Grad Did

雙重差分法 (Difference-in-Differences)

Overview

Difference-in-Differences (DID) estimates causal effects by comparing the change in outcomes over time between a treatment group (affected by an intervention) and a control group (unaffected). By differencing out both time-invariant group differences and common time trends, DID isolates the treatment effect under the parallel trends assumption.

When to Use

• Evaluating the impact of a policy, regulation, or intervention
• A natural experiment assigns treatment at a group level (state, industry, firm)
• Panel or repeated cross-section data with pre- and post-treatment periods
• Randomized experiment is infeasible but a plausible control group exists

When NOT to Use

• Parallel trends assumption is violated and cannot be remedied
• Treatment and control groups differ in ways that change over time
• Treatment is self-selected based on anticipated outcomes (anticipation effects)
• Only post-treatment data are available (no pre-treatment baseline)

Assumptions

IRON LAW: DID is valid ONLY if the parallel trends assumption holds —
without it, the estimated treatment effect is biased by differential
pre-existing trends.

Key assumptions:

• Parallel trends: absent treatment, treated and control groups would have followed the same trajectory
• No spillover effects from treated to control units (SUTVA)
• Treatment timing is sharp and exogenous
• Composition of groups is stable over time (no differential attrition)

Methodology

Step 1 — Establish Treatment and Control Groups

Define who is treated and when. Verify groups are comparable on pre-treatment observables. Document the treatment event and its timing.

Step 2 — Test Parallel Trends

Plot outcome trends for treatment vs control groups in pre-treatment periods. Run an event-study specification with leads and lags. Pre-treatment coefficients should be statistically insignificant.

Step 3 — Estimate the DID Model

Y = β₀ + β₁×Treat + β₂×Post + β₃×(Treat×Post) + Controls + ε. The coefficient β₃ is the DID estimator. Cluster standard errors at the treatment assignment level. See references/ for staggered adoption extensions.

Step 4 — Robustness Checks

Run placebo tests (fake treatment dates, fake treatment groups). Test sensitivity to control group choice. For staggered DID, use Callaway-Sant'Anna or Sun-Abraham estimators.

Output Format

## DID Analysis: [Policy / Intervention]

### Research Design
| Element | Description |
|---------|-------------|
| Treatment group | [who] |
| Control group | [who] |
| Treatment date | [when] |
| Pre-treatment periods | [range] |

### Parallel Trends Test
| Pre-period lead | Coefficient | S.E. | p-value |
|-----------------|-------------|------|---------|
| t-3 | x.xx | x.xx | x.xx |
| t-2 | x.xx | x.xx | x.xx |
| t-1 | x.xx | x.xx | x.xx |

### DID Estimate
| Specification | β (Treat×Post) | S.E. | p-value | N |
|---------------|----------------|------|---------|---|
| Baseline | x.xx | x.xx | x.xx | xxx |
| With controls | x.xx | x.xx | x.xx | xxx |

### Robustness
- Placebo test result: [pass/fail]
- Alternative control group: [result]

### Limitations
- [Note any assumption violations]

Gotchas

• Visual parallel trends are necessary but not sufficient — the assumption is about counterfactual trends
• Too few clusters for clustering standard errors inflates Type I error (use wild bootstrap if clusters < 50)
• Staggered adoption makes the standard two-way FE DID estimator biased (use recent robust estimators)
• Anticipation effects violate the sharp treatment timing assumption
• Differential pre-trends are often "fixed" by adding group-specific trends, but this is fragile
• DID estimates a local average treatment effect on the treated (ATT), not ATE

References

• Angrist, J. D., & Pischke, J.-S. (2009). *Mostly Harmless Econometrics*. Princeton University Press.
• Callaway, B., & Sant'Anna, P. H. C. (2021). Difference-in-differences with multiple time periods. *Journal of Econometrics*, 225(2), 200-230.
• Goodman-Bacon, A. (2021). Difference-in-differences with variation in treatment timing. *Journal of Econometrics*, 225(2), 254-277.