Estimating Causal Effects of Discrete and Continuous Treatments with Binary Instruments

Estimating Causal Effects of Discrete and Continuous Treatments with Binary Instruments

Introduction

Endogeneity and heterogeneity are key challenges in causal inference. Endogeneity arises because most treatments and policies of interest are the result of decisions made by economic agents. Unobserved heterogeneity also arises naturally as many of the agents' characteristics are unobserved to the researcher. Accounting for endogeneity and heterogeneity in treatment effects estimation is crucial to answer policy questions, such as how to allocate social resources and combating inequalities. This paper proposes a flexible instrumental variable (IV) modeling framework for identifying heterogeneous treatment effects under endogeneity, which yields practical estimation and inference procedures.

Without additional assumptions, IV strategies cannot point-identify meaningful treatment effects in the presence of heterogeneity. The literature has proposed different solutions to deal with this challenge that exhibit trade-offs between adding structure to the treatment assignment mechanism and potential outcomes. One line of research has restricted the structure and heterogeneity of the potential outcomes while allowing for flexible treatment assignment mechanisms (e.g., chernozhukov2005iv with binary treatments and newey2003instrumental with continuous treatments). Another line of research has shown the usefulness of restricting the treatment assignment while being flexible regarding how the potential outcomes are formed (e.g., imbens1994identification with binary treatments and imbens2009identification with continuous treatments).

We explore an intermediate route that imposes structure on the relationship between the treatment assignment and the potential outcomes to achieve point identification of treatment effects. The basis of this approach is a local Gaussian representation of the copula capturing the dependence between the potential outcomes and the unobservable determinants of treatment assignment. This representation is fully nonparametric, that is, it does not require that potential outcomes and treatment unobservables are jointly or marginally Gaussian. Indeed, the bivariate Gaussian structure always holds locally by treating the correlation parameter as an implicit function that equates the bivariate Gaussian copula with the copula of the potential outcome and selection unobservables. We use this representation to introduce an assumption that has not been previously considered for identification of treatment effects. This assumption, so-called copula invariance (CI), restricts the local dependence of the copula with respect to treatment propensity, and thus it restricts the form of endogeneity.

We show that, even with a binary IV, CI identifies quantile and average treatment effects (QTE and ATE) of binary and ordered discrete treatments and quantile and average structural functions (QSF and ASF) of continuous treatments for the entire population and subpopulations such as the treated. The results for ordered discrete and continuous treatments have particular empirical relevance, as previous identification results for global treatment parameters are scarce and typically rely on rich instrument variation; see below for the review.\footnote{There is a wide range of empirical studies estimating the effects of continuous and ordered endogenous variables using IVs with limited variation. Prominent examples include work on intergenerational mobility black2011recent, studies of the returns to schooling angrist1995twostage, analyses of the impact of air pollution chay2005does, demand analyses with discrete IVs angrist2000interpretation, and studies where discrete experimental treatments are used as IVs for ordered discrete and continuous treatments bessone2021economic.} Moreover, the same CI assumption applies without modification to continuous, discrete and mixed continuous-discrete outcomes. As a byproduct, we also identify the dependence function, which captures the direction and magnitude of endogenous selection and may be of interest in applications. When covariates are available, we impose CI conditional on these covariates, allowing for an additional source of heterogeneity in our model.

The CI-based identification strategy is constructive and leads to practical semiparametric estimation procedures based on distribution regression for both discrete and continuous treatments.\footnote{We refer to these estimators as “semiparametric” because distribution regression models involve function-valued parameters.} We establish the asymptotic Gaussianity of the estimators of the potential outcome distributions, dependence function, and functionals such as the unconditional QSF and QTE. We show that bootstrap is valid for estimating the limiting laws and provide an explicit algorithm for constructing uniform confidence bands.

The proposed method adds to the IV toolkit by expanding the directions of modeling trade-offs in identifying treatment effects. We allow for richer patterns of effect heterogeneity, compared to chernozhukov2005iv and newey2003instrumental, and more heterogeneity in the treatment assignment, compared to imbens1994identification and imbens2009identification, while imposing more restrictions on the dependence structure (i.e., the form of endogeneity), as we detail below.

We apply our method to estimating the distributional effects of sleep on well-being. In this case, sleep time is treated as a continuous treatment. We use the data from the experimental analysis of bessone2021economic, who studied the effects of randomized interventions to increase sleep time of low-income adults in India. A simple two-stage least squares analysis suggests that sleep has moderate average effects on well-being. Using our method, we document interesting patterns of heterogeneity across the distributions of sleep time and well-being, which are overlooked by standard analyses focusing on average effects. For example, the quantile treatment effects of increasing sleep on well-being at the lower tail, which are particularly policy-relevant in this context, are substantially larger than the corresponding average effect.

Related Literature

The literature on identification of heterogeneous treatment effects with endogeneity is vast. We focus the review on approaches that do not impose parametric distributional assumptions to achieve point identification.

A first strand of literature focused on imposing assumptions on the generation of the potential outcomes, such as rank invariance and rank similarity. Rank invariance imposes that the outcome equation is strictly monotonic in a scalar unobservable such that there is a one-to-one mapping between the potential outcome and unobservable, and the unobservable is the same for all potential outcomes. This assumption is very convenient for the identification analysis. It allows for identification of QTE and ATE with discrete treatments chernozhukov2005iv and identification of the ASF with continuous treatments newey2003instrumental,blundell2007semi.\footnote{ai2003efficient, chen2009efficient, chen2012estimation, chen2015sieve consider a general framework that nests these models.} chernozhukov2005iv's rank similarity is slightly weaker than rank invariance as it does not necessarily restrict the unobservable to be the same for all the potential outcomes. However, both assumptions produce the same testable restriction chernozhukov2013quantile. In subsequent work, vuong2017counterfactual showed that rank invariance and strict monotonicity are powerful enough to identify individual treatment effects (under suitable regularity conditions) in addition to the QTE and ATE. This literature remains flexible about the treatment selection process.

The proposed CI assumption and rank similarity or invariance are non-nested. The former concerns the dependence between potential outcomes and selection unobservables, whereas the latter concerns the dependence between potential outcomes. Indeed, we show that rank similarity can be viewed as another form of copula invariance assumption. However, our CI allows for more general patterns of treatment effect heterogeneity than rank invariance and similarity. Also, approaches based on rank invariance and similarity rely on strict monotonicity on the outcome equation to achieve point identification. This assumption can only hold for continuous outcomes. Moreover, these approaches rely on completeness conditions on the relationship between the treatment and instrument that rule out, for example, an ordered discrete and continuous treatment when the instrument is binary. Furthermore, when treatments are continuous, completeness conditions typically lead to ill-posedness, which complicates estimation. CI does not rely on monotonicity nor completeness and therefore can accommodate discrete and mixed discrete-continuous outcomes and ordered and continuous treatments with a binary instrument.

A second strand of literature focuses on assumptions imposed on the treatment assignment. imbens1994identification and heckman2005structural assumed that treatment assignment is determined by a scalar unobservable and combined this assumption with monotonicity of potential treatments with respect to a binary instrument to show identification of local average and marginal effects of binary treatments. abadie2002instrumental and carneiro2009estimating extended this approach to the corresponding local quantile effects, and vytlacil2006ordered extended it to the case of ordered discrete treatments. newey1999nonparametric and imbens2009identification imposed strict monotonicity of the treatment selection equation with respect to a scalar unobservable and large support of the instrument (ruling out discrete instruments) to identify global effects using a control variable approach. Due to the nature of the restrictions, their approach only applies to continuous treatments. newey2021control avoided the large support requirement on the instrument of imbens2009identification by assuming a parametric structure on the expectation of the treatment conditional on the instrument that enables extrapolation outside the instrument support. Compared to this strand of the literature, CI restricts the relationship between potential outcomes and treatment assignment, but allows for identifying global treatment effects of discrete and continuous treatments with discrete instruments without relying on a scalar unobservable in the treatment assignment mechanism.

There are also approaches that combine or modify the assumptions of the previous strands. chesher2003identification showed identification of the quantile effect of continuous treatments on continuous outcomes with continuous instruments assuming strict monotonicity of the outcome and treatment selection equations with respect to scalar unobservables. Under similar assumptions, dhaultfoeuille2015identification and torgovitsky2015identification found that quantile effects can be identified with discrete instruments. In this class of models, torgovitsky2017minimum proposes a two-step minimum distance estimator for the finite dimensional parameter of the outcome function. Again, these restrictions are different and not nested with CI. Moreover, in the case of continuous treatments, our identification strategy is constructive, yielding straightforward plug-in estimators. There are also partial identification solutions that impose less structure on the treatment assignment and potential outcomes (e.g., manski1990nonparametric and balke1997bounds for earlier references, and chesher2020generalized for a more recent survey).

The copula is a powerful tool that has been previously employed in econometrics for identification and estimation. For example, chen2006efficient used a parametric copula to achieve efficient estimation in a class of multivariate distributions.\footnote{In time series, chen2006estimation,chen2006estimation2, beare2010copulas, chen2021efficient,chen2022copula, and fan2023estimation, among others, use copulas to model temporal dependence.} In a semiparametric triangular model with binary dependent variables, han2017identification introduced a class of single-parameter copulas to model the dependence structure between the unobservables and established a condition on the copula under which the parameters are identified. They showed that many well-known copulas including the Gaussian copula satisfy the condition. When we restrict our attention to a binary treatment and binary outcome, the current paper's framework is relevant to han2017identification. However, while they assumed a parametric copula for the dependence structure, we assume CI. Assuming a Gaussian copula in han2017identification can be viewed as an extreme special case of CI. han2019estimation developed sieve estimation and inference methods based on han2017identification, and han2023semiparametric extended them to semiparametric models for dynamic treatment effects. mourifie2021layered use copula as a channel to impose assumptions in characterizing identified sets in the framework of marginal treatment effects.

arellano2017quantile and chernozhukov2018distribution studied IV identification of selection models using assumptions on the copula between the latent outcome and selection unobservable. arellano2017quantile assumed a real analytical copula and required continuous instruments. chernozhukov2018distribution used the local Gaussian representation and copula exclusion, which is a special case of CI, with a binary instrument. Therefore, our framework with binary treatment is related to their setup. However, even in the case of binary treatment, the current setting differs from chernozhukov2018distribution in several dimensions. First, our setting requires two-way sample selection due to the switching of treatment status. Second, we introduce a general selection model that does not follow the typical threshold-crossing structure, which is important to allow for rich selection patterns. Third, because of these features, the identification analysis involves local representation and copula invariance that are specific to treatment status and the value of the IV. More importantly, the use of local Gaussian representation and CI for ordered and continuous treatments is completely new to this paper. Moreover, while relying on the same CI assumption, the identification strategies and proof techniques in these two cases are distinct.

Finally, in the difference-in-differences setup, athey2006identification showed that average and quantile treatment effects on the treated (and the untreated) can be identified when the unobservable determinant of the untreated potential outcome is independent of time within groups. ghanem2023evaluating provide general identification results under a time invariance assumption on the copula between the potential outcomes and the group indicator and show that their assumption is equivalent to the assumptions in athey2006identification with continuous outcomes. Their time invariance in the copula can be viewed as a version of CI where the IV is a time indicator.

Organization of the Paper

Section (ref) introduces the key variables and parameters of interest, Section (ref) states the local Gaussian representation, and Section (ref) posits the main identifying assumptions that will be used throughout the analyses. We devote Sections (ref)--(ref) to the identification analyses with binary, ordered discrete, and continuous treatments, respectively. Section (ref) discusses copula invariance in further detail and Section (ref) discusses estimation and inference. Section (ref) provides an empirical illustration. Section (ref) presents concluding remarks. The Appendix contains the proofs and additional results.

Notation

For scalar random variables $X$ and $Y$ and possibly multivariate random variable $Z$, $F_{X,Y \mid Z}$ denotes the joint distribution of $X$ and $Y$ conditional on $Z$, $F_{X \mid Z}$ denotes the (marginal) distribution of $X$ conditional on $Z$, and $F_{Z}$ denotes the marginal (joint) distribution of $Z$. We use calligraphic letters to denote support sets of random variables. For example, $\mathcal{Z}$ denotes the support of $Z$. The symbol $\perp\!\!\!\!\perp$ denotes (stochastic) independence; for example, $X \perp\!\!\!\!\perp Y$ means that $X$ is independent of $Y$. The interior of the set $\mathcal D$ is denoted as $\operatorname{int}(\mathcal D)$.

Setup and Assumptions

We consider three classes of models depending on the type of treatment variable: binary, discrete ordered, and continuous. Before investigating identification, we introduce the setup, parameters of interest and identifying assumptions that are common to all classes of models.

Preliminaries

Let $Y\in\mathcal{Y} \subseteq \mathbb{R}$ denote the scalar outcome and $D\in\mathcal{D} \subseteq \mathbb{R}$ denote the scalar treatment. We consider binary, discrete ordered, and continuous treatments with $\mathcal{D}=\{0,1\}$, $\mathcal{D}=\{1,...,K\}$, and $\mathcal{D}$ equal to an uncountable set, respectively. The outcome is not restricted, it can be continuous, discrete or mixed continuous-discrete. Let $Z\in\{0,1\}$ be the binary IV. We focus on a binary instrument as the most challenging case; the analysis readily extends to multi-valued discrete or continuous $Z$. Let $Y_{d}$ denote the potential outcome given $d\in\mathcal{D}$ and $D_{z}$ the potential treatment given $z\in\{0,1\}$. They are related to the observed outcome and treatment through $Y=Y_{D}$ and $D=D_{Z}$.\footnote{When $D$ is continuous, we require that $Y_d$ is suitably measurable hirano2004propensity.} Let $X \in\mathcal{X} \subseteq \mathbb{R}^{d_x}$, for some positive integer $d_x$, be a vector of covariates. All the identification analysis in Section (ref) is conditional on $X$, but we keep the dependence implicit to lighten the notation. We make it explicit in Appendix (ref) and when we discuss estimation and inference in Section (ref).

We consider a general treatment assignment equation:

where $v \mapsto h(z,v)$ is weakly increasing, and we normalize $V_{z} \sim U[0,1]$. We provide examples of the function $h$ for each type of treatment below. By allowing for a different unobservable $V_{z}$ at each value of $z$, we essentially permit $D$ to be a function of the vector of unobservables $(V_{0},V_{1})$. Even this general version of a treatment assignment model may not be necessary for our analyses (see Remark (ref)) but simplifies the exposition.

We are interested in identifying the distribution of $Y_d$, $F_{Y_{d}}$, for $d\in\mathcal{D}$, and functionals of $F_{Y_{d}}$, such as QSFs and ASFs. Thus, by using appropriate operators:

where $\mathcal{Q}_{\tau}(F)\equiv\inf\{y\in\mathcal{Y}:F(y)\ge\tau\}$ and $\mathcal{E}(F)\equiv\int_{\mathcal{Y}}[1-F(y)]dy$. QTE and ATE can be expressed as $\{QSF_{\tau}(d)-QSF_{\tau}(d')\}/(d-d')$ and $\{ASF(d)-ASF(d')\}/(d-d')$, $d,d' \in \mathcal{D}$ with $d' \neq d$, and for continuous treatments we can also consider $\partial QSF_{\tau}(d)/\partial d$ and $\partial ASF_{\tau}(d)/\partial d$, $d \in \mathcal{D}$. When the treatment is binary, we may also be interested in the distribution of $Y_d$ in subpopulations such as the treated, $F_{Y_{d} \mid D}(\cdot \mid 1)$, and untreated, $F_{Y_{d} \mid D}(\cdot \mid 0)$, for $d\in\{0,1\}$, and functionals of these distributions. More generally, we may be interested in these objects for subpopulations defined by values of the covariates in $X$.

Local Gaussian Representation

Treatment endogeneity can be captured by the joint distribution of the potential outcome and unobservable of the treatment assignment equation (ref). We use a conditional version of the local Gaussian representation (LGR) to represent such a joint distribution. This representation is the basis of our identification and estimation strategies. Throughout the paper, let $C(u_{1},u_{2};\rho)$ denote the Gaussian copula with correlation coefficient $\rho$, that is $$ C(u_{1},u_{2};\rho) \equiv \Phi_2(\Phi^{-1}(u_1), \Phi^{-1}(u_2); \rho), $$ where $\Phi_2(\cdot, \cdot; \rho)$ is the standard bivariate Gaussian distribution with parameter $\rho$ and $\Phi$ is the standard univariate Gaussian distribution.

The following lemma shows that the conditional distribution of any bivariate random variable has a local Gaussian representation anjos2005representation,kolev2006copulas, chernozhukov2018distribution.

In the lemma, the LGR is fully nonparametric and is not imposing Gaussianity on the joint or marginal distribution. The distribution is equated to the Gaussian copula (with the nonparametric marginal distributions) by adjusting the value of the correlation coefficient $\rho$ for each evaluation point $(y,v,z)$. Note that the solution $\rho_{Y,V;Z}(y,v;z)$ depends on both the dependence structure and marginals. Lemma (ref) can be equivalently stated as the LGR of a copula instead of a distribution; see Section (ref).

Assumptions

We maintain the following assumptions:

(ref) is a standard exogeneity condition in IV strategies. It is weaker than $Z\perp\!\!\!\!\perp (\{Y_{d}\}_{d\in\mathcal{D}},V_0,V_1)$ or $Z\perp\!\!\!\!\perp(Y_{d},V_{z})$ for $(d,z)\in\mathcal{D}\times\{0,1\}$. Also, in (ref), a standard exclusion restriction is implicit in the notation: $Y_{d}=Y_{d,z}$ almost surely, where $Y_{d,z}$ is the potential outcome given $(d,z)$. (ref)(ii)--(iii) are the usual IV relevance and non-degeneracy conditions. (ref)(iii) for $\mathcal D=\{0,1\}$ can be formulated a special case of (ref)(iii) for $\mathcal D= \{1,...,K\}$ with $K=2$, but we state it separately for clarity.\footnote{For $K>2$, a weaker but less interpretable condition for (ref)(iii) is $\Pr[D=d\mid Z=z] > 0$ for $(z,d) \in \{0,1\}\times \mathcal D$, $F_{D \mid Z}(1 \mid 1) \neq F_{D \mid Z}(1 \mid 0)$, and either $F_{D \mid Z}(d \mid 1) \neq F_{D \mid Z}(d \mid 0)$ or $F_{D \mid Z}(d-1 \mid 1) \neq F_{D \mid Z}(d-1 \mid 0)$ for $d \in \{2,...,K\}$. Note that when $d=K$ the last condition requires that $F_{D \mid Z}(K-1 \mid 1) \neq F_{D \mid Z}(K-1 \mid 0)$ because $F_{D \mid Z}(K \mid 1) = F_{D \mid Z}(K \mid 0) =1$.}

We make the following assumption about the local dependence parameter of the LGR of $(Y_{d},V_{z})$ conditional on $Z$:

(ref) is a high-level condition that imposes a shape restriction on the dependence between $Y_d$ and $V_z$. This condition (together with the other assumptions we maintain) is sufficient for identification in all the cases that we consider, but it is not necessary. We provide weaker conditions for each case in the following section. Section (ref) provides more interpretable conditions for (ref) and compares (ref) with alternative identifying assumptions that have been used in the literature, such as rank invariance and rank similarity.

Figure (ref) shows examples of joint distributions of $(Y_d,V_z)$ that satisfy (ref). Panel (a) corresponds to a bivariate Gaussian distribution (i.e., a Gaussian copula and univariate Gaussian marginals); Panel (b) corresponds to a non-Gaussian copula (i.e., a Gaussian copula with varying correlation) and standard univariate Gaussian marginals; and Panel (c) corresponds to a Gaussian copula coupled with non-Gaussian marginals. The upper figures display contour plots of the joint distribution and the lower figures display the corresponding local dependence functions. These examples showcase that (ref) is compatible with very diverse shapes for the joint distribution including multimodality and asymmetry. Combinations of these features and more complex shapes are possible by mixing and expanding these examples (e.g., a non-Gaussian copula with non-Gaussian marginals).\footnote{We refer to wasserman-nonparanormal and chernozhukov2018distribution for more examples.} Moreover, even if Gaussian copula does not have tail dependence, (ref) allows for that because the local dependence parameter can change with the level of $Y_d$.

Identification Analysis

Binary Treatment

We start by considering the identification of the causal effects of a binary treatment $D\in\mathcal{D}=\{0,1\}$. To reflect this, we consider a treatment selection equation

with propensity score $$\Pr[D=1 \mid Z=z] = \Pr[D_z = 1 \mid Z=z] = \Pr[V_{z}\le\pi(z)] = \pi(z),$$ by (ref) and the normalization $V_{z} \sim U[0,1]$. Note that $V_{1}$ and $V_{0}$ are two distinct unobservables that do not restrict the behavior of $D_0$ and $D_1$. The LATE monotonicity assumption of imbens1994identification imposes either $D_1 \geq D_0$ or $D_0 \geq D_1$, almost surely, which corresponds to $V_{1}=V_{0}$, almost surely vytlacil2002independence.

For the identification analysis, consider

where the second equality uses equation (ref) and Lemma (ref), and the last equality follows from (ref). For each $y \in \mathcal{Y}$, this is a system of two equations in three unknowns, $F_{Y_{1}}(y)$, $\rho_{Y_{1},V_{z};Z}(y,\pi(0);0)$ and $\rho_{Y_{1},V_{z};Z}(y,\pi(1);1)$. The number of unknowns can be reduced to two by the condition

which is implied by (ref). Equation (ref) only requires that copula invariance holds at two points, $(\pi(1),1)$ and $(\pi(0),0)$. The following theorem shows that the nonlinear system of equations (ref) has a unique solution under (ref). This result follows from a global univalence theorem of gale1965jacobian, because the Jacobian of the system of equations is a P-matrix under (ref)(iii), the map defined by the system is differentiable (as the copula is differentiable) and the parameter space $(0,1)\times (-1,1)$ for $(F_{Y_{1}}(y), \rho_{Y_{1}}(y))$ is open and rectangular. A similar argument shows that the distribution of $Y_0$ and the local dependence parameter of the LGR of $Y_0$ and $V_z$ are identified.

The proof of Theorem (ref) is contained in Appendix (ref). It is interesting to compare Theorem (ref) to chernozhukov2005iv who also establish identification of $F_{Y_{d}}$ when $D$ and $Z$ are binary. Compared to them, we do not impose rank similarity and thus, as we show in Appendix (ref), allow for richer effect heterogeneity. As a trade-off, we restrict the form of endogeneity by imposing (ref). For more detailed comparison, see Section (ref) and Appendix (ref).

Ordered Discrete Treatment

We consider identification of the causal effect of a multi-valued ordered treatment $D\in\mathcal{D}=\{1,\dots,K\}$ using a binary instrument $Z\in\{0,1\}$. We assume a threshold-crossing model for the treatment selection equation, which can be viewed as a natural extension of model (ref) from two to multiple treatment levels,

where $\pi_0(z) = 0$ and $\pi_K(z) =1$. Equation (ref) generalizes the model in Section 7.2 of heckman2007chapter71 by allowing for two unobservables $(V_0,V_1)$ and a different impact of the instrument on the different cutoffs; see Remark (ref). It is not fully general, however, as it imposes that the unobservable is the same for all the treatment levels.\footnote{cunha2007identification showed that ordered choice models with multivariate unobservables are point identified under strong assumptions.}

Under the normalization $V_{z} \sim U[0,1]$ and (ref), the threshold functions $\pi_{d}(z)$ are identified by the distribution of the observed treatment conditional on the instrument, $\pi_{d}(z)=F_{D \mid Z}(d \mid z)$ for $d\in\mathcal{D}$. For the identification analysis, consider

where the first equality follows from (ref) and the second equality from (ref) and Lemma (ref). For each $d \in \mathcal{D}$ and $y \in \mathcal{Y}$, (ref) is a system of two equations in five unknowns: $F_{Y_{d}}(y)$, $\rho_{Y_{d},V_{0};Z}(y,\pi_{d-1}(0);0)$, $\rho_{Y_{d},V_{0};Z}(y,\pi_{d}(0);0)$, $\rho_{Y_{d},V_{1};Z}(y,\pi_{d-1}(1);1)$, and $\rho_{Y_{d},V_{1};Z}(y,\pi_{d}(1);1)$. (ref)(iii) guarantees that the two equations of the system are not redundant.

For $d \in \{1,K\}$, one of the terms on the right hand side drops out because either $\pi_{d-1}(z) = 0$ or $\pi_{d}(z) =1$, yielding a system of two equations on three unknowns. Consequently, the distribution of the potential outcome and local dependence parameter can be identified by combining (ref)(iii) with the condition

which is analogous to condition (ref) from the binary treatment case.

For $d\in\mathcal{D}\backslash\{1,K\}$, condition (ref) reduces the number of unknowns to three but is not sufficient to identify the unknowns. We impose additionally copula invariance between consecutive treatment levels

This condition is also implied by (ref) and reduces the number of unknowns to two: $F_{Y_{d}}(y)$ and $\rho_{Y_d}(y)$. The Jacobian of the resulting system of equations, however, does not satisfy the conditions to apply the global univalence results of gale1965jacobian even under (ref)(iii). We show uniqueness of solution using an alternative global univalence result of ambrosetti1995primer. To apply this result we impose the following sufficient condition on the distribution of the treatment conditional on the instrument:

(ref) does not necessarily follow from (ref)(iii) and imposes stochastic dominance between $F_{D \mid Z}(d \mid 0)$ and $F_{D \mid Z}(d \mid 1)$. For example, the ordered choice model considered by heckman2007chapter71 satisfies (ref). Like (ref)(iii), (ref) can be directly tested from the data. It is interesting to see what type of compliance behavior with respect to $D_0$ and $D_1$ is ruled out by this sufficient condition. To explore this, define the compliers and defiers of order $j\in\mathcal{D}\backslash\{K\}$ as

(ref) states that the distribution for $(V_{0},V_{1})$ is symmetric; most known copulas are symmetric. It holds trivially if $V_0 = V_1$, almost surely. Under (ref), we can interpret Assumption (ref) in terms of compliance behavior:

The condition about the share of compliers and defiers is reminiscent of a similar assumption used in de2017tolerating in the case of binary treatment. Another simple interpretation of Lemma (ref) can be made under the restriction $V_{0}=V_{1}$, almost surely. In this special case, we can easily see that there is no defier groups (i.e., $\Pr[D_{1}<D_{0}]=0$) if and only if $F_{D \mid Z}(d \mid 0)>F_{D \mid Z}(d \mid 1)$ for all $k\in\mathcal{D}\backslash\{K\}$. In general, when $V_{z}$ is not restricted, Assumption (ref) alone does not eliminate compliers or defiers.

We summarize the identification result in the following theorem:

In the proof of Theorem (ref), contained in Appendix (ref), we proceed as follows to show that (ref) has a unique solution. First, we show that the function that defines the system is proper (due to properties of copula) and its Jacobian has full-rank (by (ref)). Then, we apply Corollary 1.4 of ambrosetti1995primer to show that the system has a set of solutions whose cardinality is invariant over the parameter space. Since the system has a unique solution when $\rho_{Y_{d}}(y) = 0$ (locally no endogeneity), we can then conclude the solution is unique everywhere in the parameter space. A similar proof strategy was previously used in de2019identifying in the different setting of panel models with peer effects.

Continuous Treatment

Suppose $D\in\mathcal{D} \subseteq\mathbb{R}$ is an uncountable set and $d \mapsto F_{D \mid Z}(d \mid z)$ is strictly increasing on $\mathcal{D}$, for $z \in \{0,1\}$. Assume the treatment selection equation,

where $V_z \sim U(0,1)$.

For the identification analysis, consider

where the second equality holds from equation (ref) and a change of variable. By the properties of the conditional distribution, Lemma (ref), and (ref),

where $C_{2}$ and $C_{\rho}$ are the derivatives of $C(\cdot,v;\rho)$ with respect to $v$ and $\rho$, respectively. Assume that

and

where the differentiability of $v\mapsto\rho_{Y_{d},V_{z};Z}(y,v;z)$ follows by the differentiability of $C(\cdot,v;\cdot)$ with respect to $v$ and the implicit function theorem; see Section (ref) for related discussions. Note that (ref) and (ref) are implied by (ref). Then, by the properties of Gaussian copula, combining (ref) and (ref) yields

where $a_{d,y}\equiv\Phi^{-1}(F_{Y_{d}}(y))/\sqrt{1-\rho_{Y_{d}}(y)^{2}}$ and $b_{d,y}\equiv -\rho_{Y_{d}}(y)/\sqrt{1-\rho_{Y_{d}}(y)^{2}}$. Equation (ref) yields a linear system of two equations on two unknowns, $a_{d,y}$ and $b_{d,y}$, which has solution

under (ref).

This discussion implies the following identification result.

It is interesting to compare this identification result to imbens2009identification and torgovitsky2010identification,torgovitsky2015identification\footnote{Portions of torgovitsky2010identification were published in torgovitsky2015identification. Since the role of the conditional copula invariance assumption is only discussed in torgovitsky2010identification, we focus on comparing our method and assumptions to this paper.} who also provide identification results for nonseparable models with continuous $D$. Unlike imbens2009identification, our approach does not require an instrument with large support nor rank invariance in the treatment selection equation (i.e., $V_1=V_0$ almost surely), but instead imposes (ref). Unlike torgovitsky2010identification, our approach does not require rank invariance in the outcome and treatment equation, but again imposes (ref). See Appendix (ref) for more detailed comparisons.

Discussions on Copula Invariance

To further understand (ref), we provide sets of simple equivalent conditions (Sections (ref)), compare it with rank similarity (Section (ref)), and discuss its flexibility in the context of Roy models (Section (ref)).

Equivalent Conditions

Joint Independence

Here we provide equivalent conditions to (ref) and (ref) that highlight the trade-offs between copula invariance and instrument independence assumptions.

Note that $\rho_{Y_{d},V_{z};Z}(\cdot,\cdot;z)=\rho_{Y_{d},V_{z}}(\cdot,\cdot)$ by (ref) and $\rho_{Y_{d},V_{z}}(y,\cdot)=\rho_{Y_{d}}(y)$ by (ref).\footnote{In fact, not only (ref) implies $\rho_{Y_{d},V_{z};Z}(\cdot,\cdot;z)=\rho_{Y_{d},V_{z}}(\cdot,\cdot)$, but the converse is also true. See Remark (ref) below.} Conversely, (ref) and (ref) imply (ref). Then, (ref) together with (ref) imply (ref). When $D$ is binary, a sufficient condition for (ref) is $(Y_{0},Y_{1},V_{z})\perp\!\!\!\!\perp Z$, which is imposed in imbens1994identification and vytlacil2002independence with $V_0 = V_{1}$ almost surely, although it is sufficient for the LATE result to have $(Y_{d},V)\perp\!\!\!\!\perp Z$ for $d\in\{0,1\}$.

Local Single Index

We provide an equivalent condition to (ref).

Note that (ref) implies that, for $d\in\mathcal{D}$ and $z\in\{0,1\}$, $$ F_{Y_d,V_z}(y,v) = C(F_{Y_d}(y),v;\rho_{Y_d}(y)), \quad (y,v) \in \mathcal{Y} \times \mathcal{V}. $$ By the properties of the conditional distribution and Gaussian copula,

where $a_{d,y} = \Phi^{-1}(F_{Y_d}(y))/\sqrt{1-\rho_{Y_d}(y)^2}$ and $b_{d,y} = - \rho_{Y_d}(y)/\sqrt{1-\rho_{Y_d}(y)^2}$.\footnote{This is reminiscent of the derivation in Section (ref), although (ref) applies to both discrete and continuous $D$.} It can be shown that the converse is also true.

(ref) is a single index restriction on the local relationship between the potential outcome $Y_d$ and the unobservable of the treatment assignment $V_z$. This restriction does not require Gaussianity as the correlation coefficient is allowed to be a function of $y$ but implies, for example, that the sign of $(\partial/\partial v) F_{Y_d \mid V_z}(y \mid v)$ does not depend on the value of $v$, although it can change with the value of $y$. This is a stochastic monotonicity restriction in the dependence between $Y_d$ and $V_z$. chernozhukov2018distribution showed the equivalence between a special case of (ref) and a single index restriction for the selection model under the rank invariance $V_0 = V_1$, almost surely. Here, we extend the equivalence to a larger class of models that include non-binary treatments without imposing rank invariance. (ref) allows for rich forms of dependence between $Y_d$ and $V_z$, as illustrated in Figure (ref).

(ref) can also be viewed as a means of extrapolation. For example, brinch2017beyond and kowalski2023reconciling impose linearity in the marginal treatment effect to extrapolate the LATE to other treatment parameters. (ref) can be viewed as an alternative to their approach.\footnote{Other approaches exist for extrapolating from the LATE to externally valid global treatment effects. Examples include approaches based on structural models heckman2003simple, covariates angrist2013extrapolate, rank similarity wuthrich2020comparison, and the smoothness of the marginal treatment effect function mogstad2018using,han2020sharp.} Specifically, brinch2017beyond and kowalski2023reconciling impose a linear model for the conditional expectation of $Y_d $ given $V=v$, $\mathrm{E}[Y_d \mid V=v]=\alpha_d + \beta_d v$, where $V=V_0 = V_1$ under the LATE assumptions. By contrast, Proposition (ref) shows that we impose a flexible model for the entire conditional distribution, $F_{Y_{d}\mid V}(y\mid v)=\Phi\left(a_{d,y}+b_{d,y}\Phi^{-1}(v)\right)$. When $Y_d$ is binary, their approach corresponds to using linear probability model, whereas (ref) corresponds to a Probit model.

Rank Similarity

The instrumental variables quantile regression (IVQR) model of chernozhukov2005iv provides an alternative set of conditions under which the $QSF_{\tau}$ is point identified, provided that the outcome is continuous. Here, we discuss how their identifying restriction is related to our results and (ref), focusing on the case where $D$ is binary.

The IVQR model is based on the Skorohod representation for continuous variables: $Y_{d}=Q_{Y_{d}}(U_{d})$, $U_{d}\sim U[0,1]$, for $d\in\{0,1\}$. chernozhukov2005iv consider a general selection mechanism, $D=\delta(Z,V)$, where $V$ can be vector-valued and the instrument is assumed to satisfy $U_{d}\perp\!\!\!\!\perp Z$ (which is implied by Assumption (ref)). The key assumption of the IVQR model is rank similarity (RS), $U_{1}\overset{d}{=}U_{0}\mid Z,V.$ RS weakens the classical rank invariance (RI) assumption, which requires $U_{1}=U_{0}$ almost surely.

The IVQR model yields the following conditional moment restriction chernozhukov2005iv,

Under the selection model (ref) and $Y_{0}\perp\!\!\!\!\perp Z$, (ref) can be rewritten as

Using Lemma (ref), we can further rewrite it as

This shows that the IVQR model also relies on a version of copula invariance,

The IVQR model imposes restrictions across potential outcomes for each instrument level, whereas (ref) imposes restrictions across instrument levels for each potential outcome. The two conditions are non-nested. We show in Appendix (ref) that the copula invariance assumption implied by the IVQR model and RS restricts treatment effect heterogeneity.

This discussion shows that both the IVQR model and our approach rely on non-nested and complementary copula invariance assumptions to identify causal effects for the overall population using instruments. Relative to the IVQR model, the proposed identification approach has three main advantages. First, it does not rely on continuity of the outcome to achieve point identification, and it naturally accommodates discrete and mixed discrete-continuous outcomes. Second, it imposes less restrictions on the relationship across potential outcomes and thus effect heterogeneity (see Appendix (ref)). Finally, it relies on a weaker relevance condition (see Remark (ref)).

Roy Models

Figure (ref) in Section (ref) demonstrates the flexibility of (ref) in modeling the joint distribution of $(Y_d,D_z)$. Our analyses show how point identification can be achieved under much weaker modeling restrictions than Gaussianity, which can be appealing to empirical researchers. To illustrate this further, consider a generalized Roy model for two sectors indexed by binary $D$ and binary cost shock $Z$. Let the potential log wage in sector $d$ be $Y_d=\mathrm{E}[Y_d]+U_d$ for $d\in\{0,1\}$, and suppose that individuals choose sector $D=1$ if the benefit $Y_1 -Y_0$ exceeds the cost $\tilde \pi(z) +\tilde V_z$. This implies that $D_z=1\{V_z\le \pi(z)\}$, where $V_z \equiv U_0-U_1+\tilde V_z$ and $\pi(z) \equiv \mathrm{E}[Y_1] - \mathrm{E}[Y_0] -\tilde \pi(z)$.\footnote{Note that this model is more flexible than the standard generalized Roy model as the cost unobservable, $\tilde V_z$, is $z$-specific, which is the aspect we can allow for.} Suppose $Z \perp\!\!\!\!\perp (Y_d,V_z)$. Then, $\mathrm{E}[Y\mid D=1, Z=z]=\mathrm{E}[Y_1]+\mathrm{E}[U_1\mid V_z \le \pi(z)]$, where $\mathrm{E}[U_1\mid V_z \le \pi(z)]$ is the control function that captures endogenous section into sector $d=1$. We can obtain a similar expression for $d=0$.

Under joint Gaussianity of $(Y_d,V_z)$, one can show that $\mathrm{E}[U_1\mid V_z \le \pi(z)]$ is equal to the inverse Mill's ratio Heckman1979SampleSelection. This function has a specific shape, namely, strictly monotone with respect to the propensity score $\pi(z)$, which is depicted in Figure (ref)(a). Relatedly, heckman1990empirical show that Gaussianity in Roy models implies that (i) the Roy economy results in less dispersed log wages than when there is no endogenous selection of sectors (Theorem 2); (ii) the Roy economy exhibits a right-skewed distribution of aggregate log wages (Theorem 3).

On the other hand, under (ref) for the joint distribution of $(Y_d,V_z)$, one can generate general non-monotonic selection patterns in the control function, as we illustrate in Figure (ref)(b). In this sense, in CI Roy models, economic implications can be more nuanced and data-driven, unlike in Gaussian Roy models where implications are largely model-driven. Moreover, the analysis in Section (ref) implies that the parameters in the sector wage equations can be identified even with a binary cost shock.\footnote{Without any distributional assumptions or assumptions on the dependence structure, one would require IVs with large support to identify the wage parameters eisenhauer2015generalized.}

Estimation and Inference

An attractive feature of the identification results in Sections (ref)--(ref) is that they are constructive and lead to tractable estimators. Here, we propose semiparametric estimators of the potential outcome distributions for the three types of treatment based on distribution regression (DR). We also show how to construct estimators of treatment effect parameters such as unconditional QTE using the plug-in rule. We focus on target parameters that yield $\sqrt{n}$-consistent and asymptotically normal estimators. We do not consider parameters like $\partial QSF_{\tau}(d)/\partial d$, because estimating such parameters would involve non-parametric methods, which require choosing tuning parameters and exhibit slower convergence rates.

In this section, we make the role of the covariates $X$ explicit; see Appendix (ref) for a more detailed discussion of identification with covariates. We assume we have access to a random sample of size $n$ from $(Y,D,Z,X)$, $\left\{ (Y_{i},D_{i},Z_{i},X_{i})\right\} _{i=1}^{n},$ for estimation. Let $B(X_{i})$, $B(Z_i,X_{i})$, and $B(D_{i},Z_{i},X_{i})$ denote vectors of transformations of $X_{i}$, $(Z_i,X_{i})$, and $(D_{i},Z_{i},X_{i})$, respectively. Define the indicators $I_{i}(y)\equiv1\{Y_{i}\le y\}$ and $J_i(d) \equiv 1\{D_i \leq d\}$. Let $\bar{\mathcal{D}}$ and $\bar{\mathcal{Y}}$ be two finite grids covering $\mathcal{D}$ and $\mathcal{Y}$.\footnote{We can set $\bar{\mathcal{Y}} = \mathcal{Y}$ when $\mathcal{Y}$ is finite. We only use $\bar \mathcal{D}$ when $D$ is continuous.}

Binary Treatments

We consider separate DR models for the conditional potential outcome distributions,

where $y \mapsto \beta_d(y)$ is a function-valued unknown parameter, and a Probit model for the propensity score,

We model the local dependence parameter as

where $\rho(u)=\tanh(u)\in[-1,1]$, the Fisher transformation, and $y \mapsto \gamma_d(y)$ is a function-valued unknown parameter.\footnote{To simplify the notation, we use the same vector of transformations in models (ref) and (ref). This is not essential, and one can use different specifications in both models.} Together, (ref), (ref), and (ref) imply the bivariate DR model

where we have used the symmetry properties of the bivariate Gaussian distribution to simplify the previous expressions.

We propose a computationally tractable two-step maximum likelihood estimator, building on chernozhukov2018distribution.

Ordered Discrete Treatments

As for binary treatments, we model all the components using flexible generalized linear and DR models:

where $\rho(u) = \tanh(u)$.

Continuous Treatments

We construct plug-in estimators based on the closed-form solutions in Section (ref). We consider DR models for $F_{Y \mid D,Z,X}$ and $F_{D \mid Z,X}$,

where $y \mapsto \beta(y)$ and $d \mapsto \pi(d)$ are unknown function-valued parameters.

Marginal Distributions and Functionals

In the presence of covariates, the marginal distributions of the potential outcomes are identified by $$ F_{Y_d}(y) = \int F_{Y_d \mid X}(y \mid x) d F_X(x), \quad d \in \mathcal{D}, $$ where $F_X$ is the distribution of $X$. We can use this expression to construct estimators of $F_{Y_d}$ and functionals of interest, such as the QSF and QTE, by plugging in the estimators obtained above. To provide a unified approach to all the treatment cases, we set $\bar \mathcal{D} = \mathcal{D}$ when $D$ is binary or discrete ordered.

Asymptotic Theory and Inference

The target parameters in Section (ref) are function-valued. Inference on these parameters can be performed using resampling methods. To provide a unified framework, we denote the functional parameters by $u\mapsto \delta_u, u\in \mathcal{U}$, where $\mathcal{U}\subset \tilde{\mathcal{Y}}\times \tilde{\mathcal{D}}\times \mathcal{T}$, where $\mathcal{T}\subset (c,1-c)$ for $c>0$, $\tilde{\mathcal{Y}}$ is a compact subset of $\mathcal{Y}$ when $\mathcal{Y}$ is uncountable and equal to $\mathcal{Y}$ otherwise, and $\tilde{\mathcal{D}}$ is defined analogously. For example, if we are interested in $\tau \mapsto QSF_{\tau}(d)$ on $[.05,.95]$, then $u=\tau$, $\delta_u = QSF_{u}(d)$ and $\mathcal{U} = [.05,.95]$ for fixed $d$. In practice, we approximate $\mathcal{U}$ using a fine grid $\bar{\mathcal{U}}$. We denote the estimator of $\delta_u$ obtained from Algorithms (ref), (ref), (ref), and (ref) as $\widehat\delta_u$.

We show in Appendix (ref) that $\sqrt{n}(\widehat\delta_u-\delta_u)$ converges in distribution to a mean-zero Gaussian process $Z_\delta$ and the bootstrap is valid for a wide range of parameters of interest, including $F_{Y_d}$ and its functionals. These results imply that the bootstrap algorithm below is theoretically valid. When the outcomes are discrete or mixed discrete-continuous, the estimators of the QSF or QTE are not asymptotically Gaussian in general. In this case, one can use the inference methods proposed by chernozhukov2020generic.

We focus on constructing pointwise and uniform confidence bands, $CB^{pt}_{(1-\alpha)}(\delta_{\bar u})$ and $CB_{(1-\alpha)}(\delta_u)$ respectively, satisfying

Uniform confidence bands can be used to test a variety of hypotheses of interest, such as the hypotheses of no effect or constant effects when applied to QTE, or stochastic dominance.

The following algorithm provides a generic bootstrap construction of $CB_{(1-\alpha)}(\delta_u)$. A similar algorithm can be used for $CB^{pt}_{(1-\alpha)}(\delta_u)$, which we omit.

The estimation algorithms for binary and ordered treatments (Algorithms (ref) and (ref)) involve nonlinear optimization problems. Therefore, we recommend using the multiplier bootstrap in Step (ref) of Algorithm (ref), which avoids re-estimating the parameters in Algorithms (ref) and (ref) in each of the $B$ bootstrap iterations.

The estimation approach for continuous treatments in Algorithm (ref) does not involve solving a nonlinear optimization problem in the second step and is computationally less expensive than Algorithms (ref) and (ref). Therefore, the standard empirical bootstrap is a natural alternative to the multiplier bootstrap in Step (ref), provided that the sample size is not too large, as for example in Section (ref).

Empirical Illustration

We illustrate our methods by estimating the distributional effects of sleep on well-being.\footnote{The empirical results were obtained using the statistical software R R24. } We reanalyze the data from bessone2021economic, who studied the effects of randomized interventions to increase sleep time of low-income adults in India.\footnote{We downloaded the data from the Harvard Dataverse replication package bessone2021replication.} According to an expert survey conducted by bessone2021economic, increasing sleep could have large benefits in this empirical context, which is characterized by low baseline levels of sleep per night and sleep efficiency.

bessone2021economic considered two main treatments (see their Section III for details): (i) devices $+$ encouragement (information, encouragements and sleep trackers, various devices for improving sleep environment) and (ii) devices $+$ incentives (same as (i) and payments for each minute of sleep increase). In addition, they cross-randomized a nap treatment (the opportunity to nap at work).

The outcome of interest ($Y$) is an overall index of individual well-being. The treatment ($D$) is sleep time per night (in hours). We use an indicator for whether an individual received either of the main treatments as an instrument ($Z$) for sleep. The vector of covariates ($X$) includes controls for gender, three age indicators, and the baseline well-being index, as in bessone2021economic. Following dong2023nonparametric, we restrict the sample to individuals who did not receive the nap treatment, so that the total sample size is $n=226$.

The treatment $D$ takes on many values in this application and is treated as continuous. We therefore use the estimators for continuous treatments described in Algorithm (ref). We choose fully saturated (in $Z_i$) specifications for $B(d,z,x)$ and $B(z,x)$. The resulting DR models can be written as

Our flexible semiparametric estimators allow us to analyze the full distributional impact of sleep on well-being. Our results thus complement the empirical analyses of the average effect of sleep on well-being in bessone2021economic using two-stage least squares (2SLS) and in dong2023nonparametric using 2SLS and semiparametric doubly robust methods.

We begin by analyzing the (distributional) first-stage relationship between $D$ and $Z$ to shed light on the plausibility of (ref). Figure (ref) plots $\widehat{F}_{D \mid Z}(\cdot \mid 1) = n^{-1}\sum_{i=1}^n\widehat{F}_{D\mid Z,X}(\cdot \mid 1,X_i)$ and $\widehat{F}_{D \mid Z}(\cdot \mid 0) = n^{-1}\sum_{i=1}^n\widehat{F}_{D\mid Z,X}(\cdot \mid 0,X_i)$. It shows that the instrument induces a shift in the distribution of $D$. The distribution of sleep under the experimental treatments first order stochastically dominates the distribution of sleep without treatments.

In addition to (ref), our method relies on (ref) and (ref). The experimental random assignment renders the independence assumptions in (ref) plausible.\footnote{Note that random assignment does not automatically imply the (implicit) exclusion restriction, which requires that the instrument has no direct effect on the well-being.} (ref) allows the local dependence between potential well-being and the unobservable determinants of sleep to depend on the level of well-being, but not on the level of the unobservable determinants of sleep and the instrument.

Figure (ref) plots estimates of the QTE, $\widehat{QTE}_{\tau}(d,d')$, including 90% pointwise and uniform confidence intervals (CIs) computed using empirical bootstrap. We set $d=\widehat{Q}_{D}(0.75)$ and $d'=\widehat{Q}_{D}(0.25)$, where $\widehat{Q}_{D}(\tau)$ is the $\tau$-quantile of the empirical distribution of sleep. Figure (ref)(a) suggests interesting effect heterogeneity across the distribution of well-being. The QTEs are the largest and pointwise significant in the tails and smaller and insignificant around the median. The uniform CI includes zero at all quantiles. This finding is not very surprising given the relatively small sample size. In Figure (ref)(b), we “zoom-in” on the lower tail of the well-being distribution, which is of particular policy interest. The corresponding uniform CI do not include the zero QTE line, so that we can reject the null hypothesis that sleep has no impact on well-being at the lower tail.

An interesting feature of our method is that it allows for estimating the local dependence parameter $\rho_{Y_d;X}(y;x)$, which can be interpreted as a local measure of endogeneity or self-selection. Figure (ref)(a) plots the average local dependence parameter, $$ \tau \mapsto \frac{1}{n}\sum_{i=1}^n\widehat\rho_{Y_d;X}(\widehat{Q}_{Y}(\tau);X_i),\quad d\in \left\{\widehat{Q}_{D}(0.25),\widehat{Q}_{D}(0.50),\widehat{Q}_{D}(0.75)\right\}, $$ where $\widehat{Q}_{Y}(\tau)$ is the $\tau$-quantile of the empirical distribution of well-being. The average local dependence is negative at most quantiles of well-being, and there is interesting heterogeneity both across the distribution of well-being and across the three different values of sleep. This negative selection means that the unobserved propensity to sleep and the potential well-being by level of sleep are negatively correlated. In other words, individuals with relatively poor underlying health conditions sleep more hours. Figure (ref)(b) shows that the average local dependence for $d=\widehat{Q}_{D}(0.50)$ is pointwise significant at the upper tail, although the uniform CI includes zero at all quantile levels considered.

The negative local dependence suggests that estimators that ignore the endogeneity of sleep will underestimate the effect of sleep on well-being. Figure (ref)(a) demonstrates this phenomenon by comparing the QTE estimates obtained using our IV method to the corresponding estimates under conditional exogeneity, $Y_d\perp\!\!\!\!\perp D\mid X $ for $d\in \mathcal{D}$.\footnote{Under conditional exogeneity, $F_{Y_d}$ is identified as $F_{Y_d}(y)=\int F_{Y|D,X}(y|d,x)dF_{X}(x).$ For estimation, we consider the DR model $F_{Y|D,X}(y|d,x)=\Phi(d\gamma(y)+x'\beta(y)).$ The DR estimator is $\widehat{F}_{Y|D,X}(y|d,x)=\Phi(d\widehat\gamma(y)+x'\widehat\beta(y)),$ where $\widehat\gamma(y)$ and $\widehat\beta(y)$ are the coefficients obtained from a Probit regression of $1\{Y_i\le y\}$ on $D_i$ and $X_i$. The final estimator of $F_{Y_d}(y)$ under conditional exogeneity is $\widehat{F}_{Y_d}(y)=\frac{1}{n}\sum_{i=1}^n\Phi(d\widehat\gamma(y)+X_i'\widehat\beta(y)).$ } The QTE estimates under conditional exogeneity are small and insignificant at most quantiles, and thus miss the positive well-being effects of sleep in the tails of the well-being distribution. This finding demonstrates the importance of using IV methods in this empirical context.

Figure (ref)(b) further showcases the value-added that our method can bring to standard empirical analyses focusing on average effects. It compares the QTE estimates obtained from our method to standard 2SLS estimates using the same set of covariates. While the 2SLS analysis suggests that sleep has moderate average effects on well-being, our method uncovers interesting patterns of heterogeneity in the effect of sleep on well-being.

Concluding Remarks

In identifying causal effect of endogenous treatments, researchers inevitably face modeling trade-offs. This paper proposes a new direction to deal with these trade-offs by imposing assumptions on the local dependence between potential outcomes and unobservables determining treatment assignment. In doing so, we make minimal assumptions on the equations determining potential outcomes and treatments, thereby allowing for rich heterogeneity in these equations. The proposed framework applies to binary, discrete ordered, and continuous treatments, where point identification is achieved even with instruments that have limited variation (e.g., a binary instrument). In all three cases, the identification analysis leads to practical estimation and inference procedures that might appeal to practitioners.