Difference-in-Differences with a Continuous Treatment
Causal Question / Estimand
Treatment-effect parameters when the treatment is a continuous dose rather than binary: level effects ATT() — the effect of dose on those who got dose — and causal response / dose-response parameters ACRT() capturing how effects change with the dose (Causal-Estimand).
Identification Strategy
Generalizes the binary-treatment DiD logic to continuous doses. Shows that treatment-on-the-treated parameters are identified under a Parallel-Trends assumption analogous to the binary case, but comparing parameters across dose levels reintroduces selection bias unless stronger assumptions hold (e.g. “strong” parallel trends across dose groups). Proposes estimators that target the dose-response relationship while making these assumptions explicit.
Key Assumptions
A dose-specific Parallel-Trends (and a stronger version to compare across doses and rule out Treatment-Effect-Heterogeneity-driven selection bias), No-Anticipation, and SUTVA (no interference across dose levels).
Threats to Validity
Standard parallel trends does not rule out selection bias when comparing dose levels — a subtle pitfall. Popular TWFE estimands with continuous treatment admit multiple interpretations and are poor summaries of the dose-response curve under heterogeneity (Negative-Weighting-type problems).
Setting / Data
n/a — econometric methodology (R package contdid). Empirical application using
variation in treatment intensity (dose) across units and time.
Key Claims
- Continuous-treatment DiD needs a stronger parallel-trends assumption than binary DiD to make cross-dose comparisons causal.
- TWFE estimands with continuous treatment are hard to interpret as dose-response summaries.
- Provides estimators for level and causal-response parameters across the dose distribution.
Connections
- Extends: CallawaySantAnna2021-DiDMultiplePeriods
- Builds on the timing diagnosis of: GoodmanBacon2021-DiDVariationInTiming
- See also: DeChaisemartinDHaultfoeuille2020-IntertemporalTE (non-binary, non-absorbing treatments), DiD
Citation
Callaway, B., Goodman-Bacon, A., & Sant’Anna, P. H. C. (2024). Difference-in- Differences with a Continuous Treatment. Working paper, arXiv:2107.02637.