Use the Law of Iterated Expectations

Theoretical Problems

1. Researchers want to determine the effect of moving from a poor neighborhood (defined as a neighborhood in which at least 50% of families are poor) on outcomes for children.
(a) (points: 4) First, researchers gather data on a representative sample of children in Canada. For each, they estimate two variables: poornhd, which equals 1 if the child lives in a poor neighborhood, and 0 otherwise; and score, their score on a standardized test. They then regress:
score = 0 + 1poornhd + u
Will their estimate of 1 be a consistent estimate of the e↵ect of living in a poor neighborhood on this standardized test? Why or why not?
(b) (points: 4) The researchers decide to use instrumental variables. They use parental income parinc as an instrument for poornhd. Will this allow them to consistently estimate1?
(c) (points: 4) Next, the researchers run an experiment. A randomly-chosen half of participants are given a red pen; the remainder are given a blue pen. Define the variable redpen as 1 if they are given a red pen, 0 otherwise. The researchers then use redpen as an instrument for poornhd. Will this allow them to consistently estimate 1?
(d) (points: 6) The researchers run one more randomized experiment. They recruit participants who are living in public housing. A randomly chosen half of participants are given a subsidy to live anywhere they want (they call these people “untreated”, and define the variable treated = 0 for them); the other half was given a subsidy as well, but it could not be used in a poor neighborhood (they call these people “treated” and define the variable treated = 1 for them). Everyone in the experiment used the voucher; however, some people in the untreated group moved to a non-poor neighborhood. In particular, 20% of the untreated group moved to a poor neighborhood, while 0% of the treated group moved to a poor neighborhood. Among the untreated group, the average child test score was 110; among the treated group, the average child test score was 130.

Now, the researchers use treated as an instrument for poornhd. What is their estimate of the effect of living in a poor neighborhood on test score? Hint: use the law of iterated expectations. That is, if z is discrete and takes values z1, z2, …zn, then

E[xz] = E[E(xz|z)] = Xn i=1 E(xz|z = zi) ⇥ P r(z = zi)