This paper illustrates the flexibility of the ESTAR model to encompass a number of different characteristics found in economic and financial series, such as multiple equilibria, complex dynamics, chaotic-like behavior, and spurious trends. We then re-assess the power of the Kapetanios et al. (2003), Enders and Granger (1998), and Augmented Dickey Fuller unit root tests in the presence of nuisance parameters for parameter values typically encountered in the empirical literature. Our results show the lack of dominance of any particular test and that the power is not independent to priors about the nuisance parameters. Finally, we examine several asset price deviations from fundamentals and one hyper-inflation series and find contradictory results between the nonlinear fitted models and unit root tests. The findings highlight that new testing procedures with higher power are desirable.