Shape-Constrained Outcome Models for Counterfactual Bounds

Why Shape Restrictions Matter Beyond Common Support

A counterfactual mean is often required for covariate values that are rare or absent in the data used to estimate an outcome relationship. Functional-form restrictions can supply extrapolations, but their conclusions depend on how much structure is imposed. Shape restrictions offer a different approach: rather than specifying a complete formula for the outcome function, they constrain its direction or curvature. These restrictions can produce bounds without claiming that a particular linear, quadratic, or other parametric model is correct.

The key is to distinguish what the data identify from what the shape assumption implies. If the covariate values for a target population extend beyond the values observed in a treatment group, that group’s conditional outcome mean is not identified there by the data alone. Monotonicity or convexity can restrict its possible values, but ordinarily cannot determine one unique counterfactual mean.

Setup and Assumptions

Let \(D\in\{0,1\}\) denote treatment, \(X\) a scalar covariate, and \(Y(d)\) the potential outcome under treatment state \(d\). Define the conditional potential-outcome mean

Suppose the target population has covariate distribution \(F_T\). Its mean potential outcome under state \(d\) is

This tutorial concerns bounding \(m_d(x)\), and hence \(\mu_d^T\), at covariate values outside the observed support for treatment state \(d\). For an observational interpretation, consistency and conditional exchangeability, \(Y(d)\perp D\mid X\), allow the observed conditional mean in treatment group \(d\) to identify \(m_d(x)\) where \(P(D=d\mid X=x)>0\). That positivity condition may fail outside common support. The examples below treat the conditional means on the observed support as known population quantities, so they isolate the extrapolation problem rather than sampling uncertainty.

A shape restriction is an assumption about the conditional mean function, not necessarily about every individual’s potential outcome. Its credibility depends on the substantive setting and on the domain over which it is asserted. For example, a mean that rises over the observed covariate range need not continue rising outside that range; nondecreasing behavior beyond the boundary must be justified, not inferred just from a few observed values.

Bounds From Monotonicity

Suppose \(m_d(x)\) is nondecreasing and the right endpoint \(b\) of the observed support has conditional mean \(m_d(b)\). For any target value \(x>b\), monotonicity implies a lower bound:

Monotonicity alone gives no finite upper bound: a nondecreasing function can rise by an arbitrarily large amount after \(b\). If the outcome is known to lie in \([L,U]\), the bounds become \(m_d(x)\leq U\) and \(m_d(x)\geq L\), so the combined interval is \([m_d(b),U]\), provided \(m_d(b)\leq U\). If the function is nonincreasing, the direction reverses: \(m_d(x)\leq m_d(b)\), and an outcome lower bound can provide the other endpoint.

Convexity and Secant Extrapolation

Convexity can impose more than monotonicity because it restricts how slopes evolve. Suppose \(a<b\) are observed covariate values and \(x>b\). For a convex function, the slope from \(b\) to \(x\) cannot be smaller than the slope from \(a\) to \(b\). Rearranging that restriction gives a lower bound on the extrapolated mean:

This is a linear extrapolation of the last observed secant, but it is a bound implied by convexity, not an assertion that the function is linear. Convexity puts no finite upper bound on the function without further restrictions. If \(m_d\) is also nondecreasing, a valid lower bound can combine the monotonicity and convexity implications:

With a known outcome range \([L,U]\), \(U\) supplies an upper bound. If the convexity extrapolation exceeds \(U\), the assumptions are incompatible at that target value: there is no function satisfying all the restrictions there.

Concavity Gives an Upper Extrapolation Bound

For a concave function, secant slopes cannot increase as intervals move to the right. Therefore, for \(x>b\),

Concavity thus yields an upper bound, rather than the convexity lower bound. If the mean is also nondecreasing, \(m_d(b)\) is a lower bound. The combination is useful when the observed data show diminishing increases, but it depends on assuming that diminishing increases continue past the observed support. An outcome-range restriction can further tighten either endpoint.

From Pointwise Bounds to a Target-Population Mean

A target mean averages conditional means over the target covariate distribution. If valid pointwise bounds satisfy \(\ell(x)\leq m_d(x)\leq u(x)\) throughout the target support, integration preserves the inequalities:

This follows because \(u(x)-m_d(x)\geq0\) and \(m_d(x)-\ell(x)\geq0\) at every target value; expectations of nonnegative quantities are nonnegative. For a discrete target distribution with values \(x_j\) and probabilities \(w_j\), where \(w_j\geq0\) and \(\sum_jw_j=1\), the bounds are weighted sums of the corresponding pointwise bounds.

If some target covariate values lie within the support where the conditional mean is identified, use those identified means directly and bound only the unsupported portion. If bounds are available for both treatment states, subtraction gives a bound for the target average treatment effect: if \(\mu_1^T\in[L_1,U_1]\) and \(\mu_0^T\in[L_0,U_0]\), then \(\mu_1^T-\mu_0^T\in[L_1-U_0,U_1-L_0]\). This subtraction is valid but need not be the tightest possible when restrictions link the two potential-outcome functions.

Integrating pointwise bounds always gives valid bounds, but they need not be sharp. Pointwise extrema at different covariate values may require mutually incompatible functions under a common shape restriction. The resulting interval is then an outer bound: it contains the values permitted by the joint restrictions, but may be wider than the sharp identified set. Bound construction should therefore distinguish validity from sharpness.

Assumption Checks and a Common Pitfall

Shape restrictions should be checked against the observed conditional means wherever possible. For example, increasing secant slopes are required for convexity, and decreasing secant slopes for concavity. A violation in observed population means rules out the corresponding restriction on that range. In finite samples, estimates can fluctuate around the underlying shape, so sample slope violations are not by themselves a definitive population-level test; the examples here deliberately avoid that sampling issue.

• Pitfall: treating monotonicity as a two-sided bound. If \(m_d\) is nondecreasing and \(x>b\), it is correct that \(m_d(x)\geq m_d(b)\). It is not correct to conclude that \(m_d(x)\leq m_d(b)\), or that monotonicity alone supplies a finite upper endpoint. A known outcome ceiling or another defensible restriction is needed.
• Pitfall: treating a secant extrapolation as the predicted value. Under convexity, the continued last secant gives a lower bound, not necessarily the function’s value. Under concavity, it gives an upper bound.
• Pitfall: assuming observed trends prove off-support shape. A few increasing or decreasing observed means do not establish that the same pattern continues beyond common support. The extrapolation restriction is an additional assumption.
• Pitfall: confusing an outcome range with a shape restriction. Knowing \(Y\in[L,U]\) bounds the conditional mean by \([L,U]\), but does not determine how that mean changes with \(x\).