Limit Theory for Dating the Origination and Collapse of Mildly Explosive Periods in Time Series Data

Some limit theory is developed for estimators suggested in Phillips, Wu and Yu (2009) for dating bubble pheonoma in time series data. The models involve mildly explosive autoregressions and the tests rely on right sided recursive unit root tests. The estimates locate the origination and collapse dates of bubbles involving mildly explosive episodes set within longer periods where the data evolve as a stochastic trend. The dating estimators are shown to be consistent under mild regularity conditions on the process. Some simulation evidence on the performance of the estimators is reported. The proposed method works well in fi?nite samples and conforms with the limit theory.