This paper considers testing the unit root hypothesis against a smooth transition autoregressive model as the alternative. The model specification makes it possible to discriminate between nonstationary random walk and stationary nonlinear processes. Some new limit results are presented, extending earlier work, and two F type tests are proposed. Small sample simulations show some size distortions, why a bootstrap method for estimating p-values to the tests are considered. Power simulations show some gain in power, compared to the common Augmented Dickey-Fuller tests. Finally, the two proposed F type tests are applied on a number of real exchange rates. For several of the exchange rates considered the linear unit root is rejected in favor of the stationary nonlinear model, supporting the purchasing power parity hypothesis.