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

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, 775-793) derived the first derives a closed (i.e. integration-free) expression for the limiting 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 summations. 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.