8.12 Answers to Some Extra Questions

Answer to Question 4

The key condition is the parallel trends assumption that says that, in the absence of participating in the treatment, the path of outcomes that individuals in the treated group is the same, on average, as the path of outcomes that individuals in the untreated group actually experienced.

Answer to Question 9

  1. When some individuals do not comply with their treatment assignment, this approach is probably not so great. In particular, notice that the comparison in this part of the problem is among individuals who actually participated in the treatment relative to those who didn’t (the latter group includes both those assigned not to participate in the treatment along with those assigned to participate in the treatment, but ultimately didn’t actually participate). This suggests that this approach would generally lead to biased estimates of the \(ATT\). In the particular context of job training, you can see this would not be such a good idea if, for example, the people who were assigned to the job training program but who did not participate tended to do this because they were able to find a job before the job training program started.

  2. This approach is likely to be better. By construction, \(Z\) is not correlated with \(U\) (since \(Z\) is randomly assigned). \(Z\) is also likely to be positively correlated with \(Z\) (in particular, this will be the case if being randomly assigned to treatment increases the probability of being treated). This implies that \(Z\) is a valid instrument and should be able to deliver a reasonable estimate of the effect of participating in the treatment.

Answer to Question 10

While your friend’s explanation is not technically wrong, it seems to me that you are more right than your friend. There is an important issue related to external validity here. The group of people that show up to participate in the experiment could be (and likely are) quite different from the general population. Interpreting the results of the experiment as being an \(ATE\) (in the sense of across the entire population) is therefore likely to be incorrect — or at least would require extra assumptions and/or justifications. Interpreting them as an \(ATT\) (i.e., as the effect among those who participated in the treatment) is still perfectly reasonable though.