Conditional LATE Within Covariate Strata

Why Condition the LATE on Covariates?

An instrumental-variable analysis can be performed within levels of observed covariates rather than treating the population as one undivided group. This matters when the instrument’s assignment is independent of potential outcomes only after conditioning on covariates, or when the first stage and treatment effect may differ across covariate groups. The result is a separate local average treatment effect for each stratum with instrument variation and a nonzero first stage.

Let $Z$ be a binary instrument, $D$ a binary treatment received, $Y$ an outcome, and $X$ a set of observed covariates. For a discrete stratum, write $X=x$. Define $D(z)$ as the treatment that would be received under instrument value $z$, and $Y(d)$ as the outcome under treatment value $d$. Assume consistency: the observed treatment is $D=D(Z)$, and the observed outcome is $Y=Y(D)$. In particular, the outcome does not depend directly on $Z$ under the exclusion restriction. Potential outcomes and treatment receipt are understood to be defined for each unit, with no interference between units.

Definition: A complier has $D(1)=1$ and $D(0)=0$. The conditional complier average causal effect (conditional LATE) in stratum $X=x$ is $$ \operatorname{LATE}(x)=E[Y(1)-Y(0)\mid D(1)>D(0),X=x]. $$ It is the average treatment effect for compliers in that stratum, not necessarily the effect for all individuals with $X=x$.

Conditions for Identifying the Stratum-Specific Effect

The conditional LATE result requires assumptions that apply within the stratum being analyzed. Conditional independence says the instrument is independent of the potential treatment receipts and potential outcomes after conditioning on $X$. Exclusion rules out a direct effect of the instrument on the outcome. Monotonicity rules out defiers, who would take treatment when $Z=0$ but not when $Z=1$. These are causal assumptions; observed group means alone cannot establish them.

Assumptions: For a stratum $X=x$:

Conditional independence: $(Y(0),Y(1),D(0),D(1))$ is independent of $Z$ conditional on $X=x$.
Exclusion and consistency: $Y=Y(D)$, with no direct effect of $Z$ on $Y$, and $D=D(Z)$.
Monotonicity: $D(1)\geq D(0)$ for every individual in the stratum.
Instrument support: $0<P(Z=1\mid X=x)<1$, so both instrument values occur with positive population probability in the stratum.
Conditional relevance: $E[D\mid Z=1,X=x]-E[D\mid Z=0,X=x]>0$.

Instrument support and conditional relevance are distinct. Support means both instrument groups exist in the stratum in the population, allowing the conditional outcome and treatment contrasts to be defined. Relevance means those groups have different average treatment receipt. Under monotonicity, a positive first stage implies a positive share of compliers. If the first stage is zero, the Wald ratio has a zero denominator and does not identify a conditional LATE, even if both instrument values occur.

Deriving the Conditional Wald Ratio

Within $X=x$, define the reduced-form outcome contrast and the first-stage treatment contrast as $$ \Delta_Y(x)=E[Y\mid Z=1,X=x]-E[Y\mid Z=0,X=x], $$ $$ \Delta_D(x)=E[D\mid Z=1,X=x]-E[D\mid Z=0,X=x]. $$ Conditional independence lets the instrument-group contrasts be written using the potential outcomes under each instrument value. By consistency and exclusion, the outcome under instrument value $z$ is $Y(D(z))$. Therefore,

\begin{aligned} \Delta_Y(x) &=E[Y(D(1))-Y(D(0))\mid X=x],\\ \Delta_D(x) &=E[D(1)-D(0)\mid X=x]. \end{aligned}

To see what these expressions contain, classify individuals by their potential treatment receipts. Under monotonicity, the only possible types are always-takers $(D(0)=D(1)=1)$, never-takers $(D(0)=D(1)=0)$, and compliers $(D(0)=0,D(1)=1)$. Always-takers and never-takers contribute zero to both contrasts because their treatment receipt does not change with $Z$. For a complier, the treatment changes from 0 to 1, so the individual’s contribution to the outcome contrast is $Y(1)-Y(0)$ and the contribution to the first stage is 1. Averaging yields

\begin{aligned} \Delta_Y(x) &=P(C\mid X=x)\,E[Y(1)-Y(0)\mid C,X=x],\\ \Delta_D(x) &=P(C\mid X=x), \end{aligned}

where $C$ denotes being a complier. The denominator is positive by conditional relevance. Dividing the first equation by the second gives the identification result:

Theorem: Under conditional independence, exclusion and consistency, monotonicity, instrument support, and a positive conditional first stage, $$ \operatorname{LATE}(x)= \frac{E[Y\mid Z=1,X=x]-E[Y\mid Z=0,X=x]} {E[D\mid Z=1,X=x]-E[D\mid Z=0,X=x]}. $$ The ratio identifies the average effect of treatment receipt for compliers with $X=x$.

The derivation also explains why the numerator is generally not the complier effect by itself. It is the complier effect multiplied by the share of compliers in the stratum. The first stage is that share under monotonicity, so the ratio rescales the outcome contrast.

Worked Example: Conditional LATE in Two Strata

Suppose a hypothetical randomized encouragement assigns $Z=1$ or $Z=0$ within each of two covariate strata, $X=A$ and $X=B$. The following are hypothetical population conditional means. Treatment is binary, and the outcome is measured in points.

Stratum	$E[D\mid Z=0,X]$	$E[D\mid Z=1,X]$	$E[Y\mid Z=0,X]$	$E[Y\mid Z=1,X]$
A	0.20	0.50	4.00	5.20
B	0.30	0.60	6.00	6.60

Assume instrument assignment was randomized within each stratum, supporting conditional independence. Suppose the encouragement has no direct effect on the outcome, and it is not plausible that anyone is discouraged from treatment by $Z=1$; these are the exclusion and monotonicity assumptions, respectively. These assumptions require substantive justification and are not proven by the table. Both instrument values occur in each stratum, and the treatment rates increase with $Z$, so the displayed contrasts show positive first stages.

For stratum A, the first stage is $0.50-0.20=0.30$, and the reduced-form contrast is $5.20-4.00=1.20$ points. Thus $$ \widehat{\operatorname{LATE}}(A)=\frac{1.20\text{ points}}{0.30}=4.00\text{ points}. $$ Under the assumptions, this is the average treatment effect for compliers in stratum A.

For stratum B, the first stage is $0.60-0.30=0.30$, and the reduced-form contrast is $6.60-6.00=0.60$ points. Therefore $$ \widehat{\operatorname{LATE}}(B)=\frac{0.60\text{ points}}{0.30}=2.00\text{ points}. $$ The conditional effects differ: 4.00 points for compliers in A and 2.00 points for compliers in B. The results do not imply that every individual in either stratum benefits by that amount, or that the effects apply to noncompliers.

Estimating the Contrasts and Checking Their Support

In sample data, estimate each conditional mean by its sample mean within the relevant instrument group and stratum. The sample conditional Wald ratio is the difference in the two outcome means divided by the difference in the two treatment proportions. Because $D$ is binary, its conditional mean is the treatment proportion. A ratio should not be calculated when the sample first stage is zero, and a very small estimated first stage can make the ratio unstable. This sample calculation estimates a population quantity; sampling variation means the estimated contrasts need not equal their population values.

Worked Example: Calculating Conditional Ratios From Sample Counts

In a second hypothetical study, 100 people in stratum A are observed at each value of $Z$. Among those with $Z=0$, 25 receive treatment and the outcomes sum to 400 points. Among those with $Z=1$, 55 receive treatment and the outcomes sum to 520 points. In stratum B, there are also 100 people at each instrument value: the $Z=0$ group has 30 treated people and an outcome sum of 600 points; the $Z=1$ group has 60 treated people and an outcome sum of 660 points.

For A, the treatment proportions are $25/100=0.25$ and $55/100=0.55$. The outcome means are $400/100=4.00$ and $520/100=5.20$ points. Hence the estimated first stage is $0.55-0.25=0.30$, the estimated reduced form is $5.20-4.00=1.20$ points, and the ratio is $1.20/0.30=4.00$ points.

For B, the treatment proportions are $30/100=0.30$ and $60/100=0.60$. The outcome means are $600/100=6.00$ and $660/100=6.60$ points. The estimated first stage is $0.60-0.30=0.30$, and the estimated reduced form is $6.60-6.00=0.60$ points. The ratio is $0.60/0.30=2.00$ points.

Each stratum has observed members at both instrument values and a positive sample first stage, so the sample contrasts can be calculated. This does not verify conditional independence, exclusion, or monotonicity. It also does not establish that the population first stages are positive with certainty; uncertainty in the estimates must be considered in an inferential analysis.

A useful diagnostic is to tabulate the number of observations at each instrument value within every stratum, then inspect the treatment proportions and outcome means. Sparse cells or an estimated first stage near zero flag weak support or an unstable ratio. A failure to observe one instrument group means the sample cannot estimate that stratum’s contrast. It does not, by itself, show that population instrument support fails: a finite sample may omit a group even when its population probability is positive. Population support fails only if $P(Z=z\mid X=x)=0$ for one instrument value.

Worked Example: No Observed Controls in a Stratum

Suppose a hypothetical sample contains eight people with $X=B$, and all eight have $Z=1$. The sample provides no $Z=0$ outcome mean or treatment proportion for stratum B, so neither $\Delta_Y(B)$, $\Delta_D(B)$, nor the sample Wald ratio can be computed from these observations.

This is a sample-support problem, not proof that population instrument support fails. For example, if the true probability of $Z=0$ given $X=B$ is $1/2$, the probability that all eight sampled individuals have $Z=1$ is $(1/2)^8=1/256$, approximately 0.00391. That event is unlikely but possible despite positive population probability for both instrument values.

If, instead, the population has $P(Z=0\mid X=B)=0$, then instrument support truly fails in B: there is no population comparison group, and the conditional Wald contrast is not identified from the instrument in that stratum. The observed sample alone cannot distinguish that structural failure from a finite-sample omission without additional information about the assignment mechanism.

Interpretation, Alternatives, and Limitations

The conditional Wald ratio is a transparent way to estimate a separate complier effect in each stratum. It can be compared with a pooled IV analysis, which uses a broader contrast and generally targets a different effect when stratum-specific first stages or effects vary. A regression-based IV analysis can incorporate covariates and may be useful for estimation, but its coefficient’s interpretation depends on the specification and assumptions; it is not automatically the same as every stratum-specific Wald ratio. Ordinary outcome regression or matching does not, by itself, address unmeasured confounding of treatment receipt in the way a valid instrument is intended to address it.

The conditional result is limited to strata with both instrument values supported and a positive first stage. If $X$ is continuous, exact strata may contain too few observations; imposing a model or grouping values changes the practical estimation problem and can add modeling assumptions. Even with adequate sample sizes, conditional independence, exclusion, and monotonicity are substantive assumptions, not consequences of a large first stage or a favorable diagnostic plot. The estimand also remains local: it concerns compliers in the particular stratum, and should not be described as the average effect for all members of that stratum.

Common Mistake

A common mistake is to treat a missing instrument group in a finite sample as proof that instrument positivity fails in the population. The correct conclusion is narrower: the available sample does not support estimating the within-stratum contrast. Population support fails only when the conditional probability of one instrument value is actually zero. Another mistake is to divide the outcome contrast by a treatment contrast without checking that the denominator is positive and interpreting the result as the effect for everyone. The ratio has the conditional complier interpretation only under the stated assumptions and positive first stage.

Key takeaway: Within a supported covariate stratum, conditional independence and exclusion turn instrument-group differences into potential-outcome contrasts; monotonicity makes the first stage the complier share. Their ratio identifies the average treatment effect among that stratum’s compliers.

Check Your Understanding

Answer each question, then compare with the solutions below.

In one stratum, the reduced-form contrast is 0.9 outcome points and the first stage is 0.30. What is the conditional Wald estimate, and whom does it describe?
Why is a positive first stage needed in addition to instrument support?
A sample has no $Z=0$ observations in a stratum. What can be concluded about estimating the stratum’s Wald ratio, and what cannot be concluded about population support?
Under monotonicity, why does the conditional first stage equal the proportion of compliers in the stratum?

Solutions. (1) $0.9/0.30=3.00$ outcome points, interpreted as the average effect for compliers in that stratum under the assumptions. (2) Support supplies both instrument groups, but if their treatment means are equal, the first-stage denominator is zero and the ratio cannot identify a complier effect. (3) The sample cannot estimate the within-stratum contrasts. The absence alone does not establish population support failure; that requires $P(Z=0\mid X=x)=0$. (4) Under monotonicity, always-takers and never-takers have $D(1)-D(0)=0$, while each complier has $D(1)-D(0)=1$. The conditional mean of that difference is therefore exactly the complier proportion.