Angrist Imbens and Rubin 1996 — Identification of Causal Effects Using Instrumental Variables Rejoinder

Summary

The Angrist, Imbens, and Rubin (AIR) rejoinder (Journal of the American Statistical Association (JASA) Vol. 91, No. 434, pp. 468–472) responds to four commentators — Heckman, Robins and Greenland, Moffitt, and Rosenbaum — defending the Local Average Treatment Effect (LATE) framework and clarifying key points of disagreement. The central argument is that compliers are the only subpopulation for which the data are directly informative, making LATE the only directly estimable causal effect in an instrumental variables (IV) context. The rejoinder formally shows that Heckman's preferred average treatment effect on the treated (ATT) identification assumption secretly requires always-takers and compliers to have equal treatment effects, and demonstrates that LATE equals ATT in one-sided noncompliance designs where the control group cannot receive treatment.

Key Claims

• Compliers as the only identified subpopulation: Always-takers are always observed treated and never-takers are never observed treated; the data contain no direct information about their treatment effects. Estimating ATT or average treatment effect (ATE) therefore requires extrapolating beyond what the data support — extensions to groups other than compliers are extrapolations, not identification.
• Heckman's A-2' smuggles in the answer: AIR formally show that Heckman's assumption (A-2') — under random assignment, exclusion, and monotonicity — implies $E[Y(1)-Y(0)\mid\text{always-taker}] = E[Y(1)-Y(0)\mid\text{complier}]$. Identifying ATT requires assuming that volunteers have the same average effect as draftees; this is precisely the strong assumption AIR avoided.
• Mean independence vs. full independence is not meaningful in practice: If mean independence holds but full independence fails, the instrument's validity becomes tied to the functional form of the regression — valid for $Y$ but not $\log(Y)$. This reintroduces the functional-form-dependence AIR sought to escape.
• One-sided noncompliance: $\text{LATE} = \text{ATT}$: In randomized eligibility designs where $D(0)=0$ for all units (controls cannot take treatment), there are no always-takers and monotonicity holds by construction. In this special case, LATE equals the ATT. Heckman (1995) acknowledged that this design "can be placed in an instrumental variables framework" identifying $E[\Delta\mid D=1,X]$ — which is exactly LATE.
• Bounds equivalence: Under monotonicity, the two unknown components of the population ATE are $E[Y(0)]$ for always-takers and $E[Y(1)]$ for never-takers. Letting these vary over the support of $Y$ gives sharp ATE bounds — AIR show these are equivalent to both the Balke-Pearl and the Robins-Manski bounds.
• Rosenbaum's weak exclusion restriction: The exclusion restriction for non-compliers need not even require defining the counterfactual potential outcome — it suffices to assume $Y(0,D(0)) = Y(1,D(1))$ for units with $D(0)=D(1)$, a weaker form.
• Rubin Causal Model (RCM) vs. switching regression: Roy (1951) and Quandt (1958) introduced potential outcomes implicitly but neither discussed causal effects. Tinbergen (1930) and Haavelmo (1944) are cited as economists who more clearly thought in potential-outcomes terms. The "priority" debate cuts both ways.
• IV assumptions are partially testable: Against Moffitt's claim that IV assumptions are untestable, AIR note that independence assumptions impose restrictions on the joint distribution of observables (see also Balke and Pearl 1993; Pearl 1996).

Concepts Introduced or Extended

• Local Average Treatment Effect
• Instrumental Variables
• Rubin Causal Model

Entities Mentioned

• Joshua Angrist
• Guido Imbens
• Donald Rubin
• James Heckman
• James Robins
• Sander Greenland
• Robert Moffitt
• Paul Rosenbaum

Quotes

"Compliers are the only group with members observed taking the treatment and members observed not taking the treatment. Always-takers are always observed taking the treatment, so the data simply cannot be informative about average treatment effects for this group."

"The only way to get average effects for always-takers and never-takers is to assume that their average treatment effects can be deduced from those for compliers, and this is exactly what Heckman has done in his assumptions without being explicit about it."

"We hope that our article will make statisticians more appreciative of the insights offered by the IV framework invented by econometricians, while making economists more aware of the benefits of causal inference conducted in the potential outcomes framework developed by statisticians."

My Take

The most technically important contribution here is the formal proof that Heckman's A-2' assumption implies always-takers equal compliers in treatment effects — this directly rebuts the claim that ATT identification is "assumption-free" relative to LATE. The one-sided noncompliance argument is also underappreciated: it shows LATE and ATT converge in the class of designs Heckman himself finds most credible (randomized eligibility). The bounds equivalence result is a useful clarification, linking the AIR framework to the Robins-Manski and Balke-Pearl bounds without requiring additional machinery. Rosenbaum's weak exclusion restriction point is subtle but practically important for designs where one treatment arm is purely a control condition.