The Compliant Subpopulation
The LATE framework partitions any population with an instrument into a set of three instrument-dependent subgroups, defined by the manner in which members of the population react to the instrument:
Definition 4.4.1 Compliers. The subpopulation with Dii = 1 and Doi = 0.
Always-takers. The subpopulation with D1i =Doi = 1.
Never-takers. The subpopulation with D1i =Doi = 0.
LATE is the effect of treatment on the population of compliers. The term "compliers" comes from an analogy with randomized trials where some experimental subjects comply with the randomly assigned treatment protocol (e. g., take their medicine) but some do not, while some control subjects obtain access to the experimental treatment even though they were not supposed to. Those who don’t take their medicine when randomly assigned to do so are never-takers while those who take the medicine even when put into the control group are always-takers. Without adding further assumptions (e. g., constant causal effects), LATE is not informative about effects on never-takers and always-takers because, by definition, treatment status for these two groups is unchanged by the instrument (random assignment). The analogy between IV and a randomized trial with partial compliance is more than allegorical - IV solves the problem of causal inference in a randomized trial with partial compliance. This important point merits a separate subsection, below.
Before turning to this important special case, we make a few general points. First, the average causal effect on compliers is not usually the same as the average treatment effect on the treated. From the simple fact that Di =Doi + (d^— Doi)Zi, we learn that the treated population consists of two non-overlapping groups. By monotonicity, we cannot have both Doi = 1 and D1i —Doi = 1 since Doi = 1 implies D1i = 1. The treated therefore have either Doi = 1 or D1i—Doi = 1 and Zi = 1, and hence Di can be written as the sum of two mutually-exclusive dummies, Dio and (d^— Doi)Zi. The treated consist of either always-takers or compliers with the instrument switched on. Since the instrument is as good as randomly assigned, compliers with the instrument switched on are representative of all compliers. From here we get
1 'V }
effect on the treated
= E[y 1i - Yoi|Doi = 1]P[Doi = 11Di = 1]
+E [y 1i - Yoi|D1i > Doi, Zi = 1] P [D1i > Doi, Zi = 1|Di = 1]
= E[y 1i - Yoi|Doi = 1]P[Doi = 1|Di = 1]
1 - V }
effect on always-takers
+E [Y1i - Yoi |D 1i > Doi]P [D1i > Doi, Zi = 1|Di = 1]
1 - v }
effect on compliers
Since P[Doj = 1|Dj = 1] and P[Dij >Doj, Zj = 1|Dj = 1] add up to one, this means that the effect of treatment on the treated is a weighted average of effects on always-takers and compliers.
Likewise, LATE is not the average causal effect of treatment on the non-treated, E[Y1j—Yoj|Dj = 0]. In the draft-lottery example, the average effect on the non-treated is the average causal effect of military service on the population of non-veterans from the Vietnam-era cohorts. The average effect of treatment on the non-treated is a weighted average of effects on never-takers and compliers. In particular,
1 'V }
effect on the non-treated
= E [Yii - Yoj |Dij =0]P [Dii =0|Dj =0]
1 - v }
effect on never-takers
+E [Yij - Yoi|Dij > Doi]P [Dij > Doi, Zj = 0|Dj = 0] ,
4 - z }
effect on compliers
where we use the fact that, by monotonicity, those with Dij = 0 must be never-takers.
E[Yij - Yoj] = E[Yij - Yoj|Dj = 1]P[Dj = 1] + E[Yij - Yoj|Dj = 0]P[Dj = 0]
shows the overall population average treatment effect to be a weighted average of effects on compliers, always - takers, and never-takers. Of course, this is a conclusion we could have reached directly given monotonicity and the definition at the beginning of this subsection.
Because an instrumental variable is not directly informative about effects on always-takers and never - takers, instruments do not usually capture the average causal effect on all of the treated or on all of the non-treated. There are important exceptions to this rule, however: instrumental variables that allow no always-takers or no never-takers. Although this scenario is not typical, it is an important special case. One example is the twins instrument for fertility, used by Rosenzweig and Wolpin (1980), Bronars and Grogger (1994), Angrist and Evans (1998), and Angrist, Lavy, and Schlosser (2006). Another is Oreopoulos’ (2006) recent study using changes in compulsory attendance laws as instruments for schooling in Britain.
To see how this special case works with twins instruments, let Tj be a dummy variable indicating multiple second births. Angrist and Evans (1998) used this instrument to estimate the causal effect of having three children on earnings in the population of women with at least two children. The third child is especially interesting because reduced fertility for American wives in the 1960s and 1970s meant a switch from three children to two. Multiple second births provide quasi-experimental variation on this margin. Let Yoj denote potential earnings if a woman has only two children while Y ij denotes her potential earnings if she has three, an event indicated by Dj. Assuming that Tj is randomly assigned, i. e., that fertility increases by at most one child in response to a multiple birth, and that multiple births affect outcomes only by increasing fertility,
LATE using the twins instrument, Tj, is also Е[Уц—Yoj|Dj = 0], the average causal effect on women who are not treated (i. e., have two children only). This is because all women who have a multiple second birth end up with three children, i. e., there are no never-takers in response to the twins instrument.
Oreopoulos (2006) also uses IV to estimate an average causal effect of treatment on the non-treated. His study estimates the economic returns to schooling using an increase in the British compulsory attendance age from 14 to 15. Compliance with the Britain’s new compulsory attendance law was near perfect, though many teens would previously have dropped out of school at age 14. The causal effect of interest in this case is the earnings premium for an additional year of high-school. Finishing this year can be thought of as the treatment. Since everybody in Oreopoulos’ British sample finishes the additional year when compulsory schooling laws are made stricter, Oreopoulos’ IV strategy captures the average causal effect of obtaining one more year of high school on all those who leave school at 14. This turns on the fact that British teens are remarkably law-abiding people - Oreopoulos’ IV strategy wouldn’t estimate the effect of treatment on the non-treated in, say, Israel, where teenagers get more leeway when it comes to compulsory school attendance. Israeli econometricians using changes in compulsory attendance laws as instruments must therefore make do with LATE.