Sharp Balke-Pearl Bounds for Binary Instrumental Variables

Why Bound the Effect Without Exclusion?

An instrumental variable can help address confounding of treatment and outcome, but the usual point-identification argument relies on an exclusion restriction: the instrument affects the outcome only through treatment. When that restriction is not assumed, a change in the instrument can affect the outcome directly. The observed association between instrument and outcome therefore cannot be attributed solely to treatment.

This tutorial studies what can still be learned with a binary instrument, treatment, and outcome when the instrument is independent of the potential responses and observed data obey consistency. The result is an interval of effects compatible with the observed probabilities and assumptions, rather than a point estimate. The bounds are sharp: no narrower interval is guaranteed by these assumptions alone.

Definition: Let \(Z\in\{0,1\}\) be the instrument, \(D\in\{0,1\}\) the treatment, and \(Y\in\{0,1\}\) the outcome. Write \(D(z)\) for the treatment that would be received under instrument value \(z\), and \(Y(d,z)\) for the outcome if treatment were set to \(d\) and the instrument to \(z\). The observed variables satisfy \(D=D(Z)\) and \(Y=Y(D,Z)\). With no exclusion restriction, \(Y(d,0)\) need not equal \(Y(d,1)\).

Because the outcome may depend directly on the instrument, there is no single potential outcome \(Y(d)\) that represents the outcome under treatment \(d\) while leaving the instrument irrelevant. We therefore define a clear target: the average effect of setting treatment to 1 rather than 0 while holding the instrument at its observed value, averaged over the population distribution of the instrument.

Definition: The target is the instrument-distribution-standardized treatment effect \(\theta=E[Y(1,Z)-Y(0,Z)]\). Equivalently, if \(q_z=P(Z=z)\), then \(\theta=\sum_{z=0}^1q_z E[Y(1,z)-Y(0,z)]\). This is a treatment effect with the instrument held fixed for each person; it allows the effect to differ between instrument levels.

The target is not the effect of changing the instrument, nor is it automatically the effect of treatment under an exclusion restriction. The instrument’s population distribution determines the weights \(q_z\). For this tutorial, assume both instrument levels occur with positive probability, so that the relevant conditional probabilities can be observed.

Assumptions and the Group-Specific Bound

The key independence assumption is that instrument assignment is independent of the full collection of potential responses. It rules out selection into instrument levels based on the treatment or outcome responses that would occur under either level. It does not require treatment to be independent of potential outcomes.

Assumptions: (1) Instrument independence: \(Z\) is independent of the joint vector containing \(D(0),D(1)\), and all \(Y(d,z)\). (2) Consistency: the observed values satisfy \(D=D(Z)\) and \(Y=Y(D,Z)\). (3) Positivity: \(P(Z=z)>0\) for \(z=0,1\). No exclusion restriction is imposed: outcomes may depend on \(Z\) even after treatment is fixed.

Within an instrument group \(z\), the observed data identify the fraction with each combination of treatment and outcome. Define \(a_z=P(D=1,Y=1\mid Z=z)\), \(b_z=P(D=1,Y=0\mid Z=z)\), \(c_z=P(D=0,Y=1\mid Z=z)\), and \(d_z=P(D=0,Y=0\mid Z=z)\). These four nonnegative probabilities sum to 1. Consistency reveals \(Y(1,z)\) for people observed with \(D=1\), and \(Y(0,z)\) for people observed with \(D=0\). The other potential outcome at that instrument level is missing for each person.

Since outcomes are binary, the unobserved outcomes can be either 0 or 1. Consequently, among people observed with \(D=0\), the unknown \(Y(1,z)\) can contribute anywhere from 0 to their full group proportion \(c_z+d_z\). Among people observed with \(D=1\), the unknown \(Y(0,z)\) can contribute from 0 to their full group proportion \(a_z+b_z\). These possibilities yield the group-specific bounds.

a_z-(c_z+a_z+b_z) \ \leq\ E[Y(1,z)-Y(0,z)]\ \leq\ (a_z+c_z+d_z)-c_z

Simplifying gives a compact form. The lower endpoint assigns 0 to the missing treated potential outcomes among those observed untreated, and 1 to the missing untreated potential outcomes among those observed treated. The upper endpoint makes the reverse assignments.

L_z=a_z-c_z-(a_z+b_z)=-(b_z+c_z), \qquad U_z=a_z+(c_z+d_z)-c_z=a_z+d_z.

These expressions can also be written using the observed treatment rate \(P(D=1\mid Z=z)=a_z+b_z\). The first form makes clear how each endpoint uses observed joint probabilities; the equivalent second form can be useful for checking arithmetic.

Combining Instrument Groups

The target averages the group-specific effects using \(q_z=P(Z=z)\). Because each group effect lies in its own interval, a lower bound for the average is the weighted average of the lower endpoints, and an upper bound is the weighted average of the upper endpoints.

Theorem: Under instrument independence, consistency, positivity, binary \(Z,D,Y\), and no further restrictions on potential responses, the sharp bounds for \(\theta\) are \(\sum_z q_zL_z\leq\theta\leq\sum_z q_zU_z\), where \(L_z=-(b_z+c_z)\) and \(U_z=a_z+d_z\).

A short justification establishes both validity and sharpness. In each instrument group, the observed outcomes fix the contributions for the treatment actually received. The missing potential outcomes are binary and unrestricted, so their smallest and largest possible contributions give \(L_z\) and \(U_z\). Since the target is a weighted sum, applying the group-specific lower or upper endpoint in every group gives the corresponding endpoint for \(\theta\).

These endpoints are attainable, not merely conservative. For the lower endpoint, assign the missing outcomes within each group as described above; for the upper endpoint, reverse those assignments. The resulting group-specific potential-response distributions reproduce the observed data. They can be coupled across instrument levels into a joint distribution of all potential responses, and that joint distribution can be made independent of the assigned \(Z\). Thus each endpoint is compatible with the assumptions and data. Intermediate values are also attainable by mixing compatible response distributions. No narrower interval follows without adding assumptions.

Worked Example: Calculating Sharp Bounds from a Probability Table

Suppose a hypothetical encouragement study reports the following conditional probabilities. Each row is conditional on the instrument value, and the four outcome-treatment cells in a row sum to 1.

Instrument value	\(P(D=1,Y=1\mid Z)\)	\(P(D=1,Y=0\mid Z)\)	\(P(D=0,Y=1\mid Z)\)	\(P(D=0,Y=0\mid Z)\)
0	0.10	0.30	0.20	0.40
1	0.60	0.20	0.10	0.10

Assume \(P(Z=0)=0.40\) and \(P(Z=1)=0.60\). For \(Z=0\), \(a_0=0.10\), \(b_0=0.30\), \(c_0=0.20\), and \(d_0=0.40\). Therefore,

L_0=-(b_0+c_0)=-(0.30+0.20)=-0.50, \qquad U_0=a_0+d_0=0.10+0.40=0.50.

For \(Z=1\), \(a_1=0.60\), \(b_1=0.20\), \(c_1=0.10\), and \(d_1=0.10\). Thus,

L_1=-(0.20+0.10)=-0.30, \qquad U_1=0.60+0.10=0.70.

Weight the endpoints by the instrument probabilities to bound the standardized effect:

\begin{aligned} L&=(0.40)(-0.50)+(0.60)(-0.30)=-0.20-0.18=-0.38,\\ U&=(0.40)(0.50)+(0.60)(0.70)=0.20+0.42=0.62. \end{aligned}

The sharp interval is \([-0.38,0.62]\). In this hypothetical population, the assumptions and table are compatible with an average treatment effect ranging from a decrease of 0.38 to an increase of 0.62 in the probability of the binary outcome. The interval includes zero, so these assumptions and data do not determine even the sign of the effect.

Reading the Bounds and Checking the Assumptions

The bounds are expressed in probability units because \(Y\) is binary. For example, an endpoint of \(-0.38\) represents a difference of 38 percentage points in average outcome probability, not a 38 percent relative change. A wide interval means that many assignments of the missing potential outcomes remain possible. Its width is not a confidence interval: it describes identification uncertainty under the stated assumptions, not sampling uncertainty.

The calculation uses the observed distribution of \((D,Y)\) within each instrument group and the marginal distribution of \(Z\). Instrument independence is needed to interpret these group-specific observed distributions as distributions of the corresponding potential responses in the target population. Consistency links the observed treatment and outcome to the potential responses. Neither assumption says that treatment receipt is randomized within an instrument group.

Worked Example: Checking a Group’s Endpoint by Missing Outcomes

Consider a hypothetical group with \(Z=0\), \(a_0=0.25\), \(b_0=0.15\), \(c_0=0.30\), and \(d_0=0.30\). The observed proportion treated is \(0.25+0.15=0.40\), and the observed proportion untreated is \(0.30+0.30=0.60\).

For the lower endpoint, the mean of \(Y(1,0)\) is at least the observed treated successes, \(0.25\), and the mean of \(Y(0,0)\) can be as large as the observed untreated successes plus all treated people, \(0.30+0.40=0.70\). Hence,

E[Y(1,0)-Y(0,0)]\geq 0.25-0.70=-0.45.

For the upper endpoint, the mean of \(Y(1,0)\) can be as large as \(0.25+0.60=0.85\), while the mean of \(Y(0,0)\) is at least the observed untreated successes, \(0.30\). Therefore,

E[Y(1,0)-Y(0,0)]\leq 0.85-0.30=0.55.

The formula gives the same result: \(L_0=-(0.15+0.30)=-0.45\) and \(U_0=0.25+0.30=0.55\). This cross-check connects the endpoints directly to the unobserved binary outcomes.

What These Bounds Do Not Claim

When the exclusion restriction is imposed, the response \(Y(d,z)\) must not depend on \(z\) once \(d\) is fixed. That restriction links potential outcomes across instrument levels and can produce different bounds. Here those cross-level restrictions are deliberately absent. The instrument can have a direct effect, so its relationship with the outcome cannot be used as though every outcome change came through treatment.

The bounds can be wide, including the full binary-outcome range in some cases. This is a substantive conclusion: independence and consistency alone may leave substantial uncertainty about the treatment effect. A narrower interval requires defensible additional information or assumptions; it cannot be obtained merely by choosing a more elaborate calculation on the same table.

Common Mistake

Wrong reasoning: “Because the instrument is independent, treatment is effectively randomized, so the difference in outcomes across instrument groups identifies the treatment effect.”
Correction: Instrument independence concerns the assignment of \(Z\), not the receipt of \(D\). Without exclusion, \(Z\) may also affect \(Y\) directly. The outcome contrast across instrument groups is therefore not generally a treatment effect. Use the group-specific missing-outcome bounds, then average them using the distribution of \(Z\).
Wrong reasoning: “The lower and upper endpoints are confidence limits for the treatment effect.”
Correction: They are identification bounds under the assumptions. Sampling uncertainty, if relevant, is a separate issue.

Key takeaway: Without exclusion, define the treatment effect while holding the instrument fixed. For binary \(Z,D,Y\), bound the effect within each instrument group using observed cell probabilities, then average the endpoints over the instrument distribution. These bounds are sharp under instrument independence and consistency alone.

Check Your Understanding

Use the definitions and formulas above to answer the following questions.

Why does allowing a direct effect of \(Z\) make it necessary to specify how the instrument is handled in the treatment-effect estimand?
For a fixed instrument level \(z\), explain which potential outcome is observed for people with \(D=0\), and which is missing.
If \(a_z=0.20\), \(b_z=0.10\), \(c_z=0.25\), and \(d_z=0.45\), calculate \(L_z\) and \(U_z\).
What does “sharp” mean for the interval, and why does assigning the missing binary outcomes to 0 or 1 establish endpoint attainability?
Why is an interval of identification bounds not a confidence interval?