A number of recently published papers have focused on the problem of testing for a unit root inthe case where the driving shocks may be unconditionally heteroskedastic. These papers have,however, assumed that the lag length in the unit root test regression is a deterministic functionof the sample size, rather than data-determined, the latter being standard empirical practice. Inthis paper we investigate the finite sample impact of unconditional heteroskedasticity onconventional data-dependent methods of lag selection in augmented Dickey-Fuller type unit roottest regressions and propose new lag selection criteria which allow for the presence ofheteroskedasticity in the shocks. We show that standard lag selection methods show a tendency toover-fit the lag order under heteroskedasticity, which results in significant power losses in the(wild bootstrap implementation of the) augmented Dickey-Fuller tests under the alternative. Thenew lag selection criteria we propose are shown to avoid this problem yet deliver unit root testswith almost identical finite sample size and power properties as the corresponding tests based onconventional lag selection methods when the shocks are homoskedastic.