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This article uses wavelet theory to propose a frequency domain nonparametric and tuning parameter-free family of unit root tests. The proposed test exploits the wavelet power spectrum of the observed series and its fractional partial sum to construct a test of the unit root based on the ratio of the resulting scaling energies. The proposed statistic enjoys good power properties and is robust to severe size distortions even in the presence of serially correlated MA(1) errors with a highly negative moving average (MA) parameter, as well as in the presence of random additive outliers. Any remaining size distortions are e ectively eliminated using a novel wavestrapping algorithm.

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Regulated (bounded) integrated time series are of significant practical importance and a recent development in the time series literature. Although regulated integrated series are characterized by asymptotic distributions that differ substantially from their unregulated counterparts, most inferential exercises continue to be performed with complete disregard for this potential feature of time series data. To date, only Cavaliere (2005) and Cavaliere and Xu (2011) have attempted to develop a theory for regulated integrated time series, particularly in the context of unit root testing. Unfortunately, no such theory has been developed for regulated fractionally integrated series, which are particularly important in financial time series and also in some unit root testing literature. This article achieves just this: it establishes a framework for regulated fractionally integrated processes and develops their functional central limit distributions. In addition, this article presents some simulation evidence and discusses several algorithms for obtaining the limiting distributions for these processes.

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