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The limiting distribution of the t-ratio for the unit root test in an AR(1)

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  • FRANZ K. DIETRICH

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Abstract

We consider the limiting distribution of the t -statistic for testing the random walk hypothesis in the classical Gaussian AR(1) model. Abadir (1995, Econometric Theory, 11, 775­793) derived the first derives a closed (i.e. integration-free) expression for the limit-ing distribution function. This paper derives an alternative closed expression. Abadir¹s and the new expression are valid only for negative arguments and each involve two infinite summa-tions. To enable a numerical treatment, we derive inequalities that allow a suitable truncation of all series occurring in Abadir¹s and the new expression. In both expressions the outer series has a very fast convergence so that truncation after only the first summand usually suffices. The inner series of the new expression displays the numerically desirable Leibnitz property. By differentiating we obtain a new closed expression for the limiting density function. We also find an asymptotic expansion for the lower tail of the limiting distribution function.

Suggested Citation

  • Franz K. Dietrich, 2001. "The limiting distribution of the t-ratio for the unit root test in an AR(1)," Econometrics Journal, Royal Economic Society, vol. 4(2), pages 1-5.
  • Handle: RePEc:ect:emjrnl:v:4:y:2001:i:2:p:5
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