Marginal Treatment Effects from Instrumental Variation

Why Instrumental Variation Can Reveal More Than One Local Effect

A binary instrument can induce treatment for one group of people: those whose treatment status changes with that instrument. A continuously varying instrument can move treatment propensity by different amounts and at different levels. Under suitable assumptions, those movements reveal how treatment effects vary across people who differ in their latent willingness or resistance to take treatment.

The object of interest is the marginal treatment effect (MTE): the average treatment gain among people at a particular margin of latent resistance, conditional on observed covariates. It is “marginal” in the sense that it concerns people just at the boundary between taking and not taking treatment under a given propensity—not a derivative of an individual’s outcome.

Definition: Let $Y(1)$ and $Y(0)$ be a person’s potential outcomes under treatment and no treatment, and let $U$ be latent resistance to treatment, scaled to be uniform on $[0,1]$ conditional on covariates $X=x$. The marginal treatment effect at resistance level $u$ is $$ \operatorname{MTE}(u,x)=E[Y(1)-Y(0)\mid U=u,X=x]. $$ It is a conditional mean treatment effect for people at resistance margin $u$, measured in the units of the outcome.

The threshold representation makes the role of resistance precise. Write $Z$ for the instrument and $P(Z,X)=P(D=1\mid Z,X)$ for the treatment propensity, where $D$ is the observed treatment indicator. In the standard latent-index selection model,

D=1\{P(Z,X)\geq U\}.

A person with low $U$ is relatively willing to take treatment and is treated at lower propensities. A person with high $U$ requires a higher propensity. When the instrument shifts $P(Z,X)$ from one value to another, it changes treatment status for people whose $U$ lies between those values.

Assumptions Behind the Local IV Interpretation

The derivative interpretation is causal only under restrictions on the instrument and the treatment-selection process. State these restrictions before reading a response curve as evidence about treatment effects.

Consistency and no interference: the observed outcome is $Y=DY(1)+(1-D)Y(0)$, and one person’s treatment does not change another person’s potential outcomes.
Exclusion: conditional on $X$, the instrument affects the observed outcome only through treatment. The potential outcomes do not themselves vary with $Z$.
Conditional instrument independence: $(Y(0),Y(1),U)$ is independent of $Z$ given $X$. Thus, instrument changes do not select people with different potential outcomes or resistance, conditional on $X$.
Threshold selection and normalization: treatment follows $D=1\{P(Z,X)\geq U\}$, with $U\mid X=x$ uniform on $[0,1]$. This represents monotone selection into treatment as propensity rises; it rules out people whose treatment probability falls when the propensity rises.
Relevance and support: conditional on the covariates being studied, the instrument shifts the propensity over a range. To identify effects at an interior margin $u$, propensity values must be observed on both sides of $u$, locally.

These assumptions do not say that treatment is randomly assigned. People may differ in their potential outcomes and their resistance $U$. Instead, conditional instrument independence makes instrument-induced changes in propensity comparable across those latent types, while exclusion rules out a separate direct pathway from the instrument to the outcome.

From the Outcome Response to the MTE

For a fixed $x$, let $m(p,x)$ denote the conditional mean outcome associated with treatment propensity $p$. Under the assumptions above, the selection rule treats people with $U\leq p$. The conditional mean outcome can therefore be written as the no-treatment mean plus the gains of those selected into treatment:

m(p,x)=E[Y(0)\mid X=x] +\int_0^p \operatorname{MTE}(u,x)\,du.

The uniform distribution of $U$ is important here: the density of resistance over the interval is one, so the integral has no additional density weight. If resistance were expressed on a different scale, the corresponding density would appear in the integral.

Theorem: If $\operatorname{MTE}(u,x)$ is integrable over the relevant propensity range, then $m(p,x)$ is absolutely continuous in $p$, and $$ \frac{\partial m(p,x)}{\partial p}=\operatorname{MTE}(p,x) $$ for almost every $p$. At a particular interior value $p$, this pointwise equality holds if the MTE is continuous there. The derivative is identified only where the observed propensity support supplies local variation and the response curve can be differentiated.

The “almost every” qualification is substantive. An integral is unchanged if its integrand is altered at a single point, so the response curve alone cannot determine the value of a conditional-mean version of the MTE at every isolated resistance level. Continuity at the point of interest supplies the regularity needed for the pointwise derivative interpretation.

Write the outcome under a propensity threshold.
For a fixed propensity $p$, $D=1$ for people with $U\leq p$, so their outcome differs from $Y(0)$ by $Y(1)-Y(0)$.

Take the conditional expectation.
Conditional instrument independence and exclusion allow the response to be expressed using the same conditional potential-outcome distribution as the instrument varies. Uniform $U$ gives $m(p,x)=E[Y(0)\mid X=x]+\int_0^p \operatorname{MTE}(u,x)\,du$.

Differentiate where justified.
The fundamental theorem of calculus gives the integrand almost everywhere. If the MTE is continuous at an interior $p$, the derivative at that point equals $\operatorname{MTE}(p,x)$.

Worked Examples

Worked Example: Reading an MTE From a Propensity Response

Suppose, in a hypothetical population with fixed covariates $x$, the conditional outcome response is $m(p,x)=2+10p-4p^2$, where the outcome is measured in index points. Find the MTE at propensity $p=0.40$, assuming the response is generated by the threshold model and is differentiable there.

Step 1: Differentiate with respect to propensity. The derivative is $\partial m(p,x)/\partial p=10-8p$. Under the stated assumptions and the pointwise differentiability condition, this equals the MTE at that margin.

Step 2: Substitute the propensity. $\operatorname{MTE}(0.40,x)=10-8(0.40)=10-3.20=6.80$ index points.

Thus, people at resistance margin $U=0.40$, conditional on $X=x$, have an average treatment gain of 6.80 index points. This is not the average effect for everyone, nor does it imply that every individual at that margin gains exactly 6.80 points.

Worked Example: Converting an Instrument Derivative to an MTE

In a second hypothetical setting, suppose the propensity is $p(z)=0.20+0.10z$ over the relevant range of a continuous instrument $z$, and the conditional response is $m(p,x)=3+12p-5p^2$. Find the MTE at the margin reached when $z=3$, using derivatives with respect to both $z$ and $p$.

At $z=3$, the propensity is $p(3)=0.20+0.10(3)=0.50$. The MTE function obtained by differentiating with respect to propensity is $\partial m/\partial p=12-10p$, so $\operatorname{MTE}(0.50,x)=12-10(0.50)=12-5=7.00$ outcome units.

The chain rule gives the derivative with respect to the instrument: \[ \frac{\partial m(p(z),x)}{\partial z} =\frac{\partial m(p,x)}{\partial p}\frac{\partial p(z)}{\partial z}. \] Here $\partial p(z)/\partial z=0.10$, so the outcome response changes at rate $7.00(0.10)=0.700$ outcome units per instrument unit at $z=3$. Dividing this derivative by the propensity derivative recovers $0.700/0.10=7.00$ outcome units per resistance margin. The chain rule is essential: the derivative with respect to $z$ alone is not the MTE.

Worked Example: The Average Gain Over a Propensity Interval

Suppose a hypothetical MTE curve is $\operatorname{MTE}(u,x)=10-8u$ outcome units. What is the average treatment gain for people whose resistance lies between $0.20$ and $0.60$, the interval moved into treatment when propensity rises from $0.20$ to $0.60$?

The average gain over this interval is the integral of the MTE divided by the interval length:

\frac{1}{0.60-0.20}\int_{0.20}^{0.60}(10-8u)\,du =\frac{1}{0.40}\left[10u-4u^2\right]_{0.20}^{0.60} =\frac{4.56-1.84}{0.40} =\frac{2.72}{0.40} =6.80.

The arithmetic uses $10(0.60)-4(0.60)^2=4.56$ and $10(0.20)-4(0.20)^2=1.84$. The induced group’s average gain is therefore 6.80 outcome units. Because the MTE declines with resistance in this example, the gains at the lower-resistance end are larger than those at the higher-resistance end. This interval average is not the MTE at either endpoint.

What the Instrument Can and Cannot Reveal

The MTE interpretation is local in two senses. First, it concerns people at the resistance margin reached by a particular propensity, not necessarily the whole population. Second, the instrument identifies a range of margins only when the propensity has adequate support over that range. If observed propensities lie, for example, only between $0.20$ and $0.60$, derivatives outside that interval are not identified by these instrument movements without additional assumptions.

When the propensity changes by a finite amount, the induced group spans an interval of resistance values. Its average treatment effect is the average of the MTE over that interval under the uniform-resistance normalization. As the propensity interval narrows, the average approaches the MTE at the limiting margin when the MTE is continuous there. This is why small, continuous shifts can reveal a more finely localized pattern than one broad instrument contrast, but fine localization also depends on having enough support and a response curve that can be estimated reliably.

In applications, derivatives are harder to estimate than conditional means: small changes in propensity can produce noisy estimates of how the outcome response changes. A reported derivative should therefore be tied to its covariate values, propensity margin, outcome units, and available support. A smooth-looking curve is not, by itself, evidence that exclusion or instrument independence holds.

Common Mistake

Wrong reasoning: “The derivative of the outcome mean with respect to the instrument is the MTE.”
Correction: The MTE is the derivative with respect to treatment propensity, where the derivative exists under the required regularity. If the instrument changes propensity, the chain rule gives $\partial m/\partial z=\operatorname{MTE}(p,x)\,\partial p/\partial z$. Recovering the MTE requires accounting for the size of the propensity change, and the causal interpretation still depends on the IV assumptions.
Another pitfall: A differentiable response curve does not alone guarantee that its derivative equals a chosen pointwise version of the MTE at every margin. The equality holds almost everywhere for an integrable MTE; continuity of the MTE at the margin justifies the pointwise statement.

Key takeaway: Under exclusion, conditional instrument independence, threshold selection, and adequate propensity support, local IV identifies the MTE as the derivative of the conditional outcome response with respect to treatment propensity—almost everywhere, and pointwise where suitable continuity holds.

Check Your Understanding

Use the definitions and assumptions above to answer these questions.

In the threshold rule $D=1\{P(Z,X)\geq U\}$, which people are induced into treatment when propensity rises from $0.30$ to $0.50$?
Why is the derivative of the outcome response with respect to the instrument generally not equal to the MTE?
If $\operatorname{MTE}(u,x)=8-4u$, calculate the MTE at $u=0.25$, and state its units if the outcome is measured in dollars.
What does the “almost everywhere” qualification mean for the derivative result, and what condition can justify equality at a particular interior margin?
If the observed propensity support at a fixed $x$ runs only from $0.20$ to $0.60$, what prevents the local IV response from identifying an MTE at $u=0.80$ without further assumptions?