We investigate the asymptotic properties of the stopping times based on the monitoring tests for a bubble, that is, ADF‐type and CUSUM‐type tests. We derive the stochastic orders of these stopping times and find that they depend on the relative position of the emerging date of a bubble. We show that when a bubble emerges early in the monitoring period, the diverging rate of the stopping time based on the CUSUM‐type detector is slower than that based on the ADF‐type detector, which implies that the former tends to find a bubble earlier than the latter, whereas this advantage disappears for the late emerging bubble. We also derive the limiting distribution of the stopping time based on the CUSUM detector. The finite sample simulations support the theoretical result.

中文翻译：

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气泡监测试验的延迟时间的渐近行为

我们基于气泡的监视测试（即ADF型和CUSUM型测试）来研究停止时间的渐近特性。我们推导了这些停止时间的随机顺序，发现它们取决于泡沫出现日期的相对位置。我们表明，当气泡在监视期的早期出现时，基于CUSUM型检测器的停止时间的分流速率要慢于基于ADF型检测器的停止时间，这意味着前者倾向于更早地发现气泡与后者相比，这种优势在新兴泡沫的消失中消失了。我们还基于CUSUM检测器得出停止时间的极限分布。有限样本仿真支持理论结果。