The limiting distribution of the t-ratio for the unit root test in an AR(1)
AbstractWe 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, 775Â793) derived the first derives a closed (i.e. integration-free) expression for the limit-ing distribution function. This paper derives an alternative closed expression. AbadirÂ¹s 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 AbadirÂ¹s 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.
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Bibliographic InfoPaper provided by Maastricht University in its series Open Access publications from Maastricht University with number urn:nbn:nl:ui:27-15430.
Date of creation: 2001
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Publication status: Published in The econometrics journal (2001) v.4, p.242-256
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Other versions of this item:
- 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 5.
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- Abadir, Karim M, 1992. "A Distribution Generating Equation for Unit-Root Statistics," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 54(3), pages 305-23, August.
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- Dietrich, Franz, 2003. "Closed Analytical Forms and Numerical Approximation of Dickey-Fuller Probability Distributions," Open Access publications from Maastricht University urn:nbn:nl:ui:27-15429, Maastricht University.
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