Identifying Assumptions for Staggered DiD

Summary

Point-identification of $ATT(g,t)$ in the Callaway–Sant’Anna framework rests on four assumptions beyond random sampling: limited treatment anticipation (Assumption 3, with horizon $\delta \ge 0$), one of two conditional parallel trends assumptions — based on a never-treated group (Assumption 4) or on not-yet-treated groups (Assumption 5) — and an overlap/common-support condition (Assumption 6). All allow covariate-specific trends, making them strictly weaker than randomization-based or unconditional parallel-trends assumptions used elsewhere.

Overview

These conditions extend the canonical parallel-trends assumption to multiple groups and periods, and crucially allow it to hold only after conditioning on covariates $X$ — important when groups differ in observables that drive untreated outcome dynamics (e.g. job-training programs where age, education, employment history differ across participants; Heckman et al. 1997). They also accommodate (bounded) anticipation of treatment. The choice between Assumptions 4 and 5 governs which comparison group is valid; the anticipation horizon $\delta$ governs the reference period.

Main Content

Assumption 3 — Limited Treatment Anticipation

There is a known $\delta \ge 0$ such that

$$\mathbb{E}[Y_t(g)\mid X,G_g=1]=\mathbb{E}[Y_t(0)\mid X,G_g=1]\text{a.s. for all }g\in \mathcal{G},t<g-\delta .$$

When $\delta =0$ this is the standard no-anticipation condition (units do not respond before treatment starts). $\delta >0$ permits anticipation up to $\delta$ periods (e.g. $\delta =1$ if units react one period early). Under Assumption 3, $ATT(g,t)=0$ for all pre-treatment periods $t<g-\delta$. The parallel-trends assumptions become stronger as $\delta$ grows (Remark 1) — a previously-unnoticed trade-off.

Assumption 4 — Conditional Parallel Trends (Never-Treated comparison)

Let $\delta$ be as in Assumption 3. For each $g\in \mathcal{G}$ and $t\in \{2,\dots ,\mathcal{T}\}$ with $t\ge g-\delta$,

$$\mathbb{E}[Y_t(0)-Y_{t-1}(0)\mid X,G_g=1]=\mathbb{E}[Y_t(0)-Y_{t-1}(0)\mid X,C=1]\text{a.s.}$$

Conditional on covariates, group-$g$ and the never-treated group ($C=1$) would have followed parallel untreated paths. Favored when a sizable never-treated group exists that is similar to the eventually-treated. Under $\delta =0$ it places no restriction on observed pre-treatment trends.

Assumption 5 — Conditional Parallel Trends (Not-Yet-Treated comparison)

Let $\delta$ be as in Assumption 3. For each $g\in \mathcal{G}$ and $(s,t)\in \{2,\dots ,\mathcal{T}{\}}^2$ with $t\ge g-\delta$ and $t+\delta \le s<\bar{g}$,

$$\mathbb{E}[Y_t(0)-Y_{t-1}(0)\mid X,G_g=1]=\mathbb{E}[Y_t(0)-Y_{t-1}(0)\mid X,D_s=0,G_g=0]\text{a.s.}$$

Uses groups not-yet-treated by time $t+\delta$ as comparison. Favored when no/too-small never-treated group exists, as it exploits more comparison units (more informative inference). Drawback: unlike Assumption 4 it does restrict pre-treatment trends, which can fail when early periods experienced different shocks than later ones (Marcus & Sant’Anna 2020). Practitioners uncomfortable using never-treated units (who may behave differently) can drop them and proceed under Assumption 5 (Remark 2).

Assumption 6 — Overlap (common support)

For each $t\in \{2,\dots ,\mathcal{T}\}$, $g\in \mathcal{G}$, there exists $\epsilon >0$ with

$$P(G_g=1)>\epsilon \text{and}{p}_{g,t}(X)<1-\epsilon \text{a.s.}$$

A positive fraction starts treatment in $g$, and the generalized propensity score is uniformly bounded away from one. Rules out “irregular identification” (Khan & Tamer 2010). Extends the overlap conditions of Heckman et al., Abadie (2005), and Sant’Anna & Zhao (2020).

Conditional vs. unconditional. The unconditional versions of Assumptions 4–5 are still weaker than the parallel-trends conditions in de Chaisemartin & D’Haultfœuille (2020) and Sun & Abraham (2020), and weaker than the randomization-of-adoption-date assumption in Athey & Imbens (2018). Allowing conditioning on $X$ permits covariate-specific trends; ignoring them when present biases unconditional DiD. Only pre-treatment covariates may be used — post-treatment covariates can be affected by treatment (Wooldridge 2005b).

Do not pre-test to pick the assumption

It is tempting to use statistical pre-tests to choose between parallel-trends versions, but Roth (2020) shows this distorts inference. The authors recommend choosing based on the application’s context, not data-driven tests (Sec. 2.3, fn. 8).

Examples

Minimum wage: covariates that make parallel trends plausible

County characteristics used to justify conditional parallel trends: census region, county population, median income, fraction white, fraction with HS education, poverty rate. Treated counties differ markedly (much less likely Southern; population ~94k vs ~53k; 89% vs 83% white) — so unconditional parallel trends is suspect and conditioning is warranted. Sant’Anna & Song (2019) propensity-score specification tests fail to reject correct specification.

Connections

• Conditions for the identification results in Doubly-Robust Estimands for ATT(g,t).
• Apply to the target defined in Group-Time Average Treatment Effects.
• Visualized as a conditioning structure in Directed Acyclic Graphs.
• Anticipation horizon $\delta$ pins the reference period $g-\delta -1$ used by the estimands.

See Also

• The Experimental Ideal — randomization as the strongest (and stronger) benchmark
• Synthetic Control — alternative when parallel trends is implausible
• de Chaisemartin & D’Haultfœuille (2020); Sun & Abraham (2020) — stronger parallel-trends variants