Testing For Unit Roots In Autoregressions With Multiple Level Shifts

The asymptotic distributions of augmented Dickey–Fuller (ADF) unit root tests for autoregressive processes with a unit or near-unit root are discussed in the presence of multiple stochastic level shifts of large size occurring independently in time. The distributions depend on a Brownian motion and a Poisson-type jump process. Due to the latter, tests based on standard critical values experience power losses increasing rapidly with the number and the magnitude of the shifts. A new approach to unit root testing is suggested which requires no knowledge of either the location or the number of level shifts, and which dispenses with the assumption of independent shift occurrence. It is proposed to remove possible shifts from a time series by weighting its increments according to how likely it is, with respect to an ad hoc postulated distribution, a shift to have occurred in each period. If the number of level shifts is bounded in probability, the limiting distributions of the proposed test statistics coincide with those of ADF statistics under standard conditions. A Monte Carlo experiment shows that, despite their generality, the new tests perform well in finite samples.We are grateful to Pentti Saikkonen (the co-editor) and two referees for their helpful and constructive comments on earlier versions of this paper. We also thank Søren Johansen, Helmut Lütkepohl, Anders Rahbek, Robert Taylor, and seminar participants at the European University Institute, the University of Copenhagen, and the URCT Conference held in Faro, Portugal, September 29 to October 1, 2005, for useful comments. Both authors are grateful to the Danish Social Sciences Research Council, project 2114-04-001, for financial support.