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This paper studies the impact of permanent volatility shifts in the innovation process on the performance of the test for explosive …nancial bubbles based on recursive right-tailed Dickey-Fuller-type unit root tests proposed by Phillips, Wu and Yu (2011). We show that, in this situation, their supremum-based test has a non-pivotal limit distribution under the unit root null, and can be quite severely over-sized, thereby giving rise to spurious indications of explosive behaviour. We investigate the performance of a wild bootstrap implementation of their test procedure for this problem, and show it is e¤ective in controlling size, both asymptotically and in …nite samples, yet does not sacri…ce power relative to an (infeasible) size-adjusted version of their test, even when the shocks are homoskedastic. We also discuss an empirical application involving commodity price time series and …nd considerably less emphatic evidence for the presence of explosive bubbles in these data when using our proposed wild bootstrap implementation of the Phillips, Wu and Yu (2011) test.

Conventional Dickey-Fuller unit root tests have been generalized to allow for nonlinearity under the alternative hypothesis by Enders and Granger [Journal of Business Economics and Statistics, 16 (1998) 304] (EG) and Leybourne, Newbold and Vougas [Journal of Time Series Analysis, 19 (1998) 83] (LNV). EG focus on the case of asymmetric adjustment modelled as threshold autoregression, while LNV extend the concept of trend stationarity to that of stationarity around a smooth transition between deterministic linear trends. In this study, the EG and LNV methodologies are combined to develop tests of the null hypothesis of a unit root, that under the alternative hypothesis allow for stationary asymmetric adjustment around a smooth transition between deterministic linear trends. The empirical power of the combined tests is briefly investigated and an empirical application to time series on aggregate industrial production in the UK and the US is considered.

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Recent research has proposed using recursive right-tailed unit root tests to date the start and end of asset price bubbles. In this paper an alternative approach is proposed that utilises model-based minimum sum of squared residuals estimators combined with Bayesian Information Criterion model selection. Conditional on the presence of a bubble, the dating procedures suggested are shown to o¤er consistent estimation of the start and end dates of a …xed magnitude bubble, and can also be used to distinguish between di¤erent types of bubble process, i.e. a bubble that does or does not end in collapse, or a bubble that is ongoing at the end of the sample. Monte Carlo simulations show that the proposed dating approach out-performs the recursive unit root test methods for dating periods of explosive autoregressive behaviour in …nite samples, particularly in terms of accurate identi…cation of a bubble's end point. An empirical application involving Nasdaq stock prices is discussed.