Self‐Normalized KPSS Tests With Power Enhancement

ABSTRACT Much research has focused on testing the null hypothesis of stationarity against the unit root alternative. In this paper, we propose a novel class of self‐normalized KPSS tests without needing a consistent estimation of the long‐run variance. Under persistent autocorrelation, the widely used heteroskedasticity and autocorrelation consistent long‐run variance estimator proposed by Newey and West [ Econometrica (1987) Vol. 55, pp. 703–708], Newey and West [ The Review of Economic Studies (1994) Vol. 61, pp. 631–653], and Andrews [ Econometrica (1991) Vol. 59, pp. 817–858] may not be reliable and often lead to tests with size distortions and power losses in finite samples. In addition, the practitioner has to choose the truncation lag that is ultimately arbitrary. To improve the finite sample performance of the stationarity tests and make them robust to realistic amounts of dependence, we propose the use of self‐normalizing methods, for example, the range‐based self‐normalization by Hong et al. [ Journal of Econometrics (2024) Vol. 238, 105603] and the fixed‐ asymptotics by Kiefer and Vogelsang [ Econometric Theory (2005) Vol. 21, pp. 1130–1164] and Amsler, Schmidt, and Vogelsang [ Journal of Time Series Econometrics (2009) Vol. 1, pp. 1–44], to control the effect of the long‐run variance. The self‐normalized tests are inconsistent in that, under the unit root alternative, they do not diverge as the sample size approaches infinity. To recover the consistency of the tests, we devise a mechanism similar to the power enhancement mechanism proposed by Fan, Liao, and Yao [ Econometrica (2015) Vol. 83, pp. 1497–1541]. Under the null hypothesis, this mechanism is asymptotically negligible. However, under the alternative hypothesis, the mechanism diverges as the sample size increases. We show that this mechanism also enhances the power of the self‐normalized stationarity tests. A simulation study and an empirical application demonstrate the merits of the approach advocated.