Inference on Breakdown Frontiers

Abstract

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 the robustness of conclusions to relaxations of the baseline point identifying assumptions. We derive 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.

Moment inequalities in the context of simulated and predicted variables

“…Our method "regularizes" or smoothes the objective function, and we show that this method leads to a straightforward bias correction method. While the idea of regularization appears in some other contexts, such as Haile & Tamer (2003), Chernozhukov, Kocatulum & Menzel (2015) and Masten & Poirier (2017), we formally show that this approach has uniform validity in the context of inference with simulated variables. Our regularization method is based on the class of µ-smooth approximations studied in the non-smooth optimization literature (Nesterov, 2005;Beck & Teboulle, 2012).…”

“…Using these results, we provide functional forms of smooth approximations to some of the commonly used test statistics. The idea of regularizing test statistics (or estimated bounds) also appears in related contexts (Haile & Tamer, 2003;Chernozhukov et al, 2015;Kaido, 2017;Masten & Poirier, 2017). Our contribution here is to show its uniform validity in the context of inference with simulated variables.…”

A Projection Framework for Testing Shape Restrictions That Form Convex Cones

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Covariate Distribution Balance via Propensity Scores

“…Given the lack of evidence of PS misspecification, an alternative reason for these conflicting results is that the unconfoundedness assumption does not hold in this particular application. Although such an assumption is not directly testable, the sensitivity analysis in Masten and Poirier (2017) suggests that this may be the case.…”