On Quantile Treatment Effects, Rank Similarity, and Variation of Instrumental Variables

On Quantile Treatment Effects, Rank Similarity, and Variation of Instrumental Variables

Introduction

This paper investigates how certain relationship between observed and counterfactual distributions serves as an identifying condition for distributional treatment effects under endogeneity, and shows that this condition holds in a range of nonparametric models for treatment effects. To this end, we first provide a novel characterization of the prevalent assumption restricting treatment heterogeneity in the literature, namely rank similarity. Our characterization demonstrates the stringency of this assumption and allows us to relax it in a economically meaningful way, resulting in our identifying condition. It also justifies the quest of richer exogenous variations in the data (e.g., multi-valued or multiple instrumental variables) in exchange for the weaker identifying condition.

The primary goal of this investigation is to provide empirical researchers with (i) a framework where validity of identifying conditions prescribes the parameters of interest, (ii) tools for identifying and estimating treatment effects that allow for treatment heterogeneity, but that still yield tight policy evaluation and are simple to implement, and (iii) guidance on data collection that leads to drawing informative causal conclusions.

Our analysis centers on the relationship between observed and counterfactual distributions, specifically on the preservation of \enskip first-order stochastic dominance (FOSD) of one distribution over the other to their corresponding counterfactual distributions: for arbitrary compliance types $t,t'\in\mathcal{T}$ induced by induced by individuals' potential treatment responses to instrumental variables (IVs), if

then

where $Y_{d}$ denotes the counterfactual outcome given treatment $D=d$.\footnote{For r.v.'s $A$ and $B$, let $A\prec_{FOSD}B$ denotes $F_{B}(\cdot)\le F_{A}(\cdot)$ where $F_{A}$ and $F_{B}$ are CDFs of $A$ and $B$, respectively.} This condition produces a partial ordering of the $Y_{0}$'s distributions based on the partial ordering of the $Y_{1}$\textquoteright s distributions. As we demonstrate later, this condition can be interpreted as $Y_{0}$ being “noisier” than $Y_{1}$ after controlling for all the confounding variables.

We show that the proposed FOSD-preservation condition enables the identification of certain counterfactual distributions, which are essential components for identifying the treatment effects of our interest. Only for the sake of illustration, consider imbens1994identification's framework where binary instrument $Z\in\{0,1\}$ influences treatment participation monotonically. Let compliance types $C$, $AT$, and $NT$ stand for compliers, always-takers, and never-takers, respectively. Let $Y$ be the observed outcome given by $Y\equiv DY_{1}+(1-D)Y_{0}$. Suppose the observed distribution satisfies $Y_{1}|AT\prec_{FOSD}Y_{1}|C$. Then the FOSD-preservation condition implies that $Y_{0}|AT\prec_{FOSD}Y_{0}|C$. The latter provides an informative upper bound for $P[Y_{0}\le y|D=1]$ (and a symmetric analysis provides a lower bound), which is a necessary component in calculating, for example, the quantile treatment effect on the treated (QTT).\footnote{Note that $Y_{1}|AT\prec_{FOSD}Y_{1}|C$ and $Y_{0}|AT\prec_{FOSD}Y_{0}|C$ can be respectively rewritten as

}Although $Y_{1}|AT\prec_{FOSD}Y_{1}|C$ may seem restrictive, this is not generally the case when $Z$ departs from a scalar binary variable. In this sense, our approach underscores the significance of searching for richer exogenous variations of IVs, such as multi-valued or multiple instrumental variables, as a means of trading for less restrictive identifying conditions and achieving tighter bounds. Still, the benefit of our approach can be manifested without requiring continuous or large support of IVs. We also show that the proposed FOSD-preservation condition (i.e., (ref) implies (ref)) yields testable restrictions.

Nonparametric identification of treatment effects using IVs with limited support has long been a challenging goal even when the focus is on mean treatment effects, such as the average treatment effect (ATE) and the ATE on the treated (ATT). In an influential line of literature, manski1990nonparametric, manski1997monotone, and MP00, among many others, construct sharp bounds on the ATE under a set of assumptions on the directions of treatment effects and treatment selection while allowing instruments to be invalid in a specific sense. Even with valid instruments, however, bounds on the ATE are typically wide and uninformative to yield precise policy prediction. The local ATE (LATE) (imbens1994identification) and local QTE (abadie2002instrumental) have been a popular alternative when researchers are equipped with discrete IVs and impose a monotonicity assumption on the selection to treatment. However, the local group for which the treatment effect is identified may not be the group of policy interest. Therefore, the extrapolation of the local parameters becomes an important issue for policy analysis (e.g., treatment allocation), in which case the identification challenge still remains (see e.g., mogstad2018using, han2020sharp).

Another prevalent approach in the literature is to restrict the degree of treatment heterogeneity via rank similarity (or rank invariance). This assumption has been shown to have substantial identifying power for distributional treatment effects and the ATE and used in various nonparametric contexts implicitly or explicitly (heckman1997making, chesher2003identification,Che05, chernozhukov2005iv, VY07, jun2011tighter, SV11, d2015identification, torgovitsky2015identification, vuong2017counterfactual, han2021identification to name a few). However, the plausibility of this assumption can be questionable in many applications (e.g., maasoumi2019gender) and testing methods are proposed as one reaction to the skepticism (frandsen2018testing, dong2018testing, kim2022testing).

In this paper, we clarify the stringency of the rank similarity assumption by characterizing its restrictions on the relationship between observed and counterfactual distributions. In particular, we show that the strong preservation of FOSD (i.e., (ref) holds if and only if (ref) holds) is equivalent to rank linearity, a slight relaxation of rank similarity that allows for a linear transformation of an individual's rank to the counterfactual rank. By doing so, we establish the connection between the rank similarity assumption introduced by chernozhukov2005iv's structural IV model and its corresponding conditions within rubin1974estimating's counterfactual outcomes framework. Furthermore, we provide economic justifications for the weak preservation of FOSD by proposing a variety of non-separable structural IV models that imply the FOSD preservation condition, but that do not satisfy rank similarity.

Based on our identification strategy, we develop a statistical linear programming (LP) approach to estimate optimal bounds on the treatment parameters. These bounds are defined as optimal values of LP (with a discrete outcome) or semi-infinite LP (SILP) (with a continuous outcome). To address the infeasibility of the SILP problem, we transform the optimization problem by (i) randomizing the constraints or (ii) invoking duality and approximates the Lagrangian measure using sieves.

The next section formally introduces the main identifying conditions (i.e., the preservation of stochastic ordering) and establishes bounds on treatment effects. Section (ref) introduces structural models as sufficient conditions for the identifying conditions presented in the previous section. Section (ref) discusses the computation of the bounds using linear programming and, finally, Section (ref) presents numerical studies. In the Appendix, Section (ref) shows that point identification can be achieved with sufficient (but not infinite) variation of IVs. In the main text we focus on the QTE. An extension to bounding the ATE is discussed in Section (ref). Section (ref) contains more examples of structural models as sufficient conditions and Section (ref) holds further discussions on linear programming. All proofs are collected in Section (ref).

Key Conditions and Bounds on Treatment Effects

Let $D\in\{0,1\}$ be the observed treatment indicator, which represents the endogenous decision of an individual responding to IVs $Z$. We assume $Z$ is either a vector of binary IVs or a multi-valued IV, which takes $L$ distinct values: $Z\in\mathcal{Z}\equiv\{z_{1},...,z_{L}\}$. Multi-valued or multiple IVs are common in many observational studies (e.g., natural experiments typically provide more than one instrument) and experimental studies (e.g., randomized control trials where multiple treatment arms are implemented either simultaneously or sequentially).\footnote{See mogstad2021causal for a recent survey.} Let $Y_{1}$ be the counterfactual outcome of being treated and $Y_{0}$ be that of not being treated. They can be either continuously or discretely distributed. The observed outcome $Y\in\mathcal{Y}\subseteq\mathbb{R}$ satisfies $Y=DY_{1}+(1-D)Y_{0}$. Finally, $X\in\mathcal{X}\subseteq\mathbb{R}^{k}$ denotes other covariates that may be endogenous.

Define QTE and ATE for treated and untreated populations. For $d\in\{0,1\}$ and $x\in\mathcal{X}$, define

for $\tau\in(0,1)$ and

These parameters are what researchers and policymakers are potentially interested. The unconditional QTE and ATE can be recovered when these parameters are identified for all $d\in\{0,1\}$ and $x\in\mathcal{X}$. Throughout the paper, we maintain that the IVs are valid and satisfy the following exclusion restriction.

Introducing Key Conditions

Now we introduce the key condition that establishes the mapping between observed and counterfactual distributions.

Importantly, note that in (ref) the probability $P[Y_{1}\le\cdot|D=1,Z=z_{\ell},X=x]$ can be obtained from the data as $Y_{1}=Y$ given $D=1$. The mapping between observed and counterfactual distributions has been considered in vuong2017counterfactual, whose insights we share. Suppose that $Z\perp(Y_{d},D_{z})|X$ additionally holds, where $D_{z}$ is the counterfactual treatment given $Z=z$. Under this assumption, each probability term in Condition (ref) satisfies $P[Y_{d}\le\cdot|D=1,Z=z_{\ell},X=x]=P[Y_{d}\le\cdot|D_{z_{\ell}}=1,X=x]$. Note that $\sum_{\ell=1}^{L}w_{\ell}P[Y_{1}\le\cdot|D_{z_{\ell}}=1,X=x]$ is a mixture of $Y_{d}$'s distributions weighted across different compliance types defined by $\{D_{z_{\ell}}=1\}$, and thus can be viewed as a distribution for a hypothetical population with the specific composition of compliance types. Therefore, Condition (ref) posits that the FOSD ordering between the $Y_{1}$'s distributions of two compliance compositions is preserved between their counterfactual distributions of $Y_{0}$. For example, when $L=2$ and defiers are excluded from possible compliance types (e.g., by imbens1994identification's monotonicity assumption), then Condition (ref) simply describes the stochastic ordering between always-takers and compliers. When $L\ge3$, however, there are more compliance types, which composition becomes more complex as illustrated in Section (ref). Note that Condition (ref) is not an “if and only if” statement. It would be stringent to impose the preservation of ordering to hold in both directions. In fact, such a condition is closely related to the rank similarity condition (chernozhukov2005iv); see Section (ref) for full details.

Bounds on Treatment Effects

Now, we show that Condition (ref) is useful in constructing bounds on $F_{Y_{0}|D,X}(\cdot|1,x)$ and subsequently on $QTE_{\tau}(1,x)$. Let $p(z_{\ell},x)\equiv P[D=1|Z=z_{\ell},X=x]$ and

The proof of Theorem (ref) and most of other proofs are contained in the appendix. Note that there can be multiple $\gamma$ and $\tilde{\gamma}$ in $\Gamma(x)$ that satisfy (ref) and (ref), respectively. Therefore, we can further tighten the bounds as follows.

Theorem (ref) and Corollary (ref) highlight the identifying power of multi-valued IVs. The key step in Theorem (ref) to calculate the bounds is to find $\gamma$ (resp. $\tilde{\gamma}$) in $\Gamma(x)$ that satisfies (ref) (resp. (ref)), which serves as a rank condition. Note that this condition is verifiable with the data. Corollary (ref) additionally implies that the bounds can be further tightened if one increases the degree of freedom in the feasible set $\Gamma(x)$ by increasing $L$, in which case (ref)--(ref) are more likely to hold. See below and Section (ref) for related discussions.

Finally, note that \[ QTE_{\tau}(1,x)=Q_{Y|D,X}(\tau|1,x)-Q_{Y_{0}|D,X}(\tau|1,x) \] and the bounds on the second quantity on the right-hand side can be calculated using the worst case bounds for the conditional quantile (manski1994selection, blundell2007changes):

where $Q_{Y_{0}|D,X}^{LB}(\tau|1,x)$ and $Q_{Y_{0}|D,X}^{UB}(\tau|1,x)$ are the $\tau$-th quantiles of $F_{Y_{0}|D,X}^{LB}(\cdot|1,x)$ and $F_{Y_{0}|D,X}^{UB}(\cdot|1,x)$, respectively. Although the bounds on $ATE(1,x)=E[Y|D=1,X=x]-E[Y_{0}|D=1,X=x]$ can be calculated based on $E[Y_{0}|D=1,X=x]=\int_{0}^{1}Q_{Y_{0}|D,X}(\tau|1,x)d\tau$, we present later how the bounds on the $ATE(d,x)$ can be calculated under a weaker condition than Condition (ref).

If we assume the converse of Condition (ref), we can calculate bounds on the $QTE(0,x)$.

Similar to Condition (ref), Condition (ref) ensures that if the two distributions of $Y_{0}$ exhibit an FOSD ordering across different compliance compositions, this ordering is preserved in their corresponding counterfactual $Y_{1}$ distributions. In practice, depending on the specific context, Condition (ref), Condition (ref), or both may hold; Section (ref) provides some related intuitions. Not surprisingly, we can use Condition (ref) to construct bounds on $F_{Y_{1}|D,X}(\cdot|0,x)$ and subsequently on $QTE_{\tau}(0,x)$.

The proof of this theorem is analogous to that of Theorem (ref). The bounds on $QTE_{\tau}(0,x)$ can be derived symmetrically as in the case of $QTE_{\tau}(1,x)$ and thus are omitted. Notably, which treatment parameter we can obtain bounds for is determined by which identifying condition we impose (i.e., Condition (ref) or (ref)). In Section (ref), we investigate this aspect within economic structural models. Finally, in the Appendix, we introduce weaker conditions to bound average treatment effects.

Understanding Key Conditions

We further explore Conditions (ref) and (ref) to give additional interpretation and discuss testability. Suppress $X=x$ to simplify our discussions. Under $Z\perp(Y_{d},D_{z})$, the inequalities for FOSD in Conditions (ref) and (ref) can be rewritten as

Recall, Theorem (ref) relies on the existence of a sequence $\gamma=(\gamma_{1},...,\gamma_{L})$ satisfying $\sum_{\ell=1}^{L}\gamma_{\ell}=0$, $\sum_{\ell=1}^{L}\gamma_{\ell}p(z_{\ell},x)=1$, and the inequality (ref), that is, $P[Y\leq y|D=1]\leq\sum_{\ell=1}^{L}\gamma_{\ell}P[Y\leq y,D=1|Z=z_{\ell}]$ for all $y$. Note that (ref) is a special case of (ref) with $d=1$, which is the “if” part of Condition (ref). Let $p(z)\equiv(D=1|Z=z)$. Only for the purpose of this subsection, assume the generalized version of the LATE monotonicity introduced in imbens1994identification:

Under (ref), $\{D_{z_{\ell}}=1\}$ in (ref) are a mix of individuals who are compliers (C) and always-takers (AT). For $Z\in\mathcal{Z}=\{z_{1},...,z_{L}\}$, let $(z_{\ell-1},z_{\ell})$-compliers be compliers induced by the change of $Z$ from $z_{\ell-1}$ to $z_{\ell}$. When $L=3$ and $(z_{1},z_{2},z_{3})=(0,1,2)$, for example, $\{(0,1)\text{-C}\}=\{i:D_{0,i}=0,D_{1,i}=D_{2,i}=1\}$ is the set of eager compliers (E-C) and $\{(1,2)\text{-C}\}=\{i:D_{0,i}=D_{1,i}=0,D_{2,i}=1\}$ is the set of reluctant compliers (R-C), following the language of mogstad2021causal. Also, $\{\text{AT}\}=\{i:D_{0,i}=D_{1,i}=D_{2,i}=1\}$ is the set of always-takers. Let $p_{\ell}$ for $\ell=\{2,...,L\}$ is the proportion of $(z_{\ell-1},z_{\ell})$-compliers and let $p_{1}\equiv P[\text{AT}]$. We show that (ref) establishes the FOSD relationship between the mixtures of observed distributions of $Y$ conditional on various always-takers and compliers groups:

To illustrate the intuition of Lemma (ref)(i), consider $L=3$ and $(z_{1},z_{2},z_{3})=(0,1,2)$. Then,

because $\{D_{0}=1,D_{1}=1,D_{2}=0\}=\emptyset$ and $\{D_{0}=0,D_{1}=1,D_{2}=0\}=\emptyset$. Also, $\{D_{2}=1\}=\{\text{AT}\}\cup\{\text{E-C}\}\cup\{\text{R-C}\}$ and $\{D_{0}=1\}=\{\text{AT}\}$.

Lemma (ref)(ii) can be used as the basis to test (ref) and thus Condition (ref). The intuition is as follows. With a binary IV, the marginal distributions of $Y_{1}$ and $Y_{0}$ are identified for compliers (abadie2002instrumental). This result holds for any complier group defined by a pair of instrument values, such as $\{(z_{\ell-1},z_{\ell})\text{-C}\}$ in the lemma. Then, when $L\ge2$, we can find vectors $w$ and $\tilde{w}$ in $\mathbb{R}_{+}^{L}$ that assign zero weights to the distributions for AT and still make (ref) a non-trivial inequality where all the associated distributions for compliers are identified for all $d=1,0$.

Structural Models as Sufficient Conditions

We show that Conditions (ref) and (ref) can be justified in a range of nonparametric structural models for the counterfactual outcomes. To this end, it is useful to first present a stronger version of Condition (ref) (labeled as (ref)). This version of the condition is motivated by the discussion in Remark (ref). To state this condition, we introduce a general model for treatment selection:

Note that Assumption (ref) permits a more general compliance behavior than what a weakly separable model $D=1\{\eta\le h(Z,X)\}$ does (or equivalently, (ref) as shown in vytlacil2002independence). Although Assumption (ref) is not necessary for our main procedure, it is useful in defining the types of compliance behavior (i.e., treatment selection mechanism) via the unobservable $\eta$. Under this assumption, the following condition implies Condition (ref).\footnote{More precisely, it implies the condition in Remark (ref), which in turn implies Condition (ref).} Let $F_{Y_{d}|\eta,X}(y|t,x)\equiv P[Y_{d}\le y|\eta=t,X=x]$.

Because $w(\cdot,x)$ is non-negative and $\int w(t,x)dt=1$, note that $\int w(t,x)F_{Y_{d}|\eta,X}(\cdot|t,x)dt$ is a mixture of conditional CDFs (with $w(\cdot,x)$ being the mixture weight) and thus itself a CDF. In other words, defining a type distribution $W_{x}(t)=\int^{t}w(\eta,x)d\eta$, we can write $\int w(t,x)F_{Y_{d}|\eta,X}(\cdot|t,x)dt=\int F_{Y_{d}|\eta,X}(\cdot|t,x)dW_{x}(t)$.\footnote{Since $\eta$ has arbitrary dimensions, the integral with respect to $t$ is understood to be a multivariate integral.} Therefore, Condition (ref) assumes that the FOSD ordering of $Y_{1}$ distributions conditional on $\eta$ conforming to two different type distributions ($W_{x}(\cdot)$ and $\tilde{W}_{x}(\cdot)$) is preserved in the ordering of $Y_{0}$ distributions conditional on the corresponding type distributions.

The following lemma establishes the sufficiency of Condition (ref) for Condition (ref).

Symmetrically, Condition (ref) has a corresponding stronger condition, which is omitted.

Now, we relate the conditions with the structural models, which provide additional intuitions for the conditions. We present a leading model here and the rest in the Appendix. For arbitrary r.v.'s $A$ and $\tilde{A}$, let $A\stackrel{d}{=}\tilde{A}$ denote $F_{A}=F_{\tilde{A}}$.

Model 1. (i) Assumption (ref) holds and

where $q(d,x,\cdot)$ is continuous and monotone increasing and $U_{D}=DU_{1}+(1-D)U_{0}$, (ii) conditional on $(\eta,X,Z)$, $U_{d}\stackrel{d}{=}U+\xi_{d}$ where $\xi_{d}\perp(\eta,U)$, (iii) conditional on $(X,Z)$, $\xi_{0}$ is (weakly) more or less noisy than $\xi_{1}$, that is, $\xi_{0}\stackrel{d}{=}\xi_{1}+V$ for some $V$ independent of $\xi_{1}$.

Note that $U$ is the source of endogeneity in that it allowed to be dependent on $\eta$. Model 1(ii)--(iii) implies that $U_{0}\stackrel{d}{=}U_{1}+V$ conditional on $(\eta,X,Z)$. Importantly, Model 1 nests the model in chernozhukov2005iv as a special case. This can be shown as follows. First, they assume Model 1(i) and, conditional on $(X,Z)$, either rank similarity ($F_{U_{0}|\eta}=F_{U_{1}|\eta}$) or rank invariance ($U_{0}=U_{1}$).\footnote{Note that rank similarity and rank invariance are observationally equivalent under Model 1(i) in that they produce the same distribution of observables (chernozhukov2013quantile).} Then, by taking $\xi_{d}=0$ for all $d$ in Model 1(ii), we have $U_{0}\stackrel{d}{=}U_{1}\stackrel{d}{=}U$ conditional on $(\eta,X,Z)$, which proves the claim.

Model 1(iii) assumes that the unobservable under the counterfactual status of being treated are more (or less) dispersed than that under the counterfactually untreated status. Although this may seem stringent, it is substantially weaker than rank similarity (or invariance) and can be plausible in various scenarios. Before providing examples of these scenarios, we first establish the connection between Model 1 and Condition (ref) (and thus Condition (ref) by Lemma (ref)).

Analogous to Theorem (ref), one can readily show that Model 1 with $\xi_{0}$ being weakly less noisy than $\xi_{1}$ implies Condition (ref).

Now we provide examples that are consistent with Model 1.

The scenarios in Examples (ref)--(ref) justify Condition (ref) via Theorem (ref). Then, under Condition (ref), Theorem (ref) and Corollary (ref) yield bounds on $QTE_{\tau}(1,x)$, the effects of treatment for those who take the treatment. The final example illustrates the converse case.

The scenario in Example (ref) justifies Condition (ref), under which bounds on $QTE_{\tau}(0,x)$, the effects of treatment for those who abstain from it, can be obtained.

Model 1 and these examples show how a certain treatment parameter may be more relevant for policy than others depending on the plausibility of assumptions. Consider the problem of a policymaker. Assume that the policymaker concerns risk-averse individuals, which are typically the majority. For this policymaker, a candidate policy would aim at providing “insurance,” which can be either literally insurance or policies that serve as insurance (e.g., vaccination, subsidies). Therefore, she would be interested in understanding the treatment effects for the target individuals that are risk-averse. Our procedure provides a statistical tool for such a policymaker. That is, under Model 1, our procedure has the ability to bound the treatment effects for individuals with $D=d$ such that $var(\xi_{d})<var(\xi_{1-d})$. This is a unique feature of our setting: the plausibility of assumptions dictates the parameters of interest, which then can be terms as assumption-driven treatment parameters.

A remaining question one might have is as follows. How much Condition (ref) has to be strengthened to be equivalent to rank similarity? To answer this question, recall that Condition (ref) is stronger than Condition (ref) (by Lemma (ref)). We strengthen Condition (ref) further by making it an “if and only if” condition:

It turns out that we can establish the following result.

This theorem highlights the stringency of rank similarity relative to Condition (ref), which is used in our bound analysis. The proof is trivial so omitted. It is worth noting that the converse of Theorem (ref) is not true. Here is a counter-example for the converse statement.

This rank linearity implies Condition (ref), which is trivial to show. However, rank linearity is weaker than rank similarity as the latter is a special case of the former. To see this, conditional on $Z=z$ (and suppressing $X)$, (ref) with Model 1(i) yields $F_{U_{0}|\eta}(q^{-1}(0,y)|t)=\lambda(y)F_{U_{1}|\eta}(q^{-1}(1,\psi(y))|t)$. Then, by choosing $\lambda(y)=1$ and $\psi(y)=\phi(y)\equiv q(1,q^{-1}(0,y))$ (i.e., the counterfactual mapping (vuong2017counterfactual)), we have $F_{U_{0}|\eta}(\cdot|t)=F_{U_{1}|\eta}(\cdot|t)$.\footnote{Alternatively, rank similarity can be equivalently stated as $F_{Y_{0}|\eta}(y|t)=F_{Y_{1}|\eta}(\phi(y)|t)$ (where $\phi(y)$ is strictly increasing), which can be derived from (ref) by choosing $\lambda(y)=1$ and $\psi(y)=\phi(y)$.} In general, while the ranks between $Y_{0}$ and $Y_{1}$ should have the same distribution under rank similarity, their distributions can be different under rank linearity because of the multiplying term $\lambda(\cdot)$ in $F_{U_{0}|\eta}(u|t)=\lambda(q(0,u))F_{U_{1}|\eta}(u|t)$. However, note that the the difference cannot be entirely arbitrary as $\lambda(\cdot)$ does not depend on $t$, and thus rank linearity still poses substantive restrictions.

Interestingly, rank linearity is equivalent to Condition (ref). The following theorem is one of the main contributions of this paper. Suppress $(Z,X)$ for simplicity.

We prove this equivalence in the Appendix. The proof with continuous $Y_{d}$ is more involved than that with discrete $Y_{d}$; we recommend that the interested reader reads the latter first. The condition (ref) is only introduced in this theorem to establish the relationship between rank linearity (and hence rank similarity) and the range of identifying conditions of this paper, and it is not necessary for our bound analysis. This condition would be violated when there is no endogeneity (i.e., $Y_{d}\perp\eta$), which is not our focus.

Condition (ref) is crucial in bounding $QTE_{\tau}(x)=Q_{Y_{1}|X}(\tau|x)-Q_{Y_{0}|X}(\tau|x)$ unconditional with respect to $D=d$. The “only if” part (i.e., Condition (ref)) will bound $Q_{Y_{0}|D=1}(\tau)$ and thus $Q_{Y_{0}}(\tau)$ by Theorem (ref), while the “if” part (i.e., Condition (ref)) will bound $Q_{Y_{1}|D=0}(\tau)$ and thus $Q_{Y_{1}}(\tau)$ by the symmetric version of Theorem (ref). The fact that Condition (ref) is weaker than rank similarity illustrates the importance of rank similarity in the identification of the QTE and ATE.

Systematic Calculation of Bounds

In Theorem (ref), $\gamma$ is required to satisfy a set of linear inequality constraints, i.e., (ref) (respectively, (ref)), and each feasible $\gamma$ establishes an upper bound (respectively, lower bound) on $F_{Y_{1}|D,X}(\cdot|0,x)$. Therefore, it is intuitive to employ optimization methods to calculate these bounds, as detailed in Corollary (ref). For simplicity, our subsequent discussion will focus solely on the upper bound, omitting covariates $X$ for brevity.

Semi-Infinite Programming

To simplify notation, let $\boldsymbol{p}(y,d)\equiv(p(y,d|z_{1}),...,p(y,d|z_{L}))'$ where $p(y,d|z_{\ell})\equiv P[Y\le y,D=d|Z=z_{\ell}]$ and $p(y|d)\equiv P[Y\le y|D=d]$. Also, let $\boldsymbol{1}\equiv(1,...,1)'$ and $\boldsymbol{p}\equiv(p(z_{1}),...,p(z_{L}))'$ with $p(z)\equiv P[D=1|Z=z]$ so that

Consider the following linear semi-infinite programming problem for the upper bound on $P[Y_{0}\leq\bar{y}|D=1]$:

Note that the existence of $\gamma$ satisfying condition (ref) guarantees that the feasible set is non-empty. Also note that this condition is allowed to satisfy only almost everywhere (a.e.), which we suppress for simplicity. This program is infeasible to solve in practice as there are infinitely many constraints. We propose two approaches to approximate it with a linear program (LP).

Linear Program with Randomized Constraints

One approach to the semi-infinite program (ref)--(ref) is to approximate (ref) by \[ \boldsymbol{p}(\tilde{Y}_{m},1)'\gamma\ge p(\tilde{Y}_{m}|1),a.s.\ \text{for }m=1,\cdots,s_{n}, \] where $\{\tilde{Y}_{m}:m=1,\cdots,s_{n}\}$ is an i.i.d. simulated sample as is done in the literature (e.g., calafiore2005uncertain). An obvious candidate of this sample would be $\{Y_{i}\}_{i=1}^{n}$ with $s_{n}=n$. Consider a sampled LP of the following:

In Section (ref) of the Appendix, we show that the probability of violating the original constraints (ref) by using (ref) can be bounded by $O(1/n)$.

Dual Program and Sieve Approximation

Another approach to the semi-infinite program (ref)--(ref) is to invoke its dual and approximate the Lagrangian measure using sieve. With the constraint $p(\cdot|1)-\boldsymbol{p}(\cdot,1)'\gamma\le0$, the Lagrangian for (ref)--(ref) can be written as

where $\Lambda$ is a non-negative (not necessarily probability) measure (i.e., $\Lambda\succeq0$) that assigns weights to binding constraints. Moreover, let

Then, we have the following dual problem:

Note that (ref) has a finite number of constraints (i.e., $L$ constraints). It is trivial to show weak duality, $\widetilde{UB}(\bar{y})\le UB(\bar{y})$.\footnote{This is because, by (ref)--(ref) and (ref)--(ref), we have

} Strong duality also holds because of the structure of the problem (i.e., linearity in $\gamma$, continuity of $\boldsymbol{p}(\cdot,d)$ and $p(\cdot|d)$). We establish this in the following theorem.

Note that $\Lambda(y)$ is smooth as the feasible set of the primal problem is smooth due to the smoothness of $p(y|d)$ and $\boldsymbol{p}(y,d)$, which are CDFs. This motivates us to use sieve approximation for $\Lambda(y)$ to turn the dual into a linear programming problem. The smoothness class for $\Lambda(y)$ will be determined by the smoothness class of CDFs. Let $\mathcal{Y}$ is normalized to be $[0,1]$ and $\lambda(y)\equiv d\Lambda(y)/dy$ that satisfies $\int_{\mathcal{Y}}\lambda(y)dy=1$ and $\lambda(y)\ge0$ for all $y\in\mathcal{Y}$. Consider the following sieve approximation:

where $b_{j}(y)\equiv b_{j,J}(y)\equiv\left(

\right)y^{j}(1-y)^{J-j}$ is a Bernstein basis function. Then, the LP can be written as

or equivalently,

where $\theta\equiv(\theta_{1},...,\theta_{J})'$, $b_{d}\equiv(b_{d,1},...,b_{d,J})'$ with $b_{d,j}\equiv\int_{\mathcal{Y}}b_{j}(y)p(y|d)dy$, $\boldsymbol{b}_{d,j}\equiv(b_{d,j,1},...,b_{d,j,L})'$ with $b_{d,j,\ell}\equiv\int_{\mathcal{Y}}b_{j}(y)p(y,d|z_{\ell})dy$, and $B_{d}\equiv[

]$ is an $L\times J$ matrix, and by using $\int_{\mathcal{Y}}b_{j}(y)dy=\frac{1}{J+1}$ for all $j$. Note that the nonnegativity restriction on $\theta$ is imposed to reflect that $\lambda$ is a nonnegative measure. Using Bernstein polynomials to approximate infinite-dimensional decision variables is also used in han2020sharp.

Numerical Studies

To illustrate the importance of multiple IVs and the informativeness of resulting bounds, we conduct numerical exercises. We generate the data so that they are consistent with Model 1 and hence satisfy Condition (ref). The variables $(Y,D,Z)$ are generated in the following fashion:

Here, $Z$ is normalized so that the endpoints of the support are invariant regardless of the value of $L$. This is intended to understand the role of the number of values $Z$ takes while fixing the role of instrument strength. Figures (ref)--(ref) presents the bounds on $\Pr[Y_{0}\le y|D=1]$ while varying $L$. The bounds are calculated using the approach proposed in Section (ref). We only report $L\in\{2,5,6\}$ for succinctness. In these figures, the black solid line indicates the true value of $\Pr[Y_{0}\le y|D=1]$ and the red and blue crosses depict the upper and lower bounds. Although the upper bound is a trivial upper bound for the CDF when $L=2$, it quickly becomes informative as $L$ increases beyond $5$. To put this in a context, this corresponds to the number of instrument values that three binary IVs can easily surpass or a single continuous IV.