Size and power of tests of stationarity in highly autocorrelated time series

Tests of stationarity are routinely applied to highly autocorrelated time series. Following Kwiatkowski et al. (J. Econom. 54 (1992) 159), standard stationarity tests employ a rescaling by an estimator of the long-run variance of the (potentially) stationary series. This paper analytically investigates the size and power properties of such tests when the series are strongly autocorrelated in a local-to-unity asymptotic framework. It is shown that the behavior of the tests strongly depends on the long-run variance estimator employed, but is in general highly undesirable. Either the tests fail to control size even for strongly mean reverting series, or they are inconsistent against an integrated process and discriminate only poorly between stationary and integrated processes compared to optimal statistics.