Sequential Detector Statistics for Speculative Bubbles

Jörg Breitung and Max Diegel

Journal of Time Series Analysis, 2025, vol. 46, issue 5, 829-845

Abstract: We propose a heteroskedasticity‐robust locally best invariant (LBI) statistic to test the hypothesis of a unit root against the alternative of an explosive root associated with speculative bubbles. Compared to existing alternatives such as Dickey‐Fuller type tests, the LBI statistic has a standard limiting distribution and greater power, particularly in the empirically relevant scenario of a moderately explosive root. Further refinements, such as the point‐optimal linear test, approach the power envelope remarkably closely. To detect bubbles with an unknown starting date, we consider sequential (CUSUM) schemes based on constant and time‐varying boundary functions, where the exponentially weighted CUSUM detector with a constant boundary function turns out to be most powerful. We also propose a simple method for date‐stamping the start of the bubble consistently. Finally, we illustrate our methods using two empirical examples.

Date: 2025
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