Size Distortion of Bootstrap Tests: an Example from Unit Root Testing

Testing for a unit root in a series obtained by summing a stationary MA(1) process with a parameter close to -1 leads to serious size distortions under the null, on account of the near cancellation of the unit root by the MA component in the driving stationary series. The situation is analysed from the point of view of bootstrap testing, and an exact quantitative account is given of the error in rejection probability of a bootstrap test. A particular method of estimating the MA parameter is recommended, as it leads to very little distortion even when the MA parameter is close to -1. A new bootstrap procedure with still better properties is proposed. While more computationally demanding than the usual bootstrap, it is much less so than the double bootstrap.