The canonical two-period design

Consider two groups (treated $T$, control $C$) and two periods (pre $t=0$, post $t=1$). The DiD estimator is:

\[\hat{\tau}_{DiD} = \underbrace{(\bar{y}_{T,1} - \bar{y}_{T,0})}_{\text{change in treated}} - \underbrace{(\bar{y}_{C,1} - \bar{y}_{C,0})}_{\text{change in control}}\]

Equivalently, run the OLS regression:

\[y_{it} = \alpha + \gamma T_i + \lambda \text{Post}_t + \tau\, (T_i \times \text{Post}_t) + \varepsilon_{it}\]

The coefficient $\tau$ is the Average Treatment Effect on the Treated (ATT).

The parallel trends assumption

DiD identifies $\tau$ under the parallel trends assumption: in the absence of treatment, the treated group would have followed the same trend as the control group.

\[\mathbb{E}[Y_{it}(0) - Y_{i,t-1}(0) \mid T_i = 1] = \mathbb{E}[Y_{it}(0) - Y_{i,t-1}(0) \mid T_i = 0]\]

This is not testable in the post period but can be assessed with pre-treatment placebo tests.

Staggered adoption

In modern DiD settings, units receive treatment at different times. The “two-way fixed effects” (TWFE) estimator is biased in this case — it implicitly uses already-treated units as controls. Use Callaway–Sant’Anna or Sun–Abraham estimators instead.