Recent studies on general equilibrium models with transaction costs show that the dynamics of the real exchange rate are necessarily nonlinear. Our contribution to the literature on nonlinear price adjustment mechanisms is threefold. First, we model the real exchange rate by a Multi-Regime Logistic Smooth Transition Auto Regression (MR-LSTAR), allowing for both ESTAR-type and SETAR-type dynamics. This choice is motivated by the fact that even the theoretical models, which predict a smooth behavior for the real exchange rate, do not rule out the possibility of a discontinuous adjustment as a limit case. Second, we propose two classes of unit-root tests against this MR-LSTAR alternative, based respectively on the likelihood and on an auxiliary model. Their asymptotic distributions are derived analytically. Third, when applied to 28 bilateral real exchange rates, our tests reject the null hypothesis of a unit root for eleven series bringing evidence in favor of the purchasing power parity.