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In this paper, we propose a simple testing procedure to detect the presence of nonstationarity against nonlinear but globally stationary exponential smooth transition autoregressive processes. We provide an advance over the existing literature in three senses. First, we derive the limiting nonstandard distribution of the proposed tests. Second, we find via Monte Carlo simulation exercises that under the alternative of a globally stationary ESTAR process, our proposed test has better power than the standard Dickey–Fuller test, in the region of the null, where the processes are highly persistent. Third, we provide an application to ex post real interest rates and bilateral real exchange rates with the US Dollar from the 11 major OECD countries, and find our test is able to reject a unit root in many cases, whereas the linear DF tests fail, providing some evidence of nonlinear mean-reversion in both real interest and exchange rates.