ASYMPTOTIC EXPANSIONS FOR RANDOM-WALKS WITH NORMAL ERRORS

The asymptotic distribution of the least-squares estimators in the ran dom walk model was first found by White [17] and is described in terms of functionals of Brownian motion with no closed form expression know n. Evans and Savin [5,6] and others have examined numerically both the asymptotic and finite sample distribution. The purpose of this paper is to derive an asymptotic expansion for the distribution. Our approac h is in contrast to Phillips [12,13] who has already derived some term s in a general expansion by analyzing the functionals. We proceed by a ssuming that the errors are normally distributed and expand the charac teristic function directly. Then, via numerical integration, we invert the characteristic function to find the distribution. The approximati on is shown to be extremely accurate for all sample sizes greater-than -or-equal-to 25, and can be used to construct simple tests for the-pre sence of a unit root in a univariate time series model. This could hav e useful applications in applied economics.