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  • Peter C.B. Phillips (a1) and Shu-Ping Shi (a2)

Expansion and collapse are two key features of a financial asset bubble. Bubble expansion may be modeled using a mildly explosive process. Bubble implosion may take several different forms depending on the nature of the collapse and therefore requires some flexibility in modeling. This paper first strengthens the theoretical foundation of the real time bubble monitoring strategy proposed in Phillips, Shi and Yu (2015a,b, PSY) by developing analytics and studying the performance characteristics of the testing algorithm under alternative forms of bubble implosion which capture various return paths to market normalcy. Second, we propose a new reverse sample use of the PSY procedure for detecting crises and estimating the date of market recovery. Consistency of the dating estimators is established and the limit theory addresses new complications arising from the alternative forms of bubble implosion and the endogeneity effects present in the reverse regression. A real-time version of the strategy is provided that is suited for practical implementation. Simulations explore the finite sample performance of the strategy for dating market recovery. The use of the PSY strategy for bubble monitoring and the new procedure for crisis detection are illustrated with an application to the Nasdaq stock market.

Corresponding author
*Address correspondence to Peter C.B. Phillips, Cowles Foundation for Research in Economics, Yale University, Box 208281, New Haven, CT 06520-8281, USA; e-mail:
Shuping Shi, Department of Economics, Macquarie University, Building E4A, Eastern Road, North Ryde, NSW, 2109, Australia; e-mail:
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Phillips acknowledges research support from the NSF under Grant No. SES 12-58258 and the Kelly Foundation from the University of Auckland. Shi acknowledges research support from the Australian Research Council under project number DP150101716.

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Econometric Theory
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