The Credibility Revolution

Difference-in-Differences with Variation in Treatment Timing

Properties1
tagsliterature

2 min read

Difference-in-Differences with Variation in Treatment Timing

Causal Question / Estimand

What does the two-way fixed-effects (TWFE) DiD estimator actually estimate when units are treated at different times? The target is the average treatment effect on the treated (Causal-Estimand); the paper asks how the TWFE coefficient relates to it.

Identification Strategy

The Goodman-Bacon decomposition: the TWFE DiD coefficient equals a weighted average of all possible 2×2 (two-group, two-period) DiD comparisons in the data, with weights proportional to group sizes and treatment-variance. Crucially, some of these comparisons use already-treated units as controls for later-treated units. A causal interpretation requires both Parallel-Trends and treatment effects that are constant over time.

Key Assumptions

Parallel-Trends across all timing groups and time-invariant treatment effects. When effects are dynamic/heterogeneous (Treatment-Effect-Heterogeneity), the already-treated-as-control comparisons subtract the change in earlier cohorts’ effects, producing Negative-Weighting and bias. SUTVA throughout.

Threats to Validity

The “forbidden comparisons”: when treatment effects grow over time, using already-treated units as controls puts negative weights on some 2×2 terms, so the TWFE estimate can even have the wrong sign relative to every underlying effect. A diagnostic, not an estimator — it shows where TWFE bias comes from.

Setting / Data

n/a — econometric methodology. Illustrated with a re-analysis of the effect of no-fault divorce laws on female suicide (Wolfers/Stevenson-style data).

Key Claims

  • TWFE DiD = variance-weighted average of 2×2 DiDs; the weights are transparent and computable (the “Bacon decomposition”).
  • Already-treated units serving as controls is the source of bias under dynamic effects — the canonical diagnosis of the staggered-DiD problem.
  • Provides decomposition tools to see which comparisons drive an estimate and how much negative weighting is present.

Connections

Citation

Goodman-Bacon, A. (2021). Difference-in-Differences with Variation in Treatment Timing. Journal of Econometrics, 225(2), 254–277. https://doi.org/10.1016/j.jeconom.2021.03.014

Graph View

Backlinks