Rational Bubbles: Too Many to be True?

Issues that arise in the practical implementation of the Phillips, Wu, and Yu (2011) and Phillips, Shi, and Yu (2015a) recursive procedures for identifying and dating explosive episodes in time series are considered. It is argued that the use of critical values for right-tailed unit-root tests obtained under the assumption of a drift whose magnitude depends on the sample size and becomes negligible in large samples results in over-rejection of the unit-root hypothesis when, as in many financial time series, the deterministic drift effect is non-negligible relatively to the stochastic trend. In addition, the standard practice of using conventional levels of significance for critical values involved in the algorithms that locate the origination and termination dates of explosive episodes lead to false discoveries of explosiveness with high probability. The magnitude of these difficulties is quantified via simulations, using artificial data whose properties reflect closely those of real-world time series such as stock prices and dividends. The findings offer a potential explanation for the relatively largenumber of apparent explosive episodes that are often reported in applied work. Ways of overcoming the aforementioned difficulties by using bootstrap-based calibration techniques are considered. An empirical example focusing on monthly U.S. data on real stock prices and real dividends is also discussed.