Aggregating Group-Time Effects

Summary

Once the family $\{ATT(g,t)\}$ is identified, the Callaway–Sant’Anna framework aggregates it into interpretable summary parameters of the form $\theta ={\sum }_g{\sum }_{t}w(g,t)ATT(g,t)$ with researcher-chosen, non-negative weights. The main schemes are event-study/dynamic (effects by length of exposure $e=t-g$), group-specific, calendar-time, and overall ATT. Unlike TWFE coefficients, these use transparent positive weights and so cannot be negative when all underlying effects are positive; the “best” scheme is application-specific.

Overview

With many groups and periods, the raw $ATT(g,t)$‘s are hard to digest. The general aggregation is

$$\theta =\sum _{g\in \mathcal{G}}\sum _{t=2}^{\mathcal{T}}w(g,t)ATT(g,t),\text{\hspace{1em}(3.1)}$$

where $w(g,t)$ are known/estimable weights chosen to answer a specific question. The section (which assumes $\delta =0$, no anticipation, and a never-treated group available for clarity) contrasts these with the static/dynamic TWFE specs whose weights can be negative (Goodman-Bacon 2019; even leads-and-lags event-study TWFE is contaminated per Sun & Abraham 2020). Let event time $e=t-g$ = periods since adoption.

Main Content

Event-study / dynamic aggregation (by length of exposure)

Average effect $e$ periods after adoption, across cohorts observed at that exposure:

$${\theta }_{es}(e)=\sum _{g\in \mathcal{G}}1\{g+e\le \mathcal{T}\}P(G=g\mid G+e\le \mathcal{T})ATT(g,g+e).\text{\hspace{1em}(3.4)}$$

$e=0$ is the “on-impact” effect. This is the proper target for event-study plots, without the dynamic-TWFE pitfalls of ${\beta }_e$. Caveat: comparing ${\theta }_{es}(e)$ across $e$ mixes the true dynamic effect with two undesirable terms from compositional change — different cohorts enter at different $e$ (decomposition 3.5). The fix is the balanced version

$${\theta }_{es}^{bal}(e;e^{\prime })=\sum _{g\in \mathcal{G}}1\{g+e^{\prime }\le \mathcal{T}\}ATT(g,g+e)P(G=g\mid G+e^{\prime }\le \mathcal{T}),\text{\hspace{1em}(3.6)}$$

which fixes the set of cohorts observed for at least $e^{\prime }$ periods, eliminating composition effects at the cost of fewer groups (a robustness/efficiency trade-off). Resembles the common practice of only reporting event-study coefficients for non-compositionally-changing periods (Remark 8).

Group-specific aggregation

Average post-treatment effect for cohort $\tilde{g}$ across all its post periods:

$${\theta }_{sel}(\tilde{g})=\frac{1}{\mathcal{T}-\tilde{g}+1}\sum _{t=\tilde{g}}^{\mathcal{T}}ATT(\tilde{g},t).\text{\hspace{1em}(3.7)}$$

Answers: do earlier-treated groups have larger/smaller effects than later-treated ones? Building block for the recommended overall parameter.

Calendar-time aggregation

Average effect in calendar period $t$ across cohorts already treated by $t$:

$${\theta }_c(\tilde{t})=\sum _{g\in \mathcal{G}}1\{\tilde{t}\ge g\}P(G=g\mid G\le \tilde{t})ATT(g,\tilde{t}).\text{\hspace{1em}(3.8)}$$

Cumulative version ${\theta }_c^{cumu}(\tilde{t})={\sum }_{t=2}^{\tilde{t}}{\theta }_c(t)$ (3.9) gives the cumulative effect among units treated by $\tilde{t}$ — useful for business-cycle heterogeneity or e.g. cumulative COVID cases averted by shelter-in-place.

Overall treatment-effect parameters

Simple weighted average (by group size):

$${\theta }_W^O=\frac{1}{\kappa }\sum _{g\in \mathcal{G}}\sum _{t=2}^{\mathcal{T}}1\{t\ge g\}ATT(g,t)P(G=g\mid G\le \mathcal{T}),\text{\hspace{1em}(3.10)}$$

$\kappa ={\sum }_g{\sum }_{t}1\{t\ge g\}P(G=g\mid G\le \mathcal{T})$. Positive weights → cannot be negative when all effects positive (unlike TWFE $\beta$). Drawback: over-weights long-treated groups. Recommended overall ATT averages group effects first, then over groups:

$${\theta }_{sel}^O=\sum _{g\in \mathcal{G}}{\theta }_{sel}(g)P(G=g\mid G\le \mathcal{T}).\text{\hspace{1em}(3.11)}$$

${\theta }_{sel}^O$ is the average effect experienced by all units that ever participated — the natural multi-period analogue of the canonical 2x2 ATT. Overall event-study/calendar versions: ${\theta }_{es}^O=\frac{1}{\mathcal{T}-1}{\sum }_{e=0}^{\mathcal{T}-2}{\theta }_{es}(e)$ and ${\theta }_c^O=\frac{1}{\mathcal{T}-1}{\sum }_{t=2}^{\mathcal{T}}{\theta }_c(t)$ (3.12); balanced local version ${\theta }_{es}^{O,bal}(e^{\prime })=\frac{1}{e^{\prime }+1}{\sum }_{e=0}^{e^{\prime }}{\theta }_{es}^{bal}(e,e^{\prime })$ (3.13).

Weight table and equality condition

Table 1 (p. 15) gives the explicit $w(g,t)$ for ${\theta }_{es},{\theta }_{es}^{bal},{\theta }_{sel},{\theta }_c,{\theta }_c^{cumu},{\theta }_W^O,{\theta }_{sel}^O$ — all non-negative; for ${\theta }_c^{cumu}$ they sum to $\tilde{t}-1$ (it is cumulative), otherwise to one. In general none of the overall parameters equal each other except in the special case where $ATT(g,t)$ is constant across all groups and times — in which case all of them (including TWFE $\beta$) coincide.

Examples

Minimum wage: aggregations side-by-side (Table 3)

Under conditional parallel trends (DR), all clustered at the county level:

• Overall ${\theta }_{sel}^O\approx -3.1\%$ (vs. TWFE post-dummy -0.8%, statistically insignificant).
• Group-specific ${\theta }_{sel}$: $g=2004$ ≈ -4.4%, $g=2006$ ≈ -2.9%, $g=2007$ ≈ -2.9%.
• Event study ${\theta }_{es}(e)$: $e=0$ ≈ -2.4%, $e=1$ ≈ -4.1%, $e=2$ ≈ -5.0%, $e=3$ ≈ -7.1% — effects grow with exposure (a dynamic effect TWFE misses).
• Calendar-time ${\theta }_c$: t=2004 ≈ -3.0%, t=2005 ≈ -2.5%, t=2006 ≈ -3.0%, t=2007 ≈ -4.9%.
• Balanced event study (groups 2004 & 2006 only): $e=0$ ≈ -1.6%, $e=1$ ≈ -4.1%.

Connections

• Aggregates the building blocks from Group-Time Average Treatment Effects, identified via Doubly-Robust Estimands for ATT(g,t).
• Inference on these $\theta$‘s (plug-in + multiplier bootstrap) in Simultaneous Inference via Multiplier Bootstrap.
• Directly addresses the negative-weighting pitfall flagged in Difference-in-Differences with Multiple Time Periods - Overview.

See Also

• Goodman-Bacon (2019) — TWFE decomposition motivating positive-weight aggregation
• Sun & Abraham (2020) — cohort-specific (interaction-weighted) event study
• Estimands in Longitudinal Research — choosing the target estimand