4  Difference in differences analyses

Many approaches to causal inference assume exchangeability (Definition 1.10) and exploit its consequence (Theorem 1.1):

\[\text{E}{\left[Y(x) | X = x'\right]} = \text{E}{\left[Y(x) | X = x\right]}\]

Difference-in-differences makes a weaker exchangeability assumption:

\[\text{E}{\left[Y_t(0) - Y_{t'}(0) | X = 1\right]} = \text{E}{\left[Y_t(0) - Y_{t'}(0) | X = 0\right]}\]