The Wald Estimate For A Binary Outcome And Instrument

From Instrument-Specific Means to a Causal Effect

With an instrumental variable, the assignment or encouragement may change the probability of receiving treatment without determining treatment for everyone. The Wald estimate uses that change in treatment and the corresponding change in outcome to estimate an effect for people whose treatment status is changed by the instrument. When the outcome is binary, the result is expressed on the risk-difference scale.

Let $Z$ be a binary instrument, with values 0 and 1; let $D$ be a binary treatment, with 0 meaning untreated and 1 meaning treated; and let $Y$ be a binary outcome, with 1 indicating that the outcome occurred. Write $\bar D_z$ and $\bar Y_z$ for the observed treatment proportion and outcome proportion among units with $Z=z$. The treatment contrast is the first stage, and the outcome contrast is the reduced form.

Definition: If the first stage is nonzero, the sample Wald estimate is $$ \hat{\tau}_{W} = \frac{\bar Y_1-\bar Y_0}{\bar D_1-\bar D_0}. $$ The numerator is the difference in outcome risks between instrument groups; the denominator is the difference in treatment rates. Because $Y$ is binary, the numerator is a risk difference. The ratio is measured in outcome-risk units per unit change in treatment probability, and is interpreted as a complier average causal effect only under the IV assumptions described below.

The numerator alone is not the treatment effect: it is the effect of changing the instrument on the outcome, which may operate through treatment. The denominator records how strongly the instrument changes treatment uptake. Dividing the two scales the outcome contrast by the induced treatment contrast.

Assumptions and the Complier Interpretation

For a precise interpretation, define $D(z)$ as the treatment status a unit would have under instrument value $z$, and $Y(d)$ as the outcome that would occur under treatment status $d$. The following assumptions support the standard complier interpretation.

Consistency and no interference: observed treatment and outcome agree with the potential values under the conditions actually received, and one unit’s instrument or treatment does not change another unit’s outcome.
Instrument independence: $Z$ is independent of the potential treatment and outcome values. Random assignment of the instrument is one way to support this assumption.
Exclusion restriction: the instrument affects the outcome only through treatment. In potential-outcome notation, the outcome under instrument value $z$ and treatment $d$ is $Y(d)$, with no additional dependence on $z$.
Monotonicity: $D(1)\geq D(0)$ for every unit. There are no defiers who would take treatment when $Z=0$ but not when $Z=1$.
Relevance: the instrument changes treatment probability, so $E[D\mid Z=1]-E[D\mid Z=0]\neq 0$.

Under monotonicity, units can be classified by their potential treatment responses. Always-takers have $D(1)=D(0)=1$; never-takers have $D(1)=D(0)=0$; and compliers have $D(1)=1,D(0)=0$. Relevance and monotonicity imply that the instrument changes treatment for a nonzero proportion of compliers.

Theorem: Under consistency, no interference, instrument independence, exclusion, monotonicity, and relevance, the population Wald ratio equals the complier average causal effect: $$ \frac{E[Y\mid Z=1]-E[Y\mid Z=0]} {E[D\mid Z=1]-E[D\mid Z=0]} = E[Y(1)-Y(0)\mid D(1)>D(0)]. $$ For a binary outcome, this is the average treatment effect among compliers on the risk-difference scale.

The result follows by considering the two instrument groups’ potential values. Independence allows observed group means to represent means of the corresponding potential quantities. By exclusion and consistency, the outcome under instrument value $z$ is $Y(D(z))$. Therefore, the population reduced form is

\begin{aligned} E[Y\mid Z=1]-E[Y\mid Z=0] &=E[Y(D(1))-Y(D(0))]\\ &=E[(D(1)-D(0))(Y(1)-Y(0))]. \end{aligned}

The second equality holds because, for binary treatment, the outcome difference is zero when $D(1)=D(0)$; when treatment changes, it is $Y(1)-Y(0)$ if $D(1)=1,D(0)=0$, and its negative for a defier. Monotonicity rules out defiers, so the reduced form is the probability of being a complier multiplied by the average treatment effect among compliers. The first stage is

E[D\mid Z=1]-E[D\mid Z=0] =E[D(1)-D(0)] =P(D(1)>D(0)).

Dividing the reduced form by this positive complier proportion gives the stated complier effect. This derivation also shows why the estimate is generally not the average treatment effect for the whole population: the instrument identifies an effect for the group whose treatment is shifted by it.

Worked Example: Calculating a Complier Risk Difference

Suppose a hypothetical randomized encouragement study records a binary treatment and a binary outcome. Among 500 people assigned $Z=1$, 300 receive treatment and 120 experience the outcome. Among 500 assigned $Z=0$, 200 receive treatment and 80 experience the outcome. Assume for interpretation that assignment was randomized, the encouragement has no direct route to the outcome, there are no defiers, and assignment changes treatment uptake.

The instrument-specific treatment rates are $\bar D_1=300/500=0.60$ and $\bar D_0=200/500=0.40$. The first stage is $0.60-0.40=0.20$, or a 20-percentage-point increase in treatment.

The instrument-specific outcome risks are $\bar Y_1=120/500=0.24$ and $\bar Y_0=80/500=0.16$. The reduced form is $0.24-0.16=0.08$, an 8-percentage-point increase in outcome risk.

Thus, $$ \hat{\tau}_{W}=\frac{0.24-0.16}{0.60-0.40} =\frac{0.08}{0.20}=0.40. $$

Under the stated IV assumptions, the estimated complier average causal effect is a 0.40 risk difference: treatment increases the outcome probability by 40 percentage points on average among compliers. This is not the 8-percentage-point intention-to-treat contrast, and it is not an odds ratio. The assumptions are substantive; the observed means alone cannot establish exclusion or monotonicity.

Approximate Uncertainty for the Wald Ratio

A standard error is needed to describe sampling uncertainty in the estimated ratio. A usual large-sample option is the delta method, which approximates the variance of a smooth function of estimated quantities. This approximation is most dependable when the first stage is clearly separated from zero and the sample is large enough for the group means to be approximately normal.

Let $\hat A=\bar Y_1-\bar Y_0$, $\hat B=\bar D_1-\bar D_0$, and $\hat\tau_W=\hat A/\hat B$. A first-order expansion of the ratio gives the approximate variance

\widehat{\operatorname{Var}}(\hat{\tau}_W) \approx \frac{ s^2_{Y-\hat\tau_W D,1}/n_1+ s^2_{Y-\hat\tau_W D,0}/n_0 }{ \hat B^2 },

where $n_z$ is the number of units with $Z=z$, and $s^2_{Y-\hat\tau_W D,z}$ is the sample variance of $Y-\hat\tau_WD$ within instrument group $z$. This form accounts for the fact that treatment and outcome for the same person may be related. It follows by approximating $\hat A/\hat B$ around the population ratio: its first-order fluctuation is proportional to the difference between group means of $Y-\tau_WD$, divided by the first stage. Replacing unknown quantities with estimates gives the displayed standard error formula.

An approximate 95% confidence interval is $\hat\tau_W\pm1.96SE(\hat\tau_W)$. Under the approximation’s conditions, the interval procedure has about 95% coverage over repeated samples; it does not mean there is a 95% probability that the fixed population effect lies in this particular interval.

Worked Example: A Delta-Method Interval

Continue the hypothetical study above. Suppose the joint counts of treatment and outcome are as follows. Within $Z=1$, 90 people have both $D=1,Y=1$, 210 have $D=1,Y=0$, 30 have $D=0,Y=1$, and 170 have $D=0,Y=0$. Within $Z=0$, the corresponding counts are 60, 140, 20, and 280. These counts reproduce the earlier treatment and outcome margins.

The estimate is $\hat\tau_W=0.40$. In group $Z=1$, the empirical variances and covariance are $\widehat{\operatorname{Var}}(Y)=0.24(0.76)=0.1824$, $\widehat{\operatorname{Var}}(D)=0.60(0.40)=0.2400$, and $\widehat{\operatorname{Cov}}(Y,D)=0.18-(0.24)(0.60)=0.0360$. Therefore, $$ \widehat{\operatorname{Var}}(Y-0.40D\mid Z=1) =0.1824+(0.40)^2(0.2400)-2(0.40)(0.0360) =0.1920. $$

For $Z=0$, the corresponding values are $0.16(0.84)=0.1344$, $0.40(0.60)=0.2400$, and $0.12-(0.16)(0.40)=0.0560$. Hence, $$ \widehat{\operatorname{Var}}(Y-0.40D\mid Z=0) =0.1344+(0.40)^2(0.2400)-2(0.40)(0.0560) =0.1280. $$

With $n_1=n_0=500$ and $\hat B=0.20$, the estimated variance and standard error are $$ \widehat{\operatorname{Var}}(\hat\tau_W) \approx\frac{0.1920/500+0.1280/500}{(0.20)^2} =0.01600,\qquad SE(\hat\tau_W)=\sqrt{0.01600}=0.1265. $$

The approximate 95% interval is $0.40\pm1.96(0.1265)$, or $[0.1521,0.6479]$. On the risk-difference scale, this corresponds to about a 15.2- to 64.8-percentage-point increase in outcome probability among compliers under the IV assumptions. The broad interval reflects substantial sampling uncertainty; the normal approximation should not be treated as dependable when the first stage is weak.

Reading the Result and Checking the Design

Worked Example: Unequal Instrument-Group Sizes

In another hypothetical randomized encouragement study, 600 people have $Z=1$, with 300 treated and 120 experiencing the binary outcome. Of 400 people with $Z=0$, 140 are treated and 42 experience the outcome. Assume random assignment, a plausible exclusion restriction, no defiers, and a nonzero first stage.

The instrument-specific treatment rates are $300/600=0.50$ and $140/400=0.35$, so the first stage is $0.15$. The outcome risks are $120/600=0.20$ and $42/400=0.105$, so the reduced form is $0.095$. Therefore, $$ \hat\tau_W=\frac{0.20-0.105}{0.50-0.35} =\frac{0.095}{0.15}=0.6333. $$

The estimated complier effect is a 63.3-percentage-point increase in outcome risk. The calculation uses rates within each instrument group, not the raw count difference; unequal group sizes make that distinction essential. This estimate is near the upper boundary for a risk difference, so it deserves careful uncertainty reporting and scrutiny of the assumptions. The means do not show whether the instrument has a direct effect, whether assignment was independent of potential outcomes, or whether defiers exist.

Useful checks begin with the two contrasts, not just their ratio. Report the instrument-specific treatment rates and outcome risks, the first stage, and the reduced form. Verify that the coding makes $Z=1$ the instrument condition expected to increase treatment; reversing the coding changes both contrasts’ signs but not their ratio. If the first stage is close to zero, division magnifies noise and the delta-method interval may be misleading. Detailed methods for inference with weak instruments require specialized procedures.

The Wald ratio is the just-identified instrumental-variables estimate with one binary instrument and one endogenous binary treatment, without additional covariate adjustment. It agrees with the corresponding simple two-stage least-squares coefficient in this setting. The intention-to-treat contrast is instead the reduced form, which answers the effect of instrument assignment. Neither the Wald ratio nor two-stage least squares repairs an invalid instrument: random assignment, exclusion, and monotonicity remain substantive requirements.

A binary outcome does not make the Wald ratio an odds ratio or a risk ratio. It is a risk difference for compliers. Finite-sample estimates or confidence limits can also fall outside the logical range from $-1$ to $1$; that is a warning about sampling variability or approximation, not evidence that a true binary-outcome risk difference can exceed those bounds.

Common Pitfalls and Limitations

Calling the reduced form the treatment effect. The outcome contrast compares instrument groups. It includes the effect of assignment through treatment and is not the complier effect unless the first stage equals one.
Interpreting the ratio for everyone. Under the assumptions, the ratio targets compliers, not automatically the population average treatment effect or an effect among the treated.
Checking only relevance. A nonzero first stage does not establish independence, exclusion, or monotonicity. These assumptions require design knowledge and substantive justification.
Reporting only a ratio. A ratio can hide a weak first stage or imprecise component contrasts. Show the group means and component contrasts alongside the estimate and its uncertainty.
Over-trusting a normal interval. The delta-method interval is approximate and can perform poorly when the denominator is near zero. A weak first stage calls for appropriate weak-instrument inference rather than confidence in a routine Wald interval.

Key takeaway: The Wald estimate is the instrument-induced change in outcome risk divided by the instrument-induced change in treatment probability. It estimates a complier risk difference only when the IV assumptions hold; the ratio alone cannot validate those assumptions.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions below.

A binary instrument changes treatment uptake from 25% to 45% and outcome risk from 10% to 16%. Calculate the Wald estimate and state its scale.
Why is the Wald ratio not generally the population average treatment effect?
In a randomized instrument study, what does the reduced form estimate, and what additional quantity is needed to form the Wald estimate?
Name two assumptions that cannot be verified just by comparing the instrument-specific means.

Solutions. (1) The first stage is $0.45-0.25=0.20$, and the reduced form is $0.16-0.10=0.06$. The Wald estimate is $0.06/0.20=0.30$, a 30-percentage-point estimated complier risk difference if the IV assumptions hold. (2) Under independence, exclusion, monotonicity, and relevance, the ratio identifies an effect among compliers, whose treatment is changed by the instrument; that group need not represent everyone. (3) The reduced form estimates the effect of instrument assignment on the outcome. Divide it by the first stage, the assignment effect on treatment uptake, to obtain the Wald ratio. (4) For example, exclusion and monotonicity are not established by the means; instrument independence also requires design or substantive support.