The Dickey-Fuller test for exponential random walks

A common test in econometrics is the Dickey–Fuller test, which is based on the test statistic . We investigate the behavior of the test statistic if the data yt are given by an exponential random walk exp(Zt) where Zt = Zt-1 + [sigma][epsilon]t and the [epsilon]t are independent and identically distributed random variables. The test statistic DF(T) is a nonlinear transformation of the partial sums of [epsilon]t process. Under certain moment conditions on the [epsilon]t we show that tends to one as [lambda] [rightward arrow] 0. For the particular case that the [epsilon]t define a simple random walk it is shown that plimT[rightward arrow][infty infinity] DF(T)/T exists and the limit is evaluated. The theoretical results are illustrated by some simulation experiments.