What Makes Treatment Effects Identifiable? Characterizations and Estimators Beyond Unconfoundedness (Extended Abstract)

1 Introduction

Understanding cause and effect is a central goal in science and decision-making. Across disciplines, we ask: What is the effect of a new drug on disease rates? How does a policy impact growth? Is technology driving economic growth? Causal inference tackles such questions by disentangling correlation from causation. Unlike statistical learning, which predicts outcomes from data, causal inference estimates the effects of interventions that alter the data-generating process.

A fundamental challenge in causal inference is that we can never observe both potential outcomes for the same individual. For example, if a patient takes a medication and recovers, we do not know whether the patient would have recovered without it. Due to this fundamental problem of causal inference, causal effects must be inferred under certain assumptions [Holland, 1986].

To formalize this challenge, we consider the widely-used potential outcomes model introduced by Neyman [1990] (originally published in 1923) and later formalized by Rubin [1974]; see also [Hernan and Robins, 2023; Rosenbaum, 2002; Chernozhukov et al., 2024]. Here, for a unit with covariates X ∈ Rd , Y (1) and Y (0) denote potential outcomes under treatment and control, respectively. Since only the outcome Y (T) corresponding to the assigned treatment T is observed, certain assumptions are needed to estimate the average treatment effect (ATE), defined as τ := E[Y (1) - Y (0)], where (T, Y (1), Y (0)) are random variables whose distribution may depend on X. This framework underpins many modern causal inference methods - both practical and theoretical - and can capture many treatment effects, apart from τ. Two fundamental questions under this framework, studied since [Cochran, 1965; Rubin, 1974; Rubin, 1978; Heckman, 1979], are identification and estimation:

• Q1: Given infinite samples of the form (X, T, Y (T)), can we identify treatment effects?
• Q2: Given n samples (X, T, Y (T)), can we estimate treatment effects up to error ε(n)?

Due to the missingness in data (explained above), even the identification problem is unsolvable without making structural assumptions on the distribution of (X, T, Y (T )), which is a censored version of the (complete) data distribution (X, T, Y (1), Y (0)). The earliest and most widely used such assumptions are unconfoundedness and overlap.

• ▷ Unconfoundedness presumes that after conditioning on the value of the covariate X, the treatment random variable T is independent of the outcomes Y (1) and Y (0), i.e., T⊥(Y (0), Y (1)) | X.
• ▷ Overlap requires that the probability of being assigned treatment conditional on the covariate X, i.e., Pr[T=1 | X=x], is a quantity strictly between 0 and 1 for each x.

Unconfoundedness and overlap are essential for unbiased estimation of the average treatment effect and are widely studied across Statistics (e.g., [Rosenbaum, 2002]) and many other disciplines, including Medicine (e.g., [Rosenbaum and Rubin, 1983]), Economics (e.g., [Athey and Imbens, 2017]), Political Science (e.g., [Brunell and DiNardo, 2004]), Sociology (e.g., [Morgan and Harding, 2006]), and other fields (e.g., [Austin, 2008]). Despite their wide use across different disciplines, there are fundamental instances where unconfoundedness or overlap are easily violated:

• ▷ Unconfoundedness is often violated in observational studies, where treatments or exposures are not assigned by the researcher but observed in a natural setting. In a prospective cohort study, for example, individuals are followed over time to assess how exposures influence outcomes. A common violation arises when key confounders are unmeasured. For instance, in studying smoking's impact on health, omitting socioeconomic status (SES), which affects both smoking habits and health, can bias results, as lower SES correlates with higher smoking rates and poorer health, independent of smoking.
• ▷ Overlap is violated when certain covariate values make treatment assignments nearly deterministic. In a marketing study estimating the effect of personalized advertisements on purchases, covariates like demographics, browsing history, and preferences define a high-dimensional feature space. As this space grows, many user profiles either always or never receive the ad, leading to lack of overlap [D'Amour et al., 2021]. Without comparable treated and untreated units, causal inference methods struggle to estimate counterfactual outcomes, yielding unreliable effect estimates.

These examples lead us to the following question:

Is identification and estimation of treatment effects possible in any meaningful setting without unconfoundedness or overlap?

This question is not new and can be traced back to at least the work of Rubin [1977], who recognized that, without substantial overlap between treatment and control groups, identification of treatment effects necessarily requires additional prior assumptions. To the best of our knowledge, the present work provides the first formal characterization of the precise assumptions required to identify treatment effects in scenarios lacking substantial overlap, unconfoundedness, or both.

2 Framework

The main conceptual contribution of this work is a learningtheoretic approach that enables a characterization of when identification and estimation of treatment effects are possible.

Before presenting this approach, it is instructive to reconsider how unconfoundedness and overlap enable identification of the simplest and most widely used treatment effect - the average treatment effect: Given the observational study D, which is a distribution over (X, T, Y (0), Y (1)), unconfoundedness and overlap put a strong constraint on D: they require that Y (t) ⊥ T | X=x for each t ∈ {0, 1} and x ∈ Rd and that the propensity scores e(x) = Pr[T=1|X=x] are bounded away from 0 and 1. Under these assumptions, identification and estimation of ATE τ = τD is possible given censored samples (X, T, Y (T )) due to the following decomposition of τD for a fixed x ∈ Rd (we integrate over the x-marginal to get τD):

where we use overlap to divide by e(X), 1 - e(X) and unconfoundedness to obtain the equation E[Y (1) · T | X] = E[Y (1) | X] · Pr[T=1 | X] and analogously for Y (0). Note that all the quantities appearing in the RHS are identifiable and estimable 1 from the censored distribution CD [Rubin, 1978], which is defined over (X, T, Y (T )).

When no constraints are put on D, identification of ATE is impossible in general [Imbens and Rubin, 2015]. Without unconfoundedness, propensity scores Pr[T=1 | X=x] are not sufficient to identify the distribution of T , which can also depend on the outcomes Y (0) and Y (1) (conditioned on X=x). Instead, writing pt(x, y) = Pr[T=t | X=x, Y (t)=y], we can decompose the expression of τD for a fixed x ∈ Rd as follows:

If unconfoundedness holds, then we could recover (1) since then T would not depend on Y (1), Y (0) given X. However, unlike the previous decomposition of Equation (1), the above equation always holds and crucially utilizes the generalized propensity scores with t ∈ {0, 1}. 2 Unfortunately, these generalized propensity scores, in contrast to the standard propensity scores, are not always identifiable from data. To understand when these are identifiable, we need to consider the joint distribution of covariates and outcomes DX,Y (t) for each t ∈ {0, 1}.

To this end, we adopt an approach inspired by statistical learning theory [Valiant, 1984; Vapnik, 1999; Blumer et al., 1989; Hastie et al., 2013; Anthony and Bartlett, 1999; Alon et al., 1997; Lugosi, 2002; Massart and N´ed´elec, 2006;

1 We remark that the problem of estimating the propensity scores e(x) = Pr[T=1|X=x] is identical to the classical problem of learning probabilistic concepts [Kearns and Schapire, 1994].

2 Observe that we need some overlap condition to divide by p0(·) and p1(·) in the above equation. In our main results, however, we do not follow this decomposition and will not need such overlap conditions.

Vapnik, 2006; Bousquet et al., 2003; Bousquet, 2003]. We introduce concept classes for the two key quantities derived by the above discussion pt and DX,Y (t) (for each t ∈ {0, 1}) that will place some restrictions on the observational study D towards understanding which conditions enable identification and estimation. In the remainder of the paper, we assume that all distributions are continuous and have a density. (All results also extend to discrete domains by replacing densities by probability mass functions.)

We are interested in the structure of two concept classes: the class of generalized propensity scores P ⊆ {p: Rd × R → [0, 1]} and the class of covariate-outcome distributions D ⊆ ∆(Rd × R). As in classical statistical learning theory, having fixed the concept classes, our next step is to restrict the underlying distribution D to be realizable with respect to the pair of concept classes (P,D). An observational study D is said to be realizable with respect to the concept class pair (P, D) if the generalized propensity scores p0(·), p1(·) induced by D belong to P and DX,Y (t) ∈ D for each t ∈ {0, 1}. This learning-theoretic framework is quite expressive. For instance, it can capture unconfoundedness and overlap 3 by letting D be the set of all distributions over Rd × R, denoted by Dall, and restricting P to be the following class

That is, D satisfies unconfoundedness and c-overlap if and only if it is realizable with respect to the pair of classes (POU(c), Dall).

Before proceeding to our results, we introduce some further terminology. Given classes (P,D), we are particularly interested in the generalized propensity scores in P and covariate-outcome distributions in D that induce a valid observational study. To this end, we say that a tuple (p, P) ∈ P × D is compatible with classes (P,D) (henceforth, just compatible) if there exists another tuple (̂ p, ̂P) ∈ P × D such that setting (p0, p1,DX,Y (0),DX,Y (1)) = (p, ̂ p, P, ̂P) (or equivalently the re-ordered assignment (p0, p1,DX,Y (0),DX,Y (1)) = (̂ p, p, ̂P, P)) defines a valid observational study, i.e., a valid distribution over (X, T, Y (0), Y (1)).

3 Main Results on Identification

We say that a certain treatment effect ηD is identifiable from the censored distribution CD when (P, D) satisfy some Condition C, if there is a mapping f such that f(CD) = ηD for any observational study D realizable with respect to (P,D) that satisfies C. In other words, if ηD1 ̸= ηD2 then it should be CD1 ̸= CD2 . Having set the stage, we now ask our first main question:

Which conditions on (P, D) characterize the average treatment effect's identifiability?

3 We refer to overlap as c-overlap: for some absolute constant c ∈ (0, 1/2), c < p0(x, y), p1(x, y) < 1 - c.

As a first contribution, we identify a condition on the classes (P,D) that will be crucial for the results on the identification of ATE that proceed.

Condition 1 (Identifiability Condition). The concept classes (P,D) satisfy the Identifiability Condition if for any tuples (p, P), (q, Q) ∈ P × D that are compatible with (P,D), at least one of the following holds:

1. (Equivalence of Outcome Expectations) E(x,y)∼P[y] = E(x,y)∼Q[y]
2. (Distinction of Covariate Marginals) PX ̸= QX
3. (Distinction under Censoring) There exist x ∈ supp(PX) and y ∈ R such that p(x, y)P(x, y) ̸= q(x, y)Q(x, y).

Observe that in Condition 1 we only focus on tuples that are compatible. This is due to the fact that incompatible tuples of P × D cannot be part of a realization of any valid observational study and hence properties of incompatible tuples are not relevant to our characterizations.

To gain some intuition for Condition 1, consider two observational studies D1 and D2 which correspond to the pairs (p, P) and (q, Q) respectively, where P and Q are distributions of (X, Y (1)). Assume that the true observational study D is either D1 or D2. Given the censored distribution CD, we want to identify ED[Y (1)]. First, suppose that the tuples (p, P), (q, Q) satisfy Requirement 1 in Condition 1. Then we are done since we only care about the expected outcomes ED[Y (1)] = E(x,y)∼P[y] = E(x,y)∼Q[y] which are the same under both distributions. Next, let us assume that Requirement 1 is violated and, hence, the expected treatment outcome is different between the null and the alternative hypothesis. In this case, if Requirement 2 is satisfied, then we can distinguish P and Q from CD (by comparing PX and QX to the covariate marginal of CD) and, hence, distinguish between D1 and D2. Finally, if both Requirements 1 and 2 fail but Requirement 3 holds, then we can use it to identify whether the true observational study D corresponds to (p, P) or (q, Q) as follows: First, we check if both ∫ p(x, y)P(x, y)dxdy and ∫ q(x, y)Q(x, y)dxdy are identical to PrD[T=1]. If this is not true, then we are done, as if ∫ p(x, y)P(x, y)dxdy ̸= PrD[T=1], then D does not correspond to (p, P). Otherwise, if this is true, then, in particular, ∫ p(x, y)P(x, y)dxdy = ∫ q(x, y)Q(x, y)dxdy = PrD[T=1]. Now, we can distinguish between (p, P) and (q, Q) by checking which one of p(·)P(·) PrD[T=1] and q(·)Q(·) PrD[T=1] matches the density of (X, T=1, Y (1)) in the censored distribution. This is where we use Requirement 3, due to which along with the previous observation, the densities p(·)P(·) PrD[T=1] and q(·)Q(·) PrD[T=1] are not identical in at least one point (x, y). Our first result shows Condition 1 fully characterizes ATE.

Theorem 1 (Identification of ATE). The average treatment effect τD is identifiable from the censored distribution CD for any observational study D realizable with respect to (P, D) if and only if (P, D) satisfy Condition 1.

The above result adds to classical identifiability conditions in Statistics (e.g., [Everitt and Hand, 2013; Teicher, 1963]), Statistical Learning Theory 4 (e.g., [Angluin, 1980; Angluin, 1988]), and Econometrics (e.g., [Manski, 1990; Athey and Haile, 2002]). To the best of our knowledge, this is the first tight characterization of ATE identification in observational studies.

4 Applications and Estimation of ATE

For Condition 1 to be useful given the other existing conditions (such as unconfoundedness and overlap), it needs to capture interesting examples not captured by existing conditions. Below, we revisit several well-studied scenarios in causal inference or their generalizations and, for each, provide identification results based on our results.

Scenario I: Unconfoundedness and Overlap. At the end of Section 2, we mentioned that our framework can capture unconfoundedness and overlap. Identification in this scenario is standard and can also be deduced using Theorem 1. Estimation is also standard [Imbens and Rubin, 2015].

Scenario II: Overlap without Unconfoundedness. Next, we consider observational studies D which satisfy c-overlap for some c ∈ (0, 1/2) but may not satisfy unconfoundedness. We are going to use our framework to characterize the subset of these studies D for which ATE is identifiable. Since overlap holds with some parameter c ∈ (0, 1/2), it restricts the concept class P to be PO(c) where c < p(x, y) < 1 - c for any (x, y) and p ∈ PO(c). This case generalizes several models studied in the causal inference literature [Tan, 2006; Rosenbaum, 2002; Rosenbaum, 1987; Kallus and Zhou, 2021]; see the discussion below. Under this scenario, we can ask: which conditions should the covariate-outcome distributions D satisfy for τ to be identifiable, i.e., for which observational studies realizable by (PO(c),D) is the ATE identifiable?

Informal Theorem 1. Assume that for any pair P, Q ∈ D, either Item 1 or 2 of Condition 1 holds or there exist x ∈ supp(PX), y ∈ R such that P(x, y) /∈ ( c 1-c , 1-c c ) · Q(x, y). Then, τD is identifiable from the censored distribution CD for any observational study D realizable with respect to (PO(c),D). Moreover, this condition is necessary.

The above condition for identification is quite similar to Condition 1 and is satisfied by setting the outcome marginal of P ∈ D to be, e.g., Gaussian, Pareto, or Laplace, and letting the x-marginal PX be unrestricted. This captures important practical models where the outcomes are modeled as a generalized linear model with Gaussian noise.

Connections to Prior Work. Since we do not require unconfoundedness in any form, the requirements on the generalized propensity score class PO(c), in this scenario, are very mild and are already satisfied by most existing frameworks that relax unconfoundedness while retaining overlap. The restriction on the propensity score class PO(c) relaxes the model of Tan [2006] and the odds-ratio model of Rosenbaum [2002], both widely used in sensitivity analysis.

4 This mirrors language identification, whose characterizations also concern pairs of languages [Angluin, 1980].

Scenario III: Unconfoundedness without Overlap. We now consider the setting where overlap may fail but unconfoundedness holds. To rule out degenerate cases where everyone (or no one) receives the treatment, we assume that some nontrivial subset of covariates satisfies overlap. Concretely, there is a set S ⊆ Rd with Lebesgue measure vol(S) ≥ c such that for each (x, y) ∈ S × R, we have c < p0(x, y), p1(x, y) < 1 - c. This is already significantly weaker than the usual c-overlap assumption, which demands the previous inequalities pointwise for every (x, y) ∈ Rd ×R. We relax it further into the notion of c-weak-overlap by removing the upper bound on p0(x, y) and p1(x, y). Then, we define the class P = PU(c) which captures both unconfoundedness and c-weak-overlap.

Scenarios with unconfoundedness but without full overlap frequently arise in practice. Classic examples include regression discontinuity designs [Cook, 2008] and observational studies with extreme propensity scores [Crump et al., 2009; Li et al., 2018; Khan and Ugander, 2024; Kalavasis et al., 2024]; see further discussion after Informal Theorem 2. As before, we ask which conditions on D enable identification of ATE, i.e., for which observational studies realizable with respect to (PU(c), D), can one identify the ATE?

Informal Theorem 2. Assume that for any pair P, Q ∈ D, either Item 1 or 2 of Condition 1 holds or there is no set S ⊆ Rd with vol(S) ≥ c such that P(x, y) = Q(x, y) for (x, y) ∈ S×R. Then τD is identifiable from the censored distribution CD for any D realizable with respect to (PU(c), D). Moreover, this condition is necessary.

We would like to stress that the above characterization has a novel conceptual connection with an important field of statistics, called truncated statistics [Galton, 1897; Cohen, 1991]. The main task in truncated statistics concerns extrapolation: given a true density D over some domain X and a set S ⊆ X, the question is whether the structure of D can be identified from truncated samples, i.e., samples from the conditional density of D on S. The condition of the above result requires the pairs P, Q to be distinguishable on any set of the form S ×R (where S has sufficient volume). Roughly speaking, this condition holds for any family D whose elements P can be extrapolated given samples from their truncations to full-dimensional sets - which is well-studied and provides us with multiple applications [Kontonis et al., 2019; Daskalakis et al., 2021; Lee et al., 2024; Lee et al., 2026].

Connections to Prior Work. This scenario captures two important and practical settings. First, as mentioned before, it captures regression discontinuity (RD). These designs were introduced by Thistlethwaite and Campbell [1960], were independently discovered in many fields [Cook, 2008], and have found applications in various contexts from education to public health to labor economics. To the best of our knowledge in RD designs, ATE is only known to be identifiable under strong linearity assumptions on the expected outcomes [Hahn et al., 2001]. Due to that, recent work focuses on identifying certain local treatment effects [Imbens and Lemieux, 2008]. In contrast, Informal Theorem 2 enables us to achieve identification under much weaker restrictions.

Ethical Statement

There are no ethical issues.