## 8.2 Parameters of Interest

Instead of going for individual-level effects of participating in the treatment, most researchers instead go for more aggregated parameters. The two most common ones are the Average Treatment Effect (ATE) and Average Treatment Effect on the Treated (ATT).

\begin{align*} ATE = \mathbb{E}[Y(1) - Y(0)] \qquad \textrm{and} \qquad ATT = \mathbb{E}[Y(1)-Y(0) | D=1] \end{align*} $$ATE$$ is the difference between treated potential outcomes and untreated potential outcomes, on average, and for the entire population. $$ATT$$ is the difference between treated and untreated potential outcomes, on average, conditional on being in the treated group.

We will mostly focus on $$ATT$$.

It is worth considering the challenges for learning about $$ATT$$. In particular, notice that we can write \begin{align*} ATT = \mathbb{E}[Y(1)|D=1] - \mathbb{E}[Y(0)|D=1] \end{align*} and consider these term separately

• $$\mathbb{E}[Y(1)|D=1]$$ is the average treated potential outcome among the treated group. But we observe treated potential outcomes for the treated group $$\implies \mathbb{E}[Y(1)|D=1] = \mathbb{E}[Y|D=1]$$. In other words, if we want to estimate this component of the $$ATT$$, we can just look right at the data and compute the average outcome experienced by individuals in the treated group.

• $$\mathbb{E}[Y(0)|D=1]$$ is the average untreated potential outcome among the treated group. This is (potentially much) more challenging than the first term because we do not observe untreated potential outcomes among the treated group. But, in order to learn about the $$ATT$$, we will have to somehow deal with this term. I will provide a number of strategies below, but it is important to remember that this is a major challenge, and their may not be a good solution.

Side-Comment: I think it is also worth clearly pointing out that, while I am a big believer in the power/usefulness of using data to try to answer questions in economics, the above discussion suggests that there are a number of questions that we may just not be able to answer. In economics jargon, this amounts to an identification problem — in other words, there may be competing theories of the world which the available data is not able to distinguish among. I probably do not emphasize this issue enough in our class, but it is something that you should remember — there may be a large number of causal questions that we’d be interested in answering, but where it is not possible to answer them (at least given the information that we have available).