Policy Effects from Marginal Treatment Effects

From Marginal Effects to Policy Effects

A marginal treatment effect (MTE) describes the average treatment gain for people at a particular margin of willingness or resistance to treatment. A policy that changes treatment propensity does not generally affect every person equally: under a threshold representation, it adds or removes treatment for specific ranges of resistance ranks. Integrating the MTE over those ranges gives the policy’s effect on mean outcomes.

The key is to keep track of both which ranks are affected and how many people lie at each rank. The latter requires a conditional distributional assumption: a resistance rank that is uniform in the population overall need not be uniform within covariate groups. The policy formulas below therefore state conditional uniformity explicitly.

Definition: Let $Y_1$ and $Y_0$ be potential outcomes under treatment and no treatment, and let $X$ be observed covariates. Let $U_D$ be a treatment-resistance rank. A threshold policy with propensity threshold $q(x)\in[0,1]$ assigns treatment according to $D_q=1\{U_D\leq q(X)\}$. Its marginal treatment effect is $MTE(x,u)=E[Y_1-Y_0\mid X=x,U_D=u]$. The policy’s treatment share conditional on $X=x$ is $q(x)$ when $U_D\mid X=x\sim U(0,1)$.

The threshold policy is a way to describe the treatment margin: people with ranks below the threshold receive treatment, while those above it do not. A policy change from $q_0(x)$ to $q_1(x)$ affects ranks between those two thresholds for covariate value $x$. If $q_1(x)>q_0(x)$, the policy adds treatment to ranks in $(q_0(x),q_1(x)]$; if $q_1(x)<q_0(x)$, it removes treatment from ranks in $(q_1(x),q_0(x)]$.

The Policy-Effect Formula

The formula compares policies in a fixed target population: the distribution of $X$ is held constant, and the policy changes outcomes only by changing treatment assignment. The potential outcomes themselves do not change when the policy changes. We also require the MTE to be defined and integrable over every resistance rank that either policy reaches.

Theorem: Suppose treatment under policy $j$ is $D_j=1\{U_D\leq q_j(X)\}$, for $j=0,1$, and $U_D\mid X=x\sim U(0,1)$ for every relevant $x$. Suppose observed outcomes under each policy follow $Y_j=Y_0+D_j(Y_1-Y_0)$, the target population’s distribution of $X$ is unchanged, and the MTE is integrable over the policy-relevant ranks. Then $$ E[Y_1^{\mathrm{pol}}-Y_0^{\mathrm{pol}}] = E_X\left[\int_{q_0(X)}^{q_1(X)}MTE(X,u)\,du\right]. $$ Here $Y_j^{\mathrm{pol}}$ denotes the outcome under policy $j$, not the potential outcome under treatment. The integral is signed: when $q_1(x)<q_0(x)$, the policy removes treatment and the integral is negative for a positive MTE.

If the policy increases the threshold for every $x$, the formula sums the gains for all newly treated ranks. If it decreases the threshold everywhere, it sums the outcome changes from withdrawing treatment. When some thresholds rise and others fall, the signed contributions are added across covariate groups.

Why the Integral Gives the Policy Effect

For a fixed $x$, the outcome under a threshold $q$ is $Y_0+1\{U_D\leq q\}(Y_1-Y_0)$. The expected outcome therefore equals the mean untreated potential outcome plus the average treatment gains for ranks up to $q$. Conditional uniformity makes the probability weight for a rank interval equal to its width.

\begin{aligned} E[Y^{\mathrm{pol}}_q\mid X=x] &=E[Y_0\mid X=x]+E[(Y_1-Y_0)1\{U_D\leq q\}\mid X=x]\\ &=E[Y_0\mid X=x]+\int_0^q E[Y_1-Y_0\mid X=x,U_D=u]\,du\\ &=E[Y_0\mid X=x]+\int_0^q MTE(x,u)\,du. \end{aligned}

Subtract the expression for $q_0(x)$ from the expression for $q_1(x)$. The common $E[Y_0\mid X=x]$ term cancels, leaving the integral from $q_0(x)$ to $q_1(x)$. Taking expectation over $X$ gives the theorem. This argument does not require $Y_0$ to be independent of $U_D$: the untreated-outcome term is included for every person and cancels in the policy comparison.

The uniformity condition is substantive. If $U_D\mid X=x$ has density $f_{U_D\mid X}(u\mid x)$ rather than a uniform density, then the corresponding threshold-policy gain is $\int_{q_0(x)}^{q_1(x)}MTE(x,u)f_{U_D\mid X}(u\mid x)\,du$, and the treatment share is $P(U_D\leq q(x)\mid X=x)$, not necessarily $q(x)$. A marginally uniform rank alone does not justify the simpler formulas.

Worked Examples

Worked Example: Integrating MTE Over an Expanded Threshold

Suppose the target population has no relevant covariates, $U_D\sim U(0,1)$, and the MTE is $MTE(u)=2+4u$ outcome units. A policy raises the threshold from $q_0=0.20$ to $q_1=0.60$. Calculate the change in mean outcome and the average gain per newly treated person.

The policy adds treatment to ranks from $0.20$ through $0.60$. Its mean outcome effect is

\begin{aligned} \Delta E[Y] &=\int_{0.20}^{0.60}(2+4u)\,du\\ &=[2u+2u^2]_{0.20}^{0.60}\\ &=(1.20+0.72)-(0.40+0.08)\\ &=1.44\text{ outcome units}. \end{aligned}

The treatment share rises by $0.60-0.20=0.40$, or 40 percentage points, under conditional uniformity. Dividing the outcome gain by the share increase gives $1.44/0.40=3.60$ outcome units per newly treated person. This is also the average MTE over the added interval: the interval’s midpoint is $0.40$, and $2+4(0.40)=3.60$. The policy’s total mean outcome gain is 1.44 units; 3.60 units is the average gain among the added treatment margin, not the gain for the whole population.

Worked Example: Covariate-Specific Policy Changes

Suppose half the target population is in group A and half in group B. Within each group, the resistance rank is uniform conditional on group. The initial threshold is $0.20$ in both groups. A new policy raises it to $0.60$ in group A and $0.30$ in group B. Let $MTE(A,u)=1+2u$ and $MTE(B,u)=4-u$, in outcome units. Find the overall change in mean outcome, the change in treatment share, and the effect per net added treatment.

For group A, the conditional outcome gain is

\begin{aligned} \Delta_A &=\int_{0.20}^{0.60}(1+2u)\,du\\ &=[u+u^2]_{0.20}^{0.60}\\ &=(0.60+0.36)-(0.20+0.04)=0.72. \end{aligned}

For group B, it is

\begin{aligned} \Delta_B &=\int_{0.20}^{0.30}(4-u)\,du\\ &=[4u-\tfrac12u^2]_{0.20}^{0.30}\\ &=(1.20-0.045)-(0.80-0.020)=0.375. \end{aligned}

Weighting by the group proportions gives the population effect: $0.5(0.72)+0.5(0.375)=0.5475$ outcome units. The population treatment share rises by $0.5(0.60-0.20)+0.5(0.30-0.20)=0.25$, or 25 percentage points. Thus, the mean gain per net added treatment is $0.5475/0.25=2.19$ outcome units. The calculation uses each group’s own MTE curve and affected rank interval; an overall change in treatment share alone would not reveal those margins.

Worked Example: Equal Net Treatment Shares, Different Margins

Suppose groups A and B each make up half the population, with conditional uniform resistance ranks. Initially, their thresholds are $0.30$ and $0.40$, respectively. A new policy raises group A’s threshold to $0.50$ and lowers group B’s to $0.20$. Let $MTE(A,u)=2+2u$ and $MTE(B,u)=5-3u$. Calculate the overall outcome change and explain why the unchanged overall treatment share does not imply an unchanged mean outcome.

The group A policy adds treatment on ranks $0.30$ to $0.50$:

\begin{aligned} \Delta_A &=\int_{0.30}^{0.50}(2+2u)\,du\\ &=[2u+u^2]_{0.30}^{0.50}\\ &=(1.00+0.25)-(0.60+0.09)=0.56. \end{aligned}

The group B policy removes treatment on ranks $0.20$ to $0.40$. With the policy change written as the signed integral from the old threshold to the new one,

\begin{aligned} \Delta_B &=\int_{0.40}^{0.20}(5-3u)\,du\\ &=[5u-\tfrac32u^2]_{0.40}^{0.20}\\ &=(1.00-0.06)-(2.00-0.24)=-0.82. \end{aligned}

The overall change is $0.5(0.56)+0.5(-0.82)=-0.13$ outcome units. The treatment-share changes cancel: $0.5(0.50-0.30)+0.5(0.20-0.40)=0$. But the policy adds treatment to one group’s margins and removes it from another’s, where MTEs differ; consequently, the mean outcome falls by 0.13 units despite no net change in the treatment share.

Interpreting Policy Weights and Identification

For a policy that only raises thresholds, the added treatment share in group $x$ is $q_1(x)-q_0(x)$, and the group’s conditional effect per added treated person is

\frac{\int_{q_0(x)}^{q_1(x)}MTE(x,u)\,du}{q_1(x)-q_0(x)}

when the denominator is positive. This is the average MTE over the specific interval that the policy adds. Across groups, the population effect weights those conditional gains by each group’s population share and its increase in treatment propensity. If the threshold falls, the signed integral instead describes the effect of withdrawing treatment from the removed margins.

The earlier material on continuous instruments and marginal treatment effects explains how instrument-induced variation can identify MTEs over supported resistance ranks, under the relevant instrumental-variable assumptions. Policy evaluation additionally requires that every rank interval used by the policy lies within the region where the MTE is identified or otherwise credibly specified. Extrapolating an estimated MTE beyond that region is not identification; it requires extra modeling assumptions.

These formulas evaluate threshold policies. A policy that randomizes among several instrument values can induce treatment probabilities that vary with $U_D$ in a more general way; its effect must use the resulting rank-specific change in treatment probability rather than automatically treating the average propensity as a single threshold. The threshold formula applies when the policy change genuinely has the threshold representation specified above.

Common Mistakes

Using only the overall treatment-share change. Two policies can produce the same net number of treated people but affect different groups and resistance ranks. Integrate the MTE over the relevant intervals before comparing their outcome effects.
Calling the ratio the total policy effect. The integral gives the population mean outcome change. Dividing by a positive treatment-share increase gives an average gain per added treatment, a different quantity with different units of interpretation.
Ignoring the direction of a threshold change. The integral from $q_0$ to $q_1$ is signed. Reversing the limits when treatment is withdrawn is essential to represent the outcome change correctly.
Assuming marginal uniformity is enough. $U_D\sim U(0,1)$ does not ensure $U_D\mid X=x\sim U(0,1)$. Without conditional uniformity, rank-interval widths are not conditional population shares, so the threshold-share and simple MTE-integral formulas do not follow.
Claiming the policy changes outcomes directly. The derivation assumes the policy affects outcomes through treatment assignment and leaves the target population and potential outcomes fixed. If a policy also changes other inputs or the population composition, those changes need additional terms.

Key takeaway: A threshold policy’s effect is the population-weighted, signed integral of the MTE over the resistance-rank margins whose treatment status changes. Conditional uniformity makes interval width equal the affected share; covariate-specific MTEs and thresholds determine which gains and losses enter the total.

Check Your Understanding

Use the threshold-policy assumptions stated above when answering.

If a threshold rises from $0.15$ to $0.45$, which resistance ranks receive treatment because of the change?
Why does $U_D\sim U(0,1)$ alone not justify using threshold width as the treatment-share change within every covariate group?
A policy increases treatment share by 10 percentage points and raises mean outcome by 0.30 units. What is the mean gain per added treatment, and how does it differ from the total mean effect?
Two groups have equal population shares. One policy raises the threshold by 0.10 in the first group and lowers it by 0.10 in the second. Must the overall treatment share change? Must the overall mean outcome change be zero? Explain.
What additional concern arises if a proposed policy uses MTE values for resistance ranks outside the instrument-supported region?