  Given a set of baseline assumptions, a breakdown frontier is the boundary
between the set of assumptions which lead to a specific conclusion and those
which do not. In a potential outcomes model with a binary treatment, we
consider two conclusions: First, that ATE is at least a specific value (e.g.,
nonnegative) and second that the proportion of units who benefit from treatment
is at least a specific value (e.g., at least 50\%). For these conclusions, we
derive the breakdown frontier for two kinds of assumptions: one which indexes
relaxations of the baseline random assignment of treatment assumption, and one
which indexes relaxations of the baseline rank invariance assumption. These
classes of assumptions nest both the point identifying assumptions of random
assignment and rank invariance and the opposite end of no constraints on
treatment selection or the dependence structure between potential outcomes.
This frontier provides a quantitative measure of robustness of conclusions to
relaxations of the baseline point identifying assumptions. We derive
$\sqrt{N}$-consistent sample analog estimators for these frontiers. We then
provide two asymptotically valid bootstrap procedures for constructing lower
uniform confidence bands for the breakdown frontier. As a measure of
robustness, estimated breakdown frontiers and their corresponding confidence
bands can be presented alongside traditional point estimates and confidence
intervals obtained under point identifying assumptions. We illustrate this
approach in an empirical application to the effect of child soldiering on
wages. We find that sufficiently weak conclusions are robust to simultaneous
failures of rank invariance and random assignment, while some stronger
conclusions are fairly robust to failures of rank invariance but not
necessarily to relaxations of random assignment.
