  We consider treatment-effect estimation under a parallel trends assumption,
in designs where no unit is treated at period one, all units receive a strictly
positive dose at period two, and the dose varies across units. There are
therefore no true control groups in such cases. First, we develop a test of the
assumption that the treatment effect is mean independent of the treatment,
under which the commonly-used two-way-fixed-effects estimator is consistent.
When this test is rejected or lacks power, we propose alternative estimators,
robust to heterogeneous effects. If there are units with a period-two treatment
arbitrarily close to zero, the robust estimator is a difference-in-difference
using units with a period-two treatment below a bandwidth as controls. Without
such units, we propose non-parametric bounds, and an estimator relying on a
parametric specification of treatment-effect heterogeneity. We use our results
to revisit Pierce and Schott (2016) and Enikolopov et al. (2011).
