The impact of persistent cycles on zero frequency unit root tests created by Tomás del Barrio Castro; Paulo M. M. Rodrigues; A. M. Robert Taylor

In this paper we investigate the impact of persistent (nonstationary or near nonstationary) cycles on the asymptotic and finite-sample properties of standard unit root tests. Results are presented for the augmented Dickey–Fuller (ADF) normalized bias and t-ratio-based tests (Dickey and Fuller, 1979, Journal of the American Statistical Association 745, 427–431; Said and Dickey, 1984; Biometrika 71, 599–607). the variance ratio unit root test of Breitung (2002, Journal of Econometrics 108, 343–363), and the M class of unit-root tests introduced by Stock (1999, in Engle and White (eds.), A Festschrift in Honour of Clive W.J. Granger) and Perron and Ng (1996, Review of Economic Studies 63, 435–463). We show that although the ADF statistics remain asymptotically pivotal (provided the test regression is properly augmented) in the presence of persistent cycles, this is not the case for the other statistics considered and show numerically that the size properties of the tests based on these statistics are too unreliable to be used in practice. We also show that the t-ratios associated with lags of the dependent variable of order greater than two in the ADF regression are asymptotically normally distributed. This is an important result as it implies that extant sequential methods (see Hall, 1994, Journal of Business & Economic Statistics 17, 461–470; Ng and Perron, 1995, Journal of the American Statistical Association 90, 268–281) used to determine the order of augmentation in the ADF regression remain valid in the presence of persistent cycles.