Econometrics Journal (2001), volume 4, pp. 242–256. The limiting distribution of the t-ratio for the unit root test in an AR(1)

Summary 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, 775–793) derived the first derives a closed (i.e. integration-free) expression for the limit-ing distribution function. This paper derives an alternative closed expression. Abadir’s 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 Abadir’s 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