Counterfactual Bounds Under Functional-Form Restrictions

Why Functional-Form Restrictions Produce Bounds

A counterfactual mean may depend on covariate values that are not represented among the observations used to estimate it. For example, a treated population may include values of \(X\) outside the range observed for untreated units. The untreated outcome at those values is unobserved, so the data alone do not determine its conditional mean. Extending the observed relationship requires an assumption about its form.

An exact linear model gives one extrapolated value, but that value can depend heavily on treating linearity as certain. A useful alternative is to specify a class of possible functions—such as affine functions with uncertain coefficients, or smooth functions whose curvature is bounded—and report the smallest and largest counterfactual means allowed by that class. These are bounds conditional on the stated restrictions, not bounds justified by the observed data alone.

This setup needs care about what the observed outcomes identify. Consistency means that an untreated unit's observed outcome equals \(Y(0)\). On covariate values shared by treated and untreated units, one may use the untreated conditional mean \(E[Y\mid D=0,X=x]\) to anchor the treated counterfactual mean only under a conditional mean transportability assumption: \(E[Y(0)\mid D=1,X=x]=E[Y(0)\mid D=0,X=x]\) there. Outside that shared support, the functional-form restriction supplies the extrapolation. It does not follow from consistency or from the observed outcomes there.

For simplicity, the examples below treat the observed conditional means and restrictions as known population quantities. In applications, they are estimated, and sampling uncertainty is a separate issue from the width of the identified set.

Bounds from an Affine Mean Function

Suppose the admissible counterfactual mean functions are affine in a scalar covariate: \(m_0(x)=a+bx\). If observed untreated conditional means identify both coefficients exactly, this restriction yields a point value even beyond observed support. If the coefficients are only restricted to ranges, or to a joint parameter set, optimizing the target mean over that set yields bounds.

The restriction is informative only if the observed support can anchor the parameters. For an affine function, two distinct covariate values with known conditional means identify \(a\) and \(b\) in the exact model. If all observed covariate values are the same, the slope is not identified by those data; any bound then depends on additional restrictions on the slope or on the function class.

More generally, if the coefficient vector \(\theta=(a,b)\) lies in a set \(\Theta\), and the treated covariate mean \(E[X\mid D=1]\) exists, then \(E[m_0(X)\mid D=1]=a+bE[X\mid D=1]\). The bounds are the minimum and maximum of this expression over \(\Theta\). If the set couples \(a\) and \(b\), optimizing each coefficient independently can give incorrect endpoints; the joint restrictions must be respected. With several covariates and an affine mean, the target mean is still linear in the coefficients, so the same optimization principle applies.

Relaxing Linearity with a Curvature Bound

An exact affine restriction rules out any curvature. One way to relax it is to retain observed anchor values while allowing departures from the straight line, with a bound on the second derivative. This is a restriction on how rapidly the slope can change, not a claim that the function is exactly quadratic.

For distinct anchors, let \(L(x)\) be the line through \((x_0,m_0(x_0))\) and \((x_1,m_0(x_1))\). The interpolation remainder has the form \(m_0(x)-L(x)=m_0''(\xi)(x-x_0)(x-x_1)/2\) for some \(\xi\) between the three relevant points. It follows that \(|m_0(x)-L(x)|\leq K|(x-x_0)(x-x_1)|/2\). The same bound applies when \(x\) lies outside the two anchors, provided the differentiability and curvature restriction hold over the whole interval reaching \(x\). The sign of the product is not used for this absolute bound.

The derivation also shows why distance from the observed support matters. The error bound grows with \(|(x-x_0)(x-x_1)|\); a curvature limit that gives a narrow interval near the anchors may permit a much wider interval farther away. If the target covariate varies across treated units, the counterfactual mean is an average of the conditional means over their covariate distribution. Pointwise lower and upper envelopes can be integrated to give valid bounds, although they need not be sharp when restrictions jointly link function values at different covariate values.

Why Relaxing Restrictions Widens Bounds

The direction of change follows directly from set inclusion. If \(\mathcal{M}_1\) is a stricter class of admissible functions and \(\mathcal{M}_2\) is a relaxed class with \(\mathcal{M}_1\subseteq\mathcal{M}_2\), then minimizing over the larger class cannot raise the lower endpoint, and maximizing over it cannot lower the upper endpoint:

For example, allowing a larger curvature limit \(K\) enlarges the class of functions and weakly widens the counterfactual bounds. At \(K=0\), the second derivative restriction forces an affine function on the interval; a positive \(K\) allows departures. This comparison makes the assumption's role visible: narrower bounds reflect stronger functional-form commitments, not necessarily more information in the observed outcomes.

Assumptions and Common Pitfalls

Functional-form bounds are only as credible as the restrictions used to define the admissible class. An affine restriction is easy to interpret but can be fragile far beyond the observed support. A curvature bound allows more flexibility, but its numerical value \(K\) is itself a substantive sensitivity choice and must be expressed in units consistent with the outcome and covariate scales. Rescaling \(X\) changes the numerical value of a second-derivative bound.

• Confusing extrapolation with observation. The counterfactual mean outside untreated support is not learned directly from untreated outcomes there. It is constrained by the chosen functional form.
• Treating a point estimate as a bound. An exact affine model may imply one extrapolated value, but that point conclusion relies on exact linearity and known coefficients. Coefficient uncertainty or permitted departures can produce an interval.
• Ignoring target covariate weights. The target quantity averages over the treated covariate distribution. Evaluating the function at an unrelated covariate value does not generally give that average.
• Assuming that wider restrictions narrow the set. Admitting additional functions cannot make the identified set smaller. A wider model class weakly lowers the lower endpoint and weakly raises the upper endpoint.
• Omitting transportability on shared support. Equality of treated and untreated conditional counterfactual means on overlap is an assumption, not a consequence of observing both groups at the same \(X\).

A disciplined analysis states the target population, the support covered by observed anchors, the transportability assumption used on shared support, and the functional restriction imposed outside it. It then reports how the bounds change under plausible relaxations. This presents extrapolation as a transparent sensitivity analysis rather than disguising it as a purely data-driven prediction.