In this study, we extend the three-regime bubble model of Pang et al. (2021) to allow the forth regime followed by the unit root process after recovery. We provide the asymptotic and finite sample justification of the consistency of the collapse date estimator in the two-regime AR(1) model. The consistency allows us to split the sample before and after the date of collapse and to consider the estimation of the date of exuberation and date of recovery separately. We have also found that the limiting behavior of the recovery date varies depending on the extent of explosiveness and recovering.

翻译：在本研究中,我们扩展了Pang等人(2021年)的三制度泡沫模型,以允许在恢复后采用单元根过程所遵循的制度;我们提供了两个制度AR(1)模型中崩溃日期估计器一致性的零点和有限样本理由;一致性使我们能够在崩溃日期前后对样本进行分割,并分别考虑对振荡日期和复原日期的估计;我们还发现,恢复日期的限制行为视爆炸性和恢复程度而有所不同。