# Research

This paper proposes robust inference methods for difference-in-differences and event-study designs that do not require that the parallel trends assumption holds exactly. Instead, the researcher must only impose restrictions on the possible differences in trends between the treated and control groups. Several common intuitions expressed in applied work can be captured by such restrictions, including the notion that pre-treatment differences in trends are informative about counterfactual post-treatment differences in trends. Our methodology then guarantees uniformly valid ("honest'') inference when the imposed restrictions are satisfied. We first show that fixed length confidence intervals have near-optimal expected length for a practically-relevant class of restrictions. We next introduce a novel inference procedure that accommodates a wider range of restrictions, which is based on the observation that inference in our setting is equivalent to testing a system of moment inequalities with a large number of linear nuisance parameters. The resulting confidence sets are consistent, and have optimal local asymptotic power for many parameter configurations. We recommend researchers conduct sensitivity analyses to show what conclusions can be drawn under various restrictions on the possible differences in trends.

*as if*treatment were randomly assigned, but in reality treatment probabilities may depend in unknown ways on the potential outcomes. We first study the properties of the simple difference-in-means (SDIM) estimator. The SDIM is unbiased for a finite-population design-based analog to the average treatment effect on the treated (ATT) if treatment probabilities are uncorrelated with the potential outcomes in a finite population sense. We further derive expressions for the variance of the SDIM estimator and a central limit theorem under sequences of finite populations with growing sample size. We then show how our results can be applied to analyze the distribution and estimand of difference-in-differences (DiD) and two-stage least squares (2SLS) from a design-based perspective when treatment is not completely randomly assigned.

*bias reversal*. We then clarify the conditions that give rise to bias reversal. Whether a prediction algorithm reverses or inherits bias depends critically on how the decision-maker affects the training data as well as the label used in training. We illustrate our main theoretical results in a simulation study applied to the New York City Stop, Question and Frisk dataset.