Testing for unit roots in autoregressions with multiple level shifts

The asymptotic distributions of Augmented-Dickey-Fuller (ADF) unit root test statis-tics for autoregressive processes with a unit or near-unit root are derived in the presence of multiple stochastic level shifts of large size. The distributions depend on a Brown-ian motion and a Poisson-type jump process. The latter is related to the occurrence of level shifts, and induces for tests based on standard critical values power losses increasing rapidly with the number and the magnitude of the shifts. A new approach to unit root testing is suggested which does not require to know either the location or the number of level shifts. It is proposed to remove possible shifts from a time series by weighting each of its increments according to how likely it is, with respect to an ad hoc postulated distribution, a shift to have occurred in the respective period. Hence, no explicit decision rule on the occurrence of shifts is required. It is proved that if the number of level shifts is bounded in probability, the proposed test statistics have pivotal limiting distributions coinciding with those of the corresponding ADF statistics under standard conditions. A Monte Carlo experiment shows that, despite their generality, the new tests perform well in small samples