Instrument-Specific LATEs with Multiple Binary Instruments

Why the Instrument Contrast Matters

With one binary instrument, a Wald ratio can identify a local average treatment effect for people whose treatment changes when the instrument changes, under suitable assumptions. With several binary instruments, “the instrument changes” is not a complete description. One must specify which instrument changes and what values the others are held at. Different choices can induce treatment changes among different people, so they can define different causal effects.

This tutorial develops that contrast-specific interpretation. It does not presume a single complier population shared by all instruments. Instead, each effect is indexed by a particular change in one instrument at a fixed setting of the others.

Defining a Contrast-Specific Complier Population

Let \(Z=(Z_1,\ldots,Z_K)\) be a vector of \(K\) binary instruments, and let \(D\) be a binary treatment. Write \(D(z)\) for the treatment a person would receive under instrument vector \(z\). Write \(Y(1)\) and \(Y(0)\) for that person’s outcomes under treatment and control. The exclusion restriction means the instruments affect the outcome only through treatment, so the outcome under assignment \(z\) is \(Y(D(z))\).

Select an instrument \(j\), and fix a setting \(a\) of all the other instruments, denoted \(Z_{-j}=a\). The contrast compares \(z_j=0\) with \(z_j=1\), holding \(Z_{-j}=a\) constant. Define \(D(1,a)\) and \(D(0,a)\) as the potential treatments under those two instrument vectors.

The other two possible treatment-response patterns for this contrast are always-takers, with \(D(1,a)=D(0,a)=1\), and never-takers, with \(D(1,a)=D(0,a)=0\). A defier has \(D(1,a)=0\) and \(D(0,a)=1\). The contrast-specific monotonicity assumption rules out defiers for this particular contrast; it does not by itself say anything about treatment responses at other settings of the instruments.

Assumptions and Identification

The following assumptions give the contrast-specific Wald ratio its causal interpretation. They concern the instrument vector and the selected contrast, not merely the name of an instrument.

• Consistency and no interference: A person assigned \(Z=z\) receives \(D(z)\) and has observed outcome \(Y(D(z))\); one person’s assignment does not alter another person’s outcome.
• Exclusion: The outcome under an instrument assignment is \(Y(D(z))\). In particular, changing \(Z_j\) cannot directly change the outcome while treatment is held fixed.
• Joint instrument independence: \(Z\) is independent of the full collection of potential treatments \(\{D(z)\}\) and potential outcomes \(\{Y(0),Y(1)\}\). In a conditional design, the corresponding independence assumption may instead hold conditional on covariates.
• Contrast-specific monotonicity: \(D(1,a)\geq D(0,a)\) for every person. This excludes defiers for the selected contrast.
• Support and relevance: Both instrument configurations \((1,a)\) and \((0,a)\) occur with positive probability, and \(E[D(1,a)-D(0,a)]>0\). Support makes the observed contrast available; relevance ensures that it shifts treatment.

Under these conditions, the population outcome and treatment contrasts among the two instrument cells identify the LATE for \(C_j(a)=1\). The justification follows directly from the potential outcomes.

The first equality uses consistency and joint independence: within each instrument cell, the observed conditional mean equals the mean under that assignment. For the second, binary treatment and exclusion imply that an outcome changes only when the treatment changes. Under monotonicity, \(D(1,a)-D(0,a)\) is exactly the complier indicator, which gives the third equality. Applying the definition of a conditional mean gives the last equality. The same argument without the outcome replaces \(Y(D)\) by \(D\):

Dividing the outcome contrast by this positive first-stage contrast yields the identifying formula:

This formula is a population identification result. In a sample, cell means and the corresponding Wald ratio estimate its components; sampling uncertainty does not disappear simply because the assumptions hold.

Different Contrasts Can Shift Different People

The indicator \(C_j(a)\) depends on both the instrument \(j\) and the setting \(a\). Changing either can alter the set of people whose treatment responds. There is generally no reason for these sets to be identical, nested, or even overlapping. Consequently, a contrast-specific LATE is an average for its own complier population.

For example, the treatment response to \(Z_1\) may depend on \(Z_2\). A person could be a complier for \(Z_1\) when \(Z_2=0\) but not when \(Z_2=1\). Another person could respond only to \(Z_2\). These differences in treatment response are compatible with monotonicity for each contrast: monotonicity prohibits a treatment decrease when that contrast’s instrument is raised, but permits some people to respond in one setting and not another.

Reading Cell Contrasts Carefully

With two binary instruments, the instrument vector has four possible settings. A contrast for \(Z_1\) at \(Z_2=0\) uses the \((0,0)\) and \((1,0)\) cells; a contrast for \(Z_1\) at \(Z_2=1\) uses \((0,1)\) and \((1,1)\). These are separate comparisons. A comparison that changes both instruments at once is a different contrast and does not, without additional definitions and assumptions, equal either one-instrument-at-a-time LATE.

Common Pitfall: Treating All Compliers as One Group

Wrong reasoning: “Both instruments increase treatment, so their Wald ratios estimate the same effect among compliers.” This conclusion does not follow. The group that changes treatment under one instrument may differ from the group that changes treatment under another; even for one instrument, the group may depend on the settings of the other instruments.

Correction: Name the contrast before naming its complier population or LATE. Write \(C_j(a)\) and \(\tau_j(a)\), check relevance and monotonicity for that contrast, and interpret its ratio only for that population. Equality of two LATEs may occur, but it is not guaranteed by the fact that both are valid instrument contrasts.

The contrast index is substantively important, not bookkeeping. It states which variation identifies the effect and which treatment-responsive population the effect describes. Without that index, “the LATE” can conceal distinct causal estimands.