Marginal Treatment Effects (MTE)

Motivating Example#

A government is considering expanding a subsidized job training program. An earlier randomized encouragement design estimated a LATE of $3,200 in annual earnings gains for compliers -- those induced to participate by the randomized encouragement letter.

The policy question is: if the program is expanded by relaxing eligibility rules, will the next group of participants benefit as much as the original compliers? The program director assumes yes and budgets accordingly.

But the original compliers were people at the margin of participation -- those who needed only a gentle push (the encouragement letter) to enroll. The expansion targets people who did not enroll even with encouragement. These individuals have higher unobserved resistance to treatment -- perhaps they face greater barriers, have lower expected returns, or are less motivated.

The marginal treatment effect (MTE) framework, developed by Heckman and Vytlacil (2005), reveals that the treatment effect varies systematically with the propensity to participate. When MTE declines with unobserved resistance -- as it often does when individuals self-select based on expected gains -- the LATE of $3,200 overstates the benefit for the next marginal participant. The policy-relevant treatment effect for the expansion might be only $1,800.

Without MTE, the program director would have over-predicted benefits by 78%. With MTE, she can compute the correct marginal return and design the expansion accordingly.

A. Overview#

What Marginal Treatment Effects Does#

The MTE framework starts with a fundamental insight: in a world with treatment effect heterogeneity, different causal estimands -- ATE, ATT, LATE -- are all weighted averages of the same underlying object: the MTE curve.

The MTE is defined as:

where $U_D$ is the unobserved component of the selection decision, normalized to be uniform on $[0, 1]$. An individual selects into treatment when the propensity score $P(Z)$ exceeds their unobserved resistance:

Individuals with low $u_D$ (low resistance) are eager participants -- they select into treatment even when $P(Z)$ is low. Individuals with high $u_D$ (high resistance) are reluctant -- they participate only when $P(Z)$ is very high.

The MTE curve traces how the treatment effect varies across this spectrum of unobserved resistance.

The Unifying Framework#

Every conventional treatment effect parameter is a weighted average of MTE:

where $\omega^j(u)$ is the weight function specific to estimand $j$:

When MTE is flat (constant in $u_D$), all estimands are equal: LATE = ATE = ATT. This equality is the case of no essential heterogeneity. When MTE slopes downward (positive selection on gains), eager participants benefit more and ATT > ATE > ATU; LATE captures the effect somewhere in between, depending on the complier margin.

How It Differs from Standard IV#

Standard IV with a single instrument identifies a single number: the LATE for the complier subpopulation defined by that instrument. Different instruments identify different LATEs for different complier groups. MTE goes further by recovering the entire curve of treatment effects as a function of $u_D$, from which any target parameter can be computed as a weighted average.

Common Confusions#

B. Identification#

For MTE to be identified, three conditions must hold (Heckman & Vytlacil, 2005):

Assumption 1: Valid Instrument#

Plain language: The instrument $Z$ affects the outcome $Y$ only through its effect on treatment $D$. The instrument is relevant (it shifts $P(Z)$), exogenous (independent of potential outcomes and unobserved resistance), and satisfies the exclusion restriction.

Formally: $(Y(0), Y(1), U_D) \perp Z \mid X$ and $P(Z)$ is a nontrivial function of $Z$.

This assumption is the same requirement as for standard IV/LATE identification. The MTE framework does not weaken the instrument validity requirements.

Assumption 2: Threshold-Crossing Selection Model#

Plain language: Treatment take-up is determined by a threshold-crossing rule: an individual participates when the propensity score exceeds their unobserved resistance. This rule means there exists a single latent index that governs selection, and the instrument operates through this index.

Formally: $D = \mathbf{1}[P(Z) \geq U_D]$ where $U_D \sim \text{Uniform}(0, 1)$ after normalization. The selection equation can be derived from a latent utility model: $D = \mathbf{1}[\mu_D(Z) - V \geq 0]$ where $V$ represents unobserved costs/resistance, and $U_D = F_V(V)$ is the CDF transformation.

Assumption 3: Monotonicity of $P(Z)$ in $Z$#

Plain language: Increasing the instrument value (weakly) increases the probability of treatment for all individuals. There are no "defiers" -- individuals for whom a higher $Z$ reduces participation.

Formally: This condition is the standard IV monotonicity assumption, embedded in the threshold-crossing model. Because $D = \mathbf{1}[P(Z) \geq U_D]$ and $P(Z)$ is the same function for everyone, monotonicity is automatically satisfied.

The Local IV Identification Strategy#

The key identification result is:

The derivative of the conditional expectation of $Y$ with respect to the propensity score, evaluated at $p = u$, gives the MTE at $u_D = u$. Intuitively: a small increase in $P(Z)$ from $p$ to $p + dp$ induces participation by the marginal group with $U_D \approx p$. The corresponding change in the average outcome reveals the treatment effect for this marginal group.

This result means that variation in $P(Z)$ across different values of $Z$ traces out the MTE curve. Richer variation in the instrument (more values of $Z$, wider support of $P(Z)$) identifies the MTE over a larger portion of $[0, 1]$.

Estimation Procedure#

The Local IV Approach

The workhorse estimation procedure proceeds in three steps:

Step 1: Estimate the propensity score. Estimate $\hat{P}(Z) = \Pr(D = 1 \mid Z, X)$ using a probit or logit model.

Step 2: Estimate $E[Y \mid P(Z) = p]$ as a function of $p$. Use either:

• Parametric approach: regress $Y$ on $D$, $\hat{P}$, $\hat{P}^2$, $D \times \hat{P}$, $D \times \hat{P}^2$, and $X$
• Semiparametric approach: local polynomial regression of $Y$ on $\hat{P}$

Step 3: Differentiate to obtain MTE. Compute $\text{MTE}(u) = \partial E[Y \mid P = p] / \partial p$ evaluated at $p = u$:

• For the parametric approach: $\text{MTE}(u) = \hat{\beta}_D + \hat{\delta}_1 u + \hat{\delta}_2 u^2$
• For the semiparametric approach: numerical differentiation of the local polynomial fit

C. Visual Intuition#

The central object of the MTE framework is the MTE curve -- a plot of $\text{MTE}(u_D)$ against $u_D \in [0, 1]$. The shape of this curve reveals the nature of selection into treatment:

• Flat MTE: No essential heterogeneity. Treatment effects are homogeneous across the selection dimension. LATE = ATE = ATT. Standard IV is sufficient.
• Downward-sloping MTE: Positive selection on gains. Individuals who are most eager to participate (low $u_D$) benefit most. ATT > ATE > ATU. LATE overestimates ATE.
• Upward-sloping MTE: Negative selection on gains (rare). Reluctant participants benefit more. ATT < ATE < ATU.
• U-shaped or inverse-U MTE: Non-monotonic heterogeneity. The relationship between selection and gains is complex.

The weight functions determine how each estimand aggregates the MTE curve:

• ATE weights are uniform: every point on the MTE curve receives equal weight
• ATT weights tilt toward low $u_D$: the treated population is disproportionately composed of eager participants
• LATE weights are concentrated on the complier interval $[P(Z=z_0), P(Z=z_1)]$: only the margin shifted by the instrument matters
• PRTE weights depend on the specific policy: a program expansion to the next 10% weights the portion of the MTE curve corresponding to those marginal participants

D. Mathematical Derivation#

The key identification result of Heckman and Vytlacil (2005) is that the MTE is recovered as the derivative of the conditional expectation of $Y$ with respect to the propensity score $p = P(Z)$.

Begin with the outcome equation under the threshold-crossing model. Because $D = \mathbf{1}[P(Z) \geq U_D]$, the conditional expectation of $Y$ given $X = x$ and $P(Z) = p$ is:

The integral accumulates treatment effects for all individuals whose unobserved resistance $U_D$ falls below the threshold $p$ -- these are the individuals induced into treatment. Differentiating both sides with respect to $p$ yields:

Intuitively, a marginal increase in the propensity score from $p$ to $p + dp$ induces participation by individuals with $U_D \approx p$. The resulting change in the conditional expectation of $Y$ reveals the treatment effect for this marginal group -- precisely the MTE evaluated at $u_D = p$.

This derivative-based identification strategy is the foundation of the local IV estimator: estimate $E[Y \mid X, P]$ as a smooth function of $P$ (parametrically or semiparametrically), then differentiate to recover the MTE curve.

E. Implementation#

1# Using the ivmte package (Mogstad, Santos, Torgovitsky)
2library(ivmte)
3mte_fit <- ivmte(
4data = df,
5target = "ate",
6m0 = ~ x1 + x2 + u + I(u^2),
7m1 = ~ x1 + x2 + u + I(u^2),
8ivlike = y ~ treatment + x1 + x2,
9propensity = treatment ~ instrument + x1 + x2,
10instrument = ~ instrument
11)
12print(mte_fit)

F. Diagnostics#

Test for Essential Heterogeneity#

The first diagnostic question is whether MTE is flat. If it is, LATE = ATE and the MTE framework adds nothing beyond standard IV. Test by including $P(Z)$ interactions in the outcome equation:

A joint F-test on $(\delta_1, \delta_2)$ tests whether MTE varies with $u_D$. Rejection implies essential heterogeneity: LATE $\neq$ ATE.

Propensity Score Support#

The MTE is identified only over the support of $P(Z)$. Report:

• The range of $\hat{P}(Z)$ (e.g., the 1st and 99th percentiles)
• What fraction of $[0, 1]$ is covered
• Whether the target parameter (e.g., ATE, which requires full support) can be point-identified or only bounded

If the support is narrow, consider using the partial identification approach of Mogstad et al. (2018).

Visual Inspection of the MTE Curve#

Plot the estimated MTE curve with confidence bands. Look for:

• The slope: is MTE declining, rising, or flat?
• Confidence band width: is the MTE precisely estimated?
• Boundary behavior: are the endpoints of the identified region reliable?

Compare Estimands#

Compute ATE, ATT, ATU, and LATE from the estimated MTE curve. Large differences signal important treatment effect heterogeneity and indicate that LATE should not be interpreted as a general treatment effect.

Interpreting Your Results#

Reading the Output

The key outputs from an MTE analysis are:

What to Report

A well-reported MTE analysis should include:

1. The MTE curve with pointwise confidence bands
2. ATE, ATT, LATE computed from the MTE, with standard errors
3. PRTE for the specific policy change under consideration
4. Essential heterogeneity test (F-test on $P(Z)$ interactions)
5. Propensity score support -- range and coverage fraction
6. Sensitivity to polynomial order and bandwidth
7. First-stage diagnostics for the propensity score model
8. Discussion of the threshold-crossing model's plausibility

G. What Can Go Wrong#

H. Practice#

H.1 Concept Checks#

H.2 Guided Exercise#

H.3 Error Detective#

H.4 You Are the Referee#

I. Swap-In: When to Use Something Else#

• IV/2SLS: when you only need the LATE and do not need to extrapolate to other target parameters. If essential heterogeneity is not a concern (or not testable), standard IV is simpler and more robust.

• Matching: when selection is on observables and you want ATE or ATT. Matching addresses a different selection problem (selection on $X$) than MTE (selection on $U_D$).

• Causal Forests: when you want to estimate heterogeneous treatment effects as a function of observed covariates. Causal forests estimate CATE$(x)$ but do not address selection on unobservables.

• Heckman Selection Model: when the selection issue is sample selection (observing outcomes only for a non-random subsample) rather than treatment effect heterogeneity along the selection dimension. The Heckman model and MTE share the same structural foundation but answer different questions.

Limitations#

1. Requires sufficient variation in the propensity score. With a binary instrument, the MTE is identified only over the interval $[P(Z=0), P(Z=1)]$. A narrow support means ATE and ATT cannot be point-identified without parametric extrapolation.

2. Threshold-crossing model is restrictive. The model $D = \mathbf{1}[P(Z) \geq U_D]$ assumes a single latent index governs selection. This assumption may not hold with multiple treatments, strategic interactions, or complex decision processes.

3. Computationally intensive. Semiparametric MTE estimation requires local polynomial regression and numerical differentiation, which can be sensitive to bandwidth and polynomial order choices.

4. Requires a valid instrument. All the standard IV assumptions (relevance, exogeneity, exclusion restriction) must hold. MTE does not relax these requirements; it adds the threshold-crossing structure.

5. Precision can be poor. The MTE involves estimating a derivative (a second-order object), which is inherently noisier than estimating a conditional mean (a first-order object). Confidence bands on the MTE curve can be wide, especially near the boundaries of the propensity score support.

J. Reviewer Checklist#