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Identifying Marginal Treatment Effects in the Presence of Sample Selection

This article presents identification results for the marginal treatment effect (MTE) when there is sample selection. We show that the MTE is partially identified for individuals who are always observed regardless of treatment, and derive uniformly sharp bounds on this parameter under three increasingly restrictive sets of assumptions. The first result imposes standard MTE assumptions with an unrestricted sample selection mechanism. The second set of conditions imposes monotonicity of the sample selection variable with respect to treatment, considerably shrinking the identified set. Finally, we incorporate a stochastic dominance assumption which tightens the lower bound for the MTE. Our analysis extends to discrete instruments. The results rely on a mixture reformulation of the problem where the mixture weights are identified, extending Lee's (2009) trimming procedure to the MTE context. We propose estimators for the bounds derived and use data made available by Deb, Munking and Trivedi (2006) to empirically illustrate the usefulness of our approach.


Introduction
Many interesting applications in the treatment effects literature involve two simultaneous identification challenges: endogenous selection into treatment and sample selection. For instance, in labor economics, when a researcher wishes to evaluate the effect of a job training program on wages, she has to consider the individuals' decision to enroll in the training program as well as their decision to participate in the labor market. In the health sciences, the same identification challenges appear when analyzing the effect of a drug on well-being as the outcome of interest -health status -is observed only for those who survive. Moreover, in randomized control trials (RCTs), non-compliance and endogenous attrition in treated and control groups lead to the same identification concerns. This double selection problem is also present when analyzing the effect of attending college on wages, the effect of an educational intervention on short-and long-term outcomes, and the effect of procedural laws on litigation outcomes. 1 In this paper, we derive novel uniformly sharp bounds on the marginal treatment effect (MTE) for individuals who would self-select into the sample regardless of their treatment status (M T E OO ). To do so, we propose identification strategies under increasingly restrictive sets of assumptions that extend the MTE identification to scenarios with endogenous sample selection. Furthermore, the choice of treatment is allowed to be endogenous, and can be related to the sample selection mechanism. To address both identification challenges described above, we analyze a generalized sample selection model in which the realized outcome (e.g., wages) is observed only if the individual self-selects into the sample (employment status), and the treatment choice (training program participation) is observed for all individuals in the data being analyzed.
The M T E and M T E OO provide important and intuitive measures of the treatment effect 1 Training programs are studied by Heckman, LaLonde, and Smith (1999), Lee (2009) and Chen and Flores (2015). The college wage premium is analyzed by Altonji (1993), Card (1999) and Carneiro, Heckman, and Vytlacil (2011). Some education interventions are studied by Krueger and Whitmore (2001), Angrist, Bettinger, and Kremer (2006), Angrist, Lang, andOreopoulos (2009), Chetty et al. (2011) and Dobbie and Jr. (2015). Medical treatments are analyzed by CASS (1984), Sexton and Hebel (1984) and U.S. Department of Health and Human Services (2004). Litigation outcomes are discussed by Helland and Yoon (2017). RCTs with attrition are illustrated by DeMel, McKenzie, and Woodruff (2013) and Angelucci, Karlan, and Zinman (2015). and its heterogeneity across the population. For example, consider a job training program.
Training influences both the likelihood of employment and wages, which are observed only for individuals who are employed. The M T E reflects the returns to training for individuals with different levels of the (latent) cost of attending the program. As a consequence, this parameter sheds light on the heterogeneity of the training program's effects, i.e., to understand who would benefit from taking extra training. This knowledge can be used to design policies focusing on targeting of the program, affordability and services offered. Common parameters evaluated in the literature -such as the average treatment effect (AT E), the average treatment effect on the treated (AT T ), the average treatment effect on the untreated (AT U ), and the local average treatment effect (LAT E) -may not adequately capture this heterogeneity. For example, they could be positive even when most people are affected adversely by a policy, masking its effects. 2 Similarly, the M T E OO reflects the training program's effects at the intensive margin, i.e., to the group of individuals that are more attached to the labor force, and would be employed even if they had not attended the training program.
Our strategy to partially identify the M T E OO relies on standard assumptions regarding selection into treatment. Those are the same conditions usually imposed for identification of the MTE when there is no sample selection, i.e., there is an exogenous instrument excluded from the outcome determination, treatment choice is monotone in the instrument, and the propensity score is continuous. 3 We add the requirement that the instrument is also excluded from the sample selection mechanism. In the job training literature, a similar assumption is frequently imposed when analyzing RCTs with imperfect compliance, as the interest frequently lies on the effect on labor earnings and employment.
In this paper's first contribution, the partial identification result for the M T E OO leaves sample selection unrestricted, providing very general uniformly sharp bounds on that parameter. Our second result tightens the bounds around the M T E OO by exploiting a "monotonicity in selection" assumption. This condition is similar to the usual LATE monotonicity assump-tion and requires individuals to be at least as likely to be observed in the sample if they are treated. In the job training program example, this additional assumption imposes that the treatment can induce workers to join the labor force, but not the opposite.
The identified set is further reduced by imposing a stochastic dominance assumption to our third and final set of assumptions. This condition mandates that the subpopulation who self-selects into the sample regardless of the treatment status has higher potential outcomes if treated than the subpopulation who self-selects into the sample only when treated. Intuitively, workers with high attachment to the labor force (that would be employed regardless of receiving job training), would earn higher wages after the extra training than those that would choose to participate in the labor force only if they participated in the training program.
Identification relies on a reformulation of the potential outcomes' conditional probabilities as a mixture between the latent groups of individuals who are "always observed" and "observed only when treated." This reformulation extends to the MTE case the trimming procedure proposed by Imai (2008), Lee (2009) and Chen and Flores (2015) in the context of identifying the ATE and LATE. Crucially, since we are interested in the MTE, the trimming is based on the distribution of the potential outcome conditional on unobserved individual characteristics related to treatment receipt.
The results can be used to construct bounds for any treatment effect parameter that can be written as a weighted average of the M T E OO . We derive new weights to obtain sharp bounds on the ATE, the ATT, the ATU, any LATE (Imbens and Angrist, 1994) and any policy-relevant treatment effect (PRTE, Heckman and Vytlacil, 2001b) within the alwaysobserved subpopulation. Differently from the weights for the case without sample selection (Heckman and Vytlacil, 2005, Carneiro and Lee, 2009, and Carneiro, Heckman, and Vytlacil, 2011, the new weights must be integrated over the distribution of the latent heterogeneity for the always-observed subpopulation instead of its unconditional distribution. Furthermore, we show these weights are identified under the "monotonicity in selection" assumption if the support of the propensity score is the full unit interval. Moreover, we propose and discuss nonparametric and parametric estimators for the bounds 4 around the M T E OO . The usefulness and feasibility of the parametric procedure is illustrated in an empirical example using the data set organized by Deb, Munkin, and Trivedi (2006). We find that the effect of insurance plan choice on ambulatory expenditures depends negatively on the agents' relative cost of choosing managed care plans over fee for service plans. This corroborates the results found by Deb, Munkin, and Trivedi (2006), indicating that agents endogenously self-select into managed care plans.
The main identification results are extended to the case where the researcher only has access to multi-valued discrete instruments. In this context, we derive nonparametric sharp bounds on a weighted average of the MTE. This result is conceptually similar to Chen and Flores (2015), who provide an outer set for the LATE parameter when the instrument is binary. 4 This paper is organized as follows. Section 2 presents the structural model and sample selection mechanism considered, followed by a discussion of the identifying assumptions. In Section 3, we provide the identification results for the MTE bounds in the case of a continuous instrument under each set of assumptions while Section 4 discusses general aspects of estimation for the bounds. Section 5 illustrates an estimation procedure and the identifying power of each set of assumptions using data made available by Deb, Munkin, and Trivedi (2006). Section 6 presents identification results for the case of discrete instruments. Section 7 concludes. The proofs, sharp testable implications of the model, extensions to DMTE, numerical example, proposed estimation methods' details, Monte Carlo simulations for the estimator's performance, economic models illustrating our identifying assumptions, and an alternative specification for our empirical example are presented in the appendix.

Analytical Framework
Following Lee (2009), andChen andFlores (2015), we consider the generalized sample selection model, described in the potential outcomes framework: where Z is a vector of observable instrumental variables (e.g., random assignment of cash incentives for participation in a job training program) with support given by Z ⊂ R dz , D is the treatment status indicator (job training program enrollment). The variable Y * is the possibly censored realized outcome variable (wages) with support Y ⊂ R, while Y * 0 and Y * 1 are the possibly censored potential outcomes when the person is untreated and treated, respectively. Similarly, S is the realized sample selection indicator (employment status), and S 0 and S 1 are potential sample selection indicators when individuals are untreated and treated.
Finally, Y is the uncensored observed outcome (labor earnings), and V represents unobserved individual characteristics (cognitive or social costs of attending the job training program).
The researcher observes only the vector (Y, D, S, Z), while Y * 1 , Y * 0 , S 1 , S 0 and V are latent variables. 5 This model is a generalization of that considered in Heckman and Vytlacil (1999, 2001a, 2005 to the sample selection setting. The treatment status D is connected to the instrument Z and the unobserved characteristics V through the unknown function P : Z → R. In this model, we assume that the individual receives treatment when its idiosyncratic cost V is less than or equal to a threshold P (Z). This assumption is equivalent to imposing monotonicity of the treatment in the instrument Z (Imbens and Angrist, 1994) as shown by Vytlacil (2002). This setup is similar to the one proposed by Heckman and Vytlacil (2005) and leads to the definition of the MTE, for any p ∈ [0, 1].
In the setting analyzed here, the task of learning about the MTE is further complicated by the potential for nonrandom sample selection. As pointed out by Lee (2009), even in the simpler case of the ATE, point identification is no longer possible even if the treatment is randomly assigned, leading him to derive bounds for the ATE. This paper combines the insights of these literatures to develop sharp bounds for the MTE under sample selection while allowing for treatment to be endogenously determined.
Similarly to the compliance groups defined by Imbens and Angrist (1994), we define four latent groups based on the potential sample selection indicators. The subpopulations are defined as: always-observed (S 0 = 1, S 1 = 1), observed-only-when-treated (S 0 = 0, S 1 = 1), observed-only-when-untreated (S 0 = 1, S 1 = 0), and never-observed (S 0 = 0, S 1 = 0). 6 Those subgroups are summarized in Table 1.  (2008) and Lee (2009), we focus on the alwaysobserved subpopulation (S 0 = 1, S 1 = 1). Importantly, this subpopulation is the only group whose censored potential outcomes are observed in both treatment arms. For the other three subpopulations, treatment effect parameters are not point-identified or bounded in a non-trivial way without further parametric assumptions, since at least one of the potential outcomes (Y * 0 or Y * 1 ) is never observed. 7 Since our focus is on a fully non-parametric identification strategy, we do not discuss parametric identification of unconditional treatment effect parameters or for treatment effect parameters associated with the latent groups ON , N O and N N .
Our target parameter is the MTE function for the subpopulation who is always observed We also stress that the identified set can be substantially tightened by imposing that the sample selection mechanism is monotone in the treatment.
Assumption 6 (Monotone Sample Selection). Treatment has a non-negative effect on the sample selection indicator for all individuals, i.e., S 1 ≥ S 0 .
This monotonicity assumption rules out the existence of the observed-only-when-untreated subpopulation and is commonly used in the sample selection literature (Lee, 2009, Chen andFlores, 2015). 12 To obtain some intuition on the mechanisms behind this assumption, consider the job training program example. An individual is employed when her job search skills ϑ(D), a function of training take-up, are above a threshold U S so that S = 1 {ϑ(D) ≥ U S } .
Additionally, suppose that attending the job training program does not decrease someone's job search skills, i.e., ϑ(1) ≥ ϑ(0), making it more likely that program's trainees would be observed in the data. In such a case, Assumption 6 holds. However, if attending the job training program raises the agents' reservation wages or if the lost labor market experience is very costly in terms of job finding, this assumption may not hold.
Assumptions 1-6 form our second set of identification assumptions and lead to the bounds for M T E OO that are the main result of this paper, presented in Proposition 2. This second set of assumptions has a testable implication: the treatment positively affects sample selection, In other words, the share of the population for which the outcome is observed rises with p.
We discuss further testable implications in Section 3, and formally characterize sharp testable implications arising from those assumptions in Appendix B. In the job training example, this testable implication means that the likelihood of employment increases with the probability of attending the training program.
We can further shrink the identified set around the M T E OO , by adding Assumption 7 and completing the third set of identifying assumptions.
Assumption 7 (Stochastic Dominance). The distribution of the potential outcome when treated for the always-observed subpopulation first-order stochastically dominates the distribu-12 As in Lee (2009), this assumption can be stated as S1 ≥ S0 with probability 1. For the sake of simplicity, we assume it to hold for all individuals. Manski (1997) and Manski and Pepper (2000) refer to this assumption as the "monotone treatment response" assumption. All results can be stated with some straightforward changes if the inequality in Assumption 6 holds in the opposite direction. tion of the same random variable for the observed-only-when-treated subpopulation, i.e., for any y ∈ Y and any p ∈ P.
This dominance assumption imposes that the always-observed subpopulation has higher potential censored outcomes than the observed-only-when-treated group conditional on V .
This type of assumption is common in the literature (Imai, 2008, Blanco, Flores, and Flores-Lagunes, 2013, Huber and Mellace, 2015, and Huber, Laffers, and Mellace, 2017 and is intuitively based on the argument that some population sub-groups have more favorable underlying characteristics than others. 13 Naturally, the plausibility of Assumption 7 depends on the empirical context. In some cases this stochastic dominance assumption could be hard to interpret and motivate empirically. However, this challenge can be confronted constructively in a layered policy analysis (Manski, 2011), since we can offer a menu of estimates based on different assumptions, allowing the researcher to understand the continuum of information that we can gather about a specific economic parameter, as advocated by Tamer (2010). In Appendix D, we provide a simple economic model related to our job training example to illustrate that Assumption 7 may hold under plausible economic restrictions.
Remark 1. Point identification of M T E OO is achieved if, in addition to assumptions 1-5, we assume that the always-observed and never-observed subpopulations are the only existing groups, i.e., S 0 = S 1 . This "no selection effect" assumption imposes that the treatment has no impact on sample selection. This set of assumptions has a testable implication: = 0 for all p ∈ P. In the job training program context, "no selection" implies that workers' employment status would not be affected by program take-up and the likelihood of employment does not depend on the probability of attending the training program.

Identification Results
This section presents the main results of this paper, the identification for M T E OO (p) under the three different sets of assumptions described in Section 2. As stepping stones, Subsection 3.1 shows identification of the conditional joint distribution of (Y * d , S d = 1)| V for any d ∈ {0, 1}, while Subsection 3.2 shows that the distribution of the potential outcomes can be seen as a mixture of latent groups, an important feature of the model. In the following subsections, we sharply bound the M T E OO under increasingly restrictive assumptions. First, we bound the M T E OO without imposing any assumption on the selection mechanism (Subsection 3.3). We then tighten those bounds by additionally imposing monotone sample selection (Subsection 3.4) and stochastic dominance (Subsection 3.5). For completeness, we also show that the M T E OO is point-identified under the "no selection effect" assumption (Remark 1) at the end of Subsection 3.5. Finally, in Subsection 3.6, we discuss how to sharply bound treatment effect parameters that can be written as weighted averages of the M T E OO (p).

Identifying the Joint Distribution of Potential Outcome and Selection
Before we discuss the identification of the M T E OO , we point-identify the conditional joint distribution of each potential outcome and sample selection for different levels of individual Under Assumptions 1-5, for any p ∈ int P and any Borel set A ⊆ Y, we have that where the second equality follows from Assumption 1, the third and fourth equalities follow 13 from the Law of Iterated Expectations, and the last equality follows from Assumption 5. By differentiating each side with respect to p, we point-identify the conditional distribution of (Y * 1 , S 1 = 1) given V = p as follows: Similarly, we can show that Note that since equations (3.1) and (3.2) reflect probabilities, they generate two testable implications for Assumptions 1 and 5: ∂p . This effect could be of interest in itself: for example, the researcher may want to evaluate whether a training program increases employment 14 A formal characterization of the sharp testable implications implied by our model is given in Appendix B.
14 levels. We can also identify the MTE on the observed outcome, We would like to disentangle the marginal treatment on the observed outcome into the extensive margin and the intensive margin. While the extensive margin is point-identified, we show that the intensive margin (M T E OO ) is partially identified by considering that the distribution of potential outcomes is a mixture of latent groups.

Potential Outcomes as Mixtures of Latent Groups
Fundamental to our identification strategy is recognizing that the observed treated (untreated) group is composed only by OO and N O (ON ) types, as described in Table 1. Hence, the conditional distribution Y * 1 | S 1 = 1, V = p can be written as the mixture of these latent distributions. For notational simplicity, let α(p) ≡ P[OO|V =p] P[S 1 =1|V =p] be the share of always-observed individuals among those for which S 1 = 1 conditional on V = p. Naturally, the remainder, P[S 1 =1|V =p] , can be described as 1 − α(p). By the Law of Total Probability, we have that: As a consequence, E[Y * outcome within the always-observed subpopulation (3.10) (3.11) we find that M T E OO (p) lies within the interval Remark 2. The issue central to identification of the target parameter is what can be learned about the mixture weights (α(p), β(p)), and Note that the bounds on the MTE of interest will be tighter for higher values of α(p) and β(p) because we learn about by considering that the worst-and best-case outcomes of observed treated and untreated individuals are fully attributed to the always-observed. So, as α(p) increases, the share of the observed sample of treated individuals that are from our group of interest increases, providing more information about their conditional expectation of the outcomes. In the extreme case in which α(p) → 1, the E [ Y * 1 | S 0 = 1, S 1 = 1, V = p] will be point identified. Similarly, if α(p) → 0, the observed sample is uninformative about the always-observed group. A similar intuition holds regarding β(p).
In the next subsections, we investigate the bounds that are generated under the alternative sets of assumptions described in Section 2. Intuitively, those assumptions impose different restrictions on the possible values of the mixture weights (α(p), β(p)), providing different sets

Identification with No Assumption on the Sample Selection Mechanism
Initially, consider the case in which the researcher is only willing to consider Assumptions 1-5, leaving the sample selection mechanism unrestricted. To learn about M T E OO , we need information about the share of always-observed individuals in the total population, P [S 0 = 1, S 1 = 1] and, hence, the conditional joint distribution of (S 0 , S 1 )| V = p. However, we only have information about the conditional marginal distributions S 0 | V = p and S 1 | V = p based on equations (3.5) and (3.6). According to Imai (2008) and Mullahy (2018), the following Boole-Fréchet bounds are sharp around the share of always-observed individuals: (3.12) Combining this information with Equations (3.5) and (3.6), leads to Lemma 1.
Lemma 1. Under Assumptions 1-5, the share of always-observed individuals is partially identified: These bounds are sharp.
Note that, since Υ (p) provides the identified set of possible values for the share of alwaysobserved individuals, we can obtain the equivalent range of possible values for α(p) and β(p), the mixture weights described in Subsection 3.2. For brevity, let P [S 0 = 1, S 1 = 1|V = p] take any particular value, υ ∈ Υ (p). Define, Let the bounds in Equations (3.8)-(3.11), for specific values of α(p, υ) and β(p, υ) in the identified set be written as:
To bound the target parameter, we find worst-and best-case scenarios by varying the value υ. Explicitly, M T E OO (p) is partially identified and lies within the interval min υ∈Υ(p) Note that υ has a monotone relationship to the mixture weights, which define the trimming points in the bounds. As previously discussed, higher values for α(p) (β(p)) indicate that a bigger share of the observed treated (untreated) population belongs to the always-observed latent group, thus providing more information and tighter bounds for the parameter of interest.
Hence, we only need to focus on the scenario that generates the wider bounds, that is, the smallest admissible α(p) and β(p). Let υ be the lower bound of Υ(p). We have: Making the same argument to the upper bound, we can rewrite them as, min υ∈Υ(p) greatly simplifying our bounds, which need only to be evaluated at the end point of Υ(p).
We can combine these facts with equations (3.1), (3.2), (3.5) and (3.6) to propose the first identification result for M T E OO , which does not impose meaningful restrictions on the sample selection mechanism.
Proposition 1. Under Assumptions 1-5, the MTE is partially identified for the alwaysobserved, i.e., for any p ∈ P, where ∆ 1 : P → R and ∆ 1 : P → R are given by for any p ∈ P, the conditional distribution ofỸ d is given bỹ for any d ∈ {0, 1}, and Moreover, these bounds are uniformly sharp.
Remark 3. The definition of uniform sharpness (Firpo and Ridder, 2019) used in this article states that, for any function δ : P → R such that δ (p) ∈ ∆ 1 (p) , ∆ 1 (p) for any p ∈ P, it is possible to construct random variables (Ỹ * 0 ,Ỹ * 1 ,S 0 ,S 1 ,Ṽ , Z) that satisfy all the restrictions imposed on the data by the underlying assumptions, induce the joint distribution on the data (Y, S, D, Z) and achieve the candidate target parameter δ (p) = E Ỹ * 1 −Ỹ * 0 S 0 = 1,S 1 = 1,Ṽ = p for every p ∈ P.
Remark 4. In the special case in which the treatment D is independent of the potential outcomes for Y * and S, and that Υ(p) = Υ for any p ∈ P = [0, 1], the MTE on the censored outcome will be constant and equal to the ATE for the always-observed. Then, the bounds derived in Proposition 1 simplify to those derived by Imai (2008, Proposition 1).

Bounds under the Monotonicity Assumption
In this subsection, we introduce monotonicity of sample selection in the treatment (Assumption 6), which can considerably shrink the identified set for M T E OO . As discussed in Section 2, under the monotonicity assumption, individuals who self-select into the sample when untreated (S 0 = 1) would also be observed if they had been treated, ruling out the sub- This result connects the conditional share of always observed individuals to changes on the conditional mass of observed untreated individuals when the propensity score increases. Looking at the second equality, we find that the conditional probability of being always observed is the difference between the increase in the share of observed treated individuals and the increase in the share of observed individuals when the propensity score is equal to p. . (3.14) Note that the right-hand side of equation (3.14) is point-identified according to equations Then, under monotonicity, the researcher has to obtain bounds only for the expected potential outcomes under treatment, which still can be written as a mixture of the alwaysobserved and observed-only-when-treated latent subpopulations.
Remark 5. By adding the monotonicity assumption, we increase the lower bound and decrease the upper bound stated in Proposition 1. The length of the identified set here is strictly shorter than the length of the identified set in Proposition 1 when the mixture weights are not point identified. 15 This improvement shows the identifying power of Assumption 6.
Remark 6. In the special case in which the treatment D is independent of the potential outcomes for Y * and S, and α(p) = α for any p ∈ P = [0, 1], the MTE will be constant and equal to the average treatment effect for the always-observed and the bounds in Proposition 2 simplify to the ones proposed by Lee (2009, Proposition 1a).

Bounds under the Monotonicity and Dominance Assumptions
In this section, we add the stochastic mean dominance assumption to tighten the identified set for M T E OO under Assumptions 1-7. Stochastic dominance and equation (3.7) imply that for any y ∈ Y. As a consequence, the following inequality holds This tightens the lower bound for E[Y * 1 |S 0 = 1, S 1 = 1, V = p] as we no longer need to focus on the lowest α(p) mass of outcomes as the lower bound, since the stochastic dominance assumption guarantees that the expectation of outcomes for the always observed subpopulation will be larger than the one of the observed treated individuals which mixes OO and N O Moreover, these bounds are uniformly sharp.
Remark 7. The lower bound proposed here is strictly greater than the one in Proposition 2 when α (p) ∈ (0, 1) and the distribution of Y * 1 | S 1 = 1, V is not degenerate. This improvement shows the identifying power of Assumption 7.
Remark 8. In the special case in which the treatment D is independent of the potential outcomes for Y * and S, and α(p) = α for any p ∈ P = [0, 1], the MTE will be constant and equal to the average treatment effect for the always-observed and the bounds in Proposition 3 simplify to those proposed in Imai (2008, Equation (8)).

For completeness in the identification discussion, note that point identification of M T E OO
is achieved if, in addition to assumptions 1-5, we assume that the always-observed and neverobserved subpopulations are the only existing groups, i.e., S 0 = S 1 . This "no selection effect" assumption imposes that the treatment has no impact on sample selection. In the job training program context, this implies that workers' employment would not be affected by the program.
Under Assumptions 1-5 and "no selection effect", the distributions of Y * d | S 1 = 1, V for d ∈ {0, 1} are exclusively composed of always-observed individuals (α(p) = β(p) = 1). Then, where point-identification follows from equations (3.1), (3.2), (3.5) and (3.6). The expectations Alternatively, point-identification of the unconditional MTE is achieved if potential sample selection status and outcomes are independent given the unobserved characteristics, In this case, the distributions P [Y * d ≤ y|V = p] are point-identified from Equations (3.1) and (3.2), and the unconditional MTE is point-identified as:

Empirical Relevance of Bounds for the MTE OO
The partial identification results for the M T E OO are relevant for a vast array of empirical objectives. First, bounds for the M T E OO can illuminate the treatment effect's heterogeneity, allowing researchers to better understand who would benefit from a specific treatment. This feature is important because common parameters (e.g., AT E, AT T , AT U , and LAT E within the always-observed subpopulation) can be positive even when most people are adversely affected by a policy. Moreover, knowing, even partially, the M T E OO function can be useful to design policies that provide incentives to agents to take some treatment.
Second, the M T E OO bounds can be used to partially identify alternative treatment effect parameters that are described as a weighted integral of M T E OO because where t ∈ {1, 2, 3}, ∆ t and ∆ t are described in Propositions 1-3, and ω(·) is a known or identifiable weighting function. Those bounds are sharp, as summarized in Proposition 4. 16 Proposition 4. Let ω T E : P → R be a known or identifiable weighting function and define the treatment effect parameter T E := P M T E OO (p) · ω T E (p) dp. Under Assumptions 1-6, the treatment effect parameter TE is partially identified: Moreover, these bounds are sharp.
Under Assumptions 1-6, Table 2 shows the most relevant treatment effect parameters that 16 We focus on the case under the monotonicity restriction (Assumptions 1-6) for brevity. Similar results hold under our other identifying sets of assumptions.

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are partially identified using Proposition 4. The weights in Table 3 are derived in Appendix A.6 and are identified if the support of the propensity score is the full unit interval, i.e., P = [0, 1].
for two policies a and a that affect only Z (See notational details in Appendix A.6.) Importantly, note that, differently from the weights for the case without sample selection (Heckman and Vytlacil, 2005, Carneiro and Lee, 2009, and Carneiro, Heckman, and Vytlacil, 2011, these weights must be integrated over the distribution of the latent heterogeneity for the always-observed subpopulation instead of its unconditional distribution.

Estimation
This section describes the general estimation steps for the bounds proposed in Proposition 2, while details on two proposed methods are provided in Appendix E and G. For brevity, we focus on the bounds identified under monotonicity of sample selection in the treatment (Assumptions 1-6), as it is the most relevant (and feasible) case empirically. Estimators for the bounds in Proposition 1 and Proposition 3 are natural extensions. Appendix F presents a Monte Carlo Simulation that evaluates the small sample properties of the estimator.
To estimate the bounds in Proposition 2, it is useful to focus on the building blocks that are the foundation for the identification results. In particular, we need the CDFs: ∂p dp for any d ∈ {0, 1} and Consequently, we need to estimate: Furthermore, the estimation of the propensity score, P (Z), is necessary to obtain the moments of the conditional distribution of the observed outcome.

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Each of these components can be estimated by standard approaches, and multiple procedures might be valid depending on the assumptions and model structure the researcher is willing to impose. For example, one could resort to nonparametric methods to estimate P (Z), π 0 (p), π 1 (p), Γ 0 (p, y), and Γ 1 (p, y), avoiding functional form choices as discussed in Appendix E. While appealing, nonparametric estimation can be quite challenging in practice especially when covariates are added, as illustrated in the empirical application. Alternatively, we could estimate the parameters based on parametric functions , and a partition of the outcome's support The choice of estimator should be guided by the nature of the problem being studied and the data available to the researcher.

Empirical Illustration: Managed Care and Ambulatory Expenditures
To illustrate the empirical usefulness of our partial identification strategy in a concrete application, we analyze the impact of insurance plan choice on ambulatory expenditures using the data set made available by Deb, Munkin, and Trivedi (2006) through the Journal of Applied Econometrics' Data Archive. 17 We simplify their analyzes in two important dimensions. To enforce a binary treatment, we follow Papadoulos and Silva (2012)   We calculate bounds for M T E OO (p, x) based on parametric estimates for the functions . . , K} and covariate values x. These probabilities are modeled as logit functions that depend on a linear index of the covariates and a quadratic function of the propensity score, using 20 grid points for the outcome variable. The propensity score is also estimated using a logit model that depends linearly on covariate and instrumental variables. To enforce the common support assumption, we trim the top and bottom 1% of the overlapping estimated propensity score distribution. After estimating those probabilities, we estimate α (p, x), β (p, x) and the bounds around M T E OO (p, x) for each covariate value x. Then, following Deb, Munkin, and Trivedi (2006), we assume that the covariates are independent of (S 0 , S 1 , V ) and average the bounds for M T E OO (p, x) across the sample using observed covariates values. By doing so, we recover the bounds for the unconditional 18 The covariate variables are family size, age, squared age, years of schooling, income, female indicator, interaction term between age and female, African-American indicator, Hispanic indicator, marriage indicator, three geographic region dummies, metropolitan area indicator, three subjective health dummies, physical limitation indicator, number of chronic conditions, injury indicator and four year dummies. The descriptive statistics of all variables can be found in Table 1 in Deb, Munkin, and Trivedi (2006). 19 Differently from Deb, Munkin, and Trivedi (2006), we also combine the spouse's insurance plan choice in a binary variable that is equal to one if the spouse chose a managed care plan and equal to zero if the spouse chose a FFS plan.
M T E OO (p). 20 For details on this parametric approach, see Appendix G.
We estimate bounds around the M T E OO function under three sets of assumptions: (i) no restrictions on the sample selection mechanism (Assumptions 1-5), (ii) "monotonicity of sample selection in the treatment" (Assumptions 1-6), and (iii) "monotonicity of sample selection in the treatment" and "stochastic dominance" (Assumptions 1-7). We are interested in understanding how the different sets of assumptions impact the identified set for M T E OO and interpreting the heterogeneity captured by the MTE.
First, we analyze the M T E OO bounds based only on Assumptions 1-5 (Subsection 3.3).
Subfigures 1(a) and 1(b) show the estimated proportion of the always-observed subpopulation within the observed-when-treated and observed-when-untreated groups, that is, α p, υ and β p, υ in Proposition 1. Importantly, in some regions of the support those proportions are equal to one or zero, implying that M T E OO is point-identified or not-identified, respectively.
Subfigure 2(a) shows the estimated bounds. For most of the propensity score's support, the bounds without restrictions on the sample selection mechanism are very wide or identification is lost. The estimated bounds do not rule out the possibility of homogeneous treatment effects, i.e., we can still place a constant function inside the bounds in Subfigure 2(a).
In order to tighten those bounds, we consider restrictions on the sample selection mechanism. The "monotonicity of sample selection in the treatment" condition (Assumption 6) imposes that agents who would spend a positive amount of money on ambulatory services if allocated to a FFS plan would also have positive ambulatory expenditures if allocated to a managed care plan. This assumption's direction is in accordance with the results described by Deb, Munkin, and Trivedi (2006), who found that individuals enrolled in HMOs and PPOs are more likely to seek care than FFS enrollees. Note that, under Assumption 6, β(p) is always equal to 1. Subfigure 1(c) shows that, under Assumptions 1-6, the estimated proportion of 20 By averaging with respect to the observed density of the covariates, we compute bounds around a summary measure of the conditional M T E for the always-observed subgroup: , implying that the summary bounds are valid for the unconditional M T E function for the always-observed subgroup. Importantly, Deb, Munkin, and Trivedi (2006) assumed that the covariates are fully exogenous, implying that X |= (V = p, S0 = 1, S1 = 1) holds. Carneiro and Lee (2009) assume a similar exogeneity assumption. Alternatively, we can analyze the conditional M T E OO function for pre-specified values of the covariates. These results are available upon request. the always-observed subpopulation within the observed-when-treated group (α (p)) is strictly positive everywhere in this example. Consequently, the M T E OO function is at least partially identified for the entire support of the propensity score, as can be seen in the bounds reported Subfigure 2(b), illustrating the identifying power of the "monotonicity of sample selection in the treatment" assumption. As a result of tighter bounds, we can rule out the possibility of homogeneous treatment effects, i.e., we cannot fit a constant function inside the bounds in Subfigure 2(b). Interestingly, the upper bound under monotonicity is decreasing. Hence, if the M T E OO function followed the pattern from the upper bound, our results would suggest that the agents who are more likely to enroll in a managed care plan are the ones who incur larger additional ambulatory expenses due to their choice of insurance coverage. This interpretation is compatible with individuals taking into account their potential expenditures when selecting their insurance plans, at least among those for which those expenditures will be positive regardless of their plan (always-observed), and provides extra support for the selectivity result found by Deb, Munkin, and Trivedi (2006).
In order to further tighten the bounds around the M T E OO function, we impose the stochastic dominance assumption (Assumption 7). To interpret this assumption, recall that the "always-observed" (OO) individuals are those for whom ambulatory expenditures would be positive regardless of their insurance plans, while the "observed-only-when-treated"(N O) population encompasses people who would incur expenses only if enrolled in managed care plans. Assumption 7 compares the (counterfactual) expenditures that would take place if all employees were enrolled in a managed care plan (Y * 1 ) between those two groups. Formally, it says that for any particular level of expenditures, say 1000$, the share of individuals that spend less than 1000$ will be larger for the N O group compared to the OO type. Alternatively, it implies that the average expenditures among the lowest 25% (or any particular quantile) of spenders, will be smaller for the N O than for the OO subgroup. Intuitively, if everyone had a managed care plan, typical patients who would go to ambulatories only if insured by a managed care plan spend less in services than those that would go regardless of their plan. In Appendix D.2, we provide a simple theoretical framework in which this assumption holds. 21 21 As pointed out by a referee, this assumption is hard to interpret and motivate empirically. However, in a Importantly, Assumption 7 has no impact on the information about α(p). Consequently, it only impacts the results by increasing the lower bound for the M T E OO , which is much tighter as can be seen in Subfigure 2(c), illustrating the identifying power of the stochastic dominance assumption. If we consider Assumption 7 plausible, we find that the lower bound is positive for most values of the propensity score. This finding reinforces the results in Deb, Munkin, and Trivedi (2006)  In many applications, the only instruments available are discrete, e.g., treatment eligibility, number of children in the household and quarter of birth. In this section, we provide sharp layered policy analysis (Manski, 2011), we offer a menu of estimates based on different assumptions, that may or may not be plausible according to each researcher's own expertise and beliefs (Tamer, 2010).   (2015), who provide an outer set for the LATE parameter when the instrument is binary.
Assumption 8. The instrument Z is discrete with support {z 1 , z 2 , . . . , z K } and the propensity score p ≡ P [D = 1|Z = z ] satisfies 0 < p 1 < p 2 < . . . < p K < 1. We If for some values of Z, the probabilities are the same, p = p −1 , we cannot refine the partition of the unit interval describing the probabilities and, hence, cannot improve on the detail level of the MTE identified.

35
Similarly, we have Thus for A = Y, we can write for d ∈ {0, 1}. Under Assumption 6, we identify P [S 0 = 1, To implement the trimming in this setting, we define the discrete case analog of α(p), denoted byα(p −1 , p ), .

To find bounds around
Moreover, these bounds are uniformly sharp.

Conclusion
This paper derives sharp bounds for the marginal treatment effect for the always-observed individuals when there is sample selection. We achieve partial identification results under three increasingly restrictive sets of assumptions. First, we impose standard MTE assumptions without any restrictions to the sample selection mechanism. The second case, which is our main result, imposes monotonicity of the sample selection variable with respect to the treatment, considerably shrinking the identified set. Finally, we consider a strong stochastic dominance assumption which tightens the lower bound for the MTE.
All the results rely on the insight that observed individuals in the sample are a mixture of two possible groups, the ones that would always be observed regardless of treatment status and the ones that would self-select into the sample only when (un)treated. The mixture weights can be identified, leading to a trimming procedure that partially identifies the target parameter, extending Imai (2008), Lee (2009) and Chen and Flores (2015)

Supporting Information (Online Appendix)
A Proofs

A.1 Proof of Lemma 1
The validity of the bounds is proven in the main text. It remains to show that the bounds are uniformly sharp. Given the restrictions that Assumptions 1-5 impose on the data, we need to find a joint distribution on (S 0 ,S 1 ,Ṽ , Z) that satisfies these restrictions and achieves We need to define the joint density (mass) function of (S 0 ,S 1 ,Ṽ , Z). To do so, we will define the density function fṼ , define the mass functionπ (S0,S1)|Ṽ and use the density function of Z -f Z -to define f (S 0 ,S 1 ),Ṽ ,Z =π (S0,S1)|Ṽ · fṼ · f Z . Note that, by construction, Assumption 1 holds. With this goal in mind, fix (s 0 , s 1 , p, z) ∈ {0, 1} 2 × R 2 arbitrarily. Define fṼ (p) = 1 {p ∈ [0, 1]}, ensuring that Assumption 5 holds by construction. For brevity, denote the strata by OO = always observed, NO = observed only when treated, ON = observed only when untreated and NN = never observed, and the probability of the stratum k conditional onṼ = p byπ k|p . We now propose the following DGP: We are going to show that the proposed joint distribution satisfies all restrictions on the unconditional joint distribution of (S 0 , S 1 ) and the marginals of S 0 and S 1 . We define the correspondence G between the unobservables (S 0 , S 1 ) and the observables ( Similarly, we can show that π N N +π ON = 1 0 P S 1 = 0|Ṽ = p dp, Therefore,π satisfies condition (A.2). Moreover, we have: OO|p +π ON |p dp = since υ(p) ≤ min P(S 1 = 1|Ṽ = p), P(S 0 = 1|Ṽ = p) . Thus, π N N |p ≤ min P(S 1 = 0|Ṽ = p), P(S 0 = 0|Ṽ = p) , Condition (A.10) is equivalent to π OO ≤ P(S = 1, D = 1|Z = z) + P(S = 1, D = 0|Z = z).

A.2 Proof of Proposition 1
The validity of the bounds is proven in the main text. It remains to show that the bounds are uniformly sharp. Given the restrictions that Assumptions 1-5 impose on the data (i.e., equations (3.1) and (3.2)), we need to find a joint distribution on (Ỹ * 0 ,Ỹ * 1 ,S 0 ,S 1 ,Ṽ , Z) that satisfies these restrictions, induces the joint distribution on the data (Y, S, D, Z), and achieves any value δ ∈ ∆ 1 , ∆ 1 . To do so, assume that Y * is absolutely continuous and has a strictly positive density.

A.3 Proof of Proposition 2
The validity of the bounds is proven in the main text. It remains to show that the bounds are uniformly sharp. Given the restrictions that Assumptions 1-6 impose on the data (i.e., equations (3.1) and (3.2)), we need to find a joint distribution on (Ỹ * 0 ,Ỹ * 1 ,S 0 ,S 1 ,Ṽ , Z) that satisfies these restrictions, induces the joint distribution on the data (Y, S, D, Z), and achieves any value δ ∈ ∆ 2 , ∆ 2 . Assume that Y * is absolutely continuous with strictly positive density.
The probabilitiesπ k|p are given by: Notice that the lower bound in Proposition 2 is attained by the distributions ofỸ * 0 OO,Ṽ Finally, the joint distribution of (Ỹ * 0 ,Ỹ * 1 ,S 0 ,S 1 ,Ṽ , Z) induces the joint distribution on the data (Y, S, D, Z) by construction. The proof is identical to the proof in Appendix A.2.
Similar reasoning holds for the upper bound, ∆ 2 . To attain any function δ ∈ ∆ 2 , ∆ 2 , we can use convex combinations of the joint distributions that achieve the lower and upper bounds, where the weights of the convex combination may depend on the value p ∈ P.

A.4 Proof of Proposition 3
The validity of the bounds is proven in the main text. It remains to show that the bounds are sharp. Given the restrictions that Assumptions 1-7 impose on the data (i.e., equations (3.1) and (3.2)), we need to find joint distributions on (Ỹ * 0 ,Ỹ * 1 ,S 0 ,S 1 ,Ṽ , Z) that satisfy these restrictions, satisfy the stochastic dominance assumption, induce the joint distribution on the data (Y, S, D, Z), and achieve any value δ ∈ ∆ 3 , ∆ 3 .
The proof of Proposition 3 is very similar to the one in Appendix A.3. We only have to modify equations (A.15) and (A.16) to: These changes ensure that the joint distribution of (Ỹ * 0 ,Ỹ * 1 ,S 0 ,S 1 ,Ṽ , Z) satisfy the Stochastic Dominance Assumption by construction.

A.5 Proof of Proposition 4
The validity of the bounds is proven in the main text. We first prove that the bounds around T E under Assumptions (1)-(6) are sharp. Pick any treatment effect parameter T E := P M T E OO (p) · ω (p) dp, where the weighting function ω : P → R is known or identified. Given the restrictions that Assumptions (1)-(6) impose on the data (i.e., equations (3.1) and (3.2)), we need to find a joint distribution on (Ỹ * 0 ,Ỹ * 1 ,S 0 ,S 1 ,Ṽ , Z) that satisfies these restrictions, induces the joint distribution on the data (Y, S, D, Z), and achieves any value δ ∈ P (∆ 2 (p) · ω (p)) dp, P (∆ 2 (p) · ω (p)) dp . To do so, assume that Y * is absolutely continuous and has a strictly positive density.
First, we show that the lower bound ∆ 2 : P → R is attainable. The joint density (mass) function of (Ỹ * 0 ,Ỹ * 1 ,S 0 ,S 1 ,Ṽ , Z) is defined as the same distribution used for Proposition 2.
Finally, the joint distribution of (Ỹ * 0 ,Ỹ * 1 ,S 0 ,S 1 ,Ṽ , Z) induces the joint distribution on the data (Y, S, D, Z) by construction. The proof is identical to the proof in Appendix A.3.
Similar reasoning holds for the upper bound, P ∆ 2 (p) · ω (p) dp. To attain any value δ ∈ P (∆ 2 (p) · ω (p)) dp, P ∆ 2 (p) · ω (p) dp , we can use convex combinations of the joint distributions that achieve the lower and upper bounds, where the weights of the convex combination does not depend on the value p ∈ P.
The proofs that the bounds around T E under Assumptions 1-5 and 1-7 are similar to the proof above and they use the distributions described in Appendices A.2 and A.4.
A.6 Deriving Identifiable Weights for AT E OO , AT T OO , AT U OO , LAT E OO and P RT E OO (Tables 2 and 3) In this section, we refer to the Law of Iterated Expectations using its acronym, LIE.
A.6.1 ATE within the always-observed population Observe that 2) and the expectation operator) dp dp (Assumption 5) ∂p dp .

A.6.2 ATT within the always-observed population
Observe that (by the definition of a conditional expectation) by Assumption 5, definition (2.2) and the linearity of the integral operator, by the argument outlined in Appendix A.6.1, ∂p dp .

A.6.3 ATU within the always-observed population
Analogously to Appendix A.6.2, we have that A.6.4 LATE within the always-observed population Observe that, for any p, p ∈ [0, 1] 2 , The proof is similar to that of AT E OO , except that the integral is over p, p instead of [0, 1].
Consequently, we have that ω LAT E (p) =

A.6.5 PRTE within the always-observed population
First, we need to define a policy. A policy a is a reassignment of the treatment based on D a = 1 {V ≤ P a }, where the random variable P a is distributed according to the density f Pa and to the cumulative distribution function F Pa . The new random variable P a is still assumed to be independent of (V, Y * 0 , Y * 1 , S 0 , S 1 ). Moreover, we define the possibly censored outcome under policy a as Y * := D a Y * 1 + (1 − D a ) Y * 0 . Now, observe that, for a policy a, by Assumption 4 and Fubini's Theorem, Analogously, we have that, for a policy a , Combining the last two results, we have that by the argument outlined in Appendix A.6.1, implying that ω P RT E (p, a, a ) =

A.7 Proof of Proposition 5
This proof is similar to that of Proposition 2. It is given for completeness. The validity of the bounds is proven in the main text. It remains to show that the bounds are uniformly sharp. Given the restrictions that Assumptions 1, 5, 6 and 8 impose on the data, we need to find a joint distribution on (Ỹ * 0 ,Ỹ * 1 ,S 0 ,S 1 ,Ṽ , Z) that satisfies these assumptions, induces the joint distribution on the data (Y, S, D, Z), and achieves any value δ ∈ ∆ LAT E , ∆ LAT E . To do so, assume that Y * is absolutely continuous and has a strictly positive density.
For brevity, denote the strata by OO = always observed, NO = observed only when treated and NN = never observed, and the probability of the stratum k conditional onṼ = p byπ k|p .
The probabilitiesπ k|p are given bỹ Under Assumptions 1, 5, and 6, the above quantities are positive, and sum up to one.
Define the distribution on the strata: Inequalities (B.1) to (B.3) imply that the above quantities are positive and sum up to one. Define Define the joint conditional distribution of (Ỹ * 0 ,Ỹ * 1 ,S 0 ,S 1 ) given (Ṽ = p, Z = z): Finally, define We can show that (Ỹ ,S,D, Z) has the same joint distribution as (Y, S, D, Z). Finally, by construction, Assumptions 1, 5 and 6 hold.

C Numerical Illustration
In this appendix, we highlight the feasibility and usefulness of the bounds proposed in Section 3 by considering a numerical illustration of a simple structural model with endogenous treatment and sample selection. In Appendix C.1, we present the bounds for M T E OO based on Propositions 1 and 2. The illustration provides insights on the functioning of the bounds as well as the mechanisms driving how informative those bounds are. In Appendix C.2, we discuss what is identified by the local instrumental variable (LIV) estimand (Heckman and Vytlacil, 1999) when directly applied to the selected sample and explain its limitations.
Appendix C.3 details the data-generating process used in this numerical illustration.

C.1 Illustrating the Bounds around M T E OO
Consider the following data generating process (DGP): where (θ, S , Z, ξ) ∼ N (0, I), T = 1 {ξ ≥ 0}, I is the identity matrix, Φ(.) is the standard normal CDF, and Φ −1 (·) its inverse. The potential outcomes equations has random coefficients in this illustration. Intuitively, there are two sets of individuals that might face different returns to treatment due to, for example, their gender or race. From a technical standpoint, this choice guarantees reasonable overlap for treated and untreated groups in the observed population over the support of the outcome conditional on V .
We present the bounds for the M T E OO described in Proposition 1 (∆ 1 , ∆ 1 ) in Figure 3, and Proposition 2 (∆ 2 , ∆ 2 ) in Figure 4 for parameters δ 0 = 0.1, δ 1 = 0.4, β 0,0 = β 0,1 = β 1,0 = 1 and β 1,1 = 5. 23 As can be seen on Subfigure 3(a), the bounds based on Proposition 1 are not very informative for a large part of the support of V . In this DGP, when the propensity score is small (p is close to zero), υ -the lower bound on the proportion of the always-observed (P [ S 0 = 1| V = p] + P [ S 1 = 1| V = p] − 1) -approaches 1 and the M T E OO is almost pointidentified as the bounds are close to each other. On the other hand, when p is larger than 0.664, the lower bound on the proportion of the always-observed becomes exactly zero, the 23 Note that these bounds can be computed using numeric integration. See Appendix C.3 for more details.

C.2 Understanding the LIV Estimand in the Selected Sample
A researcher may believe that sample selection is not a problem and decide to apply the local instrumental variable (LIV) estimator (Heckman and Vytlacil, 1999) directly to the selected sample (S = 1). Similarly to the issues generated by sample selection when trying to identify AT E or LAT E parameters (Heckman, 1979;Lee, 2009;Chen and Flores, 2015), the LIV estimand applied directly to the selected sample will not identify our target parameter (M T E OO ). The LIV estimand in this case can be described as:

C.3 Details on the Numerical Illustration
This appendix lists the relevant densities, expectations and objects of interest implied by the DGP used in Appendices C and F. We have that D Two Economic Models Satisfying Assumptions 1-7

D.1 Model 1: Job Training Program
To better understand the intuition behind the stochastic dominance assumption, we provide a simple economic model of a worker deciding to enroll in a job training program that satisfies assumptions 1-7.
Let Y * 0 be the untreated wage whose support satisfy Y * 0 = [0, c 0 · y), where c 0 ∈ (1, +∞) and y ∈ R ++ . LetṼ ∼ FṼ be the training program's effect on wages whose support satisfy where v ∈ R ++ , and whose cumulative distribution function is strictly increasing in its support. To simplify our argument, assume thatṼ ⊥ ⊥ Y * 0 . Define Y * 1 := Y * 0 +Ṽ as the treated wage. Let Z be the training program's cash allowance whose support is given by Z = R + and whose cumulative distribution function is continuous. To further simplify our argument, assume that the support of the joint distribution ofṼ and Z satisfy supp Ṽ , Z = Z × V. The worker enrolls in the training program if its total benefits are larger than its non-random and homogeneous cost, c D ∈ R ++ . Formally, define D : any p ∈ P. Assumption 4 holds because sup |Y * 0 | = c 0 ·y < +∞ and sup |Y * 1 | = c 0 ·y+v < +∞, while Assumption 5 holds by definition.
Finally, we discuss whether the stochastic dominance assumption (Assumption 7) holds.
We can, then, conclude that Assumption 7 holds in this simple economic model.

D.2 Model 2: Managed Care and Ambulatory Expenditures
Suppose that an individual's decision to choose a managed care plan (HMO or PPO) is based on a cost-benefit analysis, such that she decides to enroll in a managed care plan if its cost V is less than its benefit P (Z), i.e., D = 1{V < P (Z)}. The individual's potential ambulatory expenditure when she enrolls in a managed care plan is Y * 1 , and her potential ambulatory expenditure when she enrolls in a fee-for-service plan is We observe the individual's ambulatory expenditure when Y * = is positive, i.e., S = 1{Y * > 0}, and the observed ambulatory expenditure We are going to show that Assumptions 6 and 7 hold in this framework. By definition, We can see that {S 0 = 1} implies {S 1 = 1} since ε ≥ 0. Therefore, Assumption 6 holds. We now show that P(Y * 1 ≤ y|V = p, S 0 = 1, S 1 = 1) ≤ P(Y * 1 ≤ y|V = p, S 0 = 0, S 1 = 1).
We have Hence, Assumption 7 holds.

E Estimation Details
This appendix presents the details on how to estimate the bounds proposed in Proposition 2, building upon the discussion on Section 4 on the main text. For brevity, we focus on the bounds identified under monotonicity of sample selection in the treatment (Assumptions 1-6), as it is the most relevant (and feasible) case empirically. Estimators for the bounds proposed in Propositions 1 and 3 are natural extensions of the estimator discussed here.
The following subsections (E.1-E.4) present the details on a suggested non(semi)parametric estimation approach that could be implemented with a large and informative dataset. 25 Nat-25 Appendix F presents a Monte Carlo Simulation that evaluates the small sample properties of this estimator.
urally, parametric alternatives are available as long as the researcher is willing to specify functional forms for the probabilities described below, imposing some structure to the model. This can be an useful alternative which greatly simplifies estimation, especially when conditioning on covariates. For concreteness, we present the details of the particular parametric estimation procedure implemented in the empirical application (Section 5) at Appendix G.
Recall from Section 4 that we need estimates for: for any d ∈ {0, 1} and Consequently, we need to estimate: Furthermore, the estimation of the propensity score P (Z) is necessary to obtain the moments of the conditional distribution of the observed outcome.
E.1 Estimating the Propensity Score P (Z) The procedures proposed by Carneiro and Lee (2009) to estimate P (z) = P(D = 1|Z = z) apply directly, since treatment status D is observed for all individuals, and are summarized here. We suggest to model the probability as a partially linear additive regression model to improve precision of the estimates while avoiding the curse of dimensionality: , where z is composed by nonparametric (z c ) and parametric (z pc ) components, z c is a continuous random vector of dimension d, ϑ is a vector of unknown parameters and ϕ j (·) are unknown functions.
Let {p κ : κ = 1, 2...} be the basis for smooth functions that we will use to approximate ϕ j (·) more closely as the number of approximating functions increases. For a given κ > 0, define P κ (z) = [z pc , p 1 (z c 1 ), . . . , p κ (z c 1 ), . . . , p 1 (z c d ), . . . , p κ (z c d )] . Then, using Carneiro and Lee (2009) notation, we have thatP( As discussed in Carneiro and Lee (2009), the estimated probabilities might fall outside of the [0, 1] interval in finite samples. As a consequence, it is preferable to use the trimmed version, , for a suitably small positive λ. Alternatively, a typical conditional probability estimator of P (·) based on a logit or probit model could be used, so that the fitted probability is always between 0 and 1.
E.2 Estimating π 1 (p), π 0 (p) and α (p) To estimate π 1 (p) and π 0 (p), consider the local polynomial estimators (Fan and Gijbels, where e g is a conformable row vector of zeros with g-th element equal to one, K(·) is a kernel function and h is a bandwidth. When implementing these estimators, we recommend using recent developments in Calonico, Cattaneo, andFarrell (2018, 2019a) for optimal coverage error bandwidth and kernel selection methods that are nonparametric robust bias-corrected (RBC). We suggest implementing the RBC version of these estimators with optimal bandwidth choice as conveniently implemented in the software R, using the package nprobust (Calonico, Cattaneo, and Farrell, 2019b).
E.3 Estimating Γ 1 (p, y) and Γ 0 (p, y) In order to estimate Γ 1 (p, y) and Γ 0 (p, y), we choose a grid for the outcome variable ({y 1 , . . . , y Kn }) and estimate the conditional density of the outcome for each bin in the grid, leading to the following local polynomial regression (Fan and Gijbels, 1996), for any k ∈ {2, . . . , K n }. Similarly to the estimation ofπ 0 andπ 1 , we implement the optimal bandwidth selector and RBC procedures mentioned previously.
The estimation ofγ d (p, k) is a crucial step and can be adversely affected by several features of the population DGP and the available data. For example, these estimators will perform well in situations for which the available data about the observed outcome covers the whole range

E.4 Estimating M T E OO (p) Bounds
The estimators for the bounds LB 2 (p) and U B 2 (p) can be obtained as where y k is the center point of each bin [y k−1 , y k ] for any k ∈ {2, . . . , K n }. Moreover, we can estimate E Ỹ 0 |S = 1, D = 0, P (Z) = p in Proposition 2 usingΞ OO,0 (p) := Kn k=2 y k ·f 0 (p, k).
Note that the estimators for LB 2 (p), U B 2 (p) andΞ OO,0 (p) rely on the estimates for den-sities and trimming points obtained in previous steps, not using on the original data for the observed outcomes.
Analyzing the inference procedures and asymptotic properties of the proposed estimators is beyond the scope of this paper and an exciting area for future work.

F Monte Carlo Simulation
In this appendix, we use the DGP described in Appendix C to produce Monte Carlo simulations using the estimator proposed in the main text. We analyze two sets of parameters: (i) δ 0 = 0.75, δ 1 = 1.5, β 00 = β 01 = β 10 = 0.1, β 11 = 0.2 and (ii) δ 0 = 0.2, δ 1 = 2.0, β 00 = β 01 = β 10 = 0.1, β 11 = 0.2. The first set of parameters is ideal for the proposed estimator in the sense that the effective sample size is large since the trimming proportion α (p) is never small and the sample selection problem is not severe. The second set of parameters intentionally decreases the trimming proportion α (p), reducing the effective sample size and worsening the sample selection problem. We find that our estimator performs adequately in both DGPs when the sample size is equal to n = 10, 000.
Based on the nonparametric estimation procedure described in the Appendix E, we need to specify the propensity score estimator, the grid points for the observed outcome variable ({y 1 , . . . , y Kn }) and the evaluation points for the unobserved characteristic V . We estimate the propensity score with a logit estimator whose index is linear in the instrument Z, implying that the propensity score estimator is misspecified. In this simulation, we focus on the performance of six estimators:α (p),Ξ OO,0 (p), LB 2 (p), U B 2 (p),∆ 2 (p) and∆ 2 (p).    (1) (3) (8)       (1) (3) Note: We define θ0 as the true population value of the estimand,θ as the estimator of θ0 and ΞOO,0 (p) := E [ Y * 0 | S0 = 1, S1 = 1, V = p].
The results are based on 1,000 Monte Carlo repetitions. The sample size in each generate data set is n = 10, 000.

G Empirical Illustration: Additional Details and Results
In this appendix, we present the parametric specification and estimation details for the results in Section 5.
As mentioned in the main text, to properly account for covariates, we calculate bounds In particular, the probabilities above are modeled as logit functions that depend on a linear index of the covariates, instruments (for the propensity score), and a quadratic function of the propensity score (for the last two equations), while using K = 20 grid points (y k ) for the outcome variable in the last case. Let Λ(·) and λ(·) be the logistic distribution's CDF and PDF respectively, then: P [ S = 1, D = d| X = x, P (Z) = p] = Λ(δ d 1 p + δ d 2 p 2 + xδ d X ), P [ y k−1 ≤ Y < y k , S = 1, D = d| X = x, P (Z) = p] = Λ(θ d 1 p + θ d 2 p 2 + xθ d X ).
The coefficients in the index function can then be estimated using a standard logit estimator, where the estimatedp i = Λ(x iψ d X + z iψZ ) replace p i in the estimation for the last two terms.
To enforce the common support assumption, we trim the bottom 1% and top 1% of the overlapping estimated propensity score distribution.
Hence, the estimates for the partial derivatives of these functions, π d (p, x) and γ d (p, x, k), can be obtained by plugging in the relevant estimates: The values for α(p, x) and β(p, x) depend on the assumptions being considered. For the case in which the sample selection mechanism is unrestricted we can obtain α(p, υ , x) and β(p, υ , x) by plugging inπ d (p, x) in the formulas in Lemma 1. In the leading case of "monotonicity of sample selection in the treatment" (Assumptions 1-6),α (p, x) =π 0 (p, x) π 1 (p, x) .
To summarize the treatment effects, we average the bounds for M T E OO (p, x) across the sample using observed covariates values. By averaging with respect to the observed density of the covariates, we compute bounds around the summary measure of the conditional M T E for the always-observed subgroup: If X |= (V = p, S 0 = 1, S 1 = 1) holds (i.e., the covariates are exogenous), then for any p ∈ P. Moreover, these bounds are sharp.