A new family of spectral shape tests was proposed recently by Durlauf (1991) for testing the martingale hypothesis. Unlike the widely used variance ratio test, spectral shape tests are consistent against all stationary non-white-noise alternatives from the martingale null. In this paper we examine the finite sample properties of the spectral shape tests and find that the tests have good size and power properties even for small samples. We apply the tests to examine the martingale hypothesis for five major currencies vis-a-vis the US dollar for the period 1974-89. The results indicate that most currencies violate the martingale hypothesis. It appears that some rejections are due to long-memory influences.