MMP.15:4.3.1 - Use a binary instrument for a specified local effect

Use this branch when a binary assignment or encouragement (Z) changes actual action (A\in{0,1}), and an instrumental-variable argument might identify an effect of (A). The effect of offering an action and the effect of performing it are different targets. If the receiving question requires the population average effect, determine whether the local quantity below answers it before using the result.

Let (A(z)) be the action a unit would take under assignment (z). Start with (Y(z,a)) for its outcome under assignment (z) and action (a). Use the simple argument only under these assumptions:

• Consistency and stable versions: observed (A=A(Z)) and (Y=Y(Z,A)); the specified interventions have fixed meanings and no interference between units.
• Assignment independence and support: (Z) is independent of the potential actions and outcomes, and both assignment levels have positive probability in the population supplying the law.
• Exclusion: (Y(z,a)=Y(a)); assignment has no effect on the outcome except through actual action.
• Monotonicity: (A(1)\geq A(0)) for each unit. Assignment 1 never discourages a unit that would act under assignment 0.
• Relevance and finite means: (\delta_A=E[A\mid Z=1]-E[A\mid Z=0]>0), and the required outcome expectations exist.

These premises can be more demanding than random assignment alone. Randomizing an offer can support assignment independence without establishing exclusion or monotonicity. This is the binary local-effect argument of Angrist, Imbens and Rubin, 1996, §§2–4.

The response pair ((A(0),A(1))) separates always-takers ((1,1)), never-takers ((0,0)), compliers ((0,1)) and defiers ((1,0)). These are potential-response groups; observing one assignment and action generally does not label the individual’s group. Monotonicity excludes defiers.

To derive the result, use exclusion and the binary-action identity (Y(a)=Y(0)+a{Y(1)-Y(0)}). Independence and consistency give

[ \delta_Y=E[Y\mid Z=1]-E[Y\mid Z=0] =E[(A(1)-A(0))(Y(1)-Y(0))]. ]

The same argument gives (\delta_A=E[A(1)-A(0)]). Under monotonicity, that difference is one for compliers and zero for the other admitted groups. Consequently,

[ \frac{\delta_Y}{\delta_A} =E[Y(1)-Y(0)\mid A(1)>A(0)]. ]

This Wald ratio identifies the local average treatment effect for compliers under the named instrument and population. It is neither automatically the effect for everyone who took the action nor the population average effect. Changing the encouragement can change the complier group.

If (\delta_A=0), the ratio is undefined. A small nonzero population first stage can still identify the local quantity under the assumptions, while finite-sample uncertainty and sensitivity to premise violations can be large; MMP.13 must supply suitable inference for the receiving use. If defiers are possible, the numerator and denominator instead mix oppositely signed response-group contributions. A direct (Z)-to-(Y) route, assignment dependence or interference also requires a different argument. Conditional instrument validity, continuous action and transport to another population need their own derivations; the simple unadjusted ratio does not supply them.