On the Use of the Flexible Fourier Form in Unit Roots Tests, Endogenous Breaks, and Parameter Instability

The possibility that a series can contain an unknown number of smooth breaks raises two distinct problems. First, even if the breaks are sharp, the number of breaks and the break dates themselves are generally unknown and need to be estimated along with the other parameters of the model. Second, even if the number of breaks is known, the possibility of a smooth break means that the functional form of the break is unknown to the researcher. A misspecification of the functional form of the breaks may be as problematic as ignoring the breaks altogether. Moreover, ever if a series contains no breaks, it may be subject to other nonlinearities or parameter instabilities. We summarize a number of papers that use a variant of Gallant’s (1981) Flexible Fourier Form to control for the unknown number and form of the breaks. The paper details and illustrates several unit root tests, stationarity tests, and tests for parameter instability that are based on a Fourier approximation.