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 DF(T) = T((Sigma(1)(T)y(t-1)y(t))/(Sigma(1)(T)y(t-1)(2)) - 1). We investigate the behavior of the test statistic if the data y(t) are given by an exponential random walk exp(Z(t)) where Z(t) = Z(t-1) + sigmaepsilon(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 e, process. Under certain moment conditions on the epsilon(t) we show that lim(T-->infinity) P(-1 less than or equal to DF(T)/T less than or equal to - lambda) tends to one as lambda --> 0. For the particular case that the c, define a simple random walk it is shown that plim(T-->infinity)DF(T)/T exists and the limit is evaluated. The theoretical results are illustrated by some simulation experiments.