A Flexible Methodological Approach for Deriving Asymptotic Distributions in Nonlinear Unit Root Tests

Abstract This paper examines the challenges associated with deriving asymptotic distributions for nonlinear unit root tests. Although the prevalence of non-linear models has increased in recent years, such complex functions make deriving analytical solutions for ergodicity conditions and asymptotic distributions more challenging. The common practice of approximating nonlinear unit root tests with linear functions results in a significant loss of information. This study proposes a novel approach that utilizes the augmented Fourier transformation of the Arctan function to overcome these limitations. The fast convergence properties of the Arctan function within the Fourier framework allow for the derivation of asymptotic distributions for nonlinear unit root tests. The effectiveness of this method is demonstrated by obtaining previously elusive asymptotic distributions for the (existing nonlinear unit root tests) Leybourne et al., in Journal of Time Series Analysis , 19 (1), 83–97 (1998) test and achieving improved approximations for the Kapetanios et al., in Journal of Econometrics , 112 (2), 359–379 (2003) test. Furthermore, we develop a new unrestricted ESTAR unit root test and demonstrate how previously unattainable asymptotic distributions can be readily derived for this novel test. An empirical application to real exchange rates, incorporating this new test alongside the existing KSS and Kılıç inft tests, reveals that our unrestricted version captures the data generating process more effectively than its restricted counterparts and demonstrates the superior performance of high-power tests that would otherwise be analytically intractable. Therefore, this approach offers a more accurate and robust way to understand the behavior of non-linear unit root tests.