# Local Average Treatment Effects

In an IV framework, the engine that drives causal inference is the instrument, zj, but the variable of interest is still Dj. This feature of the IV setup leads us to adopt a generalized potential-outcomes concept, indexed against both instruments and treatment status. Let Yj(d, z) denote the potential outcome of individual i were this person to have treatment status Dj = d and instrument value Zj = z. This tells us, for example, what the earnings of i would be given alternative combinations of veteran status and draft-eligibility status. The causal effect of veteran status given i’s realized draft-eligibility status is Yj(1,Zj)—Yj(0,Zj), while the causal effect of draft-eligibility status given i’s veteran status is Yj(Dj, 1)—Yj(Dj, 0).

We can think of instrumental variables as initiating a causal chain where the instrument, Zj, affects the variable of interest, Dj, which in turn affects outcomes, Yj. To make this precise, we need notation to express the idea that the instrument has a causal effect on Dj. Let Du be i’s treatment status when Zj = 1, while Doj is i’s treatment status when Zj = 0. Observed treatment status is therefore

Dj = D0j + (Dij _ D0j)Zj = wo + w1jZj + Cj. (4.4.1)

In random-coefficients notation, wo = E[Doj] and w1j = (D1j— Doj), so w1j is the heterogeneous causal effect of the instrument on Dj. As with potential outcomes, only one of the potential treatment assignments, D1j and Doj, is ever observed for any one person. In the draft lottery example, Doj tells us whether i would serve in the military if he draws a high (draft-ineligible) lottery number, while D1j tells us whether i would serve if he draws a low (draft-eligible) lottery number. We get to see one or the other of these potential assignments depending on Zj. The average causal effect of Zj on Dj is E[w1j].

The first assumption in the heterogeneous framework is that the instrument is as good as randomly assigned: it is independent of the vector of potential outcomes and potential treatment assignments. Formally, this can be written

[{Yj(d, z); 8 d, zg, D1j, Doj] П Zj, (4.4.2)

Independence is sufficient for a causal interpretation of the reduced form, i. e., the regression of Yj on Zj.

= E [Yi(Dii, 1) – Yi(Doi, 0)] ,

the causal effect of the instrument on Y і. Independence also means that

E [Di|Zi = 1] – E [Di|Zi =0] = E [diі|zі = 1] – E [Doi|Zi = 0]

— E [Dii D0i] ;

in other words, the first-stage from our earlier discussion of 2SLS captures the causal effect of Zi on Di.

The second key assumption in the heterogeneous-outcomes framework is the presumption that Y і (d, z) is only a function of d.[58] To be specific, while draft-eligibility clearly affects veteran status, an individual’s potential earnings as a veteran are assumed to be unchanged by draft-eligibility status; while potential earnings as a nonveteran are similarly unaffected. In general, the claim that an instrument operates through a single known causal channel is called an exclusion restriction. In a linear model with constant effects, the exclusion restriction is expressed by the omission of the instrument from the causal equation of interest, or, equivalently, E[z^] =0 in equation (4.1.14). It’s worth noting that the traditional error-term notation used for simultaneous equations models doesn’t lend itself to a clear distinction between independence and exclusion. We need Zi and ^ to be uncorrelated in this equation, but the reasoning that lies behind this assumption is unclear until we consider both the independence and exclusion restrictions.

The exclusion restriction fails for draft-lottery instruments if men with low draft lottery numbers were affected in some way other than through an increased likelihood of service. For example, Angrist and Krueger (1992) looked for an association between draft lottery numbers and schooling. Their idea was that educational draft deferments would have led men with low lottery numbers to stay in college longer than they would have otherwise desired. If so, draft lottery numbers are correlated with earnings for at least two reasons: an increased likelihood of military service and an increased likelihood of college attendance. The fact that the lottery number is randomly assigned (and therefore satisfies the independence assumption) does not make this possibility less likely. The exclusion restriction is distinct from the claim that the instrument is (as good as) randomly assigned. Rather, it is a claim about a unique channel for causal effects of the instrument.[59]

Using the exclusion restriction, we can define potential outcomes indexed solely against treatment status

using the single-index (Yj^Yoi) notation we have been using all along. In particular,

Y0i = Yi(0, 1)= Yi(0, 0). (4.4.3)

The observed outcome, Yi, can therefore be written in terms of potential outcomes as:

Yi = Yi(0, Zi) + [Yi(1, Zi) – Yi(0, Zi)]Di (4.4.4)

A random-coefficients notation for this is

Yi = ao + PiDi + ^i,

a compact version of (4.4.4) with ao = E[Yoi] and pi =Y1i—Yoi.

A final assumption needed for heterogeneous IV models is that either w1i > 0 for all i or w1i < 0 for all i. This monotonicity assumption, introduced by Imbens and Angrist (1994), means that while the instrument may have no effect on some people, all of those who are affected are affected in the same way. In other words, either D1i >Doi or D1i <Doi for all i. In what follows, we assume monotonicity holds with D1i >Doi. In the draft-lottery example, this means that although draft-eligibility may have had no effect on the probability of military service for some men, there is no one who was actually kept out of the military by being draft – eligible. Without monotonicity, instrumental variables estimators are not guaranteed to estimate a weighted average of the underlying individual causal effects, Y1i—Yoi.

Given the exclusion restriction, the independence of instruments and potential outcomes, the existence of a first stage, and monotonicity, the Wald estimand can be interpreted as the effect of veteran status on those whose treatment status can be changed by the instrument. This parameter is called the local average treatment effect ((LATE); Imbens and Angrist, 1994). Here is a formal statement:

Theorem 4.4.1 THE LATE THEOREM. Suppose

(A1, Independence) {Yi(d^, 1),Yoi(Doi, 0),D1i, Doi}HZi/

(A2, Exclusion) Yi(d, 0) =Yi(d, 1) =Y* for d = 0,1; (A3, First-stage), E[D1i—Doi] = 0 (A4, Monotonicity) D1i—Doi > 08i, or vice versa; Then

Proof. Use the exclusion restriction to write E[Yi|Zi = 1] = E[Yoi + (Y1i—Yoi)Di|Zi = 1], which equals

Е[уоі + (Yij—Yoj)Dij] by independence. Likewise E[Yj|Zj = 0] = E[Yoi + (y^—Yoi)Doi], so the numerator of the Wald estimator is E[(Y1i—Yoi)(D1i—Doi)]. Monotonicity means D1i —Doi equals one or zero, so

E[(Yii – Yoi)(Dii – Doi)] = E[Yii – Yoi|Dii > Doi]P[Dii > Doi].

A similar argument shows

E[Di|Zi = 1] – E[Di|Zi = 0] = E[Dii – Doi] = P[Dii > Doi].

This theorem says that an instrument which is as good as randomly assigned, affects the outcome through a single known channel, has a first-stage, and affects the causal channel of interest only in one direction, can be used to estimate the average causal effect on the affected group. Thus, IV estimates of effects of military service using the draft lottery estimate the effect of military service on men who served because they were draft-eligible, but would not otherwise have served. This obviously excludes volunteers and men who were exempted from military service for medical reasons, but it includes men for whom draft policy was binding.

How useful is LATE? No theorem answers this question, but it’s always worth discussing. Part of the interest in the effects of Vietnam-era service revolves around the question of whether veterans (especially, conscripts) were adequately compensated for their service. Internally valid draft lottery estimates answer this question. Draft lottery estimates of the effects of Vietnam-era conscription may also be relevant for discussions of any future conscription policy. On the other hand, while draft lottery instruments produce internally valid estimates of the causal effect of Vietnam-era conscription, the external validity – i. e., the predictive value of these estimates for military service in other times and places – is not directly addressed by the IV framework. There is nothing in IV formulas to explain why Vietnam-era service affects earnings; for that, you need a theory.[60]

You might wonder why we need monotonicity for the LATE theorem, an assumption that plays no role in the traditional simultaneous-equations framework with constant effects. A failure of monotonicity means the instrument pushes some people into treatment while pushing others out. Angrist, Imbens, and Rubin (1996) call the latter group defiers. Defiers complicate the link between LATE and the reduced form. To see why, go back to the step in the proof of the LATE theorem which shows the reduced form is

E[Yi|Zi = 1] – E[Yi|Zi = 0] = E[(yii – Yoi)(Dii – Doi)].

Without monotonicity, this is equal to

E[Yu – Yo i |Dii > D0i]P [Dii > Doi] – E [Yii – Yoi|Dii < Doi]P [dH < Doi],

We might therefore have a scenario where treatment effects are positive for everyone yet the reduced form is zero because effects on compliers are canceled out by effects on defiers. This doesn’t come up in a constant-effects model because the reduced form is always the constant effect times the first stage regardless of whether the first stage includes defiant behavior.[61]

A deeper understanding of LATE can be had by linking it to a workhorse of contemporary econometrics, the latent-index model for "dummy endogenous variables" like assignment to treatment. These models describe individual choices as determined by a comparison of partly observed and partly unknown (“latent”) utilities and costs (see, e. g., Heckman, 1978). Typically, these unobservables are thought of as being related to outcomes, in which case the treatment variable is said to be endogenous (though it is not really endogenous in a simultanenous-equations sense). For example (ignoring covariates), we might model veteran status as

1 if 7o + 7iZi > Vi 0 otherwise

where Vi is a random factor involving unobserved costs and benefits of military service assumed to be independent of Zi. This latent-index model characterizes potential treatment assignments as:

Doi = 1[7o > Vi] and Dii = 1[7o + 71 > Vi],

Note that in this model, monotonicity is automatically satisfied since 7i is a constant. Assuming 7i > 0, LATE can be written

E[yii – YoijDii > Doi] = E[yii – Yoi|7o + 71 > Vi > 7o],

which is a function of the latent first-stage parameters, 7o and 7i, as well as the joint distribution of yii—Yoi and Vi. This is not, in general, the same as the population average treatment effect, E[Yii—Yoi], or the

effect on the treated, Ey ц—Yoi|Di = 1]. We explore the distinction between different average causal effects in Section 4.4.2.

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