Abstract This paper proposes tools for robust inference in difference-in-differences and event-study designs where the parallel trends assumption may be violated. Instead of requiring that parallel trends holds exactly, we impose restrictions on how different the post-treatment violations of parallel trends can be from the pre-treatment differences in trends (“pre-trends”). The causal parameter of interest is partially identified under these restrictions. We introduce two approaches that guarantee uniformly valid inference under the imposed restrictions, and we derive novel results showing that they have desirable power properties in our context. We illustrate how economic knowledge can inform the restrictions on the possible violations of parallel trends in two economic applications. We also highlight how our approach can be used to conduct sensitivity analyses showing what causal conclusions can be drawn under various restrictions on the possible violations of the parallel trends assumption.

Abstract We show that moment inequalities in a wide variety of economic applications have a particular linear conditional structure. We use this structure to construct uniformly valid confidence sets that remain computationally tractable even in settings with nuisance parameters. We first introduce least-favorable critical values which deliver non-conservative tests if all moments are binding. Next, we introduce a novel conditional inference approach which ensures a strong form of insensitivity to slack moments. Our recommended approach is a hybrid technique which combines desirable aspects of the least favorable and conditional methods. The hybrid approach performs well in simulations calibrated to Wollmann (2018, American Economic Review, 108, 1364–1406), with favorable power and computational time comparisons relative to existing alternatives.

This paper examines the evidence from randomized evaluations of sector-focused training programs that target low-wage workers and combine up-front screening, occupational and soft-skills training, and wraparound services. The programs generate substantial and persistent earnings gains (12%–34%) following training. Theoretical mechanisms for program impacts are explored for the WorkAdvance demonstration. Earnings gains are generated by getting participants into higher-wage jobs in higher-earning industries and occupations, not just by raising employment. Training in transferable and certifiable skills (likely underprovided from poaching concerns) and reductions of employment barriers to high-wage sectors for nontraditional workers appear to play key roles.

This paper assesses when the validity of difference‐in‐differences depends on functional form. We provide a novel characterization: the parallel trends assumption holds under all strictly monotonic transformations of the outcome if and only if a stronger “parallel trends”‐type condition holds for the cumulative distribution function of untreated potential outcomes. This condition for parallel trends to be insensitive to functional form is satisfied if and essentially only if the population can be partitioned into a subgroup for which treatment is effectively randomly assigned and a remaining subgroup for which the distribution of untreated potential outcomes is stable over time. These conditions have testable implications, and we introduce falsification tests for the null that parallel trends is insensitive to functional form.

Abstract Economists are often interested in the mechanisms by which a treatment affects an outcome. We develop tests for the “sharp null of full mediation” that a treatment D affects an outcome Y only through a particular mechanism (or set of mechanisms) M. Our approach exploits connections between mediation analysis and the econometric literature on testing instrument validity. We also provide tools for quantifying the magnitude of alternative mechanisms when the sharp null is rejected: we derive sharp lower bounds on the fraction of individuals whose outcome is affected by the treatment despite having the same value of M under both treatments (“always-takers”), as well as sharp bounds on the average effect of the treatment for such always-takers. An advantage of our approach relative to existing tools for mediation analysis is that it does not require stringent assumptions about how M is assigned. We illustrate our methodology in two empirical applications.

Abstract When studying an outcome Y that is weakly positive but can equal zero (e.g., earnings), researchers frequently estimate an average treatment effect (ATE) for a “log-like” transformation that behaves like log (Y) for large Y but is defined at zero (e.g., log (1 + Y), arcsinh(Y)). We argue that ATEs for log-like transformations should not be interpreted as approximating percentage effects, since unlike a percentage, they depend on the units of the outcome. In fact, we show that if the treatment affects the extensive margin, one can obtain a treatment effect of any magnitude simply by rescaling the units of Y before taking the log-like transformation. This arbitrary unit dependence arises because an individual-level percentage effect is not well-defined for individuals whose outcome changes from zero to nonzero when receiving treatment, and the units of the outcome implicitly determine how much weight the ATE for a log-like transformation places on the extensive margin. We further establish a trilemma: when the outcome can equal zero, there is no treatment effect parameter that is an average of individual-level treatment effects, unit invariant, and point identified. We discuss several alternative approaches that may be sensible in settings with an intensive and extensive margin, including (i) expressing the ATE in levels as a percentage (e.g., using Poisson regression), (ii) explicitly calibrating the value placed on the intensive and extensive margins, and (iii) estimating separate effects for the two margins (e.g., using Lee bounds). We illustrate these approaches in three empirical applications.