Difference-in-Differences
Difference-in-Differences
DID subtracts a counterfactual change
Clearborough introduces a clean-air zone; Westborough does not.
| Mean pollution, μg/m³ | Before | After | Change |
|---|---|---|---|
| Clearborough | 42 | 30 | −12 |
| Westborough | 38 | 34 | −4 |
The DID estimate is
Equivalently, Westborough’s −4 change predicts a no-policy Clearborough value of ; observed pollution is 30, eight units lower.
Construct the DID counterfactual
The last lines turn a hidden trend violation into an explicit sensitivity parameter.
Parallel trends is about
For two periods,
The assumption allows different baseline levels. It concerns the treated group’s unobserved untreated change after policy, so it cannot be proved by pre-policy data.
Pre-trends, institutional history and alternative comparison groups can make it more or less credible.
Regression form reproduces the table
In a 2×2 saturated model, equals the four-cell DID. Controls or additional periods change the estimand unless specified carefully.
Four threats, four diagnostics
| Threat | Evidence that addresses it |
|---|---|
| differential untreated trend | multiple pre-periods, institutional argument, sensitivity bounds |
| anticipation | policy-announcement timeline and alternative treatment date |
| concurrent intervention | policy inventory and unaffected outcomes |
| spillover | geography/market links and wider treatment definition |
| composition change | stable units or repeated-cross-section composition audit |
A placebo that does not target the principal threat is decorative robustness.
Repeated cross-section versus panel
Panel DID follows the same units. Repeated cross-sections compare population means sampled each period. The latter requires stable population composition or conditional assumptions; migration caused by the policy can change who is observed.
Inference should reflect treatment assignment and serial correlation. A policy assigned to municipalities usually requires municipality-level dependence, not resident-level independence.
Classic illustration, modern reading
Card and Krueger’s minimum-wage study compares New Jersey and Pennsylvania fast-food employment before and after a policy change. Use it to identify the four-cell logic, then ask how measurement, comparison choice, one treated policy and later evidence affect the claim.
Quick check
The treated and comparison groups have identical pre-policy levels but different pre-policy trends. Is DID credible?