time series with a root of the form rho_{n} = 1+c/n^{alpha}, which represents moderate deviations from unity when alpha in (0,1). The limit theory is obtained using a combination of a functional law to a diffusion on D[0,infinity) and a central limit law to a scalar normal variate. For c 0, the serial correlation coefficient is shown to have a n^{alpha}rho_{n}^{n} convergence rate and a Cauchy limit distribution without assuming Gaussian errors, so an invariance principle applies when rho_{n} > 1. This result links moderate deviation asymptotics to earlier results on the explosive autoregression proved under Gaussian errors for alpha = 0, where the convergence rate of the serial correlation coefficient is (1 + c)^{n} and no invariance principle applies.
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Suggested Citation

  • 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.
  • Handle: RePEc:eee:econom:v:136:y:2007:i:1:p:115-130
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    References listed on IDEAS

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    1. Park, Joon, 2003. "Weak Unit Roots," Working Papers 2003-17, Rice University, Department of Economics.
    2. 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, January.
    3. Peter C.B. Phillips & Tassos Magadalinos, 2005. "Limit Theory for Moderate Deviations from a Unit Root under Weak Dependence," Cowles Foundation Discussion Papers 1517, Cowles Foundation for Research in Economics, Yale University.
    4. 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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    More about this item

    JEL classification:

    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes

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