The paper provides a general framework for investigating the effects of permanent changes in the variance of the errors of an autoregressive process on unit root tests. Such a framework - which is based on a novel asymptotic theory for integrated and near integrated processes with heteroskedastic errors - allows to evaluate how the variance dynamics affect the size and the power function of unit root tests. Contrary to previous studies, it is shown that non-constant variances can both inflate and deflate the rejection frequency of the commonly used unit root tests, both under the null and under the alternative, with early negative and late positive variance changes having the strongest impact on size and power. It is also shown that shifts smoothed across the sample have smaller impacts than shifts occurring as a single abrupt jump, while periodic variances have a negligible effect even when a small number of cycles take place over a given sample. Finally, it is proved that the locally best invariant (LBI) test of a unit root against level stationarity is robust to heteroskedasticity of any form under the null hypothesis.