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Unit Root and Cointegrating Limit Theory When Initialization Is in the Infinite Past

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Abstract

It is well known that unit root limit distributions are sensitive to initial conditions in the distant past. If the distant past initialization is extended to the infinite past, the initial condition dominates the limit theory producing a faster rate of convergence, a limiting Cauchy distribution for the least squares coefficient and a limit normal distribution for the t ratio. This amounts to the tail of the unit root process wagging the dog of the unit root limit theory. These simple results apply in the case of a univariate autoregression with no intercept. The limit theory for vector unit root regression and cointegrating regression is affected but is no longer dominated by infinite past initializations. The latter contribute to the limiting distribution of the least squares estimator and produce a singularity in the limit theory, but do not change the principal rate of convergence. Usual cointegrating regression theory and inference continues to hold in spite of the degeneracy in the limit theory and is therefore robust to initial conditions that extend to the infinite past.

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File URL: http://cowles.econ.yale.edu/P/cd/d16b/d1655.pdf
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Paper provided by Cowles Foundation for Research in Economics, Yale University in its series Cowles Foundation Discussion Papers with number 1655.

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Length: 31 pages
Date of creation: May 2008
Date of revision:
Publication status: Published in Econometric Theory (December 2009), 25(6): 1682-1715
Handle: RePEc:cwl:cwldpp:1655

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

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Keywords: Cauchy limit distribution; Cointegration; Distant past initialization; Infinite past initialization; Random orthonormalization; Singular limit theory;

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  1. Phillips, Peter C.B. & Magdalinos, Tassos, 2008. "Limit Theory For Explosively Cointegrated Systems," Econometric Theory, Cambridge University Press, vol. 24(04), pages 865-887, August.
  2. Ulrich K. M¸ller & Graham Elliott, 2003. "Tests for Unit Roots and the Initial Condition," Econometrica, Econometric Society, vol. 71(4), pages 1269-1286, 07.
  3. Elliott, Graham & Muller, Ulrich K., 2006. "Minimizing the impact of the initial condition on testing for unit roots," Journal of Econometrics, Elsevier, vol. 135(1-2), pages 285-310.
  4. Harvey, David I. & Leybourne, Stephen J. & Taylor, A.M. Robert, 2009. "Unit Root Testing In Practice: Dealing With Uncertainty Over The Trend And Initial Condition," Econometric Theory, Cambridge University Press, vol. 25(03), pages 587-636, June.
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