A Worked Partial-Identification Analysis for Binary IV

Why Binary IV May Bound Rather Than Identify the ATE

With a binary treatment and binary outcome, an instrument can reveal how outcomes change for people whose treatment responds to the instrument. That result is local: it concerns compliers, not necessarily everyone in the population. Even when the instrument identifies the complier average treatment effect, it may leave the population average treatment effect (ATE) only partially identified. The distinction matters whenever treatment effects differ across people or treatment uptake types.

This tutorial works through one hypothetical population. We first use instrument independence, exclusion, and monotonicity to bound the ATE. We then compare those bounds with wider bounds that require only consistency and a treatment-only potential-outcome model. The probabilities below are stipulated population probabilities, not estimates from a real study; sampling uncertainty is not part of the calculation.

Definition: Let \(Z\in\{0,1\}\) be an instrument, \(D\in\{0,1\}\) treatment receipt, and \(Y\in\{0,1\}\) the outcome. Write \(D(z)\) for treatment receipt if the instrument were set to \(z\), and \(Y(d)\) for the outcome if treatment were set to \(d\). The ATE is \(E[Y(1)-Y(0)]\). The complier average treatment effect, also called the local average treatment effect (LATE), is \(E[Y(1)-Y(0)\mid D(1)>D(0)]\).

The standard binary-IV assumptions used below are: consistency, so observed treatment and outcome agree with the corresponding potential values; instrument independence, \(Z\perp (Y(0),Y(1),D(0),D(1))\); exclusion, so the instrument affects the outcome only through treatment; and relevance, so treatment uptake differs across instrument values. Monotonicity adds \(D(1)\geq D(0)\) for every individual, ruling out defiers. Without monotonicity, the IV assumptions do not generally identify the LATE as a simple Wald ratio.

A Hypothetical Binary-IV Distribution

Suppose an encouragement instrument is equally common in the population. The following table gives the joint probabilities of treatment and outcome at each instrument value. Each pair of probabilities in a row sums to the probability of the displayed treatment status.

Instrument value	Treatment status	\(P(D,Y=1\mid Z)\)	\(P(D,Y=0\mid Z)\)	\(P(D\mid Z)\)
\(Z=0\)	\(D=0\)	0.28	0.42	0.70
\(Z=0\)	\(D=1\)	0.21	0.09	0.30
\(Z=1\)	\(D=0\)	0.20	0.20	0.40
\(Z=1\)	\(D=1\)	0.39	0.21	0.60

The corresponding outcome success rates are \(0.28/0.70=0.40\) among untreated people at \(Z=0\), \(0.21/0.30=0.70\) among treated people at \(Z=0\), \(0.20/0.40=0.50\) among untreated people at \(Z=1\), and \(0.39/0.60=0.65\) among treated people at \(Z=1\). These are conditional observed rates, not yet causal effects. Their causal interpretation requires the assumptions just stated.

Worked Example: Recovering the Complier Effect

Assume instrument independence, exclusion, relevance, and monotonicity. First calculate the shares of the three principal strata. Under monotonicity, the possible types are always-takers, with \(D(0)=D(1)=1\); never-takers, with \(D(0)=D(1)=0\); and compliers, with \(D(0)=0,D(1)=1\).

At \(Z=0\), treatment uptake is 0.30, which under monotonicity is the always-taker share. At \(Z=1\), uptake is 0.60, the combined always-taker and complier share. Therefore:

P(\text{always})=0.30,\qquad P(\text{complier})=0.60-0.30=0.30,\qquad P(\text{never})=1-0.60=0.40.

Among people untreated at \(Z=0\), the success probability is 0.28. They consist of never-takers and compliers. By independence, the never-takers' untreated success mass is \(0.40\times0.50=0.20\), using their observed success rate among people untreated at \(Z=1\). The remaining success mass, \(0.28-0.20=0.08\), belongs to compliers. Thus \(E[Y(0)\mid\text{complier}]=0.08/0.30=0.2667\).

At \(Z=1\), treated people consist of always-takers and compliers. The success mass among them is 0.39. Always-takers' treated success mass is \(0.30\times0.70=0.21\), using their observed treated success rate at \(Z=0\). The complier treated success mass is \(0.39-0.21=0.18\), so \(E[Y(1)\mid\text{complier}]=0.18/0.30=0.6000\). The resulting LATE is:

\text{LATE}=0.6000-0.2667=0.3333.

Under the assumptions, the instrument raises the outcome probability by about 0.333, or 33.33 percentage points, for compliers. This does not imply the same effect for always-takers or never-takers, so it does not by itself identify the ATE.

Bounding the ATE Under Monotonicity

The stratum shares and the two complier outcome means are identified under these assumptions. For always-takers, the treated outcome mean is observed, but their untreated outcome mean is not. For never-takers, the untreated outcome mean is observed, but their treated outcome mean is not. Because \(Y\) is binary, each missing mean can be anywhere from 0 to 1. Combining these limits with the identified complier effect yields ATE bounds.

Result: Under instrument independence, exclusion, relevance, monotonicity, consistency, and a binary outcome, the ATE is bounded by weighting each principal stratum's smallest and largest feasible average treatment effect by its population share. The bounds are sharp when the unobserved potential outcomes are otherwise unrestricted: each endpoint is attainable by assigning the missing binary outcomes to 0 or 1 as required.

Worked Example: ATE Bounds With Monotonicity

For always-takers, \(E[Y(1)\mid\text{always}]=0.70\), while \(E[Y(0)\mid\text{always}]\in[0,1]\). Their average effect therefore lies in \([0.70-1,\;0.70-0]=[-0.30,0.70]\). For never-takers, \(E[Y(0)\mid\text{never}]=0.50\), while \(E[Y(1)\mid\text{never}]\in[0,1]\), giving an effect in \([-0.50,0.50]\). The complier effect is 0.3333. Weighting by the stratum shares gives:

\begin{aligned} \text{ATE}_{\min} &=0.30(-0.30)+0.40(-0.50)+0.30(0.3333)\\ &=-0.0900-0.2000+0.1000=-0.1900,\\[4pt] \text{ATE}_{\max} &=0.30(0.70)+0.40(0.50)+0.30(0.3333)\\ &=0.2100+0.2000+0.1000=0.5100. \end{aligned}

Thus the sharp ATE interval under these assumptions is \([-0.190,0.510]\), in probability-point units. It includes both negative and positive effects. The data and assumptions do not determine even the sign of the population average effect, despite identifying a positive complier effect.

What Changes When IV Assumptions Are Weakened?

The width of an identified set reflects both the observed information and the restrictions placed on unobserved quantities. Here, even with the IV assumptions and monotonicity, 0.70 of the population are always-takers or never-takers. Their outcomes under the treatment status they do not take are unobserved. Those missing outcomes account for the remaining width: the always-taker contribution has width \(0.30(0.70-(-0.30))=0.30\), and the never-taker contribution has width \(0.40(0.50-(-0.50))=0.40\). Together they give \(0.70\), exactly the width of the ATE interval.

If monotonicity is removed but instrument independence and exclusion are retained, defiers may be present. The simple three-stratum decomposition above no longer applies: the observed treatment rates do not separate compliers from defiers. Sharp bounds in that model can be obtained by a linear program over the possible values of \((D(0),D(1),Y(0),Y(1))\), requiring the implied observable probabilities to match the table and minimizing or maximizing \(E[Y(1)-Y(0)]\). The bounds must contain any ATE compatible with the stronger monotonicity model, but they need not equal the broader bounds calculated below. Those broader bounds use fewer assumptions and are not claimed to be sharp for the full IV model without monotonicity.

Worked Example: Bounds With Consistency but Without Instrument Independence

Now retain a treatment-only potential-outcome model and consistency, but do not use instrument independence or monotonicity. The balanced instrument groups can be pooled to calculate the observed treated and untreated outcome masses:

\begin{aligned} P(D=1,Y=1)&=\tfrac12(0.21+0.39)=0.30,\\ P(D=1)&=\tfrac12(0.30+0.60)=0.45,\\ P(D=0,Y=1)&=\tfrac12(0.28+0.20)=0.24,\\ P(D=0)&=\tfrac12(0.70+0.40)=0.55. \end{aligned}

Consistency reveals \(Y(1)\) for treated people and \(Y(0)\) for untreated people. For binary outcomes, the missing potential outcomes are between 0 and 1. Consequently:

\begin{aligned} E[Y(1)]&\in[0.30,\;0.30+0.55]=[0.30,0.85],\\ E[Y(0)]&\in[0.24,\;0.24+0.45]=[0.24,0.69]. \end{aligned}

The lower ATE bound subtracts the largest feasible \(E[Y(0)]\) from the smallest feasible \(E[Y(1)]\); the upper bound subtracts the smallest feasible \(E[Y(0)]\) from the largest feasible \(E[Y(1)]\):

\text{ATE}\in[0.30-0.69,\;0.85-0.24]=[-0.39,0.61].

These consistency-only bounds are sharp under the stated weaker model: the missing binary potential outcomes can be set to the required extremes without contradicting the observed treatment-outcome pairs. The interval is wider than the monotonicity-IV interval. That comparison does not isolate monotonicity alone, because the weaker calculation also discards instrument independence. It does show how much the additional IV restrictions can narrow the range in this example.

Interpretation and Common Pitfalls

An identified interval is a set of population effects compatible with the data and the stated assumptions. It is not a confidence interval, and its endpoints do not describe sampling uncertainty. Here, the monotonicity-IV interval says that every ATE from \(-0.190\) to \(0.510\) is compatible with the specified model, and no ATE outside that interval is compatible with it. The interval's breadth means that the ATE's sign is not established under these assumptions.

A common mistake is to report the positive LATE of 0.3333 as the population ATE. The correction is to identify whose treatment status changes with the instrument: the LATE concerns compliers. The always-takers' and never-takers' missing potential outcomes still matter for the ATE. Another mistake is to call the consistency-only interval the sharp bound under all IV assumptions without monotonicity. It is sharp for the weaker model used in its calculation, but IV independence and exclusion impose additional restrictions; a sharp bound under those restrictions requires solving the corresponding latent-type problem.

Key takeaway: In binary IV analysis, the assumptions determine which principal-stratum outcomes can be recovered and which remain unknown. Report the estimand, the assumptions, and whether a bound is sharp for that model; do not substitute a complier effect for the ATE.

Check Your Understanding

Use the hypothetical table and results above to answer these questions.

Under monotonicity, why does the treatment rate at \(Z=0\) identify the always-taker share?
Which observed outcome cells are used to recover the complier means \(E[Y(0)\mid\text{complier}]\) and \(E[Y(1)\mid\text{complier}]\)?
Why does a positive LATE not establish that the ATE is positive in this example?
Calculate the width of the monotonicity-IV ATE interval and explain how the always-taker and never-taker contributions add to it.
Why should \([-0.39,0.61]\) not be described as the sharp bound under instrument independence and exclusion without monotonicity?