Aggregating Stratum-Specific Wald Effects

Why Aggregation Changes the Wald Effect

A covariate can divide a population into groups with different instrument effects on treatment. Even when the same instrument and outcome are used in every group, the overall Wald ratio need not give equal importance to each group’s conditional effect. The groups’ population shares matter, but so do their first-stage magnitudes: a stratum where the instrument changes treatment more strongly can contribute more to the aggregate ratio.

There is also an important qualification. A ratio of aggregated reduced forms and first stages is a weighted average of stratum-specific Wald ratios only under suitable conditions. In particular, if a stratum has a zero first stage, its reduced form cannot simply be discarded without a reason. Conditional monotonicity supplies one such reason; conditional independence and exclusion alone do not.

Conditional Contrasts and the Target Population

Let $S$ denote a discrete covariate stratum, with possible values $s=1,\ldots,K$, and let $q_s=P(S=s)$ be the proportion of the target population in stratum $s$. Let $Z$ be a binary instrument, $D$ a binary treatment, and $Y$ the outcome. Write $D(z)$ for treatment under instrument value $z$, and $Y(d)$ for outcome under treatment value $d$.

Definition: The stratum-specific reduced form and first stage are, respectively, $$ \Delta_{Y,s}=E[Y\mid Z=1,S=s]-E[Y\mid Z=0,S=s], \qquad \Delta_{D,s}=E[D\mid Z=1,S=s]-E[D\mid Z=0,S=s]. $$ When $\Delta_{D,s}\ne 0$, the stratum-specific Wald ratio is $\tau_s=\Delta_{Y,s}/\Delta_{D,s}$. The standardized aggregate reduced form and first stage, for a target population with stratum shares $q_s$, are $\Delta_Y^{\mathrm{std}}=\sum_s q_s\Delta_{Y,s}$ and $\Delta_D^{\mathrm{std}}=\sum_s q_s\Delta_{D,s}$. Their ratio, when the aggregate first stage is nonzero, is the standardized aggregate Wald ratio $\tau^{\mathrm{std}}=\Delta_Y^{\mathrm{std}}/\Delta_D^{\mathrm{std}}$.

These standardized contrasts compare the instrument conditions within each stratum and then average those contrasts over one common stratum distribution. If $Z$ is independent of $S$, the ordinary marginal contrasts $E[Y\mid Z=1]-E[Y\mid Z=0]$ and $E[D\mid Z=1]-E[D\mid Z=0]$ equal these standardized contrasts. If instrument assignment probabilities vary across strata, the distributions of $S$ among those with $Z=1$ and $Z=0$ can differ. In that case, an unadjusted marginal contrast need not equal the standardized contrast, so the target and standardization must be stated.

For the conditional contrasts to have an instrumental-variable interpretation, suppose consistency holds, the instrument affects the outcome only through treatment (exclusion), and instrument assignment is conditionally independent of the relevant potential outcomes and treatments: $Z\perp (Y(0),Y(1),D(0),D(1))\mid S$. Also require positive probability of each instrument value within every stratum being analyzed. Under these conditions, the observed conditional contrasts identify the corresponding contrasts in potential outcomes. These assumptions are stated separately from monotonicity, which will be needed for the nonnegative-weight result.

Worked Example: Combining Two Stratum-Specific Wald Ratios

Suppose a hypothetical target population has two strata, with $q_1=0.60$ and $q_2=0.40$. Assume the instrument is randomized within each stratum, exclusion holds, and treatment is monotone within each stratum. Hypothetical conditional contrasts are $\Delta_{D,1}=0.20$, $\Delta_{Y,1}=0.08$, $\Delta_{D,2}=0.50$, and $\Delta_{Y,2}=0.30$.

The stratum-specific Wald ratios are $\tau_1=0.08/0.20=0.40$ outcome units per treatment change and $\tau_2=0.30/0.50=0.60$ outcome units per treatment change. The standardized reduced form is $0.60(0.08)+0.40(0.30)=0.048+0.120=0.168$ outcome units. The standardized first stage is $0.60(0.20)+0.40(0.50)=0.120+0.200=0.320$ treatment-probability units.

Thus the standardized aggregate Wald ratio is $\tau^{\mathrm{std}}=0.168/0.320=0.525$ outcome units per treatment change. The first stratum’s weight is $0.120/0.320=0.375$; the second’s is $0.200/0.320=0.625$. Accordingly, $0.375(0.40)+0.625(0.60)=0.525$. The second stratum receives more weight despite having a smaller population share, because its first stage is larger.

The assumptions support interpreting these as conditional IV contrasts; the arithmetic alone does not establish those assumptions. In particular, randomized instrument assignment within strata supports conditional independence, while exclusion and monotonicity remain substantive assumptions of this hypothetical example.

Deriving the Aggregation Identity

For each stratum with a nonzero first stage, the definition of $\tau_s$ gives $\Delta_{Y,s}=\Delta_{D,s}\tau_s$. Substituting this identity into the standardized numerator yields the basic aggregation formula when every stratum has a nonzero first stage.

\begin{aligned} \tau^{\mathrm{std}} &=\frac{\sum_s q_s\Delta_{Y,s}}{\sum_s q_s\Delta_{D,s}}\\ &=\frac{\sum_s q_s\Delta_{D,s}\tau_s}{\sum_s q_s\Delta_{D,s}}\\ &=\sum_s w_s\tau_s, \qquad w_s=\frac{q_s\Delta_{D,s}}{\sum_j q_j\Delta_{D,j}}. \end{aligned}

The identity is algebraic, but its interpretation depends on the signs and nonzero values of the first stages. If some $\Delta_{D,s}=0$, the ratio $\tau_s=\Delta_{Y,s}/\Delta_{D,s}$ is undefined there, and the substitution $\Delta_{Y,s}=\Delta_{D,s}\tau_s$ is not available. The exact decomposition in that case is

\tau^{\mathrm{std}}= \frac{ \sum_{s:\Delta_{D,s}\ne 0}q_s\Delta_{D,s}\tau_s+ \sum_{s:\Delta_{D,s}=0}q_s\Delta_{Y,s} }{ \sum_s q_s\Delta_{D,s} }.

Thus a zero-first-stage stratum contributes nothing to the aggregate denominator, but its reduced form may still contribute to the numerator. It can be omitted from the numerator only if an additional condition ensures its reduced form is zero.

When the Weights Form an Average

Conditional monotonicity means $D(1)\ge D(0)$ for every unit within each stratum. With a binary treatment, this rules out defiers in that stratum. Under conditional independence and this monotonicity assumption, the first stage is the proportion of compliers in the stratum, and so $\Delta_{D,s}\ge 0$. If the first stage is zero, there are no compliers there: monotonicity rules out any units whose treatment changes in either direction. Exclusion then implies that changing $Z$ changes no one’s outcome through treatment, and conditional independence yields $\Delta_{Y,s}=0$.

Consequently, under these assumptions, zero-first-stage strata contribute zero to both aggregate contrasts. If the aggregate first stage is positive, the weights for strata with positive first stages are nonnegative and sum to one: $\sum_{s:\Delta_{D,s}>0}w_s=1$. The aggregate ratio is therefore a weighted average of the stratum-specific Wald ratios for those strata. A stratum with zero first stage has no defined Wald ratio and receives no weight.

Result: With conditional independence, exclusion, and conditional monotonicity, the standardized aggregate Wald ratio is a convex weighted average of the defined stratum-specific Wald ratios. The weight for stratum $s$ is proportional to its population share times its first-stage effect, $q_s\Delta_{D,s}$. Strata with zero first stages have zero reduced forms under these assumptions and make no contribution to either aggregate contrast.

Conditional monotonicity is not needed for the algebraic identity when all stratum first stages are nonzero. Without it, however, a first stage can be negative, making some weights negative; the ratio then need not lie between the stratum-specific ratios. If a zero first stage occurs without monotonicity, its reduced form may be nonzero, and the weighted-average identity over the other strata can fail altogether.

Worked Example: First-Stage Strength Determines the Weights

Consider a second hypothetical population with two equally common strata: $q_1=q_2=0.50$. Suppose conditional contrasts are $\Delta_{D,1}=0.10$, $\Delta_{Y,1}=0.08$, $\Delta_{D,2}=0.60$, and $\Delta_{Y,2}=0.18$. Assume conditional independence, exclusion, and monotonicity hold in both strata.

The conditional Wald ratios are $\tau_1=0.08/0.10=0.80$ and $\tau_2=0.18/0.60=0.30$ outcome units per treatment change. The standardized reduced form is $0.50(0.08)+0.50(0.18)=0.040+0.090=0.130$ outcome units. The standardized first stage is $0.50(0.10)+0.50(0.60)=0.050+0.300=0.350$.

The aggregate ratio is $0.130/0.350=0.3714$ outcome units per treatment change. The weights are $0.050/0.350=0.1429$ for stratum 1 and $0.300/0.350=0.8571$ for stratum 2. Checking the weighted average, $0.1429(0.80)+0.8571(0.30)\approx0.1143+0.2571=0.3714$. Although the strata are equally common, the second stratum receives much more weight because its first stage is six times as large.

The contrasts alone give the arithmetic. The weighted-average interpretation additionally relies on the stated assumptions, including monotonicity to ensure nonnegative weights.

Why a Zero First Stage Needs Care

A zero conditional first stage means the average treatment rate is the same under the two instrument values in that stratum. Without monotonicity, it does not mean that no one’s treatment changes. Compliers and defiers can offset in the treatment contrast. If those groups have different treatment effects, their outcome changes need not offset. Exclusion does not eliminate this possibility: it rules out a direct effect of the instrument, not differences in the direction and size of treatment responses across individuals.

Worked Example: A Zero First Stage With a Nonzero Reduced Form

In a hypothetical stratum A, suppose half the units are compliers, with $D(1)-D(0)=1$ and $Y(1)-Y(0)=1$, and half are defiers, with $D(1)-D(0)=-1$ and $Y(1)-Y(0)=2$. Let the instrument be randomized in the stratum and let exclusion hold. The conditional first stage is $0.50(1)+0.50(-1)=0$.

Under exclusion, the reduced-form effect is the average of $[D(1)-D(0)][Y(1)-Y(0)]$. For these hypothetical units it is $0.50(1)(1)+0.50(-1)(2)=0.50-1.00=-0.50$ outcome units. Thus $\Delta_{D,A}=0$ while $\Delta_{Y,A}=-0.50$. The stratum-specific Wald ratio is undefined, and its reduced form cannot be omitted from an aggregate numerator.

To see the implication for aggregation, suppose equally many units belong to stratum A and to stratum B. In B, suppose the first stage is $0.40$ and the reduced form is $0.20$, so its Wald ratio is $0.20/0.40=0.50$. The standardized first stage is $0.50(0)+0.50(0.40)=0.20$, while the standardized reduced form is $0.50(-0.50)+0.50(0.20)=-0.15$. The aggregate ratio is $-0.15/0.20=-0.75$, not B’s Wald ratio of $0.50$. Conditional independence and exclusion hold in this setup, but monotonicity fails in A; the nonzero reduced-form contribution from A changes the aggregate result.

Crude Contrasts, Target Shares, and Interpretation

The $q_s$ weights describe a chosen target population. If the intended target has different stratum proportions, the standardized reduced form and first stage—and generally their ratio—can change. The first-stage factor in $q_s\Delta_{D,s}$ is also essential: population shares alone do not determine the weights.

The standardized aggregate Wald ratio should not automatically be reported as the effect for every person in the target population. It combines conditional instrumental-variable contrasts under the stated assumptions and gives more weight to strata with larger instrument-induced treatment changes. Nor should it be confused with a crude marginal Wald ratio when instrument assignment and stratum membership are associated. In that setting, the two instrument groups can have different stratum compositions, so their raw outcome and treatment differences may include compositional differences as well as within-stratum contrasts.

A reliable calculation therefore proceeds in order: define the target stratum shares, compute the conditional reduced forms and first stages, inspect zero and negative first stages, and only then form the aggregate ratio. A weighted-average interpretation requires checking the assumptions that make all relevant weights nonnegative and make zero-first-stage reduced forms vanish.

Common Mistake

Wrong reasoning: “A stratum has a zero first stage, so it contributes zero to both the denominator and numerator.”
Correction: A zero first stage always makes its denominator contribution zero, but does not by itself make its reduced form zero. Under conditional monotonicity together with conditional independence and exclusion, the zero first stage implies no treatment changers and hence a zero reduced form. Without such a condition, retain $q_s\Delta_{Y,s}$ in the numerator.
Another wrong reasoning: “The aggregate effect is the population-share-weighted mean of the stratum-specific ratios.”
Correction: When the convex weighted-average result applies, the weights are proportional to $q_s\Delta_{D,s}$, not merely $q_s$. A stronger first stage gives a stratum more weight.

Check Your Understanding

Answer using the definitions and conditions in this tutorial.

Two strata have shares $0.70$ and $0.30$, and first stages $0.10$ and $0.40$. What are their normalized aggregate weights if both first stages are nonnegative and the aggregate first stage is positive?
Why can the crude marginal reduced form differ from the standardized reduced form when instrument assignment probabilities vary by stratum?
Under conditional monotonicity, why does a zero first stage imply a zero reduced form when conditional independence and exclusion also hold?
Can a zero average first stage coexist with treatment changes for individuals? State the feature of compliance behavior that permits it.
What additional contribution must be retained in the aggregate numerator if a zero-first-stage stratum has a nonzero reduced form?