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Unit Root And Cointegrating Limit Theory When Initialization Is In The Infinite Past

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

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 continue to hold in spite of the degeneracy in the limit theory and are therefore robust to initial conditions that extend to the infinite past.

Suggested Citation

  • Phillips, Peter C.B. & Magdalinos, Tassos, 2009. "Unit Root And Cointegrating Limit Theory When Initialization Is In The Infinite Past," Econometric Theory, Cambridge University Press, vol. 25(06), pages 1682-1715, December.
  • Handle: RePEc:cup:etheor:v:25:y:2009:i:06:p:1682-1715_99
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    References listed on IDEAS

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    1. Elliott, Graham & Muller, Ulrich K., 2006. "Minimizing the impact of the initial condition on testing for unit roots," Journal of Econometrics, Elsevier, pages 285-310.
    2. Ulrich K. M¸ller & Graham Elliott, 2003. "Tests for Unit Roots and the Initial Condition," Econometrica, Econometric Society, pages 1269-1286.
    3. Abadir,Karim M. & Magnus,Jan R., 2005. "Matrix Algebra," Cambridge Books, Cambridge University Press, number 9780521537469, December.
    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.
    5. repec:cup:cbooks:9780521822893 is not listed on IDEAS
    6. 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.
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    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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