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It is common practice in time series econometrics to test the null hypothesis that the generating function is integratedÐi.e. that a series is stationary only after differencingÐagainst the alternative of stationarity about either a ®xed mean or a linear trend. However, there has been considerable recent interest in the possibility of stationarity around a linear trend with an abrupt break. Here we broaden this class of alternatives to allow for a smooth transition from one trend function to another. Dickey±Fuller type tests against this alternative are developed, and their properties are explored.

SUMMARYUnit root tests, seeking mean or trend reversion, are frequently applied to panel data. We show that more powerful variants of commonly applied tests are readily available. Moreover, power gains persist when the modifications are applied to bootstrap procedures that may be employed when cross-correlation of a rather general sort among individual panel members is suspected.

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.