Techniques for testing the null hypothesis of difference stationarity against stationarity
around some deterministic function have received much attention. In particular, unit root tests
where the alternative is stationarity around a smooth transition in linear trend have recently
been proposed to permit the possibility of non-instantaneous structural change. In this paper
we extend such an approach to admit more than one structural change, allowing the model
under the alternative hypothesis to be stationary about two smooth transitions in linear trend.
Tests involving this added generality are developed and their properties investigated;
application of the tests to two interesting time series highlights the potential benefits of this
double transition extension.