Canonical DID

We consider panel data $y_{i, t}$ for $t=1,2$ and $i=1,2, \cdots, n$.

Treatment timing: some units $\left(D_i=1\right)$ are treated in period 2 , and everyone else untreated $\left(D_i=0\right)$

Potential outcomes: we observe $y_{i, t}(1) \equiv y_{i, t}(0,1)$ for treated units, and $y_{i, t}(0) \equiv y_{i, t}(0,0)$for untreated units.

Parallel trends:

$$E\left[y_{i, t=2}(0)-y_{i, t=1}(0) \mid D_i=1\right]=E\left[y_{i, t=2}(0)-y_{i, t=1}(0) \mid D_i=0\right]$$

No anticipation: $y_{i, t=1}(1)=y_{i, t=1}(0)$.