THE IMPACT OF PERSISTENT CYCLES ON ZERO FREQUENCY UNIT ROOT TESTS

Abstract

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 andt-ratio-based tests (Dickey and Fuller, 1979,Journal of the American Statistical Association745, 427–431; Said and Dickey, 1984;Biometrika71, 599–607). the variance ratio unit root test of Breitung (2002,Journal of Econometrics108, 343–363), and theMclass 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 Studies63, 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 thet-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 Statistics17, 461–470; Ng and Perron, 1995,Journal of the American Statistical Association90, 268–281) used to determine the order of augmentation in the ADF regression remain valid in the presence of persistent cycles.