We study the finite sample properties of tests for structural changes in the trend function of a time series that do not require knowledge of the degree of persistence in the noise component. The tests of interest are the quasi-feasible generalized least squares procedure by Perron and Yabu (2009b) and the weighted average of the regression t-statistics by Harvey, Leybourne and Taylor (2009), both of which have the same limit distribution whether the noise component is stationary or has a unit-root. We analyze the finite sample size and power properties of these tests under a variety of data-generating processes. The results show that the Perron-Yabu test has greater power overall. With respect to size, the Harvey-Leybourne-Taylor test exhibits larger size distortions unless a moving-average component is present. Using the Perron and Yabu procedure to test for structural changes in the trend function of long-run real exchange rates with respect to the U.S. dollar indicate that for 17 out of 19 countries the series have experienced a shift in trend since the late nineteenth centrury.