Smooth Break Detection and De-Trending in Unit Root Testing

This study explores the methods to de-trend the smooth structural break processes while conducting the unit root tests. The two most commonly applied approaches for modelling smooth structural breaks namely the smooth transition and the Fourier functions are considered. We perform a sequence of power comparisons among alternative unit root tests that accommodate smooth or sharp structural breaks. The power experiments demonstrate that the unit root tests utilizing the Fourier function lead to unexpected results. Furthermore, through simulation studies, we investigate the source of such unexpected outcomes. Moreover, we provide the asymptotic distribution of two recently proposed unit root tests, namely Fourier-Augmented Dickey–Fuller (FADF) and Fourier-Kapetanios, Shin and Shell (FKSS), which are not given in the original studies. Lastly, we find that the selection of de-trending function is pivotal for unit root testing with structural breaks.