Asymptotic properties of bubble monitoring tests

We investigate bubble monitoring tests by extending the existing sup-ADF and generalized sup-ADF tests to the monitoring scheme. We also consider applying the CUSUM detector as proposed in the literature. We derive the limiting distributions of the detecting statistics under the null hypothesis, the moderate deviation alternative, and the local alternative. We find that although the moderate deviation alternative shows the difference in the divergence rates of the ADF- and CUSUM-type detectors, the local asymptotic theory is more useful for understanding their differences in detail. We also conduct finite sample simulations and confirm that the local asymptotic theory approximates the finite sample properties of the tests very well.