Group-Time Average Treatment Effects

Summary

The group-time average treatment effect $ATT(g,t)=\mathbb{E}[Y_t(g)-Y_t(0)\mid G_g=1]$ is the disaggregated building block of the Callaway–Sant’Anna framework: the average effect at calendar time $t$ for the cohort first treated in period $g$. It imposes no restriction on treatment-effect heterogeneity across groups or over time, so it (unlike a single TWFE coefficient) always carries a well-defined causal interpretation and can be flexibly aggregated to answer many policy questions.

Overview

In the canonical two-period setup the target is $ATT=\mathbb{E}[Y_2(2)-Y_2(0)\mid G_2=1]$. With multiple periods and staggered adoption there is no single “treated” timing, so the paper indexes effects by both the cohort $g$ (when treatment starts) and the calendar period $t$ (when we measure). Fixing $g$ and varying $t$ traces effect dynamics for that cohort; varying $g$ compares cohorts. Because $ATT(g,t)$ is defined directly on potential outcomes — not as a regression coefficient — it sidesteps the TWFE negative-weighting problem entirely (see Difference-in-Differences with Multiple Time Periods - Overview).

Main Content

Setup and notation

• $\mathcal{T}$ periods, $t=1,\dots ,\mathcal{T}$. $D_{i,t}\in \{0,1\}$ = treatment status of unit $i$ at $t$.
• Group $G$ = the first period a unit is treated; $G=\infty$ if never treated. $G_{i,g}=1\{G_i=g\}$, and $C_i=1\{G_i=\infty \}=1-D_{i,\mathcal{T}}$ flags the never-treated. $\bar{g}={\max }_i{G}_i$ is the last-treated group.
• $\mathcal{G}=\text{supp}(G)\setminus \{\bar{g}\}\subseteq \{2,\dots ,\mathcal{T}\}$ is the set of identifiable groups (drops $\bar{g}$ when no never-treated group exists, since it has no valid comparison). $\mathcal{X}=\text{supp}(X)\subseteq {\mathbb{R}}^k$ is the support of pre-treatment covariates.
• Generalized propensity score $p_g(X)=P(G_g=1\mid X,G_g+C=1)$: probability of first treatment at $g$, conditional on $X$ and on being either in group $g$ or never-treated. (More generally $p_{g,s}(X)$ conditions on being in $g$ or “not-yet-treated by $s$”.)

Potential outcomes (multi-stage adoption)

Combining Robins’ dynamic potential outcomes with Heckman et al.’s multi-stage adoption: $Y_{i,t}(0)$ = untreated potential outcome (never participates through $\mathcal{T}$); $Y_{i,t}(g)$ = potential outcome if first treated in period $g$. Observed and potential outcomes link via

$$Y_{i,t}=Y_{i,t}(0)+\sum _{g=2}^{\mathcal{T}}(Y_{i,t}(g)-Y_{i,t}(0))\cdot G_{i,g}.\text{\hspace{1em}(2.1)}$$

We observe exactly one potential-outcome path per unit. Assumption 1 (Irreversibility): $D_1=0$ a.s., and $D_{t-1}=1\Rightarrow D_t=1$ a.s. (staggered adoption — units never “forget” treatment). Assumption 2 (Random sampling): the panel $\{Y_{i,1},\dots ,Y_{i,\mathcal{T}},X_i,D_{i,1},\dots ,D_{i,\mathcal{T}}{\}}_{i=1}^n$ is iid (results extend to repeated cross-sections, Appendix B). The unit index $i$ is suppressed henceforth.

The group-time average treatment effect

The main building block is

$$ATT(g,t)=\mathbb{E}[Y_t(g)-Y_t(0)\mid G_g=1],$$

the average treatment effect at period $t$ among units first treated in period $g$. It places no restriction on (a) heterogeneity across groups, (b) the timing $g$, or (c) how effects evolve over $t$. The family $\{ATT(g,t)\}$ can therefore be read directly for heterogeneity or aggregated to answer: (a) overall effect of participating by $\mathcal{T}$; (b) across-group heterogeneity; (c) effects by length of exposure $e=t-g$ (event study); (d) how effects evolve over calendar time. In the 2x2 case $ATT(g,t)$ collapses to the canonical ATT.

Why this avoids TWFE bias. A single $\beta$ in a TWFE regression must average all cohort-period effects with estimator-determined (possibly negative) weights. By contrast, each $ATT(g,t)$ is its own estimand; aggregation weights $w(g,t)$ are then chosen transparently by the researcher to match the question (see Aggregating Group-Time Effects), and the simple combinations the paper proposes use non-negative weights — ruling out the “positive-for-all-but-negative-estimate” pathology.

Examples

Minimum wage: reading $ATT(g,t)$ directly

With $\mathcal{T}$ running 2001-2007 and groups $g\in \{2004,2006,2007\}$, the cohort first raising its minimum wage in 2004 (Illinois) has $ATT(2004,2004)\approx -3.4\%$, $ATT(2004,2005)\approx -7.1\%$, $ATT(2004,2006)\approx -12.5\%$, $ATT(2004,2007)\approx -13.6\%$ — i.e. teen employment falls more the longer the higher wage is in place. Each number is interpretable on its own without any homogeneity assumption.

Connections

• Identified under conditions in Identifying Assumptions for Staggered DiD.
• Estimated via the formulas in Doubly-Robust Estimands for ATT(g,t).
• Combined into summaries in Aggregating Group-Time Effects.
• Inference on the whole family in Simultaneous Inference via Multiplier Bootstrap.

See Also

• Estimands in Longitudinal Research — potential-outcome estimands across periods
• Difference in differences — the canonical 2x2 ATT this generalizes
• Mostly Harmless Econometrics — DiD chapter