Original Article
TESTING FOR MULTIPLE BUBBLES: LIMIT THEORY OF REAL‐TIME DETECTORS
The current article and its empirical companion “Testing for Multiple Bubbles: Historical Episodes of Exuberance and Collapse in the S&P 500” build on work originally circulated in the paper “Testing for Multiple Bubbles” and its technical supplement. We are grateful to the editor and referee for valuable comments on the original version and thank many colleagues for helpful discussions. Phillips acknowledges support from the NSF under Grant Nos. SES 09‐56687 and SES 12‐58258. Shi acknowledges the Financial Integrity Research Network (FIRN) for funding support. Yu acknowledges the financial support from Singapore Ministry of Education Academic Research Fund Tier 2 under the grant number MOE2011‐T2‐2‐096. Please address correspondence to: Peter C.B. Phillips, Cowles Foundation for Research in Economics, Yale University, Box 208281, New Haven, CT 06520‐8281, U.S.A. E‐mail: peter.phillips@yale.edu.
Abstract
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.
Citing Literature
Number of times cited according to CrossRef: 117
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