What's Trending in DiD? — Slides

What’s Trending in Difference-in-Differences?

A synthesis of the recent econometrics literature

Roth · Sant’Anna · Bilinski · Poe — Journal of Econometrics (2023)

One-line thesis: start from one clean DiD model, then read the whole modern literature as relaxing one assumption at a time.

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Why this paper exists

• DiD is everywhere in the social sciences.
• 2018–2022: a dizzying wave of new DiD methods papers.
• Practitioners can’t keep up; even experts can’t see how it all fits.

This paper is the map — plus a practitioner checklist and a package guide.

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The big idea

Write down ONE canonical 2×2 model, then relax it three ways:

Canonical piece	Strand that relaxes it
Same treatment time	1. Staggered timing
Parallel trends holds	2. PT violations
Many clean clusters	3. Inference

Running example throughout: states expanding Medicaid → effect on insurance coverage.

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The canonical 2×2 setup

• Two periods; a treated group (expands Medicaid) and a comparison group (never does).
• We want the ATT: effect on coverage for the states that expanded.
• Problem: the treated group’s “no-expansion” outcome is never observed.

${\tau }_2=\mathbb{E}[Y_{i,2}(1)-Y_{i,2}(0)\mid D_i=1]$

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Two assumptions

Parallel Trends — absent treatment, both groups’ coverage would have changed by the same amount.

• Allows treated states to differ in levels (richer, bluer)…
• …but not in trends.

No Anticipation — treated states aren’t already reacting before expansion.

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Identification = impute the counterfactual

${\tau }_2=\underset{\text{treated change}}{\underbrace{(Y_{i,2}-Y_{i,1}\mid D=1)}}-\underset{\text{control change}}{\underbrace{(Y_{i,2}-Y_{i,1}\mid D=0)}}$

Take the treated group’s pre-level, add the control group’s change as “what would’ve happened anyway,” and the leftover movement is the effect.

Run as TWFE; cluster SEs at the unit level. This is the room. Now remove furniture.

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Strand 1: Staggered timing

Real policies roll out at different times (2014, 2015, never…).

Instinct: keep using TWFE (static or dynamic event-study).

It breaks — once treatment effects are heterogeneous.

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Why TWFE breaks: forbidden comparisons

• TWFE = a weighted average of all 2×2 DiDs (Goodman-Bacon 2021).
• Some use already-treated units as “controls” → forbidden.

A 2014-expansion state used as a “control” for a 2016 state — but the 2014 state is already moving from its own effect.

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The scary part: negative weights

${\beta }^{post}={\sum }_s{\omega }_s{\tau }_s,{\sum }_s{\omega }_s=1,\text{some }{\omega }_s<0$

• Every state’s true effect can be positive…
• …yet the TWFE coefficient comes out negative — the wrong sign.
• Pre-trend tests from staggered TWFE are also contaminated → unreliable.

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Strand 1: The fix

Same recipe as 2×2 DiD, done carefully:

1. Building block: group-time effect $ATT(g,t)$ — cohort $g$, time $t$.
2. Clean controls only: never- or not-yet-treated.
3. Aggregate with weights you choose (cohort size, event-study profile).

✅ Valid under any heterogeneity ✅ Transparent comparison group

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Strand 1: The estimator menu

Estimator	Control group	Pre-periods
Callaway–Sant’Anna	never / not-yet	last pre-period
Borusyak et al. (imputation)	not-yet (impute)	all pre-periods
Sun–Abraham	never / last cohort	event-study
de Chaisemartin–D’Haultfœuille	not-yet; on/off OK	adjacent
Stacking	matched clean controls	per-event

Tradeoff: more pre-periods → more efficient but needs PT over a longer horizon. In practice they usually agree.

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Strand 2: Parallel trends may fail

Why worry?

• Time-varying confounders (bluer states expand and face different shocks).
• Functional form — PT in levels ≠ PT in logs, and which is right?

Three moves: condition, test honestly, bound the violation.

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Move 1: Conditional parallel trends

Make PT hold only within covariate cells $X$ (e.g. partisan lean).

⚠️ Naive $X\times$time in TWFE is not consistent under heterogeneity. Use:

• Regression adjustment (model control outcomes)
• IPW (model the propensity score)
• Doubly-robust ← recommended default: right if either model is right

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Move 2: Pre-trends tests aren’t enough

Flat pre-trends feel reassuring, but:

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Problem 1 — parallel ≠ parallel

Boys’ and girls’ heights move in parallel until age 13, then diverge. Flat pre-trends don’t guarantee parallel post-trends. (No, bar mitzvahs don’t cause height.)

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Problem 2 — low power

A pre-trend big enough to seriously bias you is often not significant.

Detected only ~50% of the time → still produces a spurious significant effect ~50% of the time (≈10× the nominal 5%).

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Problem 3 — pre-test bias

Keeping only analyses that “pass” the pre-test selects your sample and can make bias worse.

➡️ At minimum, report the power against relevant violations.

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Move 3: Honest sensitivity analysis

Don’t assume PT holds — ask how big a violation would overturn the result?

$|{\delta }_1|\le \bar{M}\cdot {\max }_{r<0}|{\delta }_r|$

Report the breakdown $\bar{M}$:

“Significant unless post-trends were 2× larger than the biggest pre-trend.”

Wider intervals when pre-trends are noisy — imprecision honestly costs you. (Rambachan–Roth; also bracketing bounds, Ye et al.)

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Strand 3: Inference

Few clusters (e.g. 3 treated states): CLT fails; cluster shocks don’t average out.

• Donald–Lang, Conley–Taber, wild bootstrap, permutation/FRT — each needs extra assumptions. No free lunch.

Design-based view: units fixed, assignment is the randomness.

• Sampling-valid methods are usually design-valid too.
• ✅ Clear rule: cluster where treatment is assigned.

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The practitioner checklist (Table 1)

1. Same treatment time? No → heterogeneity-robust estimator.
2. Sure about parallel trends? No → condition on covariates, plot event study, report pre-test power + $\bar{M}$ sensitivity.
3. Many clusters from a super-population? Few treated → small-cluster method; can’t define it → design-based, cluster at assignment level.

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Five takeaways

1. Canonical DiD = impute the counterfactual from the control’s change.
2. Staggered TWFE → forbidden comparisons → wrong sign.
3. Build from clean $ATT(g,t)$, aggregate with your weights.
4. Flat pre-trends ≠ valid design — report power + $\bar{M}$ breakdown.
5. Cluster where treatment is assigned; go design-based when unsure.

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Where to read next

Packages (Table 2): did/csdid, did2s, did_multiplegt, fixest, staggered · DRDID · bacondecomp, TwoWayFEWeights · honestDiD, pretrends

Vault notes: RothSantAnna2023-WhatsTrendingInDiD · RothSantAnna2023-WhatsTrendingInDiD-Explainer · GoodmanBacon2021-DiDVariationInTiming · SunAbraham2021-EventStudies · CallawaySantAnna2021-DiDMultiplePeriods · SantAnnaZhao2020-DoublyRobustDiD · RothPretrends2022-PretestWithCaution · RambachanRoth2023-MoreCredibleParallelTrends · BakerEtAl2025-DiDPractitionerGuide · DiD