difference-in-differences

When a policy hits one group but not another and you have data from before and after, you can let each group serve as a partial control for the other. Difference-in-differences subtracts twice: it removes each group's fixed level differences by comparing before to after, then removes shared time trends by comparing the treated group's change to the control group's change. What survives both subtractions is, under one key assumption, the causal effect.

The estimand is the treated group's after-minus-before change minus the control group's after-minus-before change. Its validity rests on the parallel trends assumption: absent the treatment, the two groups' outcomes would have moved in lockstep, so the control's change stands in for the treated group's missing counterfactual trend. The canonical two-by-two case generalizes to a two-way fixed-effects regression with unit and time dummies, the workhorse implementation in applied econometrics.

Parallel trends is the soul and the soft spot of the method; it is fundamentally untestable, though pre-treatment trends offer circumstantial support. A recent wave of econometric work shows that the simple two-way fixed-effects estimator badly misbehaves under staggered adoption with heterogeneous effects, because already-treated units get used as controls, motivating newer estimators that handle treatment timing explicitly.