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Discussion Paper

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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 develops analytics and studies the performance characteristics of the real time bubble monitoring strategy proposed in Phillips, Shi and Yu (2014b,c, PSY) under alternative forms of bubble implosion that can be represented in terms of mildly integrated processes which capture various return paths to market normalcy. 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. Simulations explore the finite sample performance of the strategy for dating market recovery and an illustration to the Nasdaq stock market is provided. A real-time version of the strategy is provided that is suited for practical implementation.

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