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This paper develops a new exact small\u2010sample, pointwise most powerful unit root test that is invariant to the unknown mean and scale of the time series tested, that generates exact small\u2010sample critical values, powers and <jats:italic>p<\/jats:italic>\u2010values, that has power which approximates the maximum possible power, and that is highly robust to conditional heteroscedasticity. This test decisively rejects the unit root null hypothesis when applied to annual US real GNP and US real per capita GNP series. This paper also develops a modified version of the test to address whether a time series contains a permanent, unit root process in addition to a temporary, stationary process. It shows that <jats:italic>if<\/jats:italic> these GNP series contain a unit root process in addition to the stationary process, then it is most likely very small. 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