Deriving the Wald Estimator as a Complier Effect

Why Divide the Outcome Difference by the Treatment Difference?

An instrumental variable can change the probability of receiving a treatment without directly determining who takes it. Comparing outcomes across instrument levels therefore combines two changes: the instrument's effect on outcomes and its effect on treatment receipt. The Wald ratio divides the first change by the second. Under specific assumptions, this rescales the outcome contrast into the average causal effect for the people whose treatment changes when the instrument changes.

The key is to distinguish people by their potential treatment behavior. Some always receive treatment, some never receive it, and some comply with the instrument's assignment. Under monotonicity, there are no people who move in the opposite direction. The derivation below shows that always-takers and never-takers contribute nothing to either contrast, while compliers determine both.

Setup and Assumptions

Let $Z_i\in\{0,1\}$ be a binary instrument, with $Z_i=1$ denoting the instrument level expected to encourage treatment. Let $D_i\in\{0,1\}$ be the treatment and $Y_i$ an outcome. Write $D_i(z)$ for unit $i$'s treatment state if the instrument were set to $z$, and $Y_i(d)$ for the outcome if treatment were set to $d$. The observed treatment is $D_i=D_i(Z_i)$. With exclusion, the observed outcome is $Y_i=Y_i(D_i)$: it depends on the instrument only through treatment.

The argument uses the following assumptions. Independence means the instrument is independent of the joint collection of potential treatments and outcomes. Exclusion means changing the instrument while holding treatment fixed does not change the outcome. Consistency links observed values to the potential values under the realized instrument and treatment; the setup also presumes no interference between units. Monotonicity rules out defiers: no unit's treatment is reduced by moving from $Z=0$ to $Z=1$. Relevance requires that the instrument changes the population treatment rate.

Assumptions: For the binary-instrument, binary-treatment setup, assume (1) $Z_i$ is independent of $\{D_i(0),D_i(1),Y_i(0),Y_i(1)\}$; (2) exclusion, so $Y_i(z,d)=Y_i(d)$; (3) consistency and no interference; and (4) monotonicity, $D_i(1)\ge D_i(0)$ for every unit. Relevance is the condition $E[D_i\mid Z_i=1]-E[D_i\mid Z_i=0]\ne0$. In this setup monotonicity makes relevance equivalent to a positive proportion of compliers.

The ordered pair $(D_i(0),D_i(1))$ defines a unit's compliance type. Always-takers have pair $(1,1)$, never-takers have $(0,0)$, compliers have $(0,1)$, and defiers have $(1,0)$. Monotonicity rules out the last type. Define the complier average treatment effect as $E[Y_i(1)-Y_i(0)\mid D_i(1)>D_i(0)]$, provided the complier group has positive probability.

Definition: The population Wald ratio is the reduced-form outcome difference divided by the first-stage treatment difference: $$ \frac{E[Y_i\mid Z_i=1]-E[Y_i\mid Z_i=0]} {E[D_i\mid Z_i=1]-E[D_i\mid Z_i=0]}. $$ The numerator is the instrument's effect on the mean outcome, and the denominator is its effect on the mean treatment. The sample Wald estimator replaces these population conditional means with their sample counterparts.

Deriving the Complier Effect

First consider the numerator. Independence allows conditional means given $Z_i=z$ to be replaced by expectations over the same population of potential values. Exclusion and consistency then say that under instrument level $z$, the potential observed outcome is $Y_i(D_i(z))$. Thus the reduced form is the average difference between outcomes under the two instrument levels:

E[Y_i\mid Z_i=1]-E[Y_i\mid Z_i=0] = E\!\left[Y_i(D_i(1))-Y_i(D_i(0))\right].

Because treatment is binary, for any $d_0,d_1\in\{0,1\}$ and any two potential outcomes, the following identity holds:

Y_i(d_1)-Y_i(d_0) = (d_1-d_0)\{Y_i(1)-Y_i(0)\}.

If $d_1=d_0$, both sides are zero. If $(d_0,d_1)=(0,1)$, both sides equal $Y_i(1)-Y_i(0)$; if $(d_0,d_1)=(1,0)$, both sides equal its negative. Applying this identity to the potential treatment states gives

E[Y_i\mid Z_i=1]-E[Y_i\mid Z_i=0] = E\!\left[(D_i(1)-D_i(0))\{Y_i(1)-Y_i(0)\}\right].

Under monotonicity, $D_i(1)-D_i(0)$ is one for compliers and zero for always-takers and never-takers. Therefore, the expectation is the complier proportion multiplied by the average treatment effect among compliers. The first-stage difference has an analogous, simpler expression.

\begin{aligned} E[Y_i\mid Z_i=1]-E[Y_i\mid Z_i=0] &=P(C)\,E[Y_i(1)-Y_i(0)\mid C],\\ E[D_i\mid Z_i=1]-E[D_i\mid Z_i=0] &=E[D_i(1)-D_i(0)]\\ &=P(C), \end{aligned}

Here $C$ denotes the event that a unit is a complier. Relevance makes $P(C)>0$, so division is defined. Dividing the first expression by the second cancels the complier proportion.

Result: Under independence, exclusion, consistency, no interference, monotonicity, and relevance, the population Wald ratio equals the average treatment effect among compliers: $$ \frac{E[Y_i\mid Z_i=1]-E[Y_i\mid Z_i=0]} {E[D_i\mid Z_i=1]-E[D_i\mid Z_i=0]} = E[Y_i(1)-Y_i(0)\mid C]. $$

Worked Examples

Worked Example: Recovering a Complier Effect From Type Shares

Suppose a hypothetical encouragement design has 20% always-takers, 40% compliers, and 40% never-takers. Assume the instrument is independent of potential treatments and outcomes, exclusion holds, and there are no defiers. For simplicity, suppose the average outcome under treatment is 80 among always-takers and 50 under no treatment among never-takers. Among compliers, suppose the average outcome is 72 under treatment and 62 without treatment. Calculate the Wald ratio.

The conditional treatment rates are $0.20+0.40=0.60$ for $Z=1$, and $0.20$ for $Z=0$. Thus the first stage is

E[D\mid Z=1]-E[D\mid Z=0]=0.60-0.20=0.40.

For $Z=1$, always-takers receive treatment, compliers receive treatment, and never-takers do not. For $Z=0$, always-takers still receive treatment, compliers do not, and never-takers still do not. Using the stated group mean outcomes:

\begin{aligned} E[Y\mid Z=1]&=0.20(80)+0.40(72)+0.40(50)=64.8,\\ E[Y\mid Z=0]&=0.20(80)+0.40(62)+0.40(50)=60.8. \end{aligned}

The reduced form is $64.8-60.8=4.0$ outcome units. Therefore,

\text{Wald ratio}=\frac{4.0}{0.40}=10.0 \quad\text{outcome units per treatment change}.

The complier average treatment effect is $72-62=10.0$ outcome units, as the theorem predicts. The equality follows because the always-takers' and never-takers' treatment states do not change with the instrument; their mean outcomes therefore cancel from the reduced-form contrast. The assumptions in this hypothetical calculation are stipulated, not verified from the type shares alone.

Worked Example: Calculating the Sample Wald Estimator

In a hypothetical randomized encouragement study, suppose the $Z=1$ group has 100 participants, mean treatment $0.62$, and mean outcome 68.4 points. The $Z=0$ group also has 100 participants, mean treatment $0.22$, and mean outcome 64.8 points. Calculate the sample Wald estimator and state its interpretation under the assumptions.

The sample first-stage difference is $0.62-0.22=0.40$, or a 40-percentage-point increase in treatment receipt. The sample reduced-form difference is $68.4-64.8=3.6$ points. Substitution gives

\widehat{\tau}_{W} = \frac{\bar{Y}_{Z=1}-\bar{Y}_{Z=0}} {\bar{D}_{Z=1}-\bar{D}_{Z=0}} = \frac{68.4-64.8}{0.62-0.22} = \frac{3.6}{0.40} = 9.00.

The sample ratio is 9.00 outcome points per treatment change. If the population assumptions hold, it estimates the population average treatment effect for compliers. It is not, in a finite sample, generally identical to the unobserved average of individual treatment effects among the sample's compliers. The calculation alone also does not establish that the assumptions hold.

Worked Example: Compliers Can Have Different Treatment Effects

Consider a hypothetical population with 50% compliers, divided into two subgroups: 30% of the population are compliers with an average treatment effect of 6 dollars, and 20% are compliers with an average treatment effect of 14 dollars. The remaining 50% are always-takers or never-takers. Suppose the instrument is independent, exclusion holds, and monotonicity rules out defiers. Find the complier average treatment effect and the Wald ratio.

The average effect among all compliers weights each subgroup by its share of the complier population, not by its share of the full population. The two weights are $0.30/0.50=0.60$ and $0.20/0.50=0.40$. Hence

E[Y(1)-Y(0)\mid C] = 0.60(6)+0.40(14) = 3.6+5.6 = 9.2\text{ dollars}.

The reduced-form difference is the population shares of the complier subgroups multiplied by their respective effects:

0.30(6)+0.20(14)=1.8+2.8=4.6\text{ dollars}.

The first-stage difference is the total complier proportion, $0.30+0.20=0.50$. Therefore, the Wald ratio is $4.6/0.50=9.2$ dollars per treatment change, matching the complier average treatment effect. No assumption that every complier has the same effect is needed; the ratio recovers the average over the complier group.

Interpretation and Common Pitfalls

The equality is a population identification result: it says what the ratio of population contrasts represents if the assumptions hold. The sample ratio is an estimator of that quantity, not a direct observation of each person's compliance type or treatment effect. The outcome units carry through the ratio, while the treatment difference is a change in a treatment probability. For example, if outcomes are measured in dollars and the first stage is expressed as a probability difference, the Wald ratio is in dollars per unit change in treatment.

A common mistake is to call the ratio the average treatment effect for everyone. The derivation does not support that conclusion. The instrument changes treatment only for compliers under monotonicity, so the reduced form isolates their effects, and the ratio identifies their average—not the population average treatment effect unless additional conditions make those effects equal.

Another mistake is to say that monotonicity makes the outcome contrast positive for compliers. Monotonicity determines the sign of $D_i(1)-D_i(0)$; it does not determine the sign of $Y_i(1)-Y_i(0)$. Complier effects, and therefore the reduced form, may be positive, zero, or negative. If defiers are allowed, the reduced-form difference instead contains the complier share multiplied by the complier average effect, minus the defier share multiplied by the defier average effect. The signs of these terms depend on the corresponding average treatment effects; the first stage is the complier share minus the defier share. The simple cancellation giving the complier effect no longer follows.

Finally, a nonzero first stage is not a substitute for the other assumptions. Relevance permits division, but does not by itself ensure independence or exclusion. Likewise, a numerically large Wald ratio can arise when the first-stage difference is small; the algebraic identity does not guarantee precise estimation. The substantive design argument for instrument validity remains essential to interpreting the ratio causally.

Key takeaway: Independence and exclusion express the instrument contrasts in terms of potential states; monotonicity makes the first stage equal the complier proportion; relevance makes that proportion positive. Dividing then leaves exactly the average treatment effect among compliers.

Check Your Understanding

Work through these questions by tracking what changes across the two instrument levels.

Under monotonicity, what value does $D_i(1)-D_i(0)$ take for each of the three compliance types?
Which assumptions allow the reduced-form contrast to be written as $E[Y_i(D_i(1))-Y_i(D_i(0))]$?
If the complier share is 0.30 and their average treatment effect is $-4$ outcome units, what are the reduced form and first stage?
Does the Wald ratio require a constant treatment effect among compliers? Explain using the subgroup-weighting argument.
Why does relevance alone not establish that a sample Wald estimate has a causal interpretation?