Testing for multiple bubbles: Limit theory of real-time detectors

This article provides the limit theory of real-time dating algorithms for bubble detection that were suggested in Phillips, Wu, and Yu (PWY; International Economic Review 52 [2011], 201-26) and in a companion paper by the present authors (Phillips, Shi, and Yu, 2015; PSY; International Economic Review 56 [2015a], 1099-1134. Bubbles are modeled using mildly explosive bubble episodes that are embedded within longer periods where the data evolve as a stochastic trend, thereby capturing normal market behavior as well as exuberance and collapse. Both the PWY and PSY estimates rely on recursive right-tailed unit root tests (each with a different recursive algorithm) that may be used in real time to locate the origination and collapse dates of bubbles. Under certain explicit conditions, the moving window detector of PSY is shown to be a consistent dating algorithm even in the presence of multiple bubbles. The other algorithms are consistent detectors for bubbles early in the sample and, under stronger conditions, for subsequent bubbles in some cases. These asymptotic results and accompanying simulations guide the practical implementation of the procedures. They indicate that the PSY moving window detector is more reliable than the PWY strategy, sequential application of the PWY procedure, and the CUSUM procedure.