We analyze the post-float real exchange rates for a group of OECD countries using the newly developed threshold test and tests for unit roots against stationary but nonlinear alternative by Caner and Hansen ( 2001). These tools help us disentangle the nonlinearity from the nonstationarity rigorously for the first time in the literature. After applying the threshold test and unit root tests: we find evidence for non-linearity of exchange rates. Specifically real exchange rates behave like a unit root in a band and when the depreciation or appreciation of the currency against $US exceeds the boundaries of the band , the real exchange rates are mean-reverting. The threshold value is treated as unknown and estimated in the model.