We propose constructing confidence sets for the emergence, collapse, and recovery dates of a bubble separately by inverting tests for the location of the break date. We examine both likelihood ratio-type tests and the Elliott-Muller-type (2007) tests for detecting break locations. The limiting distributions of these tests are derived under the null hypothesis, and their asymptotic consistency under the alternative is established. Finite-sample properties are evaluated through Monte Carlo simulations. The results indicate that combining different types of tests effectively controls the empirical coverage rate while maintaining a reasonably small length of the confidence set.

翻译：我们提出通过逆推断点日期位置检验的方法，分别构建泡沫形成、崩溃与恢复日期的置信集。我们考察了似然比型检验以及Elliott-Muller型（2007）检验在断点位置检测中的应用。在零假设下推导了这些检验的极限分布，并验证了其在备择假设下的渐近一致性。通过蒙特卡洛模拟评估了有限样本性质。结果表明，组合不同类型的检验能有效控制经验覆盖率，同时保持置信集长度合理较小。