What?
Proposal for the local average treatment effects, discussion of its estimation and the iv estimator.
Given a valid instrument (see two conditions below), technical considerations arising from treatment effect heterogeneity need not inhibit inference.
Why?
\(Y_{0}\) is the response without treatment, and \(Y_{1}\) is the respond with treatment. \(D\) is an indicator of treatment. We observe \(D\) and \(Y = Y_{D} = D \cdot Y_{1} + (1 - D) \cdot Y_{0}\) for a random sample of individuals. The individual treatment effect is \(Y_{1} - Y_{0}\) but \(Y_{1}\) and \(Y_{0}\) is never observed for the same person.
Quite often in experiments, subjects failed to comply with their assignments. Thus the overall average treatment effects \(E[Y_{1}- Y_{0}] = E[Y_{1}|D = 1] - E[Y_{0}|D = 0]\) received biased estimation.
How?
Define an instrumental variable \(Z = {z_{1}, z_{2}, …, z_{K}}\) unrelated to the respond \(Y_{0}\) and \(Y_{1}\); \(D_{z} = 1\) if the subject comply to assignment under \(z\) and \(0\) otherwise.
- Condition 1 (instrument existance)
- \(Z\) does not directly affects \(Y_{0}\) and \(Y_{1}\).
- Condition 2 (monotonicity)
- The instrument affects the participation in a monotonic way. This assumption can never be verify, but it is reasonable.
Given the above condition, the authors showed that a conventional 2SLS etimator consistently estimates a weighted average treatment effects.
And?
The local average treatment effects (late) is a weighted average treatment effects (ate) with the weights non-negative and adding up to one: \[LATE = \frac{IIT}{IIT_{D}}\] where \(IIT\) is short for “intention to treat”, particularly:
\(IIT = E[Y_{i}|Z=1]-E[Y_{i}|Z=0]\): average for those who are assigned to treatment. Equivalent to ate in case of full compliance.
\(IIT_{D} = E[D_{i}|Z=1]-E[D_{i}|Z=0]\): the share of compliers.
Another analogy: The late is the ate for the individuals whose behaviors can be changed by changing the value of \(Z\).
Note that:
- Different instruments may lead to different estimates \(\alpha_{\lambda}\) if the treatment effect is not constant.
Some notable case study:
- Angrist (1990)
- Uses the Vietnam-era draft lottery to identify the earnings effects of veteran status on earnings. The instrument is the draft lottery number, randomly assigned to date of birth. The lottery number doesn’t affect earning but were used to determine priority for conscription.
- Angrist and Krueger (1991)
- Effect of schooling on earnings using the variation in compulsory schooling created by the variation in birth dates. Birth dates itself does not affect earning but affect the level of schooling because people born on different dates are allow to drop out of school on their birthday.
Bibliography
Angrist, Joshua D. 1990. “Lifetime Earnings and the Vietnam Era Draft Lottery: Evidence from Social Security Administrative Records.” The American Economic Review 80 (3). American Economic Association:313–36. https://www.jstor.org/stable/2006669.
Angrist, Joshua D., and Alan B. Krueger. 1991. “Does Compulsory School Attendance Affect Schooling and Earnings?” The Quarterly Journal of Economics 106 (4). Oxford University Press:979–1014. https://www.jstor.org/stable/2937954.
This post is in the collection of my public reading notes.