Explosive Asset Price Bubble Detection with Unknown Bubble Length and Initial Condition

Recent research has proposed a method of detecting explosive processes that is based on forward recursions of OLS, right‐tailed, Dickey–Fuller [DF] unit root tests. In this paper, an alternative approach using GLS DF test statistics is considered. We derive limiting distributions for both mean‐invariant and trend‐invariant versions of OLS and GLS‐based Phillips, Wu and Yu (2011, International Economic Review 52, 201–226) [PWY] test statistics under a temporary, locally explosive alternative. These limits are shown to be dependent on both the value of the initial condition and the start and end points of the temporary explosive regime. Local asymptotic power simulations show that a GLS version of the PWY statistic offers superior power when a large proportion of the data is explosive, but that the OLS approach is preferred for explosive periods of short duration as a proportion of the total sample. These power differences are magnified by the presence of an asymptotically non‐negligible initial condition. We propose a union of rejections procedure that capitalizes on the respective power advantages of both OLS and GLS‐based approaches. This procedure achieves power close to the effective envelope provided by the two individual PWY tests across all settings of the initial condition and length of the explosive period considered in this paper. These results are shown to be robust to the point in the sample at which the temporary explosive regime occurs. An application of the union procedure to NASDAQ prices confirms the empirical value of this testing strategy.