In this paper we offer a bootstrap-based version of the Cox specification test for non-nested hypothesis to discriminate between ESTAR and MSAR models. Both models are commonly used for modeling real exchange rates dynamics. We show that the test has good size and power properties in finite samples. In an application, we analyze several major real exchange rates to shed light on the question of which model describes these processes best. This allows us to draw conclusions about the driving forces of real exchange rates.