3.4 Average treatment effect on the treated (ATT)

The average treatment effect on the treated is concerned with the evaluation of treatment effects for only those units that are treated. Formally, it is defined as:

\[\begin{equation} ATT = E(Y_i(1) - Y_i(0) | W_i = 1) \end{equation}\]

ATE is only concerned with a subset of the population who received the treatment, \(E[.|W_i = 1]\). Here, ATT is comparing outcomes among the treated units in presence of the treatment versus what the outcomes would have been in absence of the treatment only for units receiving the treatment. Hence, only one segment of the counterfactual (potential outcome) is required. In our example, the counterfactual for Alia and Samaira would allow estimation of ATT, whereas ATE requires counterfactual for everyone.

The independence condition states that on average the potential outcomes for the treated group in absence of the treatment would be similar to the average outcome for the control group, i.e. \(E(Y_i(0)|W_i = 1) = E(Y_i(0)|W_i = 0)\). This allows re-writing ATT as the following:

\[\begin{align} ATT = E\{E(Y_i | W_i = 1) - E(Y_i | W_i = 0) | W_i = 1\} \\ ATT = E(Y_i | W_i = 1) - E(Y_i | W_i = 0) \end{align}\]

The second line follows from the independence assumption which allows this: \(E\{E(Y_i | W_i = 0) | W_i = 1\} = E(Y_i | W_i = 0).\) Under the independence assumption this means that we can estimate ATT by substrating the averages of exam score across the treated and control units. In purely randomized experiments, if there is a perfect case of compliance, then the ATT will be similar to ATE.