Testing Exclusion and Shape Restrictions in Potential Outcomes Models

Testing Exclusion and Shape Restrictions in Potential Outcomes Models

Introduction

The potential outcomes framework of neyman1923applications and rubin1974estimating is a workhorse of empirical economics. Within this framework, exclusion and shape restrictions on the potential response functions are commonly imposed to define causal parameters, interpret estimates, and tighten identified sets. Although such restrictions concern fundamentally unobservable objects, they have non-trivial testable implications. So far, such testable implications have been studied on a case-by-case basis in a limited set of models. Examples include testing instrumental variables (IV) validity and treatment selection monotonicity in Kitagawa15, “judge IV” designs in frandsen2023judging, encouragement designs in bai2024, and full mediation in Kwon:2024aa, among others.

In this paper, we propose a general framework for deriving sharp testable implications of exclusion and shape restrictions, and for testing these implications against data. The framework allows practitioners to empirically assess commonly imposed modeling assumptions in a broad class of potential outcome models. Formally, the observed data consist of an exogenous input $Z \in \mathcal{Z}$, typically representing instruments or exogenous treatments, and an endogenous response $R \in \mathcal{R}$, which may include an endogenous treatment, mediator, or outcome. Additional observed covariates can be accommodated but are omitted for now. The model is characterized by the potential response vector $R^* = (R^*(x))_{x\in \mathcal{X}} \in \mathcal{R}^{\mathcal{X}}$, indexed by counterfactual inputs $x \in \mathcal{X}$, which typically contain instrumental variables, treatment variables, or mediators, and a known mapping $T: \mathcal{R}^{\mathcal{X}} \times \mathcal{Z} \to \mathcal{R}$ such that the observed response satisfies $R =T(R^*, Z)$. We assume that $R^*$ and $Z$ are independent. In this notation, a wide range of restrictions studied in the literature take the form of support restrictions, i.e., $R^* \in \mathcal{R}^*$ almost surely, for a known set $\mathcal{R}^* \subseteq \mathcal{R}^{\mathcal{X}}$; See Table (ref) for the examples.

To characterize the observable implications of support restrictions, we introduce a graph-based representation that encodes the compatibility structure of the model. This representation allows us to compute the testable implications using efficient graph algorithms and facilitates interpretation of the resulting constraints. Moreover, by exploiting the correspondence between the graph and the convex polytope of distributions of potential outcomes compatible with the model, we establish that the implied restrictions are sharp, i.e., necessary and sufficient for the observed distribution to be consistent with the imposed assumptions.

Suppose, for the moment, that both $\mathcal{R}$ and $\mathcal{Z}$ are finite sets. We may represent a restriction $\mathcal{R}^*$ via an undirected graph $G = (V_G, E_G)$, in which every vertex $v_{r, z} \in V_G$ corresponds to a pair $(r, z)$ of values the response and input can take, and two vertices $v_{r, z}$ and $v_{r', z'}$ are connected by a link $(v_{r, z}, v_{r', z'}) \in E_G$ if there exists a support point $r^* \in \mathcal{R}^*$ consistent with both, i.e., $T(r^*, z) = r$ and $T(r^*, z') = r'$. In other words, the vertices $v_{r, z}$ and $v_{r', z'}$ in $G$ are connected if a pair of events “$R = r$ when $Z = z$” and “$R = r'$ when $Z = z'$” is compatible with the modeling assumptions.

With this representation, the testable implications of the modeling assumptions become transparent. If two vertices $v_{r, z}$ and $v_{r', z'}$ are disconnected in $G$, the events “$R = r$ when $Z = z$” and “$R = r'$ when $Z = z'$” cannot occur jointly, under the modeling assumptions. Thus, it must be that $P(R = r \,|\, Z = z) + P(R = r' \,|\,Z = z') \leqslant 1$. More generally, any maximal set $I \subseteq V_G$ of mutually disconnected vertices (called a Maximal Independent Set or MIS) yields an inequality

Our first main result (Theorem (ref)) shows that all such MIS inequalities provide valid testable implications. These inequalities can be efficiently computed using existing algorithms available in standard software packages and tested using a wide range of established procedures, including those that accommodate continuous response or control variables when necessary.

Next, we study the sharpness of the testable implications derived in (ref). Our second main result (Theorem (ref)) characterizes a broader class of inequalities, of which the MIS inequalities are generally a strict subset, that together yield sharp testable implications of the model. While this more general class is required in principle, many commonly used support restrictions exhibit additional structure under which the MIS inequalities alone suffice for sharpness. We refer to such restrictions as regular and show that regularity can be verified in a straightforward manner. In particular, monotonicity restrictions, encouragement designs, and LATE-type settings, including judge-IV designs, listed in Table (ref), are all regular, as are all restrictions with binary $Z$. Our third main result (Theorem (ref)) establishes that, under regularity, the MIS inequalities are sharp. By contrast, certain exclusion restrictions in IV, mediation, and interference are not regular, in which case the inequalities in (ref) are insufficient. For such cases, we provide an algorithm that constructs the additional inequalities required to achieve sharpness.

We then extend the analysis to settings in which (part of) the response vector $R$ is continuous. We obtain general analogues of Theorems (ref) and (ref), and specialize them to the canonical IV model with endogenous treatment and instrument monotonicity. This extension generalizes existing results in Kitagawa15 and Kwon:2024aa by allowing for multi-valued treatments and IVs.

Conceptually, our analysis builds on classical results in combinatorial optimization --- most notably the polyhedral characterization of clique polytopes for perfect graphs chvatal1975certain and the matroid intersection polytope theorem edmonds1979matroid. However, in our setting, the relevant object is not the classical clique polytope of $G$. Since each response type includes exactly one observable response per input $z\in \{z_1,\dots,z_K\}$, every admissible type corresponds to a $K$-clique in $G$. Thus, the set of conditional probability vectors $(P(R = r \,|\,Z = z))_{r \in \mathcal{R}, z \in \mathcal{Z}}$ compatible with the support restriction corresponds to the convex hull of incidence vectors of such $K$-cliques. We provide a novel explicit half-space characterization of this set and show how it yields sharp testable implications. Under a regularity condition, the characterization simplifies to MIS inequalities, delivering a tractable set of moment inequalities for inference.

Another useful feature of our approach is that adding or relaxing assumptions corresponds to modifying the edges of the underlying graph. This structure enables direct comparison of testable implications across different models. By systematically computing and comparing the resulting sets of testable implications, practitioners can evaluate how changes in modeling assumptions affect the implied restrictions --- an analysis that is often difficult to conduct on a case-by-case basis.

Expressing the model’s testable implications as conditional moment inequalities allows us to draw on a large literature on testing and model selection procedures that accommodate both discrete and continuous covariates. Our primary approach exploits the half-space representation of the set of conditional distributions of observables consistent with the model. We also consider an alternative procedure based on the vertex representation of this set, following bai2025inference. The choice between these representations depends on the structure of the support restriction $\mathcal{R}^*$ and the geometry of the associated polytope; Section (ref) provides practical guidance. Broadly, the half-space-based procedure is computationally attractive when the number of implied inequalities is moderate, even if the number of latent types is very large. By contrast, the vertex-based approach can be advantageous when the number of inequalities is extremely large, but the number of latent types remains manageable. To assess which regime is relevant in a given application, we recommend using the output-sensitive algorithm of tsukiyama1977new to enumerate maximal independent sets as a preprocessing step.

We assess the finite-sample performance of the proposed tests using Monte Carlo experiments and an empirical application. In simulations, we find that conditioning on exogenous covariates and exploiting instruments with rich support can substantially improve power, although some violations of the model are inherently difficult to detect. As an empirical illustration, we revisit the U.S. Lung Health Study, a randomized controlled trial of smoking cessation interventions. We test for spillover effects from treated subjects to their spouses, monotonicity of responses to treatment intensity, and persistence of treatment effects over time. Across these hypotheses, the number of maximal independent sets ranges from 1 to 726, and all are computed within three seconds, demonstrating that the proposed tests are computationally straightforward to implement in practice.

Related Literature \; This paper contributes to the vast and growing literature on identification of causal effects using instrumental variables. In a setting with binary instrument and treatment, Imbens:1994aa demonstrated that the exclusion restriction, combined with treatment selection monotonicity (which we will call IA monotonicity), suffices to identify the Local Average Treatment Effect (LATE) for compliers. This work started a large literature studying identifying power, interpretation, extensions, and testable implications of support-type restrictions in potential outcome models; see, e.g., mogstad2024instrumental for a detailed review and Table (ref) for some notable examples. Below, we discuss in more detail several papers that are most related to the present work.

balke1997bounds derived the testable implications of the exclusion restriction and instrument independence, both with and without IA monotonicity, in models with binary instruments, treatments, and outcomes. Kitagawa15 extended this analysis to continuous outcomes, showing that the resulting testable implications under IA monotonicity are sharp and proposing a corresponding statistical test.\footnote{mourifie2017testing propose an alternative test based on CLR13.} Kwon:2024aa further broadened the framework to accommodate multi-valued treatments while maintaining the same notion of monotonicity. As we discuss in Remark (ref), with binary instruments, the sharp testable implications in the above settings (and in many related ones) follow from a theorem of artstein1983distributions, which provides necessary and sufficient conditions for the existence of a joint distribution of two random elements given their marginals and joint support. We extend Artstein’s result to a multi-marginal setting, which allows us to handle multi-valued instruments and alternative treatment–selection models.\footnote{ With multi-valued instruments in the absence of IA monotonicity, the inequalities derived by balke1997bounds for binary outcomes and treatments are not sharp Bonet2001Instrumentality. The corresponding sharp testable implications were later established by KedagniMourifie2020. Related work also considers relaxations of the random-assignment assumption; see RichardsonRobins2010BinaryIV and KedagniMourifie2020.}

Several extensions of IA monotonicity to settings with multi-valued instruments have been proposed in the literature. One example is the so-called encouragement designs, where, for each potential treatment choice, there is a randomly assigned instrument that encourages an individual to take up that treatment; see, e.g., kline2016evaluating, kirkeboen2016field. In this context, bai2024 derive sharp testable implications of the modeling assumptions and discuss available testing procedures. The general approach developed here recovers the inequalities of bai2024 and applies in many other settings, including those with restrictions on the outcome response beyond exclusion.

Another common extension of the IA monotonicity assumption is the so-called “judge fixed effects” design, originally motivated by the random assignment of cases to judges in the US penitentiary system kling2006incarceration and used for identifying causal effects in a variety of settings frandsen2023judging. In addition to the standard instrument exogeneity and exclusion restrictions, the key underlying assumption is that the “judges” can be ordered from “least” to “most lenient,” so that “judges” assignment has a weakly monotone effect on each individual's treatment status. In this setting, the Two-Stage Least Squares (TSLS) estimand has a “weakly causal” interpretation of a non-negatively weighted average of LATEs for different complier groups. frandsen2023judging derived (non-sharp) testable implications of these assumptions and proposed a non-parametric test. Recent work by coulibaly2024sharptestjudgeleniency extended the analysis and derived sharp testable implications. The framework developed here allows for the derivation and testing of sharp testable implications under either class of assumptions, possibly in conjunction with other structural restrictions.

In a related setting, Mogstad:2021aa argued that IA monotonicity with multiple instruments may be overly restrictive and proposed a weaker notion, termed “partial monotonicity,” under which the TSLS estimand retains a weakly causal interpretation. This restriction is our running example discussed in detail throughout the paper. Recently, blandhol2022tsls argued that in the presence of covariates, the TSLS estimand retains a weakly causal interpretation only if covariates enter non-parametrically. Our approach accommodates covariates and can be used to test the assumptions required for such causal interpretations.

This paper also contributes to the literature on mediation, which deals with testing causal pathways between the treatment and outcomes. We refer to vanderweele2016mediation and huber2019review for detailed reviews of the literature. Kwon:2024aa developed a non-parametric test for the sharp null of full mediation --- an assumption stating that the (binary) treatment can be excluded from the potential outcomes given (multi-valued) mediators, potentially combined with additional restrictions on the effect of treatment on mediators. In this paper, we extend the analysis to accommodate any form of support restrictions on potential outcomes and mediators, which allows for testing other hypotheses of interest, accommodating multivalued treatment and continuous controls (see Example (ref)).

Moreover, this paper contributes to the literature on causal inference under interference, which explores violations of the no-interference assumption of cox1958planning and stable unit treatment value assumption in rubin1980randomization. Following hudgens2008toward, Manski:2013aa, and Aronow:2017aa, researchers commonly specify the so-called exposure maps, which restrict how each individual is affected by treatments of their peers, in order to define, identify, and estimate causal parameters of interest. This requires taking a stance on the underlying structure of interactions, which may not always be known or simple. If an exposure map is misspecified, the corresponding estimands may not have a clear causal interpretation, standard estimators may have undesirable statistical properties, and inference procedures may be invalid; see, e.g., sobel2006randomized, savje2021average, leung2022unconfoundedness, savje2024causal, auerbach2024discussion, leung2024discussion, menzel2025fixed, and auerbach2021local for detailed discussions and possible remedies. Existing tests for specification of exposure maps (applicable in a fixed population framework and based on randomization) operate on a case-by-case basis, rely on carefully constructed symmetries of the test statistic under the null hypothesis, and require a non-trivial choice of tuning parameters, including the focal units and test statistic, which greatly affect power aronow2012general, athey2018exact, basse2019randomization, puelz2022graph.\footnote{In the same design-based setting, zhong2025unconditionalrandomizationtestsinterference proposes an alternative unconditional testing framework to address these issues.} The approach we develop here (applicable in the super-population framework and valid in large samples) can be used to test the specification of any exposure map and many other hypotheses, such as monotonicity of the outcome response to treatment or endogeneity of spillover effects. In many settings, the resulting tests do not require any tuning parameters. We illustrate the capabilities of the proposed approach in a setting with interference in our empirical application.

Finally, this paper contributes to the literature on inference for linear systems and moment inequalities. With discrete inputs and responses, testing support restrictions amounts to testing whether a vector of conditional probabilities $\beta(P) = (P(R = r \,|\,Z = z))_{r \in \mathcal{R}, z \in \mathcal{Z}}$ belongs to a convex polytope $\mathbf{B}^* = \{A^*Q: Q \in \Delta(\mathcal{R}^*) \}$, where a matrix $A^* \in \{0, 1\}^{|\mathcal{R}||\mathcal{Z}| \times \mathcal{R}^*}$ encodes the support restriction $\mathcal{R}^*$, so that each column corresponds to a latent response type, and a vector $Q$ represents a distribution of latent types supported on $\mathcal{R}^*$. In this form, the hypothesis $P \in \mathbf{B}^*$ can be tested using existing methods such as KitamuraStoye2018, fang2023inference, goff2025inferencevaluelinearprogram, or bai2025inference. However, this formulation does not accommodate continuous control variables, which may be essential in applications. The reason is essentially that the polytope $\mathbf{B}^*$ is given in its vertex ($V$) rather than half-space ($H$) representation, which would correspond to testing conditional moment inequalities. In this paper, we analytically derive an $H$-representation for $\mathbf{B}^*$ (and its continuous analog) that enables using existing approaches, such as andrews2013inference, CLR13, or armstrong2015asymptotically, for inference. Additionally, the $H$-representation enables formal model selection as in shi2015model and hsu2017model.

Notation\;\; We will use the following notation thoughout. Let $V$ be a finite set with cardinality $N$. For any subset $S \subseteq V$, $\bm{1}_S \in \{0, 1\}^N$ denotes a vector with elements $(\bm{1}(v \in S))_{v \in V}$. For any $N \in \mathbb{N}$, $\bm{1}_N \in \mathbb{R}^N$ denotes a vector of ones. The cardinality of a set $S$ is denoted $|S|$. For any set $V$, $\Delta(V)$ denotes the set of all probability distributions supported on $V$. For sets $S_1, \dots, S_K$, their cartesian product is denoted by $\prod_{k = 1}^K S_k$. The subscript $a.s.$ in $\leqslant_{a.s.}$ or $\forall_{a.s.}$ indicates that the corresponding statement holds almost surely.

Outline\;\; The rest of the paper is organized as follows. Section 2 introduces the formal setup and motivating examples; Section 3 states the main result for regular support restrictions in discrete settings; Section 4 extends the discussion to non-regular settings and continuous response variables; Section 5 discusses computation and inference; Section 6 presents Monte Carlo experiments; and Section 7 contains an empirical illustration.

Basic Setup and Motivating Examples

Recalling the setup in the introduction, the observed data consists of an exogenous input $Z \in \mathcal{Z}$, an endogenous response $R\in \mathcal{R}$, and covariates $W \in \mathcal{W}$, which we omit for now. The set $\mathcal{Z} = \{z_1, \dots, z_K\}$ is assumed to be finite, and each $z\in \mathcal{Z}$ realizes with positive probability. The model specifies a vector of potential responses $R^*=(R(x))_{x\in\mathcal{X}}$, indexed by $x$ in a finite set $\mathcal{X}$, and a mapping $T:\mathcal{R}^{\mathcal{X}}\times \mathcal{Z}\to \mathcal{R}$ such that

The potential response $R(x)$ represents the outcome value that would be realized if the input is set to $x$; this quantity is generally unobservable. Instead, the researcher observes the pair $(R,Z)$ that satisfies (ref). We assume that the input is exogenous in the following sense.

Letting $Q \in \Delta(\mathcal{R}^{\mathcal{X}})$ denote the joint distribution of $R^*$, we consider general restrictions of the following type.

The following examples illustrate.

The next example will be our running example throughout the paper.

Our third example is a model of mediation.

Our final example is a model of interference.

Graphs and related objects

Our goal is to systematically derive testable implications of Assumptions (ref)--(ref). First, we focus on the case where $R$ is discrete and obtain testable restrictions on the conditional probabilities:

where $\mathcal{R}_z$ is the conditional support of $R$ given $Z=z$. The discreteness of $R$ is relaxed in Section (ref). To facilitate the derivation, we introduce the following object.

Each vertex $v_{r,z}$ represents a possible realization of $(R,Z)$, corresponding to the event “$R = r$ when $Z = z$.” For each $z_k \in \mathcal Z$, let $$ V_k = \{v_{r,z_k} : r \in \mathcal R_{z_k}\} $$ collect the vertices corresponding to $Z=z_k$. We write $V_G = \bigcup_k V_k$ throughout. Below, we illustrate the potential response graphs using examples. We will provide more details on the computation of $G$ in Section (ref).

For illustration, consider a special case of Example (ref) in which $R^* = D^*$ (i.e., treatment selection only) and both $D$ and $Z$ are binary. The vertex set \[ V_G = \{v_{0,0}, v_{1,0}, v_{0,1}, v_{1,1}\} \] enumerates all possible values of $(D, Z)$. Figure (ref) displays the potential response graph for the model that imposes the monotonicity restriction $D(1) \geqslant_{a.s.} D(0)$. An edge connects $v_{0,0}$ and $v_{1,1}$ because they are jointly consistent with the complier type: \[ r^* = (D(0), D(1)) = (0,1)\in \mathcal{R}^*, \] which satisfies monotonicity and generates the observable pairs $(D(Z), Z) = (0,0)$ and $(D(Z'), Z') = (1,1)$. Similarly, the other two edges correspond to the remaining latent types, commonly referred to as the never-taker (nt) and always-taker (at).

When $Z$ takes more than two values, the potential response graph uses a generalized notion of adjacent vertices to represent each support point. To see this, consider Example (ref). Recall that $Z\in\{0,1\}^2$, and $\mathcal{R}_z=supp(D\,|\,Z=z)=\{0,1\}$ for any $k$. Hence, $V_G=\{v_{r,z}\,|\,r\in \{0,1\},z\in\{0,1\}^2\}$. Consider $r^*=(0,0,0,0)$, the never taker in (ref). The presence of this support point ensures the following vertices

are mutually adjacent to each other. Indeed, each support point $r^*$ can be represented by a maximal clique $C$ of $G$ (defined below), a set of mutually adjacent vertices.\footnote{Figure (ref) in the Appendix show examples of such maximal cliques} We note that a maximal clique representing a support point takes a vertex from each $V_k$. Collecting such maximal cliques across all support points yields the graph in Figure (ref)-(a). Figure (ref)-(b) shows the graph's set representation, where each vertex $v_{r,z}$ is identified by the set of maximal cliques that contain $v_{r,z}$.

The following notion plays a central role in deriving testable restrictions.

For any $z$, vertices $v_{r,z}$ and $v_{r',z}$ are not adjacent because $R=r$ and $R=r'$ cannot occur simultaneously given $Z=z$. Hence, $V_G$ can be partitioned into $|\mathcal{Z}| =K$ independent sets $V_k,k=1,\dots,K$. Such a graph is called a $K$-partite graph.

Testable Implications for Discrete Outcomes

Maximal Independent Set Inequalities

This section presents the first main result: a testable implication of support restrictions for discrete outcomes. We begin with an intuitive argument based on Example (ref). Consider the set $\{v_{1,(0,0)}, v_{0,(1,0)}\}$. These two vertices are not adjacent because no latent type is compatible with both events --- namely, “$D = 1$ when $Z = (0,0)$” and “$D = 0$ when $Z = (1,0)$.” The former event is consistent only with type $at$, whereas the latter is consistent with types $nt$, $rc$, or $2c$; see Figure (ref)-(b).\footnote{Another equivalent way to describe this separation is to represent $D(z)$ by a structural random coefficient model and consider the sets of latent variable values consistent with either of the events; See Appendix (ref).} Since these two events cannot occur jointly under the modeling assumptions, it must be that \[ P(D=1 \mid Z=(0,0)) + P(D=0 \mid Z=(1,0)) \leqslant 1. \] The potential response graph provides a systematic way to identify all combinations of mutually exclusive events—and, equivalently, the sets of latent types that cannot co-occur. Such mutually exclusive events correspond to independent sets in the graph, with maximal independent sets generating the tightest inequalities. Figure (ref) lists all maximal independent sets of $G$ in Example (ref).

Theorem (ref) below formalizes this intuition.

Theorem (ref) applies to any support-type restriction as in Assumption (ref), which covers most restrictions considered in the literature. One can use well-established algorithms to find all maximal independent sets of a graph (see Section (ref)) and compute the inequalities above. Hence, the graph is not merely illustrative. It provides a complete model representation in which latent types correspond to $K$-cliques and testable restrictions correspond to independent sets, yielding an algorithmic route from economic assumptions to sharp moment inequalities.

Applying Theorem (ref) to Example (ref) gives the following set of testable implications:

These restrictions correspond to MISs 1--5 in Figure (ref) Panels (a)--(e).\footnote{The independent sets, which form a $K$-partition of $V_G$, also provide restrictions, but these are trivial because each restriction is based on a single conditioning event of $Z$. For example, MIS6 in Panel (f) of Figure (ref) implies that $P(D=1|Z=(0,0))+P(D=0|Z=(0,0))\leqslant 1$, which is trivial by the definition of a conditional probability.} For this example, the inequalities above can be shown to be sharp, meaning that they are also sufficient for the existence of the joint distribution of $D^*$ satisfying the partial monotonicity assumption.

More generally, in settings with discrete responses, Theorem (ref) recovers the inequalities derived in Kitagawa15 and bai2024, and provides a simple characterization of the sharp testable implications for the setting in Kwon:2024aa.\footnote{Continuous outcomes are discussed in Section (ref).} Natural questions are “Is (ref) always sharp? If not, what are the sharp testable implications?” The next section addresses these questions.

Sharpness

We now provide a general characterization of sharp testable implications and conditions under which (ref) is sharp. Recall that each $r^* \in \mathcal{R}^*$ induces a maximal clique of size $K$ in $G$, which contains exactly one vertex from each part $V_k$. Let $\mathcal{C}^*_G$ denote the set of such maximal cliques. For each $V\subseteq V_G$, let $\ell_G(V)=\max_{C \in \mathcal{C}_G^*}|C \cap V|\in \{1,\dots,K\}$ be the level of $V$, which counts the maximum number of components $V$ shares with the elements of $\mathcal{C}^*_G$. The following theorem characterizes the sharp testable implication of Assumptions (ref)--(ref).

The inequalities in Theorem (ref) fully characterize the empirical content of the underlying model. Note that, for any $I\in\mathcal{I}_G$, $\ell_G(I)=1$ because each clique intersects any independent set in at most one vertex. Hence, Theorem (ref) provides a subset of the inequalities in (ref), which may not be sharp in general. As discussed earlier, the MIS inequalities (ref)–(ref) in Example (ref) are sharp and have far lower cardinality than (ref) --- only five inequalities compared to $2^{|V_G|}=256$. Many potential outcome models (including Example (ref)) possess additional structure that guarantees this reduced set of inequalities remains sharp.

We now formalize this structure as a regularity condition on $\mathcal{R}^*$, which directly translates into a structural property of $G$. Recall that an (induced) subgraph $G'=G[V]$ of $G$ is a subset $V$ of the vertices of $G$ and all edges between them. A graph $G$ is perfect if, for every induced subgraph $G'$, the chromatic number $\chi(G')$ (the minimum number of independent sets needed to partition $V_{G'}$) equals the clique number $\omega(G')$ (the size of the largest clique).

The first condition can be verified either by checking any of the known analytical sufficient conditions discussed in the next section or by existing numerical algorithms (see Section (ref)). The second condition ensures a one-to-one correspondence between the model’s support points and the maximal cliques in $G$; it rules out cases where the graph implies more possible types than actually exist in $\mathcal{R}^*$. This condition can also be checked analytically or numerically. Notably, when $Z$ is binary, these regularity conditions are automatically satisfied.

The following is the third main result of the paper, which shows that the regularity of a support restriction can significantly simplify the analysis by limiting the class of constraints that must be checked.

Discussion

Several comments are in order. First, the framework easily accommodates additional control variables. Let $W \in \mathcal{W}$ denote a vector of observed covariates such that $R^* \perp Z \mid W$. This conditional independence assumption leaves the support restriction $\mathcal{R}^*$, and hence the independent sets of the potential response graph $G$, unchanged. Repeating the argument of Theorems (ref)--(ref), and conditioning on $W = w$, we obtain a richer set of testable implications:\footnote{If $\mathcal{R}^*$ varies with $w$, let $G_w$ represent the support restriction for a given $w$. Then, $\mathcal I_G$ in (ref) can be replaced with $\mathcal I_{G_w}$. Such a restriction appears in heiler2024treatmentevaluationintensiveextensive.}

When $W$ includes continuous variables, these inequalities provide a computationally tractable way to characterize sharp testable implications of the modeling assumptions. The implementation of corresponding testing procedures is discussed in Section (ref).

Second, verifiable sufficient conditions for Assumption (ref) exist. For example, if $Z$ is binary, all support restrictions are regular because the potential response graph is bipartite. Indeed, bipartite graphs can only have even cycles, and each support point (maximal clique) corresponds to a single link in a graph. If $Z$ is multi-valued, a useful sufficient condition for Assumption (ref) (i) is as follows.

This condition can be verified from the underlying restrictions. For example, consider Example (ref) with multi-valued $Z$ in an ordered set and assume the following exclusion and monotonicity: $Y(d, z) =_{a.s.} Y(d, z'), \forall d \in \mathcal{D}, \forall z, z' \in \mathcal{Z}$ and $D(z_1) \leqslant_{a.s.} \dots \leqslant_{a.s.} D(z_K)$. This model satisfies Condition (ref) with the following order:\footnote{See Section (ref) and Appendix (ref) for further details.}

More generally, there are many known subclasses of perfect graphs hougardy06, some of which could be verified on a case-by-case basis. Otherwise, one can use the numerical algorithm discussed in Section (ref).

Next, consider Assumption (ref) (ii). For each $v \in V_G$, let $S_v = \{ r^* \in \mathcal R^\mathcal X : v = (r,z),\; r = T(r^*,z)\}$ be the set of latent types compatible with observing $v$. We say that the family $\{S_v\}$ satisfies the pairwise incompatibility criterion if the following condition holds.

In words, if no latent type $r^*$ is compatible with observing $v_1,\dots,v_K$ jointly, then at least one pair $(v_i,v_j)$ must already be mutually incompatible with the support restriction. The criterion is equivalent to Assumption (ref) (ii); see Proposition (ref). Its appeal is that it can often be verified directly, without analyzing higher-order interactions among vertices. In particular, it rules out situations in which a support restriction is violated only through comparisons involving more than two vertices. In the example above, if $(v_1,\dots,v_K)$ does not correspond to a feasible support point, then there must exist a pair that violates either the exclusion restriction or monotonicity. Hence, the example satisfies the criterion. More generally, the pairwise incompatibility criterion can be verified for commonly used restrictions (see Appendix (ref)).

Third, our results on sharp characterization follow from a novel duality theorem for a class of convex polytopes, which may be of independent interest. To elaborate, denote $K = |\mathcal{Z}|$, $N = \sum_{z \in \mathcal{Z}}|\mathcal{R}_z|$ and $M = |\mathcal{R}^*|$. The support restriction $\mathcal{R}^*$ can be represented as a binary matrix $A^* \in \{0, 1\}^{N \times M}$, which we call a support matrix. The rows of $A^*$ are indexed by pairs $\{(r, z): r \in \mathcal{R}_z, z \in \mathcal{Z}\}$, the columns are indexed by support points $r^* \in \mathcal{R}^*$, and the entries $\bm{1}(r = T(r^*,z))$ indicate whether a value of the observables $(r,z)$ is consistent with a support point $r^*$. For each $k \in \{1, \dots, K\}$, let $P_k =(P(R = s\,|\,Z = z_k))_{s \in \mathcal{R}_{z_k}}$ denote the observed conditional distributions, and $\beta(P) = (P_1', \dots, P_K')'$. Any joint distribution $Q$ on $\mathcal{R}^*$ induces a vector $\beta(P)$ via $\beta(P) = A^*Q$. Such vectors form a convex polytope

given in its vertex ($V$) representation. Theorem (ref) provides a half-space ($H$) representation of $\mathbf{B}^*$, establishing that it coincides with the set of vectors $\beta(P)$ satisfying (ref).

Under Assumption (ref), $\ell_G(V) = \omega(G[V]) = \chi(G[V])$, for any subset $V \subseteq V_G$.\footnote{The relation $\ell_G(V) \leqslant \omega(G[V]) \leqslant \chi(G[V])$ always holds because $\mathcal{C}^*_G\subseteq \mathcal C_G$, and coloring the vertices of the largest clique requires at least $\omega(G[V])$ distinct colors. Assumption (ref)-2 implies that $\ell_G(V)=\omega(G[V])$. Assumption (ref)-1 implies $ \omega(G[V])=\chi(G[V])$.} This equality permits a partition of $V$ into independent sets $\{I_1, \dots, I_{\chi(G[V])}\}$ and rewrite (ref) as

The MIS inequalities ensure $\sum_{v_{r,z} \in I_j} P(R = r \mid Z = z)\leqslant 1$ for any $I_j$, which in turn implies the inequality above. Hence, under Assumption (ref), the inequalities in (ref) fully characterize the sharp testable implication.

Fourth, the question we consider can also be stated as follows: When does there exist a joint distribution $Q$ with a given set of marginals $P_1, \dots, P_K$ and a support $\mathcal{R}^*$? The case of $K = 2$ has been resolved in artstein1983distributions. In Theorem (ref) in the Appendix, we provide an extension to $K > 2$ for finite $\mathcal{R}^*$. We also give examples of support restrictions such that $\mathbf{B}^*$ is a strict subset of a polytope characterized by (ref) when either one of the two conditions in Assumption (ref) is violated. To the best of our knowledge, Theorem (ref) is new and may be of independent interest, e.g., in multi-marginal optimal transport problems; see pass2015multi. We discuss extensions to continuous settings in Section (ref).

Fifth, we have found that most support restrictions considered in the literature are regular. A notable exception is the IV model with multi-valued instruments ($K\geqslant 3$), where only exclusion is assumed: $Y(d,z)=Y(d,z'), \forall (d,z,z')\in \mathcal{D}\times\mathcal{Z}^2$, without any additional shape restrictions. This support restriction is non-regular because one can show its potential response graph $G$ is imperfect.\footnote{Interestingly, adding a monotonicity assumption to the exclusion restriction leads to a perfect potential response graph. Hence, the joint support restriction of exclusion and monotonicity is regular. See Section (ref).} Hence, while regularity holds for many structured restrictions (e.g., monotonicity/exposure maps), it may fail in some economically important models. In those cases, Theorem (ref) still characterizes sharp content, and our computational approach provides a way to generate additional inequalities beyond MIS. Indeed, for the example above with binary $Y$, one can show that the inequalities in (ref) with $\ell_G(V)=1$ recover those in Pearl1995Testability, and the inequalities with $\ell_G(V)=2$ and $\ell_G(V)=3$ recover the additional sharp inequalities found by KedagniMourifie2020 (see Appendix (ref)). Furthermore, we provide an algorithm that identifies collections of higher-level vertex sets $V$ with $\ell_G(V)=k$ for $k\geqslant 2$, which yield the additional inequalities needed to complete the characterization.

We conclude the discussion here with two remarks pointing out connections to the literature on partial identification using random sets galichon2011set, chesher2017generalized, molchanov2018random.

Extensions

This section provides extensions of the main theoretical results in two directions. Section (ref) extends Theorems (ref)--(ref) to accommodate general (e.g., continuous) outcome variables. As an application of this result, we characterize the sharp testable implications of IV validity with multi-valued treatment and instruments. Section (ref) presents a result that is useful for the practitioner to compare the testable implications of multiple models.

Testable Implications for General Outcome Variables

Let $\mathcal{R}$ be a subset of a Polish space, and let $\mathcal{X}$ be a finite set. Let $\mathcal{R}^*\subseteq \mathcal{R}^{\mathcal{X}}$ be the support of the potential response variable $R^*=(R(x))_{x\in\mathcal{X}}$. To characterize testable implications, we construct a sequence of partitions of the outcome space. For each $n\in\mathbb N$ and $k\in\{1,\dots,K\}$, let $\mathcal{A}_{n,k}\equiv\{A_{1,k},\dots,A_{n,k}\}$ be a partition of $\mathcal{R}_{z_k}$, consisting of $n$ nonempty sets. For any $m\leqslant n$, let $\mathcal{A}_{n,k}$ be a refinement of $\mathcal{A}_{m,k}$. That is, every element $A_{j,k}$ of $\mathcal{A}_{n,k}$ is contained in some element of $\mathcal{A}_{m,k}$. One can construct such a sequence by recursively splitting the cells in $\mathcal{A}_{m,k}$ into smaller disjoint sets.

For each $n$, we define the potential response graph $G_n=(V_{G_n},E_{G_n})$ associated with $\mathcal{R}^*$ as an undirected graph with vertices $V_{G_n} = \{v_{s,k}: s\in \{1,\dots,n\}, k \in \{1,\dots,K\} \}$ and edges $E_{G_n}=\{(v_{s,k},v_{s',k'})\in V_{G_n}\times V_{G_n}:\exists \, r^* \in \mathcal R^*: T(r^*,z_k) \in A_{s,k}, T(r^*,z_{k'}) \in A_{s',k'}\}.$ This construction naturally generalizes the definition of $G$ for discrete variables to the current setting. The following theorem characterizes the sharp testable implication of $\mathcal{R}^*$.

This theorem shows that sharp testable implications similar to those in the discrete case can be derived by replacing observable outcome values with a sequence of partitions. The following section illustrates an application of Theorem (ref).

Exclusion and Monotonicity with Multivalued Treatments and IV

We now revisit Example (ref). Let $Y$ be a continuous outcome, $D$ be a treatment in a finite, totally ordered set, and let $Z$ take $K$ different values. For example, $Y$ is healthcare utilization, $D$ indicates levels of insurance generosity (e.g., rates of coinsurance), and $Z$ is a random assignment to insurance plans with varying generosity AronDine2013RAND. The setting also includes the judge fixed effects design where $D$ is binary.

Consider testing the assumption that $z$ is excluded from the potential outcome, and $D$ is weakly monotonic in $z$:

To our knowledge, sharp testable restrictions for this setting have been unknown outside an important special case where both treatment and instrument are binary (see Remark (ref)).

For any $d \in \mathcal{D}$ and $S_d \subseteq \{1, \dots, K\}$, define $\underline{\ell}_d = \min_{\ell \in S_d}\ell$, $\overline{\ell}_d = \max_{\ell \in S_d} \ell$. Consider any finite set of tuples $\{ \{(B_{\ell, d}, d, z_\ell)\}_{\ell \in S_d} \}_{d \in \mathcal{D}}$ such that: (i) $\{B_{\ell, d}\}_{\ell \in S_d}$ form a partition of $\mathcal{Y}$, for each $d \in \mathcal{D}$; and (ii) $d < d' \implies \underline{\ell}_d \geqslant \overline{\ell}_{d'}$. Figure (ref) illustrates with $\mathcal{D} = \{1, 2, 3\}$ and $\mathcal{Z} = \{z_1, z_2, z_3, z_4\}$. Here, different colors represent the values $z_k$ of the instrument, and colored regions correspond to the sets $B_{k,d}, k\in S_d$. On the left side, $S_1 = \{4\}$, $S_2 = \{1, 2, 3, 4\}$, $S_3 = \{1\}$, and on the right side, $S_1 = \{3, 4\}$, $S_2 = \{2, 3\}$, $S_3 = \{1, 2\}$. It is clear that, for each $d$, $\{B_{\ell, d}\}_{\ell \in S_d}$ form a paritition of $\mathcal{Y}$. Furthermore, they satisfy (ii). For example, on the right side of Figure (ref), the index sets $S_d,d\in\mathcal{D}$ are constructed so that $\underline{\ell}_1=3=\overline{\ell}_2\ge \underline{\ell}_2=2=\overline{\ell}_3.$

Viewing each $(B_{\ell, d}, d, z_\ell)$ as a vertex of a graph, the vertices satisfying (i) and (ii) are independent of each other. This is because, for any pair of such vertices, they are independent in $G$ if either $z_\ell \ne z_{\ell'}$, $d = d'$ and $B_{\ell,d}\cap B_{\ell',d'} = \varnothing$ (i.e., $Y(d,z_\ell)$ and $Y(d,z_{\ell'})$ take values in disjoint sets despite $d$ being common), violating the exclusion, or $z_\ell \leqslant z_{\ell'}$ and $d_\ell > d_{\ell'}$ (i.e., $D(z_\ell)>D(z_{\ell'})$ despite $z_\ell\leqslant z_{\ell'}$), violating the monotonicity. Forming maximal independent sets from such vertices gives sharp testable restrictions. The following proposition summarizes the argument above. The restrictions' validity and sharpness follow from Theorem (ref).

Comparing and Interpreting Testable Implications of Multiple Models

Consider a collection of models characterized by different sets of support restrictions. The framework developed thus far can be used to derive testable implications and to apply existing model selection methods shi2015model,hsu2017model. When the model restrictions are nested, we can further show that adding or relaxing support restrictions has systematic effects on the implied inequalities. We provide a formal result and illustrative examples below.

Consider two models where Model 1 imposes a stronger support restriction than Model 2. The following proposition shows that every inequality implied by Model 1 is either a (weakly) tightened version of an inequality implied by Model 2 or a new inequality that does not appear among the testable implications of Model 2.

The tightening of the inequalities occurs because adding support restrictions deletes some of the edges present in the graph of the baseline model. This result allows practitioners to evaluate how adding or removing assumptions impacts the resulting restrictions. The following examples illustrate this point. \setcounter{example}{0}

Computation and Inference

This section discusses computational aspects of the proposed framework and inference procedures. Section (ref) outlines how to construct potential response graphs, compute the testable implication, and check their sharpness. A pseudo-code for the main steps is given in Algorithm (ref). To facilitate implementation, we provide a Python library for these tasks.\footnote{The library is available at \url{https://github.com/hkaido0718/SupportRestriction}. It contains functions for defining graphs, obtaining maximal independent sets, and checking the regularity condition, along with their applications to the examples in this paper.} Section (ref) shows how to conduct inference, and Section (ref) discusses alternative approaches.

Computing Testable Implications and Checking Regularity

A potential response graph $G$ can be constructed by one of two methods: 1. By checking pairwise incompatibility of the support restriction, or 2. By enumerating all $r^* \in \mathcal{R}^*$.

To elaborate on Method 1, consider Example (ref), in which the model imposes the exclusion restriction: \[ Y(d,z)=Y(d,z'), \qquad \forall (d,z,z')\in \mathcal{D}\times\mathcal{Z}^2. \] Let $v_{y,d,z}$ and $v_{y',d',z'}$ denote a pair of vertices such that the treatment status is identical, $d=d'$, but $z\neq z'$. If $y\neq y'$, no potential response function can be compatible with this pair, as doing so would violate the exclusion restriction. Hence, no edge exists between $v_{y,d,z}$ and $v_{y',d',z'}$. By contrast, if $y=y'$, the pair is compatible and an edge exists. This observation motivates a first method for graph construction: checking, for each pair of vertices, whether they are jointly compatible with the imposed support restrictions. This method is applicable to any model for the pairwise incompatibility criterion (Condition (ref)) holds.

An alternative approach (Method 2) is to enumerate all $r^* \in \mathcal{R}^*$ and construct the support matrix $A^*$. Given $A^*$, the graph $G$ is constructed by forming an edge between $v_{r,z}$ and $v_{r',z'}$ whenever there exists a column of $A^*$ in which the entries corresponding to rows $(r,z)$ and $(r',z')$ are both equal to 1. The matrix $A^*$ is the vertex--clique incidence matrix of $G$, and the associated adjacency matrix $A_G \in \{0,1\}^{N \times N}$ is given by $[A_G]_{kk} = 0$ and \[ [A_G]_{kl} = \mathbf{1}\!\left( \big[ A^* (A^*)' \big]_{kl} > 0 \right), \qquad k \neq l. \] When the number of support points is moderate, this procedure is straightforward to implement. Moreover, it applies to models in which the pairwise incompatibility criterion does not hold. That said, enumerating all support points can be computationally demanding in some settings. In the example above, the cost of enumerating all support points grows exponentially in $|\mathcal{D}|$ and $|\mathcal{Z}|$, whereas the cost of the pairwise construction grows only polynomially in these quantities; see Appendix (ref).

Next, we discuss how to compute the inequalities derived in Theorem (ref). Since most support restrictions $\mathcal{R}^*$ of interest are regular, we focus on the characterization in (ref). A discussion of non-regular restrictions is deferred to Appendix (ref). The potential response graph $G$ has $|\mathcal{Z}| = K$ parts with $|\mathcal{R}_k| = J$ vertices in each, so $|V_G| = KJ$ vertices in total. The number of edges is at most $|E_G| \leqslant J^2K(K - 1)/2$.

First, to compute all maximal independent sets (MIS), we can use the algorithm of tsukiyama1977new. The algorithm is output-sensitive: its time complexity is proportional to the total number of MIS and is of order $O(|V_G| |E_G| |\mathcal{I}_G|)$. Depending on the structure of the graph $G$, the number of MIS may be drastically different, ranging from $O(1)$ to $O(2^{|V(G)|})$, so the computing time will scale accordingly. The algorithm is implemented in the functions maximal_ivs in an R package igraph and maximal_independent_vertex_sets in the Python interface of the same package.

Second, to guarantee that the obtained testable implications are sharp, we need to verify that a given support restriction $\mathcal{R}^*$ is regular, that is, (i) $G$ is a perfect graph; and (ii) Every maximal clique of $G$ is listed in $\mathcal{C}_G$. Both conditions can be verified either analytically, as discussed in Section (ref), or numerically, as discussed below. Condition (i) can be verified in polynomial time using the fact that a graph $G$ is perfect if and only if none of its induced subgraphs is an odd cycle of length five or more, or a complement of one, a deep combinatorial result known as the Strong Perfect Graph Theorem chudnovsky2006strong, chudnovsky2020detecting. An implementation is available in SageMath as a built-in function is_perfect or, alternatively, using our Algorithm (ref) in Appendix (ref). In turn, Condition (ii) can be verified by finding all maximal cliques of size $K$ in $G$ and comparing them with the support points. This step can be performed efficiently using a version of bron1973algorithm algorithm as in eppstein2010listing, implemented, for example, in the max_cliques function in an R package igraph.

Inference Based on Moment Inequalities

Recall that $N = \sum_{k \in \mathcal{Z}}|\mathcal{R}_k|$ and $M = |\mathcal{R}^*|$. Denote $\beta(r\,|\,z) = P(R = r\,|\,Z = z)$ and $\beta_0(P) = ( (\beta(r \,|\,z))_{r \in \mathcal{R}_z})_{z \in \mathcal{Z}}$. If $R$ contains continuous components, let $\beta(s\,|\,z)=P(R \in A_{s,z}\,|\,Z = z)$ and define $\beta_0(P)= ( (\beta(s \,|\,z))_{A_{s,z} \in \mathcal{A}_{n,z}})_{z \in \mathcal{Z}}$ as in Section (ref). All objects introduced below are understood to be redefined accordingly.

Denoting $\mu(P) = (\bm{1}_I'\beta_0(P) - 1)_{I\in \mathcal{I}_G}$ and given an i.i.d. sample $\{(R_i, Z_i)\}_{i = 1}^n$ from a distribution $P \in \mathbf{P}$, the goal is to test $H_0: P \in \mathbf{P}_0$ against $H_1: P \in \mathbf{P} \backslash \mathbf{P}_0$, where

The restrictions of this form are known as moment inequalities without any nuisance parameters. Testing hypotheses of this form is an extensively studied problem; see canay2017practical for a technical review and canay2023user for a user's guide. In the following two sections, we provide a step-by-step implementation of the tests based on (ref), which we employ in our Monte Carlo experiments and empirical application.

Inference Based on Instruments and Discrete Covariates

Denote \[

\] A straightforward calculation shows that, for each $r \in \mathcal{R}_z$ and $z \in \mathcal{Z}$, \[ \sqrt{n}(\hat{\beta}_{n}(r\,|\,z) - \beta(r\,|\,z)) = \frac{1}{\sqrt{n}} \sum_{i = 1}^n \frac{\pi(z) \bm{1}(D_i = r, Z_i = z) - p(d, z) \bm{1}(Z_i = z)}{\pi(z)^2} + o_P(1). \] Thus, the asymptotic covariance matrix of $\hat{\beta}_{0, n}$ can be consistently estimated by

where \[ \hat{b}(R_i, Z_i) = \left(\left(\frac{\hat{\pi}_n(z) \bm{1}(R_i = r, Z_i = z) - \hat{p}_n(r, z) \bm{1}(Z_i = z)}{\hat{\pi}_n(z)^2} \right)_{r \in \mathcal{R}_z} \right)_{z \in \mathcal{Z}}. \] Next, let $\bm{A} \in \{0, 1\}^{|\mathcal{I}_G| \times N}$ denote the matrix with rows $a_I' = \bm{1}_I'$. Let $\mu_{I}(P) = a_I'\beta_0(P) - 1$, and \[

\] Consider the test statistic:\footnote{Other choices are possible, see andrews2010inference. We focus on the maximum test statistic since theoretical guarantees for the resulting tests are also available in high-dimensional settings; see chernozhukov2019inference and bai2022two.} \[ T_n = \max \left\{ \max \limits_{I \in \mathcal{I}_G} \frac{\sqrt{n}\hat{\mu}_{I, n} }{\hat{\sigma}_{I, n}},\; 0\right\}. \] The existing tests differ in how they construct the critical value with which $T_n$ is compared. The difficulty lies in bounding the slackness parameter $\sqrt{n}\mu_I(P)$, for $P \in \mathbf{P}_0$, which cannot be consistently estimated. Letting $\hat{u}_{I, n}$ denote a suitable upper bound on $\sqrt{n}\mu_I(P)$, we set $\hat{c}_{n, \alpha}$ to be a consistent estimator of the $(1-\alpha)$-th quantile of the distribution of \[ \max \left\{ \max \limits_{I \in \mathcal{I}} \frac{\sqrt{n}(\hat{\mu}_{I, n} - \mu_{I}(P)) + \hat{u}_{I,n}}{\hat{\sigma}_{I, n}} + , \;0 \right\}. \] Such an estimator may be obtained using bootstrap or Normal approximation, conditional on the data. Depending on the number of inequalities in (ref) relative to the sample size, different testing procedures may be appropriate. One may use, for example, andrews2010inference or romano2014practical when $|\mathcal{I}_G|$ is small, and chernozhukov2019inference or bai2022two when $|\mathcal{I}_G|$ is large. Then, under regularity conditions, the test \[ \phi_n = \bm{1}(T_n > \hat{c}_{n, \alpha}) \] is uniformly asymptotically valid over a large class of distributions.

If discrete covariates $W$ are present, one can use the same inference procedure, while replacing each conditioning statement $Z=z$ with $Z=z,W=w$, e.g., $\beta(r\,|\,z,w)=P(R =r\,|\,Z = z,W=w)$.

Inference with Continuous Covariates

Suppose $W$ contains finitely supported components $W_d$ and continuous components $W_c$, distributed over $[0,1]^{|w_c|}$. For such settings, one can use methods developed by andrews2013inference, CLR13, armstrong2015asymptotically, cox2023simple, and andrews2023inference.

We outline below the test of CLR13 (CLR, henceforth). Let $w=(w_d',w_c')'$. For each $w\in \mathcal{W}$, let $\beta_0(w) = (\beta(r \,|\,z,w)_{r \in \mathcal{R}})_{z \in \mathcal{Z}}$. Let $v=(w,I)\in \mathcal V=\{(w,I):w\in\mathcal W,I\in\mathcal{I}\}$ and define $\mu(v)=a_I'\beta_0(w) - 1$. Then, a testable implication can be formulated as an intersection bound:

Let $\hat\beta_{0,n}(w)=((\hat \beta_n(r\,|\,z,w))_{r\in\mathcal{R}})_{z\in \mathcal{Z}}$ be a series estimator of $\beta_0(w)$ such that

where $b_n(w)=(b_{n1}(w),\dots,b_{nm_n}(w))'$ is a vector of basis functions. For example, $b_{nk}(w)=\bm{1}(w_d=\tilde w)\times \psi_{nh}(w_c)$, where $\tilde w$ is a support point of the discrete component, and $\psi_{nh}$ is a tensor-product $B$-spline of a fixed order. Let $\hat\mu_n=(\hat\mu_n(v),v\in \mathcal{V})$ be an estimator of $\mu$ defined pointwise by $\hat \mu_n(v)= a_I'\hat\beta_{0,n}(w) - 1.$ Define the precision-corrected bound: $$S_{n,\alpha}=\sup_{v\in \mathcal{V}}(\hat\mu_n(v) - \hat c_{n,\alpha}\hat \sigma_n(v)), $$ where $\hat c_{n,\alpha}$ is a critical value (computed by CLR's Algorithm 1), and $\hat \sigma_n(v)$ is an estimator of the standard error of $\hat\mu_n(v)$. Then, under regularity conditions, the test \[ \phi_n = \bm{1}(S_{n,\alpha} > 0) \] is uniformly asymptotically valid (CLR, Theorem 5). The details of how to compute $\hat\sigma_n$ are summarized in Appendix (ref).

Alternative Approaches

The set of distributions $\mathbf{P}_0$ can alternatively be formulated as follows:

where $A^*$ denotes the support matrix. The set $\mathbf{B}^*$ is a convex polytope given in its vertex representation: each column of $A^*$ corresponds to a vertex of $\mathbf{B}^*$. The characterization in (ref) can be viewed as one based on the half-space representation of $\mathbf{B}^*$. Denoting \[ A =

; \;\;\;\; \beta(P) =

,\;\; \] the null set of distributions can be written as

In this form, the null hypothesis can be tested using any of the available methods for testing existence of solutions in linear systems with known coefficients, such as KitamuraStoye2018, fang2023inference, and goff2025inferencevaluelinearprogram. For example, fang2023inference computes a test statistic by solving linear programs (LP) and compares it to a critical value obtained by repeatedly solving LPs across bootstrap replications.\footnote{If $\hat\beta_{n,0}$ is not in the range of $A^*$, one also needs to additionally solve quadratic programs.} Details of this procedure are discussed in Appendix (ref).

Another approach relies on random set theory molchanov2018random. Although it is applicable more broadly, for simplicity, we use the notation appropriate for discrete $(R, X, Z)$. The modeling assumptions can be summarized by $R^* \in F(R, X)$, a.s., for a correspondence $F: \mathcal{R} \times \mathcal{X} \rightrightarrows \mathcal{R}^{\mathcal{X}}$ defined as $F(R, X) = B_X(R) \cap \mathcal{R}^*$, where $B_x(R) = \prod_{x' \in \mathcal{X}} ( \{R\}\bm{1}(x' = x) + \mathcal{R} \bm{1}(x' \ne x) ).$ By the theorem of artstein1983distributions, $R^* \in F(R, X)$, a.s., if and only if \[ P(R^* \in C \,|\, Z = z) \geqslant P(F(R, X) \subseteq C \,|\,Z = z), \;\;\;\; \forall \,C \subseteq \mathcal{R}^{*}, \;\; \forall \, z \in \mathcal{Z}. \] Since $R^*$ and $Z$ are assumed independent, $P(R^* \in C \,|\,Z = z) = Q(C)$, so we can express

This characterization may involve redundant inequalities and can often be substantially simplified; see luo_ponomarev_wan. In some cases (e.g., when testing an exclusion restriction in an IV model), all relevant inequalities in (ref) are binding, so the characterizations in (ref) and (ref) coincide. When $Z$ has rich support, (ref) typically delivers a simpler characterization.

The choice between testing procedures depends on the structure of the support restriction $\mathcal{R}^*$ and the associated polytope $\mathbf{B}^*$. When $\mathbf{B}^*$ has many vertices (i.e., latent types) but relatively few faces (i.e., inequalities), tests based on (ref) are typically preferable. As discussed earlier, a broad class of established testing and model selection procedures is available. Conversely, when $\mathbf{B}^*$ has few vertices and many faces, tests based on (ref) may be more suitable. To assess the relative complexity, we suggest applying the output-sensitive algorithm of tsukiyama1977new. If $|I_G|$ is moderate, it will quickly enumerate all MISs, making implementation of tests based on (ref) straightforward. The hypotheses considered in our empirical application have $|I_G|$ ranging from 1 to 726. For each hypothesis, our library found all MISs within 3 seconds.\footnote{The library was run on Google Colab with Python 3 Google Compute Engine with 12.67 GB RAM.} If $|I_G|$ is prohibitively large, we suggest checking the number of vertices of $\mathbf{B}^*$. If this number is moderate, tests based on (ref) are feasible.

Another possibility is to treat the testable implications as moment inequalities involving a nuisance parameter, and to apply inference methods designed for such settings andrews2023inference. For instance, the equality in (ref) can be reinterpreted as a pair of opposing moment inequalities with nuisance parameter $x \in \mathbb{R}^M_+$. Similarly, (ref) can be expressed as a family of moment inequalities indexed by $(z, C)$, with nuisance parameter $Q \in \Delta(\mathcal{R}^*)$. In contrast to the approach based on (ref), however, this method introduces additional parameters, which can make inference computationally intensive when $M$ is large.\footnote{andrews2023inference's andrews2023inference method uses the vertices of a feasibility set defined by a linear program. As they note (p. 2781), “Enumerating all of the vertices is, however, computationally prohibitive when there are many moments or nuisance parameters,” and they offer two computational shortcuts. One applies only to special cases, while the other, a more general method, requires solving a linear program at each step of a bisection algorithm, which becomes increasingly costly as $M$ (or $N$) grows.}

Finally, the nature of the available covariates is also an important consideration. An advantage of the half-space representation of $\mathbf{P}_0$ is its flexibility in accommodating covariates with rich, potentially continuous, support. For discrete covariates with $L$ support points, the computational cost of tests based on (ref) scales linearly in $L$. Moreover, our framework naturally accommodates continuous covariates through conditional moment inequality tests, as discussed in Section (ref). By contrast, tests based on (ref) require explicit discretization of covariates. While discretization is not inherently problematic, it introduces additional computational complexity. In particular, discretizing a covariate $W$ into $L$ bins expands the linear system in (ref) to dimension $(N+1)L \times ML$. This increase substantially raises the computational burden, especially when the resulting optimization problems must be solved repeatedly across bootstrap replications.\footnote{Although computational shortcuts may be available in specific implementations, the time complexity of the commonly used interior-point method is $O!\left(\sqrt{d},\log!\left(\tfrac{1}{\epsilon}\right)(p^{2}d + d^{2}p)\right)$, where $p=(N+1)L$ and $d=ML$, to achieve accuracy $\epsilon$. This cost is incurred in each bootstrap replication.}

Monte Carlo Experiments

In this section, we investigate the power of testing procedures based on the testable implications derived in Theorem (ref).

Partial Monotonicity

Let $D \in \{0, 1\}$ and $Z_1, Z_2 \in \{0, 1\}$, and consider the null hypothesis of partial monotonicity: \[ D(z_1, z_2) \leqslant_{a.s.} D(z_1', z_2') \iff z_1 \leqslant z_1' \text{ and } z_2 \leqslant z_2'. \] This support restriction is regular (by verifying Conditions (ref) and (ref) or numerical checks), so the testable implications of Theorem (ref) are sharp.

Suppose potential treatments are generated from a binary threshold model $D(z_1, z_2) = \bm{1}(B_0 + B_1z_1 + B_2z_2 \geqslant 0)$, where $(B_0, B_1, B_2)$ represent individual heterogeneity. Further, suppose $B_j = \beta_jW + U_j, j \in \{1, 2\}$, where $W \geqslant 0$, $\beta_j \geqslant 0$, $U_j \geqslant 0$, and $(W, U_1, U_2) \perp (Z_1, Z_2)$. This DGP satisfies full independence, $((D(z_1, z_2))_{z_1, z_2 \in \{0, 1\}}, W) \perp Z$, so we can compare the performance of conditional (on $W$) and unconditional tests. The null hypothesis corresponds to $B_1, B_2 \geqslant_{a.s} 0$, and we construct a family of DGP's under the alternative hypothesis as follows. Fix $\delta, \gamma \geqslant 0$ with $\delta + \gamma \leqslant 1$, and consider a family of DGPs under which $B_1, B_2 \geqslant 0$ are positive for share $h$ of the population, $B_1 \geqslant 0$ and $B_2 < 0$ for share $\gamma(1-h)$, $B_2 < 0$ and $B_2 \geqslant 0$ for share $\delta(1-h)$, and both $B_1, B_2 < 0$ for the remaining share $(1-\gamma-\delta)(1-h)$. We take $Z_1, Z_2, W \sim \text{i.i.d.}\; \text{Bernoulli}(1/2)$, $B_0 \sim N(1, 1)$, $\beta_1 = \beta_2 = 1/2$, and $U_1, U_2 \sim \text{i.i.d.} \; U[0, 1]$. We conduct $5000$ simulations with sample size $200$, four different values of $(\delta, \gamma)$, and $h \in [0, 1]$.

Figure (ref) depicts the power functions of two tests based on the inequalities derived in Theorem (ref). Since $W$ is discrete, we stack the corresponding conditional moment inequalities and use the GMS test of andrews2010inference. We find that conditioning on the exogenous covariates $W$ substantially improves power, and that power changes dramatically across different alternatives. For example, in Panel (a), the power of the conditional test reaches 0.5 already when a third of the population violates monotonicity with respect to both $z_1$ and $z_2$. However, in Panel (d), even though 100% of the population violates partial monotonicity with respect to $z_1$ or $z_2$, the power of the conditional test is still below 0.5. Since the test exhausts all information available in the data, these results imply that certain violations of the support restriction are fundamentally hard to detect.

Monotonicity

Let $D = \{1, \dots, J\}$ and $\mathcal{Z} = \{1, \dots, K\}$, and consider the null hypothesis \[ D(z) \leqslant D(z'), \;\; \forall z \leqslant z'. \] This support restriction is regular (by verifying Conditions (ref) and (ref) or numerical checks), so the testable implications of Theorem (ref) are sharp.

Suppose the potential treatments are generated from the multiple thresholds model \[ D(z) = \textstyle \sum_{j = 1}^{J} j\cdot \bm{1}\left( t_{j-1} \leqslant B_0 + \sum_{k = 1}^K B_k \bm{1}(z = k) \leqslant t_j \right) \] where $(B_0, B_1, \dots, B_K)$ represent individual heterogeneity, and $t_0, \dots, t_J$ is a set of thresholds. The null hypothesis corresponds to $B_k \leqslant_{a.s.} B_{k+1}$, for all $k \in\{ 1, \dots, K-1\}$. We focus on alternative DGPs such that, for each individual, monotonicity either holds for all $k$ or is violated for all $k$. Specifically, let $U_1, \dots, U_{K} \sim \text{i.i.d. } F_U$ and $U_{1:K}, \dots, U_{K:K}$ denote the order statistics ($U_{1:K}$ is the smallest). Let $U_0 \sim U[0, 1]$ be drawn independently of $U_1, \dots, U_K$, and \[ (B_1, \dots, B_K) = (U_{1:K}, \dots, U_{K:K})\cdot \bm{1}(U_0 \leqslant h) + (U_{K:K}, \dots, U_{1:K})\cdot \bm{1}(U_0 > h). \]

We set the thresholds $t_0, \dots, t_J$ uniformly on a grid within $[-1, 1]$ and $U_k \sim \exp(1)$. We conduct $S = 2000$ simulations with sample size $n = 200$, for $J \in \{3, 4\}, K \in \{2, 5, 10\}$ and $h \in [0, 1]$. Since the number of inequalities in some specifications exceeds the sample size, we use the test of bai2022two with the pre-test level $\beta = \alpha/10$. Figure (ref) presents the results. We find that instruments with richer support lead to substantially more powerful tests, at the expense of testing many moment inequalities.

Empirical Illustration

As an empirical illustration, we study the testable implications of various support restrictions in a potential outcome model of smoking cessation interventions. The purpose of this exercise is threefold: to demonstrate that empirically relevant hypotheses can be formulated as support restrictions, to show that the proposed tests are computationally feasible in practice, and to illustrate how they can yield economically meaningful insights.

We use data from the US Lung Health Study (LHS), a randomized clinical trial that enrolled subjects aged 35 to 59 with early chronic obstructive pulmonary disease. The study assigned subjects to three treatment groups: control ($C$), intensive smoking cessation therapy with an inhaled bronchodilator ($SIA$), and the same therapy with a placebo bronchodilator ($SIP$). The therapy intervention consisted of physician-led counseling on the health consequences of smoking, along with orientation sessions involving family members and friends. The sample size is $n=5,887$.

Interference

The LHS followed subjects and recorded both their smoking status and that of their spouses for up to five years after the intervention. Using a linear regression framework, Fletcher:2017aa document evidence of spillover effects of the SIA and SIP interventions on spouses’ smoking behavior. Motivated by these findings, we adopt the following formulation.

Let $i \in \{1,2\}$ index individuals, where $i=1$ denotes the subject and $i=2$ denotes the subject’s spouse. Let $Y_i$ be a binary outcome indicating whether individual $i$ smoked after the intervention. Let $z=(z_1,z_2)$ denote the vector of treatment assignments for individuals 1 and 2, and let $Y_i(z)$ denote individual $i$’s potential outcome under treatment assignment $z$.

For a given treatment assignment $z$, let $D_i(z)$ denote the level of exposure experienced by individual $i$. We define

where $z_i$ denotes individual $i$’s treatment status and $z_{-i}$ denotes the treatment status of individual $i$’s spouse. Here, $D_i=3$ corresponds to receiving the SIA treatment, $D_i=2$ corresponds to receiving the SIP treatment, $D_i=1$ indicates exposure through the spouse receiving either SIA or SIP, and $D_i=0$ indicates no treatment exposure. For now, we maintain the exposure mapping assumption in (ref), and we test this restriction in the next section.

Consider testing for the absence of spillover effects from a subject to their spouse. Using the exposure map, the restriction states\footnote{Under (ref), we can define a potential outcome $\tilde Y_i(d)$ such that $\tilde Y_i(d)=Y_i(z)$ for any $z$ with $D_i(z)=d$. The no spillover effect restriction can also be expressed more succinctly as $\tilde Y_i(1)= \tilde Y_i(0), ~a.s.$ Similarly, (ref) is equivalent to $\tilde Y_i(1)\le \tilde Y_i(0), ~a.s.,$ which can be viewed as a monotone treatment response assumption on the exposure-based potential outcome. }

We also test a weaker restriction that allows non-positive spillover effects:

Applying Theorem (ref) and conditioning on covariates $W_i$, the testable implications of the no spillover restriction are

Similarly, the testable implication of the non-positive spillover restriction is as follows.

We numerically verify that Assumption (ref) holds for both restrictions. Hence, by Theorem (ref), the model yields no additional testable implications.

We restrict the sample to married units with no missing observations, yielding 3,874 observations. Along with the exposure level $D_i$, we condition on survey wave $W_i \in \{1,2,3,4,5\}$. Since $W_i$ is discrete, we stack the conditional inequalities together and apply the tests of andrews2010inference (AS).\footnote{As a comparison, we have also implemented the test of romano2014practical and obtained the same conclusion.} Panel (A) of Table (ref) reports the results. The test rejects the null hypothesis of no spillover effects at the 5% significance level, but it does not reject the null of non-positive spillover effects. The test statistic for the latter restriction is zero, indicating that the sample satisfies the empirical analog of (ref). This finding suggests that a subject’s participation in the smoking cessation program can reduce a spouse’s likelihood of smoking. Hence, the smoking cessation programs may generate broader benefits while remaining cost-effective. One possible explanation is the spouse’s attendance at orientation sessions, as well as mutual support within the couple for quitting smoking and participating in the program.

Following Fletcher:2017aa, we next include an extended set of covariates: age, sex, education, BMI, survey wave, and an indicator for the spouse’s baseline smoking status, treating age and BMI as continuous. To handle continuous covariates, we use the test of CLR13 (CLR). The second row of Panel (A) of Table (ref) shows that the CLR test rejects the no-spillover null because the precision-corrected intersection bound is strictly positive, but does not reject the non-positive spillover null. These findings confirm that the earlier conclusions are robust to controlling for additional covariates.

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Exposure Mapping and Semi-monotonicity

So far, we have assumed that the exposure level determined the outcome. We now test this assumption. Let $Y=(Y_1,Y_2)$. By Theorems (ref)-(ref) (and ensuring Assumption (ref) numerically), the following inequalities are the sharp testable implications of the specified exposure mapping (EM) and imposing (ref):

and

In addition to the exposure mapping assumption, we consider imposing the following semimonotonicity restriction:

This restriction simultaneously imposes (i) administering an inhaled bronchodilator (as opposed to a placebo) weakly reduces $Y_i$, (ii) receiving the therapy as a subject weakly reduces $Y_i$ compared to receiving it as a subject's spouse, and (iii) being a subject's spouse weakly reduces $Y_i$ relative to the baseline. Imposing them adds 20 conditional moment inequalities (for each $w$) to (ref)-(ref).

Panel (B) of Table (ref) summarizes the results of the AS and CLR tests. When implementing the AS test, we used a sample of married units and included only the survey wave as a conditioning variable. The AS test does not reject the exposure mapping assumption. However, when conditioning on the extended set of covariates, the CLR test rejects the specified exposure mapping, indicating that a more refined definition of exposure levels may be needed for some $w$. The joint hypothesis of semimonotonicity and the exposure mapping is rejected by a large margin by both tests.

Persistence of the Cessation Program

Finally, we examine the persistence of the cessation program on the subject's smoking status. Let $Y(z)=(Y_1(z),\dots,Y_5(z))$ be the subject's potential outcome across the five survey waves at the treatment status $z\in \{C, SIP, SIA\}$. Let $L(z)$ denote the potential length of cessation, defined as the duration of the initial spell where $Y_t(z)=0$. Specifically, $L(z)=\max\{t\in\{0,1,\dots,5\}:Y_1(z)=Y_2(z)=\cdots=Y_t(z)=0\}$. We first consider the null hypothesis that treatment has no effect on cessation length:

Next, we examine a monotonicity restriction that assumes more intensive treatments weakly extend cessation length:

The restrictions allow many possible support points for $Y(\cdot)$: 4,682 types satisfy (ref) and 12,494 satisfy (ref), posing a computational challenge for the alternative approaches. Nevertheless, both satisfy Assumption (ref), and verifying this took under 10 seconds using despite the graph size.\footnote{The graph perfectness is checked by applying is_perfect() in SageMath.} Thus, it suffices to check inequalities implied by the maximal independent sets --- 726 for (ref) and 25 for (ref) --- making the AS test computationally feasible.

Panel (C) of Table (ref) reports results based on 5,738 subjects. The AS test rejects the null hypothesis of no effect, while failing to reject the monotonicity restriction. These findings are consistent with the interpretation that the intervention prolongs smoking cessation and that its effect is (weakly) larger when combined with a medical component.

Conclusion

This paper develops a systematic approach for deriving sharp testable implications of exclusion and shape restrictions in potential outcome models. The proposed framework covers a broad class of restrictions studied in the literature and accommodates continuous outcomes as well as control variables. A result of independent interest is a simple necessary and sufficient condition for the existence of a joint distribution with a given finite collection of marginals and support. The class of distributions to which our sharp characterization applies can be further intersected with sets imposing additional distributional assumptions on potential outcomes. Investigating the resulting testable implications in such settings is a promising direction for future research.