Covariate-Conditional LATE and Its Aggregation

Why Condition LATE on Covariates?

A binary instrument may shift treatment differently across groups defined by pretreatment characteristics. The effect for compliers in one group can also differ from the effect for compliers in another. A single unconditional LATE can conceal both patterns. Conditioning on covariates makes the relevant comparison within groups, while aggregation clarifies which groups of compliers contribute to the overall effect.

The key distinction is between the effect among compliers in a covariate stratum and the average effect among all compliers in the target population. The latter is not generally the unweighted average of the stratum-specific effects: strata contribute according to how prevalent they are and how many compliers they contain.

Setup and Assumptions

Let \(Z\in\{0,1\}\) be a binary instrument, \(D\in\{0,1\}\) a binary treatment, \(Y\) an outcome, and \(X\) a vector of pretreatment covariates. Write \(D(z)\) for the treatment that would be received under instrument value \(z\), and \(Y(d)\) for the outcome under treatment value \(d\). The observed treatment and outcome satisfy consistency: \(D=D(Z)\) and \(Y=Y(D)\). Assume no interference between units and a well-defined treatment and instrument.

Assumptions for covariate-conditional LATE: For covariate values \(x\) in the target population:

Conditional instrument independence: \((Y(0),Y(1),D(0),D(1))\) is independent of \(Z\) given \(X=x\).
Exclusion: the instrument affects the outcome only through treatment, so the relevant potential outcomes are \(Y(0)\) and \(Y(1)\), not outcomes indexed directly by \(Z\).
Monotonicity: \(D(1)\geq D(0)\) for every unit; the instrument does not make anyone less likely to receive treatment in the potential-treatment sense.
Instrument positivity: \(0<P(Z=1\mid X=x)<1\), so both instrument values occur within the stratum.
Conditional relevance: \(E[D\mid Z=1,X=x]-E[D\mid Z=0,X=x]>0\) for strata where a conditional LATE is to be identified.

The principal strata are defined by \((D(0),D(1))\). Under monotonicity, units are never-takers \((0,0)\), always-takers \((1,1)\), or compliers \((0,1)\); defiers \((1,0)\) are ruled out. Denote the complier event by \(C=\{D(1)>D(0)\}\). Conditional instrument independence ensures that comparisons by \(Z\) within \(X=x\) compare the same distribution of these potential-treatment types and potential outcomes.

Conditional Wald Ratio

Within a stratum \(X=x\), define the conditional reduced-form contrast and first-stage contrast as

\Delta_Y(x)=E[Y\mid Z=1,X=x]-E[Y\mid Z=0,X=x], \qquad \Delta_D(x)=E[D\mid Z=1,X=x]-E[D\mid Z=0,X=x].

The first contrast measures how much the instrument changes the mean outcome in the stratum; the second measures how much it changes the treatment rate. Their ratio identifies the conditional LATE when the first-stage contrast is positive.

Definition: The covariate-conditional LATE is \( \tau_C(x)=E[Y(1)-Y(0)\mid C,X=x] \), the average treatment effect for compliers among units with covariates \(X=x\). Under the assumptions above and conditional relevance, \( \tau_C(x)=\Delta_Y(x)/\Delta_D(x) \).

The ratio is local in two respects: it concerns compliers, rather than everyone in the stratum, and it concerns compliers at the specified covariate value. It is not automatically the conditional average treatment effect for the whole stratum.

Why the Ratio Identifies a Complier Effect

Let \(\pi_C(x)=P(C\mid X=x)\) be the conditional complier share. Under monotonicity, treatment changes with the instrument only for compliers. Conditional independence and consistency therefore imply

\Delta_D(x) =E[D(1)-D(0)\mid X=x] =P(C\mid X=x) =\pi_C(x).

For the outcome contrast, exclusion means that changing \(Z\) can change \(Y\) only when it changes \(D\). Never-takers and always-takers have the same treatment under either instrument value, so they contribute zero to the conditional outcome contrast. Compliers contribute their treatment effect. Thus

\begin{aligned} \Delta_Y(x) &=E\!\left[(Y(D(1))-Y(D(0)))\mid X=x\right]\\ &=P(C\mid X=x)E[Y(1)-Y(0)\mid C,X=x]\\ &=\pi_C(x)\tau_C(x). \end{aligned}

Dividing the outcome contrast by the treatment contrast cancels the complier share:

\frac{\Delta_Y(x)}{\Delta_D(x)} = \frac{\pi_C(x)\tau_C(x)}{\pi_C(x)} =\tau_C(x), \qquad \pi_C(x)>0.

This also shows why relevance is essential: if \(\pi_C(x)=0\), the instrument induces no treatment change in that stratum, and the ratio is undefined. A conditional effect for compliers cannot be recovered there from that instrument.

Worked Example: Conditional LATE in One Stratum

Suppose \(X=x\) denotes a particular pretreatment risk group. In a hypothetical population, the outcome means are 12.0 units for \(Z=1\) and 10.5 units for \(Z=0\). Treatment rates are 0.45 and 0.20, respectively. Assume conditional instrument independence, exclusion, monotonicity, and instrument positivity hold in this group.

The conditional reduced form is \( \Delta_Y(x)=12.0-10.5=1.50 \) outcome units. The conditional first stage is \( \Delta_D(x)=0.45-0.20=0.25 \), a 25-percentage-point increase in treatment probability.

Therefore the conditional Wald ratio is \( \tau_C(x)=1.50/0.25=6.00 \) outcome units. Under the stated assumptions, this is the mean treatment effect for compliers in this risk group. The first stage also identifies the group’s complier share as 0.25. These numbers are hypothetical population contrasts, not results from a real study.

Aggregating Across Covariate Strata

Suppose first that \(X\) has a finite set of strata \(x_1,\ldots,x_K\). Write \(P(X=x_k)\) for the target population share in stratum \(k\), \(\pi_C(x_k)\) for its complier share, and \(\tau_C(x_k)\) for its conditional LATE. The share of the entire population who are compliers in stratum \(k\) is \(P(X=x_k)\pi_C(x_k)\). Consequently, the overall mean effect among all compliers is

\tau_C^{\mathrm{pop}} = \frac{\displaystyle\sum_{k=1}^K P(X=x_k)\pi_C(x_k)\tau_C(x_k)} {\displaystyle\sum_{k=1}^K P(X=x_k)\pi_C(x_k)}.

The denominator is the population share of compliers. The numerator adds each stratum’s contribution to the total complier effect, weighting its conditional effect by the stratum’s population share and its conditional probability of being a complier. Equivalently, the weight on \(\tau_C(x_k)\) among all compliers is \(P(X=x_k)\pi_C(x_k)\big/\sum_j P(X=x_j)\pi_C(x_j)\).

The formula also follows by standardizing the conditional contrasts to the target distribution of \(X\). The population-average first stage is the overall complier share; the population-average reduced form is the complier-share-weighted sum of conditional LATEs. Their ratio is the overall complier effect. For continuously distributed \(X\), replace the sums by expectations:

\tau_C^{\mathrm{pop}} = \frac{E[\pi_C(X)\tau_C(X)]}{E[\pi_C(X)]} = \frac{E[\Delta_Y(X)]}{E[\Delta_D(X)]}, \qquad E[\pi_C(X)]>0.

This is an aggregation over a specified target distribution of covariates. The standardization matters: changing the target distribution can change the aggregate effect, even if the conditional effects and complier shares stay the same.

Worked Example: Combining Two Strata

Consider a hypothetical target population with two equally prevalent strata, \(A\) and \(B\). Suppose the conditional complier shares and conditional LATEs are \(\pi_C(A)=0.20\), \(\tau_C(A)=4.00\) outcome units, \(\pi_C(B)=0.50\), and \(\tau_C(B)=10.0\) outcome units. Assume the binary-instrument assumptions hold in both strata.

The population share of stratum \(A\) compliers is \(0.50(0.20)=0.10\), and their contribution to the numerator is \(0.10(4.00)=0.40\). The corresponding values for stratum \(B\) are \(0.50(0.50)=0.25\) and \(0.25(10.0)=2.50\). Thus the total complier share is \(0.10+0.25=0.35\), and

\tau_C^{\mathrm{pop}} = \frac{0.40+2.50}{0.10+0.25} = \frac{2.90}{0.35} = 8.286\text{ outcome units}.

Among all compliers, the weight on stratum \(A\) is \(0.10/0.35=0.2857\), and the weight on stratum \(B\) is \(0.25/0.35=0.7143\). The same result is \(0.2857(4.00)+0.7143(10.0)=8.286\) outcome units. A simple average, \((4.00+10.0)/2=7.00\), would give each stratum equal weight even though most compliers come from stratum \(B\).

Why an Unadjusted Wald Ratio Can Differ

Conditional identification does not by itself guarantee that the raw, unconditional difference in outcomes by \(Z\), divided by the raw difference in treatment by \(Z\), identifies the target-population complier effect. If instrument assignment probabilities differ across covariate strata, the \(Z=1\) and \(Z=0\) groups can have different covariate distributions. Their raw outcome difference can then mix instrument effects with differences in stratum composition. Conditional contrasts avoid that comparison by comparing instrument values within \(X\); those contrasts can then be standardized to a stated target distribution.

Worked Example: Composition Can Distort a Raw Contrast

In a hypothetical population, \(P(X=A)=P(X=B)=0.50\). Within each stratum the instrument assumptions hold. Stratum \(A\) has complier share 0.20 and LATE 2.00 units; stratum \(B\) has complier share 0.80 and LATE 10.0 units. Let every unit’s untreated outcome be 0 in stratum \(A\) and 100 in stratum \(B\); never-takers and compliers are the only types, with compliers receiving the stated treatment effect. Assignment probabilities are \(P(Z=1\mid A)=0.90\) and \(P(Z=1\mid B)=0.10\), so both instrument values occur in both strata.

The conditional first stages are 0.20 in \(A\) and 0.80 in \(B\). The conditional reduced forms are \(0.20(2.00)=0.40\) units in \(A\) and \(0.80(10.0)=8.00\) units in \(B\). Standardizing to the equally weighted target population gives first stage \(0.50(0.20)+0.50(0.80)=0.50\) and reduced form \(0.50(0.40)+0.50(8.00)=4.20\) units. The target-population complier effect is therefore \(4.20/0.50=8.40\) units.

But the instrument groups have very different covariate compositions: \(P(A\mid Z=1)=0.90\) and \(P(A\mid Z=0)=0.10\). Their mean outcomes are \(E[Y\mid Z=1]=0.90(0.40)+0.10(108)=11.16\) and \(E[Y\mid Z=0]=0.10(0)+0.90(100)=90.0\). Their treatment means are \(E[D\mid Z=1]=0.90(0.20)+0.10(0.80)=0.26\) and \(E[D\mid Z=0]=0\). The raw Wald ratio is therefore \((11.16-90.0)/(0.26-0)=-303.2\) outcome units, not 8.40. The large negative raw contrast is driven by the different proportions of high-baseline-outcome stratum \(B\) in the two instrument groups; it is not the standardized complier effect.

Common Pitfall and Interpretation

Wrong reasoning: “There are two strata, so the overall LATE is the average of their two conditional LATEs.” This silently assigns equal weight to the strata, regardless of how many people they contain or what share of each stratum complies. The correction is to weight each conditional LATE by \(P(X=x)\pi_C(x)\), then divide by the total complier share. The resulting estimand is the average treatment effect among compliers in the specified target population, not the average effect for everyone and not necessarily the effect for compliers in a different population.

In applications, conditional LATEs may be difficult to estimate precisely in strata with few observations or weak first stages. That practical concern does not change the identification formula, but it is a reason to report the stratum-specific first stages and complier shares alongside the effects. A reported aggregate should state its target covariate distribution and make clear that its weights reflect the distribution of compliers, not just the distribution of the population.

Key takeaway: Under conditional independence, exclusion, monotonicity, positivity, and relevance, the conditional Wald ratio identifies the LATE for compliers at \(X=x\). The population complier effect is the average of those conditional LATEs weighted by each stratum’s population share times its conditional complier share.

Check Your Understanding

Use the definitions and aggregation results to answer the following.

What does conditional instrument relevance mean, and why is a conditional LATE not identified by the Wald ratio when the first stage is zero?
In a stratum, the reduced form is 2.4 outcome units and the first stage is 0.30. What is the conditional LATE, and what does it describe?
Two equally prevalent strata have complier shares 0.10 and 0.40, with conditional LATEs of 3 and 9 units. Which stratum receives more weight among all compliers, and what is the aggregate complier effect?
Why can the raw unconditional Wald ratio fail when instrument assignment probabilities vary with \(X\)?
Does a conditional LATE describe all units in its covariate stratum? Explain the role of compliance type.