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Difference-in-Differences with Variation in Treatment Timing

TL;DR: In this article, the authors derived an expression for the general difference-in-differences estimator and showed that it is a weighted average of all possible two-group/two-period estimators in the data.
Abstract: The canonical difference-in-differences (DD) model contains two time periods, “pre” and “post”, and two groups, “treatment” and “control”. Most DD applications, however, exploit variation across groups of units that receive treatment at different times. This paper derives an expression for this general DD estimator, and shows that it is a weighted average of all possible two-group/two-period DD estimators in the data. This result provides detailed guidance about how to use regression DD in practice. I define the DD estimand and show how it averages treatment effect heterogeneity and that it is biased when effects change over time. I propose a new balance test derived from a unified definition of common trends. I show how to decompose the difference between two specifications, and I apply it to models that drop untreated units, weight, disaggregate time fixed effects, control for unit-specific time trends, or exploit a third difference.
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TL;DR: In this paper, the authors consider identification, estimation, and inference procedures for treatment effect parameters using Difference-in-Differences (DiD) with multiple time periods, variation in treatment timing, and when the "parallel trends assumption" holds potentially only after conditioning on observed covariates.
Abstract: In this article, we consider identification, estimation, and inference procedures for treatment effect parameters using Difference-in-Differences (DiD) with (i) multiple time periods, (ii) variation in treatment timing, and (iii) when the "parallel trends assumption" holds potentially only after conditioning on observed covariates. We show that a family of causal effect parameters are identified in staggered DiD setups, even if differences in observed characteristics create non-parallel outcome dynamics between groups. Our identification results allow one to use outcome regression, inverse probability weighting, or doubly-robust estimands. We also propose different aggregation schemes that can be used to highlight treatment effect heterogeneity across different dimensions as well as to summarize the overall effect of participating in the treatment. We establish the asymptotic properties of the proposed estimators and prove the validity of a computationally convenient bootstrap procedure to conduct asymptotically valid simultaneous (instead of pointwise) inference. Finally, we illustrate the relevance of our proposed tools by analyzing the effect of the minimum wage on teen employment from 2001--2007. Open-source software is available for implementing the proposed methods.

831 citations

Posted Content
TL;DR: In this article, the authors proposed an alternative estimator that is free of contamination, and illustrate the relative shortcomings of two-way fixed effects regressions with leads and lags through an empirical application.
Abstract: To estimate the dynamic effects of an absorbing treatment, researchers often use two-way fixed effects regressions that include leads and lags of the treatment. We show that in settings with variation in treatment timing across units, the coefficient on a given lead or lag can be contaminated by effects from other periods, and apparent pretrends can arise solely from treatment effects heterogeneity. We propose an alternative estimator that is free of contamination, and illustrate the relative shortcomings of two-way fixed effects regressions with leads and lags through an empirical application.

727 citations

Journal ArticleDOI
TL;DR: A simple analytical model calibrated with empirical estimates demonstrated that the “loss from anarchy” in uncoordinated state policies is increasing in the number of noncooperating states and the size of social and geographic spillovers.
Abstract: Social distancing is the core policy response to coronavirus disease 2019 (COVID-19). But, as federal, state and local governments begin opening businesses and relaxing shelter-in-place orders worldwide, we lack quantitative evidence on how policies in one region affect mobility and social distancing in other regions and the consequences of uncoordinated regional policies adopted in the presence of such spillovers. To investigate this concern, we combined daily, county-level data on shelter-in-place policies with movement data from over 27 million mobile devices, social network connections among over 220 million Facebook users, daily temperature and precipitation data from 62,000 weather stations, and county-level census data on population demographics to estimate the geographic and social network spillovers created by regional policies across the United States. Our analysis shows that the contact patterns of people in a given region are significantly influenced by the policies and behaviors of people in other, sometimes distant, regions. When just one-third of a state's social and geographic peer states adopt shelter-in-place policies, it creates a reduction in mobility equal to the state's own policy decisions. These spillovers are mediated by peer travel and distancing behaviors in those states. A simple analytical model calibrated with our empirical estimates demonstrated that the "loss from anarchy" in uncoordinated state policies is increasing in the number of noncooperating states and the size of social and geographic spillovers. These results suggest a substantial cost of uncoordinated government responses to COVID-19 when people, ideas, and media move across borders.

134 citations

Journal ArticleDOI
TL;DR: In this article, the authors study the estimation of and inference for average treatment effects in a setting with panel data, and show that under random assignment of the adoption date, the standard Difference-In-Differences (DID) estimator is an unbiased estimator of a particular weighted average causal effect.

134 citations

Journal ArticleDOI
04 Jan 2021
TL;DR: In this paper, the authors evaluated the association between receipt of unemployment insurance, including a $600/wk federal supplement between April and July, and food insecurity among people who lost their jobs during the coronavirus disease 2019 (COVID-19) pandemic.
Abstract: Importance More than 50 million US residents have lost work during the coronavirus disease 2019 (COVID-19) pandemic, and food insecurity has increased. Objective To evaluate the association between receipt of unemployment insurance, including a $600/wk federal supplement between April and July, and food insecurity among people who lost their jobs during the COVID-19 pandemic. Design, Setting, and Participants This cohort study used difference-in-differences analysis of longitudinal data from a nationally representative sample of US adults residing in low- and middle-income households (ie, Exposure Receipt of unemployment insurance benefits. Main Outcomes and Measures Food insecurity and eating less due to financial constraints, assessed every 2 weeks by self-report. Results Of 2319 adults living in households earning less than $75 000 annually and employed in February 2020, 1119 (48.3%) experienced unemployment during the COVID-19 pandemic and made up our main sample (588 [53.6%] White individuals; mean [SD] age 45 [15] years; 732 [65.4%] women). Of those who lost employment, 415 (37.1%) reported food insecurity and 437 (39.1%) reported eating less due to financial constraints in 1 or more waves of the study. Among people who lost work, receipt of unemployment insurance was associated with a 4.3 (95% CI, 1.8-6.9) percentage point decrease in food insecurity (a 35.0% relative reduction) and a 5.7 (95% CI, 3.0-8.4) percentage point decrease in eating less due to financial constraints (a 47.8% relative reduction). Decreases in food insecurity were larger with the $600/wk supplement and for individuals who were receiving larger amounts of unemployment insurance. Conclusions and Relevance In this US national cohort study, receiving unemployment insurance was associated with large reductions in food insecurity among people who lost employment during the COVID-19 pandemic. The $600/wk federal supplement and larger amounts of unemployment insurance were associated with larger reductions in food insecurity.

90 citations