Asymptotic Expansions for Random Walks with Normal Errors

The asymptotic distribution of the least-squares estimators in the random walk model was first found by White [17] and is described in terms of functional of Brownian motion with no closed form expression known. 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 approach is in contrast to Phillips [12,13] who has already derived some terms in a general expansion by analyzing the functionals. We proceed by assuming that the errors are normally distributed and expand the characteristic function directly. Then, via numerical integration, we invert the characteristic function to find the distribution. The approximation is shown to be extremely accurate for all sample sizes ≥25, and can be used to construct simple tests for the presence of a unit root in a univariate time series model. This could have useful applications in applied economics.