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Difference-in-difference (DD) is a version of fixed effects estimation. It uses longitudinal data to estimate the effect of a treatment administered at a certain point in time. The idea is to follow units having a common trend in the outcome variable $Y$ in period $t_1$ before the treatment. Some units are then treated ($T$) and some remain untreated controls ($C$). All units are observed again at time $t_2$ after the treatment. The difference-in-difference treatment effect $\delta$ is the average change experienced by the treatment units compared to the average change experienced by the control units,
$$\delta = (E[Y_{T,t_{2}}] - E[Y_{T,t_{1}}]) - (E[Y_{C,t_{2}}] - E[Y_{C,t_{1}}]).$$
The common pre-treatment trend assumption on $Y$ is important to make the argument that a different rate of change for the treated group after the treatment compared to the control group is due to the treatment itself and not due to unobserved factors. Differencing removes fixed but not time-invariant effects.