THE LIMITING DISTRIBUTION OF THE T RATIO UNDER A UNIT-ROOT

An encompassing formula to calculate density and distribution functions for unit root statistics was given in Abadir (1992, Oxford Bulletin of Economics and Statistics 54, 305-323) and was applied there to computing the exact limiting density and distribution of the Studentized t ratio. That formula was not a closed form, and it included a double sum and an integral. The purpose of the present paper is to develop a complete analytical study of the t ratio. To do so, closed forms (which are free of integrals) and a simple summation-free integral are derived for the exact limiting density and distribution functions of the t ratio, The density of t being asymmetric, different forms emerge for different signs of t. The forms have in common the use of confluent hypergeometric functions such as the incomplete gamma and the parabolic cylinder functions, thus departing from the simple standard Normal distribution that arises in the cases of a stable or an explosive root. It is also shown that, depending on the sign and value of t, the density and distribution may be well approximated by different proportions of the standard Normal. The shifted Normal distribution is also considered as an approximation. Numerical results available from previous studies are extended and refined using the new formulae, whose relative merits are then analyzed. These merits include analytical features of the distribution that other methods could not uncover. The paper also uses these features for an analytical comparison of the densities of the t ratio and the normalized autocorrelation coefficient.