Combining Monotone Response and Selection Restrictions

Why Combine Response and Selection Restrictions?

When treatment is not randomly assigned, treated and untreated people may differ even before treatment. The observed difference in their outcome rates therefore need not equal the average treatment effect. Partial identification provides a disciplined alternative: impose explicit restrictions on potential outcomes, then determine which treatment effects remain compatible with those restrictions and the observed data.

This tutorial combines two such restrictions for a binary outcome. Monotone treatment response says treatment cannot lower anyone’s outcome. Monotone treatment selection says that, under either treatment state, the treated group’s potential outcomes are at least as favorable in distribution as the untreated group’s. Together, these assumptions constrain both within-person response and differences between the groups who select into treatment.

Model and Assumptions

Let \(D\in\{0,1\}\) indicate treatment, and let \(Y(1)\) and \(Y(0)\), each in \(\{0,1\}\), denote the potential outcomes under treatment and no treatment. The observed outcome is \(Y=DY(1)+(1-D)Y(0)\). Write \(\pi=P(D=1)\), and assume \(0<\pi<1\), so both treatment groups are represented. The target is the population average treatment effect \(E[Y(1)-Y(0)]\).

The observed outcome rates are \(a=E[Y\mid D=1]\) and \(b=E[Y\mid D=0]\). By consistency, \(a=E[Y(1)\mid D=1]\) and \(b=E[Y(0)\mid D=0]\). The other two group-specific potential-outcome means are not observed.

For a binary outcome, first-order stochastic dominance reduces to a comparison of success probabilities: the treated group has at least as high a probability of outcome 1 under each fixed treatment state. MTR is an individual-level assumption, not merely a statement that the average response is nonnegative. It implies a within-group ordering of potential-outcome probabilities, which is what enters the bounds below.

These restrictions are substantive, not consequences of the observed data. MTR rules out treatment harm for every person. MTS rules out a treated group that is less likely to have outcome 1 under either treatment state. Neither assumption says treatment was randomly assigned, and neither alone establishes that the observed difference is causal.

Translating the Assumptions into Inequalities

Define the two unobserved success probabilities \(x=E[Y(0)\mid D=1]\) and \(y=E[Y(1)\mid D=0]\). MTR implies that the treated group’s untreated success probability cannot exceed its treated success probability, so \(x\leq a\). Among untreated people, it implies \(b\leq y\). MTS compares the same potential outcome across treatment groups: for \(Y(0)\), it gives \(x\geq b\); for \(Y(1)\), it gives \(a\geq y\).

Thus the joint assumptions require \(a\geq b\). If the observed treated outcome rate is below the untreated rate, no population distribution can satisfy both MTR and MTS as defined here. If \(a\geq b\), the inequalities restrict the two unobserved rates to the same interval. The restrictions do not require \(x=y\): the two unknown rates refer to different potential outcomes.

Sharp Bounds for the Average Treatment Effect

The population effect is a weighted average of the treatment effects in the two treatment groups. Substituting the observed and unobserved rates gives

Both terms \(a-x\) and \(y-b\) are nonnegative under MTR. Their largest values under the restrictions are \(a-b\), because \(x\geq b\) and \(y\leq a\). Since \(\pi\) and \(1-\pi\) are nonnegative and sum to one, this yields the candidate bounds.

Why the Bounds Are Sharp

The inequalities above establish that no compatible effect can lie outside \([0,a-b]\). To show that the interval is sharp, it remains to show that its endpoints—and every value between them—can actually occur. For binary outcomes, potential outcomes can be coupled monotonically whenever their success probabilities are ordered: draw \(U\) uniformly on \([0,1]\), and set \(Y(t)=1\) when \(U\) is no greater than the desired success probability. If the probability for \(Y(1)\) is at least that for \(Y(0)\), this construction ensures \(Y(1)\geq Y(0)\).

This argument establishes sharpness for the population identified set, not a confidence interval. In an application, \(a\) and \(b\) are estimated from a sample; uncertainty about those rates must be addressed separately. The identification result itself describes what the assumptions and population rates permit.

Interpretation and Common Pitfall

The restrictions tighten the unrestricted range by linking potential outcomes across people and treatment groups. MTR rules out negative individual effects, which supplies the lower bound of zero. MTS, together with MTR, limits how much of the observed treated–untreated outcome gap can be attributed to treatment, giving the upper bound \(a-b\). The treatment share \(\pi\) matters in the decomposition, but cancels from these endpoints because each group’s effect can attain the same endpoint.

Common mistake: “The treated success rate exceeds the untreated rate by 20 percentage points, so the treatment effect is 20 percentage points.” This treats the observed difference as causal without accounting for selection. Under the stated assumptions, that difference is the upper bound; the lower bound is zero. A different mistake is to report bounds when \(a<b\). That reverses the restrictions rather than producing a valid interval: the observed rates then contradict the joint MTR and MTS assumptions, so at least one assumption or the model must be reconsidered.

These conclusions are conditional on the assumptions being credible for the application. In particular, MTS is a strong distributional claim about both potential outcomes across the groups defined by actual treatment receipt, including outcomes that are counterfactual for each group. The bounds do not establish that claim; they make its implications explicit.