The Limiting Distribution of the Autocorrelation Coefficient under a Unit Root

Abstract

The limiting distribution of the normalized autocorrelation coefficient in the case of a unit root is given in a closed form. It is found that high order transcendental functions such as the parabolic cylinder functions are indispensable to express this distribution, thus departing from the simple standard normal distribution that arises in the case of a stable root. Using the formulae derived in this paper, some numerical results available from previous studies are then extended and refined. Finally, the formulae are manipulated analytically to explain the unusual shape of the distribution.