Document Type

Discussion Paper

Publication Date


CFDP Number


CFDP Pages


Journal of Economic Literature (JEL) Code(s)

C15, C22


In the presence of bubbles, asset prices consist of a fundamental and a bubble component, with the bubble component following an explosive dynamic. The general idea for bubble identification is to apply explosive root tests to a proxy of the unobservable bubble. Three notable proxies are the real asset prices, log price-payoff ratios, and estimated non-fundamental components. The rationale for all three proxy choices rests on the definition of bubbles, which has been presented in various forms in the literature. This chapter provides a theoretical framework that incorporates several definitions of bubbles (and fundamentals) and offers guidance for selecting proxies. For explosive root tests, we introduce the recursive evolving test of Phillips et al. (2015b,c) along with its asymptotic properties. This procedure can serve as a real-time monitoring device and has been shown to outperform several other tests. Like all other recursive testing procedures, the PSY algorithm faces the issue of multiplicity in testing that contaminates conventional significance values. To address this issue, we propose
a multiple-testing algorithm to determine appropriate test critical values and show its satisfactory performance in finite samples by simulations. To illustrate, we conduct a pseudo real-time bubble monitoring exercise in the S&P 500 stock market from January 1990 to June 2020. The empirical results reveal the importance of using a good proxy for bubbles and addressing the multiplicity issue.

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