Asymptotic Behavior of Delay Times of Bubble Monitoring Tests.

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. [ABSTRACT FROM AUTHOR]