# Sensitivity Analysis in Unconditional Quantile Effects

Julián Martínez-Iriarte
Department of Economics
UC Santa Cruz Email<EMAIL_ADDRESS>I am deeply indebted to my advisor
Yixiao Sun for his constant support and guidance. I thank Jeffrey Clemens,
Dimitris Christelis, Antonio Galvao, Ying-Ying Lee, Michael Leung, Jessie Li,
Xinwei Ma, Augusto Nieto-Barthaburu, Joel Sobel, Pietro Spini, Kaspar
Wuthrich, Ying Zhu, and seminar participants at various universities for
helpful comments. All errors remain my own.

(June 15, 2023)

###### Abstract

This paper proposes a framework to analyze the effects of counterfactual
policies on the unconditional quantiles of an outcome variable. For a given
counterfactual policy, we obtain identified sets for the effect of both
marginal and global changes in the proportion of treated individuals. To
conduct a sensitivity analysis, we introduce the quantile breakdown frontier,
a curve that $(i)$ indicates whether a sensitivity analysis if possible or
not, and $(ii)$ when a sensitivity analysis is possible, quantifies the amount
of selection bias consistent with a given conclusion of interest across
different quantiles. To illustrate our method, we perform a sensitivity
analysis on the effect of unionizing low income workers on the quantiles of
the distribution of (log) wages.

Keywords: unconditional quantile effects, partial identification, sensitivity
analysis.

##  1 Introduction

In this paper we propose a sensitivity analysis on the effect of
counterfactual policies that change the proportion of treated individuals.
Consider a situation where a policy maker is interested in treating non-
treated individuals. The key identification challenge is that we do not the
observe the counterfactual outcome of individuals who switch groups, that is,
the _newly_ treated individuals. In some cases, however, it is still possible
to recover the distribution of the unobserved counterfactual outcome. For
example, suppose that treatment status is randomly assigned, and a policy
maker increases the proportion of treated individuals by randomly selecting
non-treated individuals.111We assume full compliance in both randomizations.
Although we do not observe the counterfactual outcome of the _newly_ treated
individuals, we know it is drawn from the same distribution as the _already_
treated individuals. Hence, we identified the counterfactual distribution of
_newly_ treated individuals.

When treatment status is _not_ randomly assigned in the first place, the
identification strategy previously described breaks down. The reason is that
due to the selection bias in the original treatment status, a random selection
of individuals from the control group will be drawn from a different
distribution. Thus, in the presence of selection bias, identification of the
counterfactual distribution requires that the policy maker has enough
information to devise a policy such that the (unobservable) distribution of
the _newly_ treated “matches” the distribution of the _already_ treated
individuals. This is usually infeasible. Even if the policy maker has this
information, such as when treatment status is randomly assigned, they might
not be interested in a policy that merely selects the _newly_ treated
individuals at random.

The previous discussion highlights that identification of counterfactual
distributions results in either very stringent information requirements, or in
policies that might not be interesting. In both cases, the distribution of the
_newly_ treated individuals is restricted. From the point of view of the
policy maker, this can rule out many interesting policies. To see this,
consider the following example. A policy maker might like to know if an
increase in the unionization rate reduces inequality. If unionized workers are
relatively high-skilled, and a policy expands unionization with low-skilled
workers, then the distribution of wages conditional on being in the union, is
likely to change.

In order to analyze a richer set of counterfactual policies, we drop the
restrictions on the distribution of the _newly_ treated individuals and
provide partial identification results for two effects. The first one is a
global effect that compares the quantiles of the observed outcome, to those of
the counterfactual outcome, where the proportion of treated individuals has
been increased by $\delta$. The second one is a marginal effect where we let
$\delta$ go to zero, and analyze its limiting effect on the unconditional
quantiles of the outcome.

Another important contribution of this paper is to propose a framework for a
sensitivity analysis on certain conclusions of interest. To do this, we
quantify the departure from point identification by the vertical distance
between the distributions of the _newly_ treated individuals and the _already_
treated individuals. We introduce a curve called the _quantile breakdown
frontier_ , which first indicates whether a sensitivity analysis is possible,
and second, it quantifies the maximum departure from point identification such
that a given set of conclusions holds across different quantiles. Using this
curve, we bound the global effects curve using this maximum departure derived
from the quantile breakdown frontier. In this way, we obtain an identified
region for the global effect curve consistent with the desired conclusions.
Estimation of both the quantile breakdown frontier and the bounds on the
global effect are based on empirical distribution functions and empirical
quantiles, and are $\sqrt{n}$-consistent.

The departure from point identification is due to the selection bias induced
by the counterfactual policy. We call this the _policy selection bias_. The
usual selection bias states that treated and non-treated individuals are
different in a sense, and that is what explains the selection in the first
place. Instead, the policy selection bias is the difference between the
distributions of the _newly_ treated individuals and the _already_ treated
individuals. Returning to the unionization example, the policy selection bias
arises because the union wages of _newly_ unionized workers may not be drawn
from distribution of the _already_ unionized workers. We do not know the
distribution of union wages of newly unionized workers, hence we can only
partially identify the global and marginal effects.

The policy selection bias can be non-negligible even if the original selection
into treatment is randomly assigned. The reason is that, for the policy
selection bias, what matters is who the _newly_ treated individuals are.
Conversely, if there is selection bias initially, but the distribution of the
_newly_ treated “matches” the distribution of the _already_ treated
individuals, then there will be no policy selection bias. Thus, the policy
selection bias depends on the particular counterfactual policy being analyzed,
not whether there is selection bias in the original selection mechanism.

We apply these methods to the study of unions and inequality, which has long
been of interest to labor economics. A recent comprehensive review of this
extensive literature is provided by Farber et al. (2020). Using the data in
Firpo, Fortin and Lemieux (2009), our empirical application considers the
effect of expanding unionization on the quantiles of the distribution of (log)
wages. Our approach allows us to tackle the question from a different
perspective. Using the tools developed in this paper, we can quantify the
amount of policy selection bias that is consistent with a policy that
increases the unionization rate by unionizing low earnings workers. By looking
at the global effect in the $30$th quantile of the distribution of wages we
investigate the amount of policy selection bias consistent with unions
reducing overall inequality. To this end, we examine the following conclusion:
whether the $30$th quantile increases by more than 10%. We find that this is
consistent with moderate values of policy selection bias. The bounds on the
global effect for other quantiles reveals that this policy can have a bigger
effect on quantiles below the $30$th quantile, without hurting those at the
middle and top of the distribution of income.

Related Literature There is an extensive literature devoted to the analysis of
counterfactual distributions. A good reference is Firpo, Fortin and Lemieux
(2011). In this paper, we focus on counterfactual distributions that arise as
a result of a counterfactual policy that changes the proportion of treated
individuals. The Policy Relevant Treatment Effect (PRTE) of Heckman and
Vytlacil (2001, 2005), and the Marginal PRTE (MPRTE) of Carneiro, Heckman and
Vytlacil (2010, 2011) are examples of the aforementioned global and marginal
effects. The difference is that they analyze the unconditional mean of the
outcome. Identification relies on the a separable threshold model for the
selection equation, and the availability of a continuous instrumental
variable. In this setting, the proportion of treated individuals is changed by
manipulating the instrumental variable. Our analysis does not make any
assumptions on the selection equation. We do not require an instrumental
variable either.

The marginal effect on the unconditional quantiles of an outcome was first
studied by Firpo, Fortin and Lemieux (2009). The identification arguments of
Firpo, Fortin and Lemieux (2009) are based on a distributional invariance
assumption: the distribution of the outcome for the original treatment group
(under the original policy regime) is the same as that for the new treatment
group (under the new policy regime), and this also holds for the control
groups under the two policy regimes.222See the proof to Corollary 3 of the
working paper version Firpo, Fortin and Lemieux (2007). For the case of an
endogenous binary covariate, where distributional invariance might not hold,
Martinez-Iriarte and Sun (2022) achieve identification by generalizing the
Marginal Treatment Effect framework. Kasy (2016) also analyzes counterfactual
policies which assign a binary treatment, but focuses on a welfare ranking.

Rothe (2012) provides a general treatment for functionals of the unconditional
distribution of the outcome. What we call a global effect, Rothe (2012) refers
to as a _Fixed Partial Policy Effect_ , and what we call a marginal effect,
Rothe (2012) refers to as a _Marginal Partial Distributional Policy_. However,
Rothe (2012) imposes different identifying assumptions, namely a form of
conditional exogeneity, which also yield a partial identified set. We do not
impose such assumptions in order to broaden the types of policies we can
analyze.

It is important to highlight that we do not estimate a quantile treatment
effect. The quantile treatment effect is the difference between the
$\tau$-quantile under treatment and the $\tau$-quantile under control, and
depends on the distribution of the covariates. In a recent contribution, Hsu,
Lai and Lieli (2020) investigate the changes in this effect when the
distribution of the covariates is manipulated. Aside from treatment status, we
do not manipulate the distribution of covariates.

Our sensitivity analysis is based on the breakdown analysis of Kline and
Santos (2013) and Masten and Poirier (2020). Kline and Santos (2013) perform a
sensitivity analysis in a different context: departures from a missing (data)
at random assumption. In a manner similar to us, this departure is measured as
the Kolmogorov-Smirnov distance between the distribution of observed outcomes
and the (unobserved) distribution of missing outcomes. Our quantile breakdown
frontier builds on the breakdown frontier introduced by Masten and Poirier
(2020). However, instead of relaxing two parameters, we relax just one, and
plot it against different quantiles. Another recent application of the
breakdown analysis is Noack (2021) in the context of LATE.

Notation All the CDFs are denoted by $F$ with a subscript indicating the
random variable. So, the CDF of $Y$ is $F_{Y}(y)$. Conditional CDFs are
denoted similarly. For example, the CDF of $Y$ conditional on $D=1$ and $X=x$
is denoted by $F_{Y|D=1,X=x}(y)$. The $\tau$-quantile of $Y$ is denoted by
$F^{-1}_{Y}(\tau)$. Weak convergence is denoted by $\rightsquigarrow$.

##  2 Counterfactual Policies and Unconditional Effects

We will work with the potential outcomes framework. For some unknown functions
$h_{0}$ and $h_{1}$

$\displaystyle Y(0)$ $\displaystyle=h_{0}(X,U_{0}),$ $\displaystyle Y(1)$
$\displaystyle=h_{1}(X,U_{1}),$

where $X$ are observed covariates and $U_{0}$ and $U_{1}$ consist of
unobservables. We do not impose any restriction on the dimension of the
unobservables. The observed outcome is thus

$\displaystyle Y$ $\displaystyle=D\cdot h_{1}(X,U_{1})+(1-D)\cdot
h_{0}(X,U_{0}).$ $\displaystyle:$ $\displaystyle=h(D,X,U),$

for a general nonseparable function $h$, where $D$ is a binary random variable
taking values $0$ and $1$, and $U:=(U_{0},U_{1})^{\prime}$. The variable $D$
can be interpreted as the treatment status, and $p:=\Pr(D=1)$ is the
proportion of treated individuals.

A counterfactual policy is an alternative assignment of individuals to
treatment. It is given by a binary random variable $D_{\delta}$, such that
$\Pr(D_{\delta}=1)=p+\delta$ for a fixed $\delta\in(-p,1-p)$. It is called
counterfactual because it may assign $D_{\delta}=1$ to an individual whose
$D=0$. As $\delta$ varies over $(-p,1-p)$, we obtain a collection of
(counterfactual) policies which is denoted by $\mathcal{D}$. When a particular
counterfactual policy $D_{\delta}$ belongs to $\mathcal{D}$ we write
$D_{\delta}\in\mathcal{D}$. The counterfactual outcome we would observe for a
given $D_{\delta}\in\mathcal{D}$ is

$\displaystyle Y_{D_{\delta}}$ $\displaystyle=h(D_{\delta},X,U),$

where we implicitly assumes that the potential outcomes are not affected by
the manipulation of $D$.

Strictly speaking, the counterfactual outcome $Y_{D_{\delta}}$ is not well
defined until we define $\mathcal{D}$, the collection of counterfactual
policies. We will restrict ourselves to policies that shift a portion of
individuals in the control group to the treatment group. We refer to such
individuals as _newly treated_. This means that for every individual,
$D_{\delta}-D\geq 0$. This is shown in Figure 1.

_counterfactual policy_
$D=1$$D=0$$D=1$$D_{\delta}=1$$D=0$$D_{\delta}=1$$D=0$$D_{\delta}=0$ Figure 1:
_A counterfactual policy where $D_{\delta}-D\geq 0$._

######  Assumption 1 (Counterfactual Policies).

The collection of policies $\mathcal{D}$ satisfies

  1. 1.

$\Pr(D_{\delta}=1)=p+\delta$ for $\delta\in[0,1-p)$ and
$D_{\delta}\in\mathcal{D}$;

  2. 2.

Monotonicity: $D_{\delta}-D\geq 0$;

The monotonicity assumption $D_{\delta}-D\geq 0$ is mainly for expositional
simplicity. We can do without this assumption, but we need to make some minor
changes to our approach. However, there is also a practical purpose. In a
context where $D$ is union status, and $D=1$ denotes unionized individuals,
Assumption 1 requires that we increase the unionization rate by unionizing
previously nonunionized workers. It would probably be hard to simultaneously
unionize and deunionize different workers.

Another way to look at the monotonicity assumption is by inspecting the joint
distribution of $D$ and $D_{\delta}$ it induces:

| $D_{\delta}=0$ | $D_{\delta}=1$
---|---|---
$D=0$ | $1-p-\delta$ | $\delta$
$D=1$ | $0$ | $p$

In other words, Assumption 1 rules out the presence of _newly untreated_
individuals. Also, in the limit, when $\delta=0$, we return to the original
distribution of individuals. We will evaluate the effect of a counterfactual
policy with two parameters: the global and the marginal effects. Let
$F_{Y}^{-1}(\tau)$ and $F_{Y_{D_{\delta}}}^{-1}(\tau)$ denote the
$\tau$-quantiles of $Y$ and $Y_{D_{\delta}}$ respectively.

######  Definition 1 (Global and Marginal Effects).

For a given collection of policies $\mathcal{D}$, the unconditional global
effect at the $\tau$-quantile of $D_{\delta}\in\mathcal{D}$ is

$\displaystyle G_{\tau,D_{\delta}}$
$\displaystyle:=F_{Y_{D_{\delta}}}^{-1}(\tau)-F_{Y}^{-1}(\tau),$

and the unconditional marginal effect at the $\tau$-quantile is

$M_{\tau,\mathcal{D}}:=\lim_{\delta\to
0}\frac{F_{Y_{D_{\delta}}}^{-1}(\tau)-F_{Y}^{-1}(\tau)}{\delta}$

whenever this limit exists.

The global effect $G_{\tau,D_{\delta}}$ is the comparison of quantiles of the
counterfactual distribution vs. the observed distribution for a fixed policy
$D_{\delta}$. Naturally, for a collection $\mathcal{D}$, we have a
corresponding collection on global effects. The marginal effect
$M_{\tau,\mathcal{D}}$ can be interpreted as an ordinary derivative: for small
$\delta$, it provides an approximation to the direction of the change in a
given $\tau$-quantile.

######  Remark 1 (Firpo, Fortin and Lemieux (2009)).

The marginal effect $M_{\tau,\mathcal{D}}$ was originally studied by Firpo,
Fortin and Lemieux (2009). Instead of Assumption 1, Firpo, Fortin and Lemieux
(2009) assume a form of distributional invariance:
$F_{Y_{D_{\delta}}|D_{\delta}=d}=F_{Y|D=d}$ and obtain point identification.
See the proof to Corollary 3 of the working paper version Firpo, Fortin and
Lemieux (2007). When both $D$ and $D_{\delta}$ are independent of $U$ and $X$,
then distributional invariance will be satisfied. In this particular case, a
policy maker can randomize $D_{\delta}$ so that for a given $\delta$, a
fraction $p+\delta$ of individuals is randomly assigned to treatment. However,
if we allow for $D$ to be endogenous, and if, as is usually the case, the
structural form of endogeneity is unknown, then it may be impossible for the
policy maker to design a sequence $\mathcal{D}$, such that for every
$D_{\delta}\in\mathcal{D}$, $F_{Y_{D_{\delta}}|D_{\delta}=d}$ “matches”
$F_{Y|D=d}$. From the point of view of the policy maker, this is a significant
restriction on the types of counterfactual policies they can consider.

######  Remark 2 (Policy Relevant Treatment Effect).

Heckman and Vytlacil (2001, 2005) and Carneiro, Heckman and Vytlacil (2010,
2011) investigate the effect on the unconditional mean of the outcome. Using
our notation, the Policy Relevant Treatment Effect (PRTE) of Heckman and
Vytlacil (2001, 2005) is

$\text{{PRTE}}_{D_{\delta}}=\frac{E(Y_{D_{\delta}})-E(Y)}{\delta}$

and taking the limit $\delta\rightarrow 0$ yields the Marginal PRTE (MPRTE) of
Carneiro, Heckman and Vytlacil (2010, 2011):

$\mathrm{MPRTE_{\mathcal{D}}}=\lim_{\delta\rightarrow
0}\text{{PRTE}}_{\delta}.$

Martinez-Iriarte and Sun (2022) show how to generalize the MPRTE to cover the
case of Firpo, Fortin and Lemieux (2009) as well.

######  Remark 3 (Rothe (2012)).

Rothe (2012) also studies the global and marginal effects but under a
different identifying assumption, namely a form of conditional exogeneity.
This assumption also yields an identified set. Let the outcome be
$Y=h(D,X,U)$. For uniformly distributed random variables $\tilde{U}_{1}$ and
$\tilde{U}_{2}$, the outcome can be represented as
$Y=h(Q_{D}(\tilde{U}_{1}),Q_{X}(\tilde{U}_{2}),U)$ where $Q_{D}$ and $Q_{X}$
are the quantile functions. Then $Q_{D}$ is changed to another quantile
function $Q_{D}^{*}$, generating a counterfactual distribution, which is
identified when $\tilde{U}_{1}\perp U\|X$ and $D$ is continuous. When $D$ is
discrete, $\tilde{U}_{1}$ is not uniquely determined, so that a range of
possible counterfactual distributions is possible resulting in partial
identification.

The next task is to define _who_ are the _newly treated_ individuals, that is,
how does $D_{\delta}$ determine who receives treatment among the individuals
whose $D=0$? In this paper we will focus on two types of policies: a policy
that simply chooses individuals whose $D=0$ at random and assigns them to
$D_{\delta}=1$, and a policy that chooses individuals based on a user-
specified criterion. We will refer to these two types of policies as
_randomized policy_ and _non-randomized policy_ respectively.

######  Example 1 (Randomized policy).

A randomized policy satisfies: for any $\delta\in[0,1-p)$

$\displaystyle D_{\delta}=\begin{cases}1&\quad\text{if }D=1\\\ 0\text{ or
}1&\quad\text{if }D=0\\\ \end{cases}$

and the _newly treated_ are selected at random. Using the conditional
independence notation333Dawid (1979) writes $X\perp Y\|Z$ to denote that $X$
and $Y$ are independent conditional on $Z=z$ for any $z$. Here, we require
independence to hold conditionally only on $D=0$. we write $D_{\delta}\perp
Y(1),Y(0)\|D=0$:

$\displaystyle
Pr(D_{\delta}=1|D=0)=\Pr(D_{\delta}=1|D=0,Y(1),Y(0))=\frac{\delta}{1-p}.$ (1)

######  Example 2 (Non-randomized policy).

An example of a non-randomized policy is the following: for any
$\delta\in[0,1-p)$

$\displaystyle D_{\delta}=\begin{cases}1&\quad\text{if }D=1\\\ 1&\quad\text{if
}D=0\text{ and }Z\leq F^{-1}_{Z|D=0}\left(\frac{\delta}{1-p}\right)\\\
0&\quad\text{otherwise}\\\ \end{cases}$ (2)

for some observable random variable $Z$. In this case, the individuals in the
group $\left\\{D=0\right\\}$ whose $Z$ is less than the
$\frac{\delta}{1-p}$-quantile of this group are shifted to $D_{\delta}=1$.
This rule guarantees that, in expectation, a proportion $\delta$ of
individuals is shifted.

For a collection of policies $\mathcal{D}$ that satisfies Assumptions 1, the
counterfactual distribution $F_{Y_{D_{\delta}}}(y)$ can be decomposed as

$\displaystyle F_{Y_{D_{\delta}}}(y)$
$\displaystyle=pF_{Y|D=1}(y)+(1-p-\delta)F_{Y|D_{\delta}=0}(y)+\delta
F_{Y(1)|D=0,D_{\delta}=1}(y)$ (3)

for each $D_{\delta}$. Here, $F_{Y(1)|D=0,D_{\delta}=1}(y)$ corresponds to the
distribution of the _newly treated_. This distribution cannot be identified
from the data because it requires observing $Y(1)$ for a subpopulation for
which we only observe their $Y(0)$. Consequently, $F_{Y_{D_{\delta}}}(y)$ is
not identified either. The goal is to bound the quantiles of
$F_{Y_{D_{\delta}}}(y)$. To that end, we make the following regularity
assumptions.

######  Assumption 2 (Regularity Assumptions).

  1. 1.

For $d=0,1$, $F_{Y|D=d,X=x}(y)$ is continuous and strictly increasing for all
$y$ such that $0<F_{Y|D=d,X=x}(y)<1$, and for all $x\in\mathcal{X}$.

  2. 2.

For every $D_{\delta}\in\mathcal{D}$, $F_{Y|D_{\delta}=0}(y)$ is continuous
and strictly increasing for all $y$ such that $0<F_{Y|D_{\delta}=0}(y)<1$.

  3. 3.

For every $D_{\delta}\in\mathcal{D}$, $F_{Y(1)|D=0,D_{\delta}=1}(y)$ is
continuous and strictly increasing for all $y$ such that
$0<F_{Y(1)|D=0,D_{\delta}=1}(y)<1$.

  4. 4.

For every $D_{\delta}\in\mathcal{D}$, the support of $X$ conditional on $D=0$
and $D_{\delta}=1$ is included in the support of $X$ conditional on $D=1$.

The next assumption is our main working assumption to obtain the bounds on the
quantiles of $F_{Y_{D_{\delta}}}(y)$.

######  Assumption 3 (KS-distance).

For a given $D_{\delta}\in\mathcal{D}$, there exists a known $c\in[0,1]$ such
that

$\displaystyle
c:=\sup_{y\in\mathbb{R}}\left|F_{Y(1)|D=0,D_{\delta}=1}(y)-\int_{\mathcal{X}}F_{Y|D=1,X=x}(y)dF_{X|D=0,D_{\delta}=1}(x)\right|.$
(4)

We refer to $c$ as the _policy selection bias_. The idea is that in the
absence of policy selection bias, _i.e._ , $c=0$, we can recover
$F_{Y(1)|D=0,D_{\delta}=1}(y)$ by matching _already_ unionized individuals
with _newly_ unionized individuals and integrating against the characteristics
of the _newly_ unionized individuals, using $F_{X|D=0,D_{\delta}=1}(x)$. This
is akin to a joint unconfoundedness assumption: $D\perp Y(1)\|X$ and
$D_{\delta}\perp Y(1)\|X$.444If $D,D_{\delta}\perp Y(1)\|X$, then
$\displaystyle F_{Y|D=1,X=x}(y)=F_{Y(1)|D=0,D_{\delta}=1,X=x}(y)$ so that
integrating against $F_{X|D=0,D_{\delta}=1}(x)$ yields
$F_{Y(1)|D=0,D_{\delta}=1}(y)$. By allowing $c$ to be non-zero, we are
relaxing this particular type of conditional independence assumption (Masten
and Poirier (2018)). Here, Assumption 2.4 becomes relevant. Kline and Santos
(2013) also use the Kolmogorov-Smirnov distance to bound the quantiles of the
outcome when data might not be missing at random.

######  Remark 4.

For now we take $c$ as known. In the next section it will be the quantity on
which will be base the sensitivity analysis.

######  Theorem 1 (Bounds on the Global Effect).

If a policy $D_{\delta}$ satisfies assumptions 1, 2, and 3, then for
$\tau\in(\delta,1-\delta)$

$\displaystyle\max\left\\{\tilde{F}_{A}^{-1}(\tau-\delta),F_{A}^{-1}(\tau-\delta
c)\right\\}-F_{Y}^{-1}(\tau)\leq
G_{\tau,D_{\delta}}\leq\min\left\\{\tilde{F}_{A}^{-1}(\tau),F_{A}^{-1}(\tau+\delta
c)\right\\}-F_{Y}^{-1}(\tau)$

where $\tilde{F}_{A}(y):=pF_{Y|D=1}(y)+(1-p-\delta)F_{Y|D_{\delta}=0}(y),$ and

$\displaystyle F_{A}(y)$
$\displaystyle:=\tilde{F}_{A}(y)+\delta\int_{\mathcal{X}}F_{Y|D=1,X=x}(y)dF_{X|D=0,D_{\delta}=1}(x).$

These bounds are sharp.

######  Remark 5.

When $c=0$, $G_{\tau,D_{\delta}}$ is point identified because the
counterfactual distribution in (3) becomes $F_{A}(y)$, which is identified. We
refer to $F_{A}(y)$ as the apparent distribution, hence the subscript $A$,
because it is the distribution that “appears” to be the counterfactual
distribution. Naturally, for $c=0$, we have that the lower and upper bound are
identical, and equal to $F_{A}^{-1}(\tau)$.555 For $c=0$, the upper bound is
$F_{A}^{-1}(\tau)$ because $F_{A}^{-1}(\tau)\leq\tilde{F}_{A}^{-1}(\tau)$. For
the lower bound we need to show that
$\max\left\\{\tilde{F}_{A}^{-1}(\tau-\delta),F_{A}^{-1}(\tau)\right\\}=F_{A}^{-1}(\tau)$.
Note that $\tilde{F}_{A}^{-1}(\tau-\delta)$ satifies:  $\displaystyle
pF_{Y|D=1}(\tilde{F}_{A}^{-1}(\tau-\delta))+(1-p-\delta)F_{Y|D_{\delta}=0}(\tilde{F}_{A}^{-1}(\tau-\delta))=\tau-\delta$
On the other hand, $F_{A}^{-1}(\tau)$ when $c=0$ satisfies  $\displaystyle
pF_{Y|D=1}(F_{A}^{-1}(\tau))+(1-p-\delta)F_{Y|D_{\delta}=0}(F_{A}^{-1}(\tau))+\delta\int_{\mathcal{X}}F_{Y|D=1,X=x}(F_{A}^{-1}(\tau))dF_{X|D=0,D_{\delta}=1}(x)=\tau$
Comparing these two expression, we obtain  $\displaystyle
pF_{Y|D=1}(\tilde{F}_{A}^{-1}(\tau-\delta))+(1-p-\delta)F_{Y|D_{\delta}=0}(\tilde{F}_{A}^{-1}(\tau-\delta))\leq
pF_{Y|D=1}(F_{A}^{-1}(\tau))+(1-p-\delta)F_{Y|D_{\delta}=0}(F_{A}^{-1}(\tau))$
which implies that we must have $\tilde{F}_{A}^{-1}(\tau-\delta)\leq
F_{A}^{-1}(\tau)$.  Thus, for $c=0$,
$G_{\tau,D_{\delta}}=F_{A}^{-1}(\tau)-F_{Y}^{-1}(\tau)$.

######  Remark 6.

By construction, we have that $\tilde{F}_{A}(y)\leq F_{A}(y)$, which implies,
for $\tau\in(\delta,1-\delta)$, that
$F_{A}^{-1}(\tau)\leq\tilde{F}_{A}^{-1}(\tau)$. Therefore, whether $F_{A}$ or
$\tilde{F}_{A}$ bounds the global effect depends on $\tau$, $\delta$, and $c$.
In other words, the $\min$ and $\max$ are not redundant.

######  Remark 7.

The bounds are sharp in the sense that for any possible value of the global
effect within the bounds, there is a corresponding newly treated distribution
$F_{Y(1)|D=0,D_{\delta}=1}(y)$ which delivers that same global effect.

######  Remark 8.

The rage of $\tau$ is restricted to $(\delta,1-\delta)$ in order to ensure
that $c$ to plays a role in the bounds. For $\tau$ outside this range, the
bounds might not depend on $c$, if $c$ is close enough to 1.

Before we attempt to bound the marginal effect, we will first investigate the
issue of existence. To that end, we introduce the following assumption.

######  Assumption 4 (Limiting Distributions).

For a given sequence of policies $\mathcal{D}$, the following maps exist and
are continuous on $y\in\mathbb{R}$:

$\displaystyle y\mapsto\frac{\partial
F_{Y|D_{\delta}=0}(y)}{\partial\delta}\bigg{|}_{\delta=0}:=\lim_{\delta\to
0}\frac{F_{Y|D_{\delta}=0}(y)-F_{Y|D=0}(y)}{\delta}$

and

$\displaystyle y\mapsto F_{Y(1)|\partial D}(y):=\lim_{\delta\to
0}F_{Y(1)|D=0,D_{\delta}=1}(y).$

Moreover, convergence takes place uniformly:

$\displaystyle\lim_{\delta\to
0}\sup_{y\in\mathbb{R}}\left|\frac{F_{Y|D_{\delta}=0}(y)-F_{Y|D=0}(y)}{\delta}-\frac{\partial
F_{Y|D_{\delta}=0}(y)}{\partial\delta}\bigg{|}_{\delta=0}\right|=0,$

and

$\displaystyle\lim_{\delta\to
0}\sup_{y\in\mathbb{R}}\left|F_{Y(1)|D=0,D_{\delta}=1}(y)-F_{Y(1)|\partial
D}(y)\right|=0.$

######  Theorem 2 (Existence of Marginal Effect).

Let $\mathcal{D}$ be a sequence of policies such that Assumptions 1, 2, and 4
hold. If $f_{Y}(F_{Y}^{-1}(\tau))>0$, then $M_{\tau,\mathcal{D}}$ exists and
is given by

$M_{\tau,\mathcal{D}}=-\frac{\dot{F}_{Y,\mathcal{D}}(F_{Y}^{-1}(\tau))}{f_{Y}(F_{Y}^{-1}(\tau))}.$

where $\dot{F}_{Y,\mathcal{D}}(y)$ satisfies

$\displaystyle\lim_{\delta\downarrow
0}\sup_{y\in\mathbb{R}}\left|\frac{F_{Y_{D_{\delta}}}(y)-F_{Y}(y)}{\delta}-\dot{F}_{Y,\mathcal{D}}(y)\right|=0.$

The conditions and the proof of Theorem 2 come from viewing the marginal
effect as a Hadamard derivative. Inspection of (3) shows that the assumptions
of the theorem ensure that $\lim_{\delta\to 0}F_{Y_{D_{\delta}}}=F_{Y}$. That
is, in the limit, the sequence of policies lead to the observed distribution
of $Y$. To better understand the smoothness conditions imposed by Assumption 4
we provide an example and a counterexample.

######  Example 3.

Suppose that $D=\mathds{1}\left\\{V\geq 0\right\\}$ for $V$ continuously
distributed around 0. For a sequence $\delta\downarrow 0$, consider
$D_{\delta}=\mathds{1}\left\\{V\geq F_{V}^{-1}(F_{V}(0)-\delta)\right\\}$.
Then, for $p:=\Pr(D=1)$, we have that $\Pr(D_{\delta}=1)=p+\delta$. Under mild
conditions

$\displaystyle\lim_{\delta\to
0}\frac{F_{Y|D_{\delta}=0}(y)-F_{Y|D=0}(y)}{\delta}=-\left(\frac{1}{f_{V}(0)}+\frac{1}{p}\right)F_{Y|D=0}(y)$

and

$\displaystyle\lim_{\delta\to 0}F_{Y(1)|D=0,D_{\delta}=1}(y)=F_{Y(1)|V=0}(y)$

We can interpret $V=0$ as those individuals who are indifferent with respect
to treatment. That is why we denote them with $\partial D$. See the appendix
for detailed calculations.

For a counterexample, consider instead $D_{\delta}=\mathds{1}\left\\{V\leq
F_{V}^{-1}(F_{V}(0)+\delta)\right\\}$, where the inequality sign is reversed.
Here, $\Pr(D_{\delta}=1)=p+\delta$ provided $V$ is symmetric around 0:
$F_{V}(0)=1-F_{V}(0)$. However,

$\displaystyle F_{Y|D_{\delta}=0}(y)=\Pr(Y\leq y|V\geq
F_{V}^{-1}(F_{V}(0)+\delta))\to\Pr(Y\leq y|V\geq 0)=F_{Y|D=1}(y).$

which contradicts the existence of the derivative $\frac{\partial
F_{Y|D_{\delta}=0}(y)}{\partial\delta}\bigg{|}_{\delta=0}$ required in
Assumption 4.

To provide sharp bounds on the marginal effect we need the following
assumption.

######  Assumption 5 (More Limiting Distributions).

For a given sequence of policies $\mathcal{D}$,

  1. 1.

$\delta\mapsto F_{Y|D_{\delta}}(y)$ is differentiable at $\delta=0$ for every
$y\in\mathbb{R}$.

  2. 2.

$\delta\mapsto F_{X|D=0,D_{\delta}=1}(x)$ is differentiable at $\delta=0$ for
every $x\in\mathcal{X}$. Moreover, we denote $F_{X|\partial
D}(x):=\lim_{\delta\downarrow 0}F_{X|D=0,D_{\delta}=1}(x).$

Here, $F_{X|\partial D}(x)$ can be seen as the observable characteristics of
the those individuals who are “indifferent” to treatment. Indeed, for the case
of Example 3, we have $F_{X|\partial D}(x)=F_{X|V}(x|0)$.

######  Theorem 3 (Bounds on the Marginal Effect).

If Assumptions 1, 2, 4, and 5 hold, Assumption 3 holds for all $\delta$ in a
neighborhood of 0, and $F_{X|\partial D}$ is known, then, for $\tau\in(0,1)$,

$\displaystyle\theta_{L,M}\leq M_{\tau,\mathcal{D}}\leq\theta_{U,M}$

where

$\displaystyle\theta_{L,M}:=\max\left\\{\frac{F_{Y|D=0}(F^{-1}_{Y}(\tau))-1}{f_{Y}(F^{-1}_{Y}(\tau))},\frac{F_{Y|D=0}(F^{-1}_{Y}(\tau))-\int_{\mathcal{X}}F_{Y|D=1,X=x}(F^{-1}_{Y}(\tau))dF_{X|\partial
D}(x)-c}{f_{Y}(F^{-1}_{Y}(\tau))}\right\\},$

and

$\displaystyle\theta_{U,M}:=\min\left\\{\frac{F_{Y|D=0}(F^{-1}_{Y}(\tau))}{f_{Y}(F^{-1}_{Y}(\tau))},\frac{F_{Y|D=0}(F^{-1}_{Y}(\tau))-\int_{\mathcal{X}}F_{Y|D=1,X=x}(F^{-1}_{Y}(\tau))dF_{X|\partial
D}(x)+c}{f_{Y}(F^{-1}_{Y}(\tau))}\right\\}.$

These bounds are sharp.

######  Remark 9.

The bounds for $M_{\tau,\mathcal{D}}$ require knowledge of $F_{X|\partial
D}(x)$, _i.e._ , the covariates for the individuals along the margin of
indifference. This depends crucially on the type of counterfactual policy
considered. Different policies may result in different individuals along the
margin of indifference. One option is to set $F_{X|\partial
D}(x)=F_{X|D=1}(x)$: marginal individuals have the (observable)
characteristics of those with $D=1$. In this case, covariates no longer play a
role in the bounds because

$\displaystyle\int_{\mathcal{X}}F_{Y|D=1,X=x}(F^{-1}_{Y}(\tau))dF_{X|\partial
D}(x)=\int_{\mathcal{X}}F_{Y|D=1,X=x}(F^{-1}_{Y}(\tau))dF_{X|D=1}(x)=\tau.$

A more general approach is to consider, for some $\alpha(x)\in[0,1]$

$\displaystyle F_{X|\partial
D}(x)=\alpha(x)F_{X|D=1}(x)+(1-\alpha(x))F_{X|D=0}(x).$ (5)

It must be kept in mind that any choice of $F_{X|\partial D}(x)$ entails a
risk of misspecification. This makes the problem of bounding the marginal
effect much harder.

######  Remark 10.

If we set $c=0$, and if Assumption 3 holds for all $\delta$ in a neighborhood
of 0, upon taking the limit in (4) we obtain

$\displaystyle F_{Y(1)|\partial
D}(y)=\int_{\mathcal{X}}F_{Y|D=1,X=x}(y)dF_{X|\partial D}(x)$

for all $y\in\mathbb{R}$. This opens a number of ways to point identify the
marginal effect. For example, we could require either $F_{Y(1)|\partial
D}(y)=F_{Y|D=1}(y)$ or $F_{X|\partial D}(x)=F_{X|D=1}(x)$. In either case, we
obtain

$\displaystyle
M_{\tau,\mathcal{D}}=\frac{F_{Y|D=0}(F^{-1}_{Y}(\tau))-F_{Y|D=1}(F^{-1}_{Y}(\tau))}{f_{Y}(F^{-1}_{Y}(\tau))},$

which is precisely the estimand of Firpo, Fortin and Lemieux (2009). We could
use (5) for a given $\alpha(x)$ to obtain yet a different value for the
marginal effect.

######  Remark 11.

The bounds are sharp in the sense that for any possible value of the marginal
effect within the bounds, there is a corresponding sequence of newly treated
distribution $F_{Y(1)|D=0,D_{\delta}=1}(y)$ which delivers that same marginal
effect.

##  3 Quantile Breakdown Frontier

The quantile breakdown frontier is a curve that allows to perform to a
sensitivity analysis with respect to $c$, the policy selection bias. Suppose
we are interested in a target conclusion $G_{\tau,D_{\delta}}\geq g_{\tau,L}$
for some $g_{\tau,L}$. If the conclusion does not hold under point
identification, i.e. when $c=0$, then there is no point in performing a
sensitivity analysis. On the other hand, if the conclusion does hold under
point identification, we would like to know the maximum amount of such that
$c$ such that the conclusion continues to hold. This is the sensitivity
analysis. The quantile breakdown frontier for the global effect tackles both
of these issues. The frontier is the map

$\displaystyle\tau\mapsto c_{\tau,L}=\frac{\tau-
F_{A}\left(F_{Y}^{-1}(\tau)+g_{\tau,L}\right)}{\delta}$ (6)

for $\tau\in(\delta,1-\delta)$.666We can also look at conclusions of the type
$G_{\tau,D_{\delta}}\leq g_{\tau,U}$. In this case, the quantile breakdown
frontier is the map  $\displaystyle\tau\mapsto
c_{\tau,U}=\frac{F_{A}\left(F_{Y}^{-1}(\tau)+g_{\tau,U}\right)-\tau}{\delta}.$
When $c_{\tau,L}<0$, the desired target conclusion does not hold. If
$c_{\tau,L}>0,$ then for $c\leq c_{\tau,L}$ then target conclusion holds under
point identification. If $c>c_{\tau,L}>0$, then the target conclusion might
not hold. In this sense, when $c_{\tau,L}>0,$ the frontier provides an amount
of policy selection bias which is compatible with the conclusion.

The next lemma contains some properties of the quantile breakdown frontier.

######  Lemma 1.

Under the assumptions of Theorem 1, the quantile breakdown frontier for
$G_{\tau,D_{\delta}}$ defined in (6) satisfies the following: (i) if
$\tau\mapsto g_{\tau,L}$ is continuous, then $\tau\mapsto c_{\tau,L}$ is
continuous; (ii) if $c_{\tau,L}<0$, then the conclusion does not hold under
point identification; (iii) if $c_{\tau,L}\geq 0$, then for any $c\geq 0$ such
that $c\leq c_{\tau,L}$, the conclusion holds, and for $c>c_{\tau,L}$ the
conclusion might not hold; and (iv) if $\tilde{F}_{A}^{-1}(\tau-\delta)\geq
F_{A}^{-1}(\tau-\delta c_{\tau,L})$, then the conclusion holds for any
$c\in[0,1]$.

Part $(i)$ of the lemma puts a restriction on the types of conclusion by
requiring continuity of the family of target conclusions. This is no
essential, but illustrates the fact that the smoothness of the frontier
depends, among other things, on the conclusions. For notational simplicity,
later on we assume $g_{\tau,L}\equiv g$. Part $(iv)$ addresses a potential
“conservadurism” in the breakdown analysis. This stems from the fact there is
a possibility that the lower bound for the global effect does not depend on
$c$. In the notation of Theorem 1, this means that we cannot rule out

$\displaystyle\max\left\\{\tilde{F}_{A}^{-1}(\tau-\delta),F_{A}^{-1}(\tau-\delta
c)\right\\}=\tilde{F}_{A}^{-1}(\tau-\delta).$

This is related to part $(ii)$. Indeed, if the opposite is true,
$\tilde{F}_{A}^{-1}(\tau-\delta)<F_{A}^{-1}(\tau-\delta c_{\tau,L})$, then we
can modify the language of part $(ii)$ to say that for $c>c_{\tau,L}$ the
conclusion will not hold. In the empirical application we estimate
$\tilde{F}_{A}^{-1}(\tau-\delta)\geq F_{A}^{-1}(\tau-\delta c_{\tau,L})$ and
we rule out this case.

For the marginal effect we can construct the quantile breakdown frontier in a
similar fashion. However, in order to avoid the presence of the density
$f_{Y}$ in the denominator,777Avoiding $f_{Y}$ allows us to retain a
$\sqrt{n}$-consistent estimator. we focus on conclusions
$M_{\tau,\mathcal{D}}\leq 0$.888The case $M_{\tau,\mathcal{D}}\geq 0$ can be
dealt with in the same way after some minor modifications. It is therefore
omitted. Thus, instead of a collection of conclusions that may vary with
$\tau$, here we have a common conclusion across $\tau$. Since the marginal
effect is a derivative, this amounts to looking at conclusions of whether the
global effect in a neighborhood of $\delta=0$ is increasing or decreasing as
$\delta$ increases. The quantile breakdown frontier for the marginal effect is
the map

$\displaystyle\tau\mapsto
c_{\tau,U}^{M}=\int_{\mathcal{X}}F_{Y|D=1,X=x}(F^{-1}_{Y}(\tau))dF_{X|\partial
D}(x)-F_{Y|D=0}(F^{-1}_{Y}(\tau)),$ (7)

for $\tau\in(0,1)$. The next lemma is analogous to Lemma 1.

######  Lemma 2.

Under the assumptions of Theorem 3, the quantile breakdown frontier for
$M_{\tau,\mathcal{D}}$ defined as the map in (7) satisfies the following: (i)
$\tau\mapsto c_{\tau,U}^{M}$ is continuous; (ii) if $c_{\tau,U}^{M}<0$, then
the conclusion does not hold under point identification; (iii) if
$c_{\tau,U}^{M}\geq 0$, then for any $c\geq 0$ such that $c\leq
c_{\tau,U}^{M}$, the conclusion holds, and for $c>c_{\tau,U}^{M}$ the
conclusion might not hold; and (iv) if
$F_{Y|D=0}(F^{-1}_{Y}(\tau))\leq\int_{\mathcal{X}}F_{Y|D=1,X=x}(F^{-1}_{Y}(\tau))dF_{X|\partial
D}(x)$, then the conclusion holds for any $c\in[0,1]$.

######  Remark 12.

While the frontier for the global effect is valid for
$\tau\in(\delta,1-\delta)$, for the marginal effect it is valid for
$\tau\in(0,1)$, provided that $Y$ has a bounded support. This difference
arises because in the marginal effect we are taking the limit as $\delta\to
0$.

###  3.1 Bounds derived from the QBF

Suppose that a policy maker choose a particular quantile $\tau^{*}$ and a
target conclusion $g_{\tau^{*},L}$. Then, provided that
$c_{\tau^{*},L}\in[0,1]$, we can obtain bounds on the global effect other
quantiles $\tau\neq\tau^{*}$. The interpretation is that, as long as the
policy selection bias satisfies $c\leq c_{\tau^{*},L}$, the global effect will
be bounded across quantiles by the bounds of Theorem 1 evaluated at
$c_{\tau^{*},L}$. In a sense, this allows us to extend the sensitivity
analysis to other quantiles. That is for $\tau\in(\delta,1-\delta)$, we have

$\displaystyle\max\left\\{\tilde{F}_{A}^{-1}(\tau-\delta),F_{A}^{-1}(\tau-\delta
c_{\tau^{*},L})\right\\}-F_{Y}^{-1}(\tau)\leq
G_{\tau,D_{\delta}}\leq\min\left\\{\tilde{F}_{A}^{-1}(\tau),F_{A}^{-1}(\tau+\delta
c_{\tau^{*},L})\right\\}-F_{Y}^{-1}(\tau).$ (8)

##  4 Estimation and Inference

We work in the space $\ell^{\infty}(\delta,1-\delta)$ of bounded real-valued
functions defined on $(\delta,1-\delta)$. As usual, we endow this space with
the supremum norm: $\|x\|_{\infty}:=\sup_{t\in(\delta,1-\delta)}|x(t)|$.999The
reason we restrict the space to be $\ell^{\infty}(\delta,1-\delta)$ and not
$\ell^{\infty}(0,1)$ is due to the fact that for a given $\delta$, we cannot
reach quantiles below $\delta$ or above $1-\delta$. See Remark 8 above. In
order to simplify notation, and ensure the continuity of the quantile
breakdown frontier, we are going to focus on the case where the threshold is
constant across $\tau$.

######  Assumption 6 (Constant Threshold).

For some scalar $g$, the threshold $g_{\tau,L}$ satisfies $g_{\tau,L}=g$ for
any $\tau\in(\delta,1-\delta)$.

This assumption can be relaxed at the expense of more complicated notation.
However, we still require smoothness in the map $\tau\mapsto g_{\tau}.$ For
the case of the quantile breakdown for the sign of the marginal effect, we
will set $g=0$.

######  Assumption 7 (DGP).

We observe an i.i.d. sample $\\{Y_{i},D_{i},X_{i}\\}_{i=1}^{n}$, where the
support of $X$ is finite.

###  4.1 Global Effect

To estimate $c_{\tau,L}$ given in (6) we need to specify two components: $(i)$
$\delta$, the increase in the treated proportion, and $(ii)$ $D_{i,\delta}$,
the counterfactual treatment assignment, as in examples 1 and 2. Once this has
been done, we use sample analogs. The estimator of the quantile breakdown
frontier for the global effect is

$\displaystyle\hat{c}_{\tau,L}=\frac{\tau-\hat{F}_{A}\left(\hat{F}_{Y}^{-1}(\tau)+g\right)}{\delta},$
(9)

where $\hat{F}_{Y}^{-1}(\tau)$ is the empirical $\tau$-quantile of $Y$:
$\hat{F}_{Y}^{-1}(\tau):=\inf\left\\{y:\hat{F}_{Y}(y)\geq\tau\right\\}$, and
$\hat{F}_{A}(y)$ is the empirical counterpart of $F_{A}(y)$ given in Theorem
1. Detailed expressions are provided in Appendix A. This is similar to a
quantile-quantile transformation (see Exercise 4 in Chapter 3.9 in van der
Vaart and Wellner (1996)). We base our proof of the asymptotic distribution of
$\sqrt{n}(\hat{c}_{\tau,L}-c_{\tau,L})$ on the proof of Lemma A.1 in Beare and
Shi (2019).101010Beare and Shi (2019) also offer some interesting historical
context for the result. We view the map $\tau\mapsto\hat{c}_{\tau,L}$ as a
random element of $\ell^{\infty}(\delta,1-\delta)$. In that case, we denote it
simply by $\hat{c}_{L}.$

The main assumption is the following.

######  Assumption 8 (Functional CLT).

The following multivariate functional central limit theorem holds

$\displaystyle\sqrt{n}\begin{pmatrix}\hat{F}_{Y}-F_{Y}\\\
\hat{F}_{A}-F_{A}\end{pmatrix}\rightsquigarrow\begin{pmatrix}\mathbb{G}_{Y}\\\
\mathbb{G}_{A}\end{pmatrix},$

where $\mathbb{G}_{Y}$ and $\mathbb{G}_{A}$ are Brownian bridges in
$\ell^{\infty}(\mathcal{Y})$.

######  Remark 13.

More primitive assumptions in terms of the CDFs that comprise $F_{A}$ can be
stated that lead to Assumption 8. For brevity, we leave it as it is.

The following assumption is needed to establish the Hadamard differentiable of
different functions used in the construction of $\theta$.

######  Assumption 9 (Conditions for Hadamard Differentiability).

  1. 1.

For some $\varepsilon>0$, $F_{Y}$ is continuously differentiable in
$[F_{Y}^{-1}(\delta)-\varepsilon,F_{Y}^{-1}(1-\delta)+\varepsilon]\subset\mathcal{Y}$
with strictly positive derivative $f_{Y}$.

  2. 2.

$y\mapsto F_{A}(y)$ is differentiable, with uniformly continuous and bounded
derivatives.

The first item in Assumption 9 concerns the support $\mathcal{Y}$ and the
smoothness of $F_{Y}$. It is used to guarantee the Hadamard differentiability
of the quantile process $\tau\mapsto F_{Y}^{-1}(\tau)$ for
$\tau\in(\delta,1-\delta)$. The second item ensures that $F_{A}(y)$ has a
uniformly continuous and bounded derivative which we denote by $f_{A}(y)$. It
is needed to establish the Hadamard differentiability of the composition map
$(F_{A},F_{Y}^{-1})\mapsto F_{A}\circ(F_{Y}^{-1}+g)$.111111Section 3.9 in van
der Vaart and Wellner (1996) studies the Hadamard differentiability of
composition maps.

######  Theorem 4.

Under Assumptions 6, 7, 8, and 9

$\displaystyle\sqrt{n}(\hat{c}_{L}-c_{L})\rightsquigarrow\frac{1}{\delta}\mathbb{G}_{A}\circ(F_{Y}^{-1}+g)-\frac{1}{\delta}f_{A}\circ(F_{Y}^{-1}+g)\frac{\mathbb{G}_{Y}\circ
F_{Y}^{-1}}{f_{Y}\circ F_{Y}^{-1}},$

a tight Gaussian element in $\ell^{\infty}(\delta,1-\delta)$.

Instead of providing a closed form expression and a consistent estimator for
the variance of the quantile breakdown frontier, we note that, by Theorem 23.9
in van der Vaart (1998), the empirical bootstrap is valid. Confidence
intervals can be constructed following the usual resampling scheme.

###  4.2 Marginal Effect

For the marginal effect, using sample analogs we obtain the following
estimator of (7).

$\displaystyle\hat{c}_{\tau,U}^{M}=\sum_{x\in\mathcal{X}}\hat{F}_{Y|D=1,X=x}(\hat{F}^{-1}_{Y}(\tau))\hat{p}_{x|\partial
D}-\hat{F}_{Y|D=0}(\hat{F}^{-1}_{Y}(\tau))$ (10)

Here, $\hat{p}_{x|\partial D}$ is the empirical counterpart of
$\Pr(X=x|\partial D)$, which, as mentioned above, is the distribution of
covariates for individuals at the margin of indifference. By Assumption 5,
this is $F_{X|\partial D}(x):=\lim_{\delta\downarrow
0}F_{X|D=0,D_{\delta}=1}(x).$ Following (5), for a _user-supplied_
$\alpha(x)\equiv\alpha$,121212Note that $\alpha(x)$ is fixed across $x$ at
$\alpha$ for simplicity. we take $F_{X|\partial D}(x)=\alpha
F_{X|D=1}(x)+(1-\alpha)F_{X|D=0}(x).$ Thus,

$\displaystyle\hat{p}_{x|\partial
D}=\alpha\hat{F}_{X|D=1}(x)+(1-\alpha)\hat{F}_{X|D=0}(x).$

Details are given in Appendix A. The main assumption is

######  Assumption 10 (Functional CLT).

The following multivariate functional central limit theorem holds

$\displaystyle\sqrt{n}\begin{pmatrix}\sum_{x\in\mathcal{X}}\hat{F}_{Y|D=1,X=x}\hat{p}_{x|\partial
D}-E[F_{Y|D=1,X}|\partial D]\\\ \hat{F}_{Y|D=0}-F_{Y|D=0}\\\
\end{pmatrix}\rightsquigarrow\begin{pmatrix}\mathbb{G}_{\partial D}\\\
\mathbb{G}_{0}\end{pmatrix},$

where $\mathbb{G}_{\partial D}$ and $\mathbb{G}_{0}$ are Brownian bridges in
$\ell^{\infty}(\mathcal{Y})$.

Here, $E[F_{Y|D=1,X}|\partial D]$ implicitly denotes the map $y\mapsto
E[F_{Y|D=1,X}(y)|\partial D]$. That is, the expectation is taken with respect
to $X$ using $F_{X|\partial D}(x)$. By Assumption 7, this expectation is just
a finite convex combination of the distribution functions $F_{Y|D=1,X=x}(y)$

######  Remark 14.

More primitive conditions, in terms of $F_{Y|D=1,X=x}(y)$, $F_{X|D=0}(x)$, and
$F_{X|D=1}$ can be given that result in Assumption 10.

The next assumption is needed to establish the Hadamard differentiability of
the composition map, and the quantile process.

######  Assumption 11 (Conditions for Hadamard Differentiability).

  1. 1.

The distribution functions $F_{Y|D=0}(y)$ and $E[F_{Y|D=1,X}(y)|\partial D]$
are differentiable, with uniformly continuous and bounded derivatives on their
support $\mathcal{Y}$. The derivatives are denoted by $f_{Y|D=0}(y)$ and
$E[f_{Y|D=1,X}(y)|\partial D]$ respectively.

  2. 2.

The support $\mathcal{Y}$ is the compact set $[y_{l},y_{u}]$.

  3. 3.

$F_{Y}(y)$ is continuously differentiable on $\mathcal{Y}$ with strictly
positive derivative $f_{Y}$.

The next theorem establishes the asymptotic distribution of
$\hat{c}_{\tau,U}^{M}$ as a process in $\ell^{\infty}(0,1)$. In such a case,
we denote the process simply by $\hat{c}_{U}^{M}$.

######  Theorem 5 (Asymptotic Distribution of QBF for Marginal Effect).

Under Assumptions 7, 10 and 11

$\displaystyle\sqrt{n}(\hat{c}_{U}^{M}-c_{U}^{M})$
$\displaystyle\rightsquigarrow\mathbb{G}_{\partial D,Y}-\mathbb{G}_{0,Y},$

where

$\displaystyle\mathbb{G}_{\partial D,Y}$ $\displaystyle=\mathbb{G}_{\partial
D}\circ F_{Y}^{-1}-E[f_{Y|D=1,X}|\partial D]\circ
F_{Y}^{-1}\cdot\frac{\mathbb{G}_{Y}\circ F_{Y}^{-1}}{f_{Y}\circ F_{Y}^{-1}},$
$\displaystyle\mathbb{G}_{0,Y}$ $\displaystyle=\mathbb{G}_{0}\circ
F_{Y}^{-1}-f_{Y|D=0}\circ F_{Y}^{-1}\cdot\frac{\mathbb{G}_{Y}\circ
F_{Y}^{-1}}{f_{Y}\circ F_{Y}^{-1}}.$

As in the case of the quantile breakdown frontier for the global effect, the
empirical bootstrap is valid and can be used to construct confidence
intervals.

###  4.3 Bounds derived from the QBF

To estimate the bounds derived the from the QBF for the global effect, we use
the sample counterpart of (8). Besides the target conclusion $g$ for the
global effect, we need to supply, a quantile level $\tau^{*}$ of interest. For
$\tau\in(\delta,1-\delta)$, the estimator of the derived bounds is

$\displaystyle\max\left\\{\hat{\tilde{F}}_{A}^{-1}(\tau-\delta),\hat{F}_{A}^{-1}(\tau-\delta\hat{\tilde{c}}_{\tau^{*},L})\right\\}-\hat{F}_{Y}^{-1}(\tau)\leq
G_{\tau,D_{\delta}}\leq\min\left\\{\hat{\tilde{F}}_{A}^{-1}(\tau),\hat{F}_{A}^{-1}(\tau+\delta\hat{\tilde{c}}_{\tau^{*},L})\right\\}-\hat{F}_{Y}^{-1}(\tau),$

where

$\displaystyle\hat{\tilde{c}}_{\tau^{*},L}=\max\left\\{\min\left\\{\hat{c}_{\tau^{*},L},1\right\\},0\right\\}.$
(11)

Here, $\hat{c}_{\tau^{*},L}$ is the estimator given in (9) evaluated
$\tau=\tau^{*}$.

The asymptotic distribution of the bounds can be found in Section C of the
Appendix. The main feature of the asymptotic distribution is that it is not
Gaussian due to being the composition of maps which are not Hadamard fully
differentiable, only directional. Hence, by Corollary 3.1 in Fang and Santos
(2019), the standard bootstrap will fail. This means that if we attempt to
construct confidence intervals in the usual way by resampling, we will not
obtain correct asymptotic coverage. An alternative is to use the numerical
delta method of Hong and Li (2018).

##  5 Empirical application: What do unions do?

There is an extensive literature that studies unions and inequality. A recent
contribution by Farber et al. (2020) contains a review of the literature. In
our empirical application, in particular, we look at how unions affect the
distribution of wages for _all_ workers. Unions can have a variety of effects
on the distribution of wages. As argued by Freeman (1980), unions can raise
the wages of unionized workers relative to non-unionized workers, possibly
through more bargaining power. So, if higher paid workers unionize, the
dispersion of wages can increase, but if lower paid workers unionize, the
dispersion of wages can decrease. Furthermore, within a given industry, the
union can reduce the dispersion of wages by standardizing the wages. This will
impact the distribution of wages more or less depending on the size of the
industry and the wages it pays.

A key difficulty in identifying the causal effect of unions on wages is that
selection into unions is non-random. Hence, any measurement of the union
premium–the difference in wages between similar union and nonunion
workers–will be biased for the causal effect. Indeed, this has been a long
standing concern of labor economists.131313Indeed, the opening words of Card
(1996) are:  _Despite a large and sophisticated literature there is still
substantial disagreement over the extent to which differences in the structure
of wages between union and nonunion workers represent an effect of trade
unions, rather than a consequence of the nonrandom selection of unionized
workers._ With respect to selection into unions, Card (1996) argues that
unionized workers with low observed skills, tend to have high unobserved
skills. The reverse happens with high skilled unionized workers: they tend to
have low unobservable skills. Due to this selection bias, it might be
impossible for a policy maker to devise a policy where the _newly_ unionized
workers are selected in a way such that they are drawn from the distribution
of the _already_ unionized workers.

Using the techniques developed in this paper, we are going to consider the
effect of both globally and marginally expanding union coverage. We will
explicitly allow for non-random selection into unions. This allows for the
_newly_ unionized and _already_ unionized workers to be drawn from different
distributions. We do not use any imputation method to impute the union premium
of the newly unionized workers.

Following Freeman (1980), Card (2001) and Card, Lemieux and Riddell (2004) we
consider a two sector economy. Each worker has a well-defined pair of
potential (log) wages: $Y_{i}(1)$ for the unionized sector and $Y_{i}(0)$ for
the nonunionized sector. Under Assumption 1, and for any policy $D_{\delta}$,
we have the following classification of individuals:

| $D_{\delta}=0$ | $D_{\delta}=1$
---|---|---
$D=0$ | nonunionized | newly unionized
$D=1$ | - | unionized

The relevant unobserved distribution is then $F_{Y(1)|D=0,D_{\delta}=1}$: the
union wages of the newly unionized workers. Following (4), we look at
departures of $F_{Y(1)|D=0,D_{\delta}=1}$ from

$\displaystyle\int_{\mathcal{X}}F_{Y|D=1,X=x}(y)dF_{X|D=0,D_{\delta}=1}(x),$

which is observed. This difference is what we refer to as the policy selection
bias.

Using the data in Firpo, Fortin and Lemieux (2009) we estimate the quantile
breakdown frontier for marginal and global effects of different type of
policies on the distribution of real log hourly wages. We use the 1983-1985
Outgoing Rotation Group (ORG) Supplement of the Current Population Survey. Our
sample consists of 266,956 observations on U.S. males. See Lemieux (2006) for
more details about the data. The covariates are: years of education, age,
marital status, dummy for race (nonwhite), years of experience, and union
status indicator.

The unionization rate in the dataset is $0.26$. Figure 2 shows the typical
hump-shaped pattern of the unionization rates by quantiles of the distribution
of wages. For lower quantiles, unionization rates are quite low. They peak in
the past the middle of the distribution and then drop at the higher quantiles.

Figure 2: _Unionization rates by quantiles of the distribution of wages._

###  5.1 Global effect and bounds

Consider a policy that increase in the unionization rate by $10\%$. It
consists of unionizing workers whose wages are below the $.10/(1-p)$-quantile
$\approx 0.14$-quantile of the wages of the nonunionized sector.141414This
guarantees that the unionization rate increases by roughly 10%. Indeed, the
mean of $D_{\delta}$ is now $0.36$. In the notation of this paper, we have
$D=1$ if a worker is unionized, $D_{\delta}=1$ if a worker is unionized under
the policy, $Y$ is (log) wage, and $\delta=0.1$. That is, $D_{\delta}$ is
given by

$\displaystyle D_{\delta}=\begin{cases}1&\quad\text{if }D=1\\\ 1&\quad\text{if
}D=0\text{ and }Y\leq F^{-1}_{Y|D=0}(0.14)\\\ 0&\quad\text{otherwise}\\\
\end{cases}$

Figure 3 shows the quantile breakdown frontiers for $g=0.1$ for a grid of
$\tau\in(0.15,0.85)$. Confidence intervals are obtained via 1,000 bootstrap
replications. Since the outcome variable is log wages, this means
approximately a $10\%$ increase in wages. First, we can see that for lower
quantiles, the conclusion will hold as long as $c$ is below the frontier. On
the other hand, for upper quantiles, the conclusion does not hold under point
identification. Thus there is no point in doing a sensitivity analysis for
upper quantiles.

Suppose the policy maker is interested in $\tau^{*}=.3$. In this case,
$c_{\tau^{*},L}=c_{.3,L}=0.29$, and $[0.27,0.30]$ is the $95\%$ confidence
interval. Importantly, $\tilde{F}_{A}^{-1}(\tau-\delta)<F_{A}^{-1}(\tau-\delta
c_{\tau^{*},L})$, meaning that the conclusion does not hold for any
$c\in[0,1]$ by part $(iv)$ of Lemma 1, and hence the quantile breakdown
frontier is sharp at $\tau=.3$. Using equation (8), we can find the bounds for
the global effect for all quantiles which are consistent with this amount of
policy selection bias. This is shown in Figure 4. Note that at $\tau=0.3$, the
lower bound is $.10$ by construction. At upper quantiles, the lower bound is
slightly negative.

Figure 3: _Quantile Breakdown Frontier for $G_{\tau,D_{\delta}}\geq 0.1$ with
$95\%$ pointwise confidence intervals._ Figure 4: _Bounds derived from the
QBF, and (dashed) point identified global effect._

##  6 Conclusion

This paper provides a way to perform a sensitivity analysis on the
unconditional quantiles effects of policies that manipulate the proportion of
treated individuals. To do so, we introduce the quantile breakdown frontier as
a tool to examine, across quantiles, whether a sensitivity analysis is
possible, and how much policy selection bias is compatible with a given
conclusion. Next, we use the information from the curve at a particular
quantile to provide bounds on the effect of a policy across quantiles. Our
empirical application looks at the effect of increasing the proportion of
unionized workers by unionizing lower earners. We find that an increase of
$10\%$ for the $30$th quantile is consistent under moderate values of
selection bias. Moreover, this implies that the effect on lower quantiles
would be even bigger, and that lower quantiles would not be hurt.

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## Appendices

###  A Notation for Estimation and Inference

For $F_{A}(y)$ given in Theorem 1, $\hat{F}_{A}(y)$ is its empirical
counterpart. It is given by

$\displaystyle\hat{F}_{A}(y)$
$\displaystyle:=\hat{p}\hat{F}_{Y|D=1}(y)+(1-\hat{p}-\delta)\hat{F}_{Y|D_{\delta}=0}(y)+\delta\sum_{x\in\mathcal{X}}\hat{F}_{Y|D=1,X=x}(y)\hat{p}_{x}$
(A.1)

where $p_{x}:=\Pr(X=x|D=0,D_{\delta}=1)$ and

$\displaystyle\hat{p}_{x}$
$\displaystyle=\frac{\sum_{i=1}^{n}\mathds{1}\left\\{X_{i}=x\right\\}(1-D_{i})(1-D_{i,\delta})}{\sum_{i=1}^{n}(1-D_{i})(1-D_{i,\delta})},$
$\displaystyle\hat{p}$ $\displaystyle=\frac{1}{n}\sum_{i=1}^{n}D_{i},$
$\displaystyle\hat{F}_{Y|D=1}(y)$
$\displaystyle=\frac{\sum_{i=1}^{n}\mathds{1}\left\\{Y_{i}\leq
y\right\\}D_{i}}{\sum_{i=1}^{n}D_{i}},$
$\displaystyle\hat{F}_{Y|D_{\delta}=0}(y)$
$\displaystyle=\frac{\sum_{i=1}^{n}\mathds{1}\left\\{Y_{i}\leq
y\right\\}(1-D_{i,\delta})}{\sum_{i=1}^{n}(1-D_{i,\delta})},$
$\displaystyle\hat{F}_{Y|D=1,X=x}(y)$
$\displaystyle=\frac{\sum_{i=1}^{n}\mathds{1}\left\\{Y_{i}\leq
y\right\\}D_{i}\mathds{1}\left\\{X_{i}=x\right\\}}{\sum_{i=1}^{n}D_{i}\mathds{1}\left\\{X_{i}=x\right\\}}.$

Moreover, we have

$\displaystyle\hat{\tilde{F}}_{A}(y)$
$\displaystyle:=\hat{p}\hat{F}_{Y|D=1}(y)+(1-\hat{p}-\delta)\hat{F}_{Y|D_{\delta}=0}(y).$
(A.2)

For the marginal effect, we have the following empirical cdf:

$\displaystyle\hat{F}_{Y|D=0}(y)=\frac{\sum_{i=1}^{n}\mathds{1}\\{Y_{i}\leq
y\\}(1-D_{i})}{\sum_{i=1}^{n}((1-D_{i}))}.$

and

$\displaystyle\sum_{x\in\mathcal{X}}\hat{F}_{Y|D=1,X=x}(y)\hat{p}_{x|\partial
D}=\sum_{x\in\mathcal{X}}\frac{\sum_{i=1}^{n}\mathds{1}\left\\{Y_{i}\leq
y\right\\}D_{i}\mathds{1}\left\\{X_{i}=x\right\\}}{\sum_{i=1}^{n}D_{i}\mathds{1}\left\\{X_{i}=x\right\\}}\hat{p}_{x|\partial
D}.$

###  B Proofs

###### Proof of Theorem 1.

Recall that, for a given $D_{\delta}$, the counterfactual distribution is

$\displaystyle F_{Y_{D_{\delta}}}(y)$
$\displaystyle=pF_{Y|D=1}(y)+(1-p-\delta)F_{Y|D_{\delta}=0}(y)+\delta
F_{Y(1)|D=0,D_{\delta}=1}(y).$

Since $F_{Y(1)|D=0,D_{\delta}=1}$ is a distribution function, we naturally
have $0\leq F_{Y(1)|D=0,D_{\delta}=1}(y)\leq 1$ for every $y\in\mathbb{R}$.
Moreover, by Assumption 3, we also have that

$\displaystyle\int_{\mathcal{X}}F_{Y|D=1,X=x}(y)dF_{X|D=0,D_{\delta}=1}(x)-c\leq
F_{Y(1)|D=0,D_{\delta}=1}(y)\leq\int_{\mathcal{X}}F_{Y|D=1,X=x}(y)dF_{X|D=0,D_{\delta}=1}(x)+c$

$y\in\mathbb{R}$. Thus combining both bounds, we obtain

$\displaystyle\max\left\\{0,\int_{\mathcal{X}}F_{Y|D=1,X=x}(y)dF_{X|D=0,D_{\delta}=1}(x)-c\right\\}$
$\displaystyle\leq F_{Y(1)|D=0,D_{\delta}=1}(y)$
$\displaystyle\leq\min\left\\{1,\int_{\mathcal{X}}F_{Y|D=1,X=x}(y)dF_{X|D=0,D_{\delta}=1}(x)+c\right\\}.$

for every $y\in\mathbb{R}$. To help with the notation, define

$\displaystyle\tilde{F}_{A}(y)$
$\displaystyle:=pF_{Y|D=1}(y)+(1-p-\delta)F_{Y|D_{\delta}=0}(y),$ (A.3)

and

$\displaystyle F_{A}(y)$
$\displaystyle:=\tilde{F}_{A}(y)+\delta\int_{\mathcal{X}}F_{Y|D=1,X=x}(y)dF_{X|D=0,D_{\delta}=1}(x).$
(A.4)

Note that, while $F_{A}(y)$ is a proper distribution function,
$\tilde{F}_{A}(y)$ is not. We refer to $F_{A}$ as the apparent distribution,
and to $\tilde{F}_{A}$ as the incomplete apparent distribution. By Assumption
2, both $F_{A}(y)$ and $\tilde{F}_{A}(y)$ are strictly increasing and
continuous on for all $y$ such that $0<F_{A}(y)<1$ and $0<\tilde{F}_{A}(y)<1$.
The bounds on $F_{Y(1)|D=0,D_{\delta}=1}(y)$, allow us to bound
$F_{Y_{D_{\delta}}}(y)$. The upper bound is

$\displaystyle F_{Y_{D_{\delta}}}(y)$ $\displaystyle\leq
pF_{Y|D=1}(y)+(1-p-\delta)F_{Y|D_{\delta}=0}(y)$
$\displaystyle+\delta\min\left\\{1,\int_{\mathcal{X}}F_{Y|D=1,X=x}(y)dF_{X|D=0,D_{\delta}=1}(x)+c\right\\}$
$\displaystyle=\min\left\\{\tilde{F}_{A}(y)+\delta,F_{A}(y)+\delta
c\right\\}.$

Similarly, the lower bound is

$\displaystyle F_{Y_{D_{\delta}}}(y)$ $\displaystyle\geq
pF_{Y|D=1}(y)+(1-p-\delta)F_{Y|D_{\delta}=0}(y)$
$\displaystyle+\delta\max\left\\{0,\int_{\mathcal{X}}F_{Y|D=1,X=x}(y)dF_{X|D=0,D_{\delta}=1}(x)-c\right\\}$
$\displaystyle=\max\left\\{\tilde{F}_{A}(y),F_{A}(y)-\delta c\right\\}.$

Let $F_{Y_{D_{\delta}}}^{-1}(\tau)$ be that $\tau$-quantile of
$Y_{D_{\delta}}$ which is unique by Assumption 2. Then

$\displaystyle\max\left\\{\tilde{F}_{A}(F_{Y_{D_{\delta}}}^{-1}(\tau)),F_{A}(F_{Y_{D_{\delta}}}^{-1}(\tau))-\delta
c\right\\}\leq\tau\leq\min\left\\{\tilde{F}_{A}(F_{Y_{D_{\delta}}}^{-1}(\tau))+\delta,F_{A}(F_{Y_{D_{\delta}}}^{-1}(\tau))+\delta
c\right\\}.$

For two strictly increasing increasing continuous functions $f$ and $g$, we
have that (this is best seen graphically):

$\displaystyle\left[\min\left\\{f,g\right\\}\right]^{-1}(\tau)=\max\left\\{f^{-1}(\tau),g^{-1}(\tau)\right\\}.$

and

$\displaystyle\left[\max\left\\{f,g\right\\}\right]^{-1}(\tau)=\min\left\\{f^{-1}(\tau),g^{-1}(\tau)\right\\}.$

Moreover, for $F_{A}$, define the inverse to be $F_{A}^{-1}(t)$, where
$F_{A}(F_{A}^{-1}(t))=t$ for $t\in(0,1)$, $F_{A}^{-1}(t)=-\infty$ if $t\leq
0$, and $F_{A}^{-1}(t)=\infty$ if $t\geq 1$. For the “incomplete” apparent
distribution $\tilde{F}_{A}$, we have that
$\tilde{F}_{A}:\mathbb{R}\mapsto(0,1-\delta)$. Thus, $\tilde{F}_{A}^{-1}(t)$
is defined as usual for $t\in(0,1-\delta)$. Whereas,
$\tilde{F}_{A}^{-1}(t)=-\infty$ if $t\leq 0$, and
$\tilde{F}_{A}^{-1}(t)=\infty$ if $t\geq 1-\delta$.

Thus, we can bound $F_{Y_{D_{\delta}}}^{-1}(\tau)$ by

$\displaystyle\theta_{L}:=\max\left\\{\tilde{F}_{A}^{-1}(\tau-\delta),F_{A}^{-1}(\tau-\delta
c)\right\\}\leq
F_{Y_{D_{\delta}}}^{-1}(\tau)\leq\min\left\\{\tilde{F}_{A}^{-1}(\tau),F_{A}^{-1}(\tau+\delta
c)\right\\}:=\theta_{H}.$

Since we restrict $\tau$ to $(\delta,1-\delta)$, for any $c\in[0,1)$, then
$\theta_{L}>-\infty$ and $\theta_{H}<\infty.$ To show that the bounds are
sharp we follow the approach of the proof of Lemma 2.1 of Kline and Santos
(2013). For any value that $F_{Y_{D_{\delta}}}^{-1}(\tau)$ may take in
$[\theta_{L},\theta_{H}]$, we need to show that there exists a function
$F^{*}_{\delta}:\mathbb{R}\mapsto(0,1)$ such that:

  1. 1.

$\tau=pF_{Y|D=1}(F_{Y_{D_{\delta}}}^{-1}(\tau))+(1-p-\delta)F_{Y|D_{\delta}=0}(F_{Y_{D_{\delta}}}^{-1}(\tau))+\delta
F^{*}_{\delta}(F_{Y_{D_{\delta}}}^{-1}(\tau))$;

  2. 2.

$\sup_{y\in\mathbb{R}}\left|F^{*}_{\delta}(y)-\int_{\mathcal{X}}F_{Y|D=1,X=x}(y)dF_{X|D=0,D_{\delta}=1}(x)\right|\leq
c$;

  3. 3.

$F^{*}_{\delta}(y)$ is continuous and strictly increasing in $\mathbb{R}$ for
all $y$ such that $0<F^{*}_{\delta}(y)<1.$

By condition 1., we can express
$F^{*}_{\delta}(F_{Y_{D_{\delta}}}^{-1}(\tau))$ as

$\displaystyle F^{*}_{\delta}(F_{Y_{D_{\delta}}}^{-1}(\tau))=\frac{\tau-
pF_{Y|D=1}(F_{Y_{D_{\delta}}}^{-1}(\tau))-(1-p-\delta)F_{Y|D_{\delta}=0}(F_{Y_{D_{\delta}}}^{-1}(\tau))}{\delta}$

To verify condition 2., we start by checking that
$\left|F^{*}_{\delta}(F_{Y_{D_{\delta}}}^{-1}(\tau))-\int_{\mathcal{X}}F_{Y|D=1,X=x}(F_{Y_{D_{\delta}}}^{-1}(\tau))dF_{X|D=0,D_{\delta}=1}(x)\right|\leq
c$. We write

$\displaystyle
F^{*}_{\delta}(F_{Y_{D_{\delta}}}^{-1}(\tau))-\int_{\mathcal{X}}F_{Y|D=1,X=x}(F_{Y_{D_{\delta}}}^{-1}(\tau))dF_{X|D=0,D_{\delta}=1}(x)$
$\displaystyle=\frac{\tau-
pF_{Y|D=1}(F_{Y_{D_{\delta}}}^{-1}(\tau))-(1-p-\delta)F_{Y|D_{\delta}=0}(F_{Y_{D_{\delta}}}^{-1}(\tau))}{\delta}$
$\displaystyle-\int_{\mathcal{X}}F_{Y|D=1,X=x}(F_{Y_{D_{\delta}}}^{-1}(\tau))dF_{X|D=0,D_{\delta}=1}(x)$
$\displaystyle=\frac{\tau-F_{A}(F_{Y_{D_{\delta}}}^{-1}(\tau))}{\delta}.$

Because $F_{A}$ is increasing, and
$F_{Y_{D_{\delta}}}^{-1}(\tau)\in[\theta_{L},\theta_{H}]$, we need to check
all the possible limiting cases.

  1. 1.

$F_{Y_{D_{\delta}}}^{-1}(\tau)=F_{A}^{-1}(\tau-\delta c)$. Then,

$\displaystyle\frac{\tau-F_{A}(F_{A}^{-1}(\tau-\delta c))}{\delta}=c.$

  2. 2.

$F_{Y_{D_{\delta}}}^{-1}(\tau)=\tilde{F}_{A}^{-1}(\tau-\delta)$. Then,

$\displaystyle\frac{\tau-
F_{A}(\tilde{F}_{A}^{-1}(\tau-\delta))}{\delta}\leq\frac{\tau-
F_{A}(F_{A}^{-1}(\tau-\delta c))}{\delta}=c,$

where the inequality follows from $\tilde{F}_{A}^{-1}(\tau-\delta)\geq
F_{A}^{-1}(\tau-\delta c)$.

  3. 3.

$F_{Y_{D_{\delta}}}^{-1}(\tau)=F_{A}^{-1}(\tau+\delta c)$. Then,

$\displaystyle\frac{\tau-F_{A}(F_{A}^{-1}(\tau+\delta c))}{\delta}=-c.$

  4. 4.

$F_{Y_{D_{\delta}}}^{-1}(\tau)=\tilde{F}_{A}^{-1}(\tau)$. Then,

$\displaystyle\frac{\tau-
F_{A}(\tilde{F}_{A}^{-1}(\tau))}{\delta}\geq\frac{\tau-
F_{A}(F_{A}^{-1}(\tau+\delta c))}{\delta}=-c,$

where the inequality follows from $\tilde{F}_{A}^{-1}(\tau)\leq
F_{A}^{-1}(\tau+\delta c)$.

Thus, we have that
$\left|F^{*}_{\delta}(F_{Y_{D_{\delta}}}^{-1}(\tau))-\int_{\mathcal{X}}F_{Y|D=1,X=x}(F_{Y_{D_{\delta}}}^{-1}(\tau))dF_{X|D=0,D_{\delta}=1}(x)\right|\leq
c$ whenever $F_{Y_{D_{\delta}}}^{-1}(\tau)\in[\theta_{L},\theta_{H}]$.

Now we will construct $F^{*}_{\delta}(y)$ such that conditions 2. and 3. also
hold. The starting point is the given value
$F^{*}_{\delta}(F_{Y_{D_{\delta}}}^{-1}(\tau))$. To alleviate notation, define

$\displaystyle\tilde{F}(y):=\int_{\mathcal{X}}F_{Y|D=1,X=x}(y)dF_{X|D=0,D_{\delta}=1}(x)$

Suppose first that

$\displaystyle
F^{*}_{\delta}(F_{Y_{D_{\delta}}}^{-1}(\tau))\leq\tilde{F}(F_{Y_{D_{\delta}}}^{-1}(\tau))$

Then, we define $F^{*}_{\delta}(y)$ as

$\displaystyle
F^{*}_{\delta}(y)=\begin{cases}\max\left\\{0,\tilde{F}(y)-\left[\tilde{F}(F_{Y_{D_{\delta}}}^{-1}(\tau))-F^{*}_{\delta}(F_{Y_{D_{\delta}}}^{-1}(\tau))\right]\right\\}\text{
for }y\leq F_{Y_{D_{\delta}}}^{-1}(\tau)\\\
\tilde{F}(y)-\omega(y-F_{Y_{D_{\delta}}}^{-1}(\tau))\left[\tilde{F}(F_{Y_{D_{\delta}}}^{-1}(\tau))-F^{*}_{\delta}(F_{Y_{D_{\delta}}}^{-1}(\tau))\right]\text{
for }y>F_{Y_{D_{\delta}}}\\\ \end{cases}$

where $\omega(y)$ is a continuous function that satisfies: $\omega(0)=1$,
$\lim_{y\to\pm\infty}\omega(y)=0$, and it is strictly decreasing for $y>0$,
and strictly increasing for $y<0$. We can take $\omega(y)=\exp(-y^{2})$. The
role of $\omega(y)$ is that of a weighting function. It brings
$F^{*}_{\delta}(y)$ closer to $\tilde{F}(y)$ as we move to the right of
$F_{Y_{D_{\delta}}}^{-1}(\tau)$. Since $\omega(y)\to 0$ as $y\to 0$,
eventually the gap disappears. For the values of $y$ to the left of
$F_{Y_{D_{\delta}}}^{-1}(\tau)$, we simply subtract the gap and use the $\max$
to keep the function non-negative. This function $F^{*}_{\delta}(y)$ satisfies
1., 2., and 3.

If, instead,

$\displaystyle
F^{*}_{\delta}(F_{Y_{D_{\delta}}}^{-1}(\tau))\geq\tilde{F}(F_{Y_{D_{\delta}}}^{-1}(\tau))$

then

$\displaystyle
F^{*}_{\delta}(y)=\begin{cases}\tilde{F}(y)-\omega(y-F_{Y_{D_{\delta}}}^{-1}(\tau))\left[\tilde{F}(F_{Y_{D_{\delta}}}^{-1}(\tau))-F^{*}_{\delta}(F_{Y_{D_{\delta}}}^{-1}(\tau))\right]\text{
for }y\leq F_{Y_{D_{\delta}}}\\\
\min\left\\{1,\tilde{F}(y)-\left[\tilde{F}(F_{Y_{D_{\delta}}}^{-1}(\tau))-F^{*}_{\delta}(F_{Y_{D_{\delta}}}^{-1}(\tau))\right]\right\\}\text{
for }y>F_{Y_{D_{\delta}}}^{-1}(\tau)\\\ \end{cases}$

Here, $F^{*}_{\delta}(y)$ lies above $\tilde{F}(y)$ and we use the weight
function to close the gap as $y\to-\infty.$ For the values of
$y>F_{Y_{D_{\delta}}}^{-1}(\tau)$ we simply add the gap (we are subtracting a
negative number) and use the $\min$ to avoid the function from being greater
than 1. This function $F^{*}_{\delta}(y)$ satisfies 1., 2., and 3.

Therefore, sharp bounds for the global effect
$G_{\tau,D_{\delta}}:=F_{Y_{D_{\delta}}}^{-1}(\tau)-F_{Y}^{-1}(\tau)$, are
given by

$\displaystyle\max\left\\{\tilde{F}_{A}^{-1}(\tau-\delta),F_{A}^{-1}(\tau-\delta
c)\right\\}-F_{Y}^{-1}(\tau)\leq
G_{\tau,D_{\delta}}\leq\min\left\\{\tilde{F}_{A}^{-1}(\tau),F_{A}^{-1}(\tau+\delta
c)\right\\}-F_{Y}^{-1}(\tau).$

∎

###### Proof of Theorem 2.

Recall that by (3), the counterfactual distribution of a given policy
$D_{\delta}$ that satisfies assumption 1 is

$\displaystyle F_{Y_{D_{\delta}}}(y)$
$\displaystyle=pF_{Y|D=1}(y)+(1-p-\delta)F_{Y|D_{\delta}=0}(y)+\delta
F_{Y(1)|D=0,D_{\delta}=1}(y)$

Then, we can write the quotient as

$\displaystyle\frac{F_{Y_{D_{\delta}}}(y)-F_{Y}(y)}{\delta}$
$\displaystyle=(1-p)\frac{F_{Y|D_{\delta}=0}(y)-F_{Y|D=0}(y)}{\delta}-F_{Y|D_{\delta}=0}(y)+F_{Y(1)|D=0,D_{\delta}=1}(y).$

By Assumption 4, we can take the limit $\delta\downarrow 0$ to obtain

$\displaystyle\lim_{\delta\downarrow
0}\frac{F_{Y_{D_{\delta}}}(y)-F_{Y}(y)}{\delta}$
$\displaystyle=(1-p)\frac{\partial
F_{Y|D_{\delta}=0}(y)}{\partial\delta}\bigg{|}_{\delta=0}-F_{Y|D=0}(y)+F_{Y(1)|\partial
D}(y)$ $\displaystyle=:\dot{F}_{Y,\mathcal{D}}(y).$

The map $y\mapsto\dot{F}_{Y,\mathcal{D}}(y)$ is continuous by Assumption 4.
Now we aim to strengthen the pointwise result to a uniform result. For that,
we write:

$\displaystyle\sup_{y\in\mathbb{R}}\left|\frac{F_{Y_{D_{\delta}}}(y)-F_{Y}(y)}{\delta}-\dot{F}_{Y,\mathcal{D}}(y)\right|$
$\displaystyle\leq(1-p)\sup_{y\in\mathbb{R}}\left|\frac{F_{Y|D_{\delta}=0}(y)-F_{Y|D=0}(y)}{\delta}-\frac{\partial
F_{Y|D_{\delta}=0}(y)}{\partial\delta}\bigg{|}_{\delta=0}\right|$
$\displaystyle+\sup_{y\in\mathbb{R}}\left|F_{Y|D_{\delta}=0}(y)-F_{Y|D=0}(y)\right|$
$\displaystyle+\sup_{y\in\mathbb{R}}\left|F_{Y(1)|D=0,D_{\delta}=1}(y)-F_{Y(1)|\partial
D}(y)\right|$

Assumption 4 ensures the uniform convergence of the first and third summands.
The second summand convergence uniformly because it is analogous to weak
convergence to a continuous distribution function (see Exercise 5.(b) in
Chapter 8 of Resnick (2005)).

Finally, let $\Gamma_{\tau}[F]$ be the $\tau$-quantile of $F$. The Hadamard
derivative at $F$ is (see Lemma 21.3 in van der Vaart (1998))

$\displaystyle\Gamma_{\tau,F}^{\prime}[h]=-\frac{h(F^{-1}(\tau))}{f(F^{-1}(\tau))}.$

for any $h\in\mathbb{D}[-\infty,\infty]$ continuous at $F^{-1}(\tau)$. We
write the marginal effect as

$\displaystyle M_{\tau,\mathcal{D}}$ $\displaystyle=\lim_{\delta\to
0}\frac{\Gamma_{\tau}\left[F_{Y_{D_{\delta}}}\right]-\Gamma_{\tau}[F_{Y}]}{\delta}=\lim_{\delta\to
0}\frac{\Gamma_{\tau}\left[F_{Y_{D_{0}}}+\delta\left(\frac{F_{Y_{D_{\delta}}}-F_{Y_{D_{0}}}}{\delta}\right)\right]-\Gamma_{\tau}[F_{Y}]}{\delta}$
$\displaystyle=\lim_{\delta\to
0}\frac{\Gamma_{\tau}\left[F_{Y}+\delta\left(\frac{F_{Y_{D_{\delta}}}-F_{Y}}{\delta}\right)\right]-\Gamma_{\tau}[F_{Y}]}{\delta}$
$\displaystyle=\Gamma_{\tau,F_{Y}}^{\prime}[\dot{F}_{Y,\mathcal{D}}]=\frac{\dot{F}_{Y,\mathcal{D}}(F_{Y}^{-1}(\tau))}{f_{Y}(F_{Y}^{-1}(\tau))}.$

The third equality follows from $F_{Y_{D_{0}}}=F_{Y}$. This is because by (3),
and the fact we have assumed that $\lim_{\delta\to
0}F_{Y|D_{\delta}=0}=F_{Y|D=0}$ and that $\lim_{\delta\to
0}F_{Y(1)|D=0,D_{\delta}=1}$ is well defined. The fourth equality follows from

$\displaystyle\lim_{\delta\downarrow
0}\sup_{y\in\mathcal{Y}}\left|\frac{F_{Y_{D_{\delta}}}(y)-F_{Y}(y)}{\delta}-\dot{F}_{Y,\mathcal{D}}(y)\right|=0,$

which is required by Lemma 21.3 in van der Vaart (1998).

∎

###### Details of 3.

Let $D=\mathds{1}\left\\{V\geq 0\right\\}$ for $V$ continuously distributed
around 0, and, for a sequence $\delta\downarrow 0$, let
$D_{\delta}=\mathds{1}\left\\{V\geq F_{V}^{-1}(F_{V}(0)-\delta)\right\\}$. It
follows that $p:=\Pr(D=1)=1-F_{V}(0)$ and that $\Pr(D_{\delta}=1)=p+\delta$.
Suppose that $\lim_{\delta\to 0}\Pr(Y\leq y|V\leq
F_{V}^{-1}(F_{V}(0)-\delta))=\Pr(Y\leq y|V\leq 0)$. Then

$\displaystyle\lim_{\delta\to 0}F_{Y|D_{\delta}=0}(y)$
$\displaystyle=\lim_{\delta\to 0}\Pr(Y\leq y|V\leq
F_{V}^{-1}(F_{V}(0)-\delta))$ $\displaystyle=\Pr(Y\leq y|V\leq 0)$
$\displaystyle=F_{Y|D=0}(y).$

Now, we have

$\displaystyle F_{Y|D_{\delta}=0}(y)-F_{Y|D=0}(y)$
$\displaystyle=\frac{\Pr(Y\leq y,V\leq
F_{V}^{-1}(F_{V}(0)-\delta))}{p+\delta}-\frac{\Pr(Y\leq y,V\leq 0)}{p}$
$\displaystyle=$ $\displaystyle\frac{p\Pr(Y\leq y,V\leq
F_{V}^{-1}(F_{V}(0)-\delta))-(p+\delta)\Pr(Y\leq y,V\leq 0)}{(p+\delta)p}$
$\displaystyle=\frac{1}{p+\delta}\left[\Pr(Y\leq y,V\leq
F_{V}^{-1}(F_{V}(0)-\delta))-\Pr(Y\leq y,V\leq 0)\right]$
$\displaystyle-\frac{\delta}{p(p+\delta)}\Pr(Y\leq y,V\leq 0)$

Consider the first term:

$\displaystyle\frac{\Pr(Y\leq y,V\leq F_{V}^{-1}(F_{V}(0)-\delta))-\Pr(Y\leq
y,V\leq 0)}{\delta}$
$\displaystyle=\frac{F_{V}^{-1}(F_{V}(0)-\delta)}{\delta}$
$\displaystyle\times\frac{\Pr(Y\leq y,V\leq
F_{V}^{-1}(F_{V}(0)-\delta))-\Pr(Y\leq y,V\leq
0)}{F_{V}^{-1}(F_{V}(0)-\delta)}$

Now, provided that $f_{V}(0)\neq 0$,

$\displaystyle\lim_{\delta\to
0}\frac{F_{V}^{-1}(F_{V}(0)-\delta)}{\delta}=-\frac{1}{f_{V}(0)}.$

For the other term we have

$\displaystyle\frac{\Pr(Y\leq y,V\leq F_{V}^{-1}(F_{V}(0)-\delta))-\Pr(Y\leq
y,V\leq 0)}{F_{V}^{-1}(F_{V}(0)-\delta)}$
$\displaystyle=\frac{\int^{F_{V}^{-1}(F_{V}(0)-\delta)}_{0}\int_{-\infty}^{y}f_{Y,V}(u,v)dudv}{F_{V}^{-1}(F_{V}(0)-\delta)}$
$\displaystyle\to\int_{-\infty}^{y}f_{Y,V}(u,0)du$

Combining all together, we obtain

$\displaystyle\lim_{\delta\to
0}\frac{F_{Y|D_{\delta}=0}(y)-F_{Y|D=0}(y)}{\delta}=-\left(\frac{1}{f_{V}(0)}+\frac{1}{p}\right)F_{Y|D=0}(y)$

Thus we can take

$\displaystyle\frac{\partial
F_{Y|D_{\delta}=0}(y)}{\partial\delta}\bigg{|}_{\delta=0}=-\left(\frac{1}{f_{V}(0)}-\frac{1}{p}\right)F_{Y|D=0}(y)$

which will be continuous as long as $F_{Y|D=0}(y)$ is continuous (which we
assumed). Now, consider

$\displaystyle F_{Y(1)|D=0,D_{\delta}=1}(y)$ $\displaystyle=\Pr(Y(1)\leq
y|V\leq 0,V\geq F_{V}^{-1}(F_{V}(0)-\delta))$
$\displaystyle=\frac{\Pr(Y(1)\leq y,V\leq 0,V\geq
F_{V}^{-1}(F_{V}(0)-\delta))}{\Pr(V\leq 0,V\geq F_{V}^{-1}(F_{V}(0)-\delta))}$
$\displaystyle=\frac{\int_{F_{V}^{-1}(F_{V}(0)-\delta)}^{0}\int_{-\infty}^{0}f_{Y(1),V}(u,v)dudv}{\delta}$
$\displaystyle=\frac{F_{V}^{-1}(F_{V}(0)-\delta)}{\delta}\frac{\int_{F_{V}^{-1}(F_{V}(0)-\delta)}^{0}\int_{-\infty}^{0}f_{Y(1),V}(u,v)dudv}{F_{V}^{-1}(F_{V}(0)-\delta)}.$

Under some mild regularity conditions:

$\displaystyle\lim_{\delta\to 0}F_{Y(1)|D=0,D_{\delta}=1}(y)=F_{Y(1)|V=0}(y)$

Thus, we can take

$\displaystyle F_{Y(1)|\partial D}(y)=F_{Y(1)|V=0}(y).$

∎

###### Proof of Theorem 3.

To bound the marginal effect, which we assume that exists, we aim to take the
limit when $\delta\downarrow 0$ of the rescaled bounds of the global effect.
That is, we aim to bound the marginal effect by the limiting versions of

$\displaystyle\frac{\max\left\\{\tilde{F}_{A}^{-1}(\tau-\delta),F_{A}^{-1}(\tau-\delta
c)\right\\}-F_{Y}^{-1}(\tau)}{\delta},$

and

$\displaystyle\frac{\min\left\\{\tilde{F}_{A}^{-1}(\tau),F_{A}^{-1}(\tau+\delta
c)\right\\}-F_{Y}^{-1}(\tau)}{\delta}.$

We would like to show that, when $\delta\downarrow 0$, we have

$\displaystyle\lim_{\delta\downarrow
0}\frac{\max\left\\{\tilde{F}_{A}^{-1}(\tau-\delta),F_{A}^{-1}(\tau-\delta
c)\right\\}-F_{Y}^{-1}(\tau)}{\delta}\leq
M_{\tau,\mathcal{D}}\leq\lim_{\delta\downarrow
0}\frac{\min\left\\{\tilde{F}_{A}^{-1}(\tau),F_{A}^{-1}(\tau+\delta
c)\right\\}-F_{Y}^{-1}(\tau)}{\delta}$

The reason we take the limit from the right, _i.e._ $\delta\downarrow 0$, is
because twofold the ordinary limit might fail to exist. This is the reason why
we assume that the sequence of policies consists of $\delta>0$ moving towards
0. This is part of Assumption 1. For $\delta>0$, we can write

$\displaystyle\frac{\max\left\\{\tilde{F}_{A}^{-1}(\tau-\delta),F_{A}^{-1}(\tau-\delta
c)\right\\}-F_{Y}^{-1}(\tau)}{\delta}$
$\displaystyle=\frac{\max\left\\{\tilde{F}_{A}^{-1}(\tau-\delta)-F_{Y}^{-1}(\tau),F_{A}^{-1}(\tau-\delta
c)-F_{Y}^{-1}(\tau)\right\\}}{\delta}$
$\displaystyle=\max\left\\{\frac{\tilde{F}_{A}^{-1}(\tau-\delta)-F_{Y}^{-1}(\tau)}{\delta},\frac{F_{A}^{-1}(\tau-\delta
c)-F_{Y}^{-1}(\tau)}{\delta}\right\\}.$

The last step, bringing $1/\delta$ into the $\max$, is only valid because
$\delta>0$. Similarly, for $\delta>0$, we have

$\displaystyle\frac{\min\left\\{\tilde{F}_{A}^{-1}(\tau),F_{A}^{-1}(\tau+\delta
c)\right\\}-F_{Y}^{-1}(\tau)}{\delta}=\min\left\\{\frac{\tilde{F}_{A}^{-1}(\tau)-F_{Y}^{-1}(\tau)}{\delta},\frac{F_{A}^{-1}(\tau+\delta
c)-F_{Y}^{-1}(\tau)}{\delta}\right\\}$

Now, since the $\max$ and $\min$ are continuous, to pass the limit as
$\delta\downarrow 0$ we need to show that the maps

$\displaystyle\delta\mapsto\tilde{F}_{A}^{-1}(\tau-\delta),$
$\displaystyle\delta\mapsto F_{A}^{-1}(\tau-\delta c),$
$\displaystyle\delta\mapsto\tilde{F}_{A}^{-1}(\tau),$
$\displaystyle\delta\mapsto F_{A}^{-1}(\tau+\delta c),$

are differentiable at $\delta=0$. We need to show differentiability because it
follows from equations (A.3) and (A.4) that all these maps evaluated at
$\delta=0$ are equal to $F_{Y}^{-1}(\tau)$. Provided they are differentiable
(something we show below), we obtain:

$\displaystyle\max\left\\{\frac{\tilde{F}_{A}^{-1}(\tau-\delta)}{\partial\delta}\bigg{|}_{\delta=0},\frac{\partial
F_{A}^{-1}(\tau-\delta c)}{\partial\delta}\bigg{|}_{\delta=0}\right\\}\leq
M_{\tau,\mathcal{D}}\leq\min\left\\{\frac{\partial\tilde{F}_{A}^{-1}(\tau)}{\partial\delta}\bigg{|}_{\delta=0},\frac{\partial
F_{A}^{-1}(\tau+\delta c)}{\partial\delta}\bigg{|}_{\delta=0}\right\\}$

We start with the map $\delta\mapsto F_{A}^{-1}(\tau-\delta c)$. By equations
(A.3) and (A.4), when $\delta=0$ , $F_{A}=F_{Y}$. So, we want to find the
limit as $\delta\downarrow 0$ of

$\frac{F^{-1}_{A}(\tau-\delta c)-F_{Y}^{-1}(\tau)}{\delta}.$ (A.5)

We note that $\delta$ plays a triple role in the previous expression. Using
(A.4), we write $F_{A}(y)$ as

$\displaystyle
F_{A}(y;\delta_{1},\delta_{2})=pF_{Y|D=1}(y)+(1-p-\delta_{1})F_{Y|D_{\delta_{2}}=0}(y)+\delta_{1}\int_{\mathcal{X}}F_{Y|D=1,X=x}(y)dF_{X|D=0,D_{\delta_{2}}=1}(x)$
(A.6)

and we define

$\displaystyle
g(\delta_{1},\delta_{2},\delta_{3})=F^{-1}_{A}(\tau-\delta_{3}c;\delta_{1},\delta_{2})$

The map $\delta_{1}\mapsto g(\delta_{1},\delta_{2},\delta_{3})$ for a fixed
$\delta_{2}$ and $\delta_{3}$ is the composition151515For
$[a,b]\subset[-\infty,\infty]$, $\mathbb{D}[a,b]$ is the set of all real-
valued cadlag functions: right continuous with left limits everywhere in
$[a,b]$. $\mathbb{D}[a,b]$ is equipped with the uniform norm
$\|\cdot\|_{\infty}$.

$\displaystyle\delta_{1}\in\mathbb{R}\overset{h}{\mapsto}F_{A}(\cdot;\delta_{1},\delta_{2})\in\mathbb{D}[-\infty,\infty]\overset{\Gamma}{\mapsto}F^{-1}_{A}(\tau-\delta_{3}c;\delta_{1},\delta_{2})\in\mathbb{R}.$

The first map $h:\delta_{1}\in\mathbb{R}\mapsto
F_{A}(\cdot;\delta_{1},\delta_{2})\in\mathbb{D}[-\infty,\infty]$ has Hadamard
derivative given by

$\displaystyle\int_{\mathcal{X}}F_{Y|D=1,X=x}(\cdot)dF_{X|D=0,D_{\delta_{2}}=1}(x)-F_{Y|D_{\delta_{2}}=0}(\cdot).$

The second map has Hadamard derivative given by (See Lemma 21.3 in van der
Vaart (1998))

$\displaystyle\Gamma^{\prime}_{F_{A}(\cdot;\delta_{1},\delta_{2})}[G]=-\frac{G(F^{-1}_{A}(\tau-\delta_{3}c;\delta_{1},\delta_{2}))}{f_{A}(F^{-1}_{A}(\tau-\delta_{3}c;\delta_{1},\delta_{2});\delta_{1},\delta_{2})}.$

for $G\in\mathbb{D}[-\infty,\infty]$ continuous at
$F^{-1}_{A}(\tau-\delta_{3}c;\delta_{1},\delta_{2})$. Then, the derivative of
the composite map $\delta_{1}\mapsto\Gamma\circ h(\delta_{1})$ is
$\Gamma^{\prime}_{F_{A}(\cdot;\delta_{1},\delta_{2})}[h^{\prime}(\delta_{1})]$,
which is at $\delta_{2}=\delta_{3}=0$

$\displaystyle\frac{\partial
F^{-1}_{A}(\tau;\delta_{1},0)}{\partial\delta_{1}}=-\frac{\int_{\mathcal{X}}F_{Y|D=1,X=x}(F^{-1}_{A}(\tau;\delta_{1},0))dF_{X|\partial
D}(x)-F_{Y|D=0}(F^{-1}_{A}(\tau;\delta_{1},0))}{f_{A}(F^{-1}_{A}(\tau;\delta_{1},0);\delta_{1},0)}.$

which is continuous at $\delta_{1}=0$.

The derivative of the second map $\delta_{2}\mapsto
g(\delta_{1},\delta_{2},\delta_{3})$, for a fixed $\delta_{1}=\delta_{3}=0$ is
exactly 0 because as (A.4) shows, $\delta_{1}$ multiplies the expression

$\displaystyle-
F_{Y|D_{\delta_{2}}}(y)+\int_{\mathcal{X}}F_{Y|D=1,X=x}(y)dF_{X|D=0,D_{\delta_{2}}=1}(x).$

All of the terms in this expression are differentiable at $\delta_{2}=0$ by
Assumption 5.

The derivative of the third map $\delta_{3}\mapsto
g(\delta_{1},\delta_{2},\delta_{3})$, for a fixed $\delta_{1}=\delta_{2}=0$,
can be obtained via the identity

$\displaystyle
F_{A}\left(F^{-1}_{A}(\tau-\delta_{3}c;\delta_{1},\delta_{2})\delta_{1},\delta_{2}\right)=\tau-\delta_{3}c.$

Differentiating through with respect to $\delta_{3}$, we obtain at
$\delta_{1}=\delta_{2}=0$

$\displaystyle\frac{\partial
F^{-1}_{A}(\tau-\delta_{3}c;0,0)}{\partial\delta_{3}}=-\frac{c}{f_{Y}(F_{Y}^{-1}(\tau-\delta_{3}c))}.$

which is continuous with respect to $\delta_{3}$.

Therefore, all the partial derivatives of the map
$(\delta_{1},\delta_{2},\delta_{3})\mapsto
g(\delta_{1},\delta_{2},\delta_{3})$ exist and are continuous, hence the limit
in (A.5) exists and is equal to

$\displaystyle\lim_{\delta\to 0}\frac{F^{-1}_{A}(\tau-\delta
c)-F_{Y}^{-1}(\tau)}{\delta}$ $\displaystyle=\frac{\partial
g(\delta_{1},0,0)}{\partial\delta_{1}}\bigg{|}_{\delta_{1}=0}+\frac{\partial
g(0,\delta_{2},0)}{\partial\delta_{2}}\bigg{|}_{\delta_{2}=0}+\frac{\partial
g(0,0,\delta_{3})}{\partial\delta_{3}}\bigg{|}_{\delta_{3}=0}$
$\displaystyle=-\frac{\int_{\mathcal{X}}F_{Y|D=1,X=x}(F^{-1}_{Y}(\tau))dF_{X|\partial
D}(x)-F_{Y|D=0}(F^{-1}_{Y}(\tau))}{f_{Y}(F^{-1}_{Y}(\tau))}-\frac{c}{f_{Y}(F_{Y}^{-1}(\tau))}.$

Analogous arguments can be used to obtain that

$\displaystyle\lim_{\delta\to 0}\frac{F^{-1}_{A}(\tau+\delta
c)-F_{Y}^{-1}(\tau)}{\delta}$
$\displaystyle=-\frac{\int_{\mathcal{X}}F_{Y|D=1,X=x}(F^{-1}_{Y}(\tau))dF_{X|\partial
D}(x)-F_{Y|D=0}(F^{-1}_{Y}(\tau))}{f_{Y}(F^{-1}_{Y}(\tau))}+\frac{c}{f_{Y}(F_{Y}^{-1}(\tau))}.$

Now we turn our attention to the incomplete apparent distribution
$\tilde{F}_{A}$ and the map $\delta\mapsto\tilde{F}^{-1}_{A}(\tau-\delta)$. By
equation (A.3), the incomplete apparent distribution is given by

$\displaystyle\tilde{F}_{A}(y;\delta_{1},\delta_{2})$
$\displaystyle:=pF_{Y|D=1}(y)+(1-p-\delta_{1})F_{Y|D_{\delta_{2}}=0}(y),$

where, again, we index by $\delta_{1}$ and $\delta_{2}$ to emphasize the dual
role played by $\delta.$ As before, we define

$\displaystyle
g(\delta_{1},\delta_{2},\delta_{3})=\tilde{F}^{-1}_{A}(\tau-\delta_{3};\delta_{1},\delta_{2}).$

The map $\delta_{1}\mapsto g(\delta_{1},\delta_{2},\delta_{3})$ for a fixed
$\delta_{2}$ and $\delta_{3}$ is the composition

$\displaystyle\delta_{1}\in\mathbb{R}\overset{h}{\mapsto}\tilde{F}_{A}(\cdot;\delta_{1},\delta_{2})\in\mathbb{D}[-\infty,\infty]\overset{\Gamma}{\mapsto}\tilde{F}^{-1}_{A}(\tau-\delta_{3};\delta_{1},\delta_{2})\in\mathbb{R}.$

The first map
$h:\delta_{1}\in\mathbb{R}\mapsto\tilde{F}_{A}(\cdot;\delta_{1},\delta_{2})\in\mathbb{D}[-\infty,\infty]$
has Hadamard derivative given by

$\displaystyle-F_{Y|D_{\delta_{2}}=0}(\cdot).$

The second map has Hadamard derivative given by

$\displaystyle\Gamma^{\prime}_{\tilde{F}_{A}(\cdot;\delta_{1},\delta_{2})}[G]=-\frac{G(\tilde{F}^{-1}_{A}(\tau-\delta_{3}c;\delta_{1},\delta_{2}))}{\tilde{f}_{A}(\tilde{F}^{-1}_{A}(\tau-\delta_{3};\delta_{1},\delta_{2});\delta_{1},\delta_{2})}.$

Then, the derivative of $\delta_{1}\mapsto\Gamma\circ h(\delta_{1})$ is
$\Gamma^{\prime}_{\tilde{F}_{A}(\cdot;\delta_{1},\delta_{2})}[h^{\prime}(\delta_{1})]$,
which is at $\delta_{2}=\delta_{3}=0$

$\displaystyle\frac{\partial\tilde{F}^{-1}_{A}(\tau;\delta_{1},0)}{\partial\delta_{1}}=\frac{F_{Y|D=0}(\tilde{F}^{-1}_{A}(\tau;\delta_{1},0))}{\tilde{f}_{A}(\tilde{F}^{-1}_{A}(\tau;\delta_{1},0);\delta_{1},0)}$

which is continuous at $\delta_{1}=0$.

The derivative of the second map $\delta_{2}\mapsto
g(\delta_{1},\delta_{2},\delta_{3})$, for a fixed $\delta_{1}=\delta_{3}=0$ is
exactly 0 because, as in the case of the (complete) apparent distribution, as
(A.3) shows, $\delta_{1}$ multiplies the expression

$\displaystyle F_{Y|D_{\delta_{2}}=0}(y)$

which is differentiable at $\delta_{2}=0$ by Assumption 5.

The derivative of the third map $\delta_{3}\mapsto
g(\delta_{1},\delta_{2},\delta_{3})$, for a fixed $\delta_{1}=\delta_{2}=0$,
can be obtained via the identity

$\displaystyle\tilde{F}_{A}\left(\tilde{F}^{-1}_{A}(\tau-\delta_{3};\delta_{1},\delta_{2})\delta_{1},\delta_{2}\right)=\tau-\delta_{3}.$

Differentiating through with respect to $\delta_{3}$, we obtain at
$\delta_{1}=\delta_{2}=0$

$\displaystyle\frac{\partial\tilde{F}^{-1}_{A}(\tau-\delta_{3};0,0)}{\partial\delta_{3}}=-\frac{1}{f_{Y}(F_{Y}^{-1}(\tau-\delta_{3}))}.$

which is continuous with respect to $\delta_{3}$. Therefore, all the partial
derivatives of the map $(\delta_{1},\delta_{2},\delta_{3})\mapsto
g(\delta_{1},\delta_{2},\delta_{3})$ exist and are continuous, we can take the
limit

$\displaystyle\lim_{\delta\to
0}\frac{\tilde{F}^{-1}_{A}(\tau-\delta)-F^{-1}_{Y}(\tau)}{\delta}$
$\displaystyle=\frac{\partial
g(\delta_{1},0,0)}{\partial\delta_{1}}\bigg{|}_{\delta_{1}=0}+\frac{\partial
g(0,\delta_{2},0)}{\partial\delta_{2}}\bigg{|}_{\delta_{2}=0}+\frac{\partial
g(0,0,\delta_{3})}{\partial\delta_{3}}\bigg{|}_{\delta_{3}=0}$
$\displaystyle=\frac{F_{Y|D=0}(F^{-1}_{Y}(\tau))-1}{f_{Y}(F^{-1}_{Y}(\tau))}$

because $F_{Y|D_{0}=0}=F_{Y|D=0}$ by Assumption 4.

For the remaining map, $\delta\mapsto\tilde{F}^{-1}_{A}(\tau)$, similar
arguments show that

$\displaystyle\lim_{\delta\to
0}\frac{\tilde{F}^{-1}_{A}(\tau)-F^{-1}_{Y}(\tau)}{\delta}$
$\displaystyle=\frac{F_{Y|D=0}(F^{-1}_{Y}(\tau))}{f_{Y}(F^{-1}_{Y}(\tau))}.$

Therefore, we have that

$\displaystyle\theta_{L,M}\leq M_{\tau,\mathcal{D}}\leq\theta_{U,M}$

where

$\displaystyle\theta_{L,M}:=\max\left\\{\frac{F_{Y|D=0}(F^{-1}_{Y}(\tau))-1}{f_{Y}(F^{-1}_{Y}(\tau))},\frac{F_{Y|D=0}(F^{-1}_{Y}(\tau))-\int_{\mathcal{X}}F_{Y|D=1,X=x}(F^{-1}_{Y}(\tau))dF_{X|\partial
D}(x)-c}{f_{Y}(F^{-1}_{Y}(\tau))}\right\\},$ (A.7)

and

$\displaystyle\theta_{U,M}:=\min\left\\{\frac{F_{Y|D=0}(F^{-1}_{Y}(\tau))}{f_{Y}(F^{-1}_{Y}(\tau))},\frac{F_{Y|D=0}(F^{-1}_{Y}(\tau))-\int_{\mathcal{X}}F_{Y|D=1,X=x}(F^{-1}_{Y}(\tau))dF_{X|\partial
D}(x)+c}{f_{Y}(F^{-1}_{Y}(\tau))}\right\\}.$ (A.8)

Now, the question is whether these bounds are sharp. As in the proof of
Theorem 1, for any $\theta\in[\theta_{L,M},\theta_{U,M}]$, we need to find a
sequence of CDFs $F_{\delta}^{*}(y)$ such that for a counterfactual
distribution constructed as

$\displaystyle
F_{Y_{\delta}}^{*}(y)=pF_{Y|D=1}(y)+(1-p-\delta)F_{Y|D_{\delta}=0}(y)+\delta
F_{\delta}^{*}(y)$

we have that

$\displaystyle\lim_{\delta\downarrow
0}\frac{F_{Y_{\delta}}^{*-1}(\tau)-F^{-1}(\tau)}{\delta}=\theta$

For a given $\theta$, we can find (more than) a sequence of global effects
that converge to $\theta$. For each $\delta$, these global effects are within
the bounds provided in Theorem 1. For each $\delta$ these bounds are sharp. So
that for each $\delta$ we can find $F_{\delta}^{*}(y)$ that delivers that
particular value of the global effect. Thus, that particular sequence will
deliver a marginal effect of $\theta$. Hence, the bounds in (A.7) and (A.8)
are sharp.

∎

###### Proof of Lemma 1.

The Lemma has four statements. We prove each one at a time.

  1. $(i)$

Continuity of the quantile breakdown frontier is ensured by the continuity of
each of its components: $F_{Y}^{-1}(\tau)$ is continuous by Assumption 2.1;
continuity of $F_{A}$ is assured by Assumptions 2.1, and 2.2, and an
application of the Dominated Convergence Theorem; and continuity of the
collection of conclusions of $\tau\mapsto g_{\tau,L}$ is assumed.

  2. $(ii)$

We have $c_{\tau,L}<0$ when
$\tau<F_{A}\left(F_{Y}^{-1}(\tau)+g_{\tau,L}\right)$, which in turn implies
that $F_{A}^{-1}(\tau)-F_{Y}^{-1}(\tau)<g_{\tau,L}$. By footnote 5, when
$c=0$, the lower and upper bounds collapse to
$F_{A}^{-1}(\tau)-F_{Y}^{-1}(\tau)$, but
$F_{A}^{-1}(\tau)-F_{Y}^{-1}(\tau)<g_{\tau,L}.$ Hence, the target conclusion
does not hold under point identification.

  3. $(iii)$

To show that $c_{\tau,L}\geq 0$, we note that if the conclusion holds under
point identification, by Theorem 1 we have by setting $c=0$ that

$\displaystyle
g_{\tau,L}\leq\max\left\\{\tilde{F}_{A}^{-1}(\tau-\delta),F_{A}^{-1}(\tau)\right\\}-F_{Y}^{-1}(\tau).$

By footnote 5, $\tilde{F}_{A}^{-1}(\tau-\delta)\leq F_{A}^{-1}(\tau)$, so that

$\displaystyle g_{\tau,L}\leq F_{A}^{-1}(\tau)-F_{Y}^{-1}(\tau).$

which implies that $F_{A}(F_{Y}^{-1}(\tau)+g_{\tau,L})\leq\tau$. Hence,
$c_{\tau,L}\geq 0$. To show that for $c\leq c_{\tau,L}$, the conclusion holds,
we invoke Theorem 1, and show that the lower bound of the global effect is
decreasing in $c$, and when evaluated at $c=c_{\tau,L}$ is greater than
$g_{\tau,L}$. By footnote 5, $\tilde{F}_{A}^{-1}(\tau-\delta)\leq
F_{A}^{-1}(\tau)$, and $\delta$ and $c$ are positive, then
$c\mapsto\max\left\\{\tilde{F}_{A}^{-1}(\tau-\delta),F_{A}^{-1}(\tau-\delta
c)\right\\}$ is weakly decreasing. Now, if $c_{\tau,L}=\frac{\tau-
F_{A}\left(F_{Y}^{-1}(\tau)+g_{\tau,L}\right)}{\delta}$, then

$\displaystyle\max\left\\{\tilde{F}_{A}^{-1}(\tau-\delta),F_{A}^{-1}(\tau-\delta
c_{\tau,L})\right\\}-F_{Y}^{-1}(\tau)$
$\displaystyle=\max\left\\{\tilde{F}_{A}^{-1}(\tau),F_{Y}^{-1}(\tau)+g_{\tau,L}\right\\}-F_{Y}^{-1}(\tau)$
$\displaystyle=\max\left\\{\tilde{F}_{A}^{-1}(\tau)-F_{Y}^{-1}(\tau),g_{\tau,L}\right\\}$
$\displaystyle\geq g_{\tau,L}.$

This shows that if $c_{\tau,L}=\frac{\tau-
F_{A}\left(F_{Y}^{-1}(\tau)+g_{\tau,L}\right)}{\delta}<1$, the conclusion
holds. And if $c_{\tau,L}=1$, then it also works because the bounds are
decreasing in $c$, and in this case we have $c_{\tau,L}=1<\frac{\tau-
F_{A}\left(F_{Y}^{-1}(\tau)+g_{\tau,U}\right)}{\delta}$.

  4. $(iv)$

Let $c^{*}$ be implicitly defined as:
$\tilde{F}_{A}^{-1}(\tau-\delta)=F_{A}^{-1}(\tau-\delta c^{*})$. Then, for
$c\geq c^{*}$ (see footnote 5) we have $\tilde{F}_{A}^{-1}(\tau-\delta)\geq
F_{A}^{-1}(\tau-\delta c)$, and $c$ plays no role in the bound. So, if
$c_{\tau,L}\geq c^{*}$, then the conclusion holds irrespective of the value of
$c$. This means that $\tilde{F}_{A}^{-1}(\tau-\delta)\geq
F_{A}^{-1}(\tau-\delta c_{\tau,L})$.

∎

###### Proof of Lemma 2.

The Lemma has four statements. We prove each one at a time.

  1. $(i)$

The continuity of $F^{-1}_{Y}(\tau)$, $F_{Y|D=0}(y)$, and $F_{Y|D=1,X=x}(y)$
guaranteed by Assumption 2.1, plus a straightforward application of the
Dominated Convergence Theorem ensure that $\tau\mapsto c_{\tau,U}^{M}$ is
continuous.

  2. $(ii)$

$c_{\tau,U}^{M}$ is negative whenever
$\int_{\mathcal{X}}F_{Y|D=1,X=x}(F^{-1}_{Y}(\tau))dF_{X|\partial
D}(x)<F_{Y|D=0}(F^{-1}_{Y}(\tau))$. By Theorem 3, when $c=0$, the upper and
lower bounds collapse to

$\displaystyle\frac{F_{Y|D=0}(F^{-1}_{Y}(\tau))-\int_{\mathcal{X}}F_{Y|D=1,X=x}(F^{-1}_{Y}(\tau))dF_{X|\partial
D}(x)}{f_{Y}(F^{-1}_{Y}(\tau))}.$

If $c_{\tau,U}^{M}<0$, then

$\displaystyle\frac{F_{Y|D=0}(F^{-1}_{Y}(\tau))-\int_{\mathcal{X}}F_{Y|D=1,X=x}(F^{-1}_{Y}(\tau))dF_{X|\partial
D}(x)}{f_{Y}(F^{-1}_{Y}(\tau))}>0$

which implies that the target conclusion $M_{\tau,\mathcal{D}}\leq 0$ does not
hold under point identification.

  3. $(iii)$

Suppose that $0\leq c\leq c_{\tau,U}^{M}$. We want to show that this implies
that $\theta_{U,M}\leq 0$. By Theorem 3, the upper bound on the marginal
effect is (see (A.8))

$\displaystyle\theta_{U,M}:=\min\left\\{\frac{F_{Y|D=0}(F^{-1}_{Y}(\tau))}{f_{Y}(F^{-1}_{Y}(\tau))},\frac{F_{Y|D=0}(F^{-1}_{Y}(\tau))-\int_{\mathcal{X}}F_{Y|D=1,X=x}(F^{-1}_{Y}(\tau))dF_{X|\partial
D}(x)+c}{f_{Y}(F^{-1}_{Y}(\tau))}\right\\}.$

Since $\frac{F_{Y|D=0}(F^{-1}_{Y}(\tau))}{f_{Y}(F^{-1}_{Y}(\tau))}$ is always
strictly positive under Assumption 2.1, then we need the second term inside
the $\min$ in $\theta_{U,M}$ to be non-positive. For $c=0$, it holds
trivially, since we assumed that the conclusions hold at $c=0$. For $0<c\leq
c_{\tau,U}^{M}$, we have

$\displaystyle
F_{Y|D=0}(F^{-1}_{Y}(\tau))-\int_{\mathcal{X}}F_{Y|D=1,X=x}(F^{-1}_{Y}(\tau))dF_{X|\partial
D}(x)+c$ $\displaystyle\leq
F_{Y|D=0}(F^{-1}_{Y}(\tau))-\int_{\mathcal{X}}F_{Y|D=1,X=x}(F^{-1}_{Y}(\tau))dF_{X|\partial
D}(x)+c_{\tau}$ $\displaystyle\leq 0,$

because
$c_{\tau}\leq\int_{\mathcal{X}}F_{Y|D=1,X=x}(F^{-1}_{Y}(\tau))dF_{X|\partial
D}(x)-F_{Y|D=0}(F^{-1}_{Y}(\tau))$.

  4. $(iv)$

The second term inside the $\min$ in $\theta_{U,M}$ is increasing in $c$. So,
if at $c=0$ we have that

$\displaystyle\frac{F_{Y|D=0}(F^{-1}_{Y}(\tau))-\int_{\mathcal{X}}F_{Y|D=1,X=x}(F^{-1}_{Y}(\tau))dF_{X|\partial
D}(x)}{f_{Y}(F^{-1}_{Y}(\tau))}\leq 0,$

which is equivalent to

$\displaystyle
F_{Y|D=0}(F^{-1}_{Y}(\tau))\leq\int_{\mathcal{X}}F_{Y|D=1,X=x}(F^{-1}_{Y}(\tau))dF_{X|\partial
D}(x).$

So, if the previous displays holds, then the conclusion holds for any
$c\in[0,1]$.

∎

###### Proof of Theorem 4.

First we establish the asymptotic distribution of the apparent distribution:
$\sqrt{n}(\hat{F}_{A}-F_{A})$, as a process in $\ell^{\infty}(\mathcal{Y})$.
The estimator $\hat{F}_{A}(y)$ given in (A.1) is

$\displaystyle\hat{F}_{A}(y)$
$\displaystyle:=\hat{p}\hat{F}_{Y|D=1}(y)+(1-\hat{p}-\delta)\hat{F}_{Y|D_{\delta}=0}(y)+\delta\int_{\mathcal{X}}\hat{F}_{Y|D=1,X=x}(y)d\hat{F}_{X|D=0,D_{\delta}=1}(x)$

Since the support of $X$ is finite, then this can be written as

$\displaystyle\hat{F}_{A}(y)$
$\displaystyle:=\hat{p}\hat{F}_{Y|D=1}(y)+(1-\hat{p}-\delta)\hat{F}_{Y|D_{\delta}=0}(y)+\delta\sum_{x\in\mathcal{X}}\hat{F}_{Y|D=1,X=x}(y)\hat{p}_{x}$

where $p_{x}:=\Pr(X=x|D=0,D_{\delta}=1)$ and $\hat{p}_{x}$ is its empirical
counterpart. The apparent counterfactual is

$\displaystyle F_{A}(y)$
$\displaystyle:=pF_{Y|D=1}(y)+(1-p-\delta)F_{Y|D_{\delta}=0}(y)+\delta\sum_{x\in\mathcal{X}}F_{Y|D=1,X=x}(y)p_{x}$
(A.9)

and can be written as the map
$\mathbb{D}(\mathcal{Y})^{2+dim(\mathcal{X})}\times(0,1)^{1+dim(\mathcal{X})}\mapsto\mathbb{D}(\mathcal{Y})$
given by

$\displaystyle\psi(F_{Y|D=1},F_{Y|D_{\delta}=0},F_{Y|D=1,X},p,p_{X};\delta)$
$\displaystyle=pF_{Y|D=1}(\cdot)+(1-p-\delta)F_{Y|D_{\delta}=0}(\cdot)$
$\displaystyle+\delta\sum_{x\in\mathcal{X}}F_{Y|D=1,X=x}(\cdot)p_{x}$

Here, $F_{Y|D=1,X}$ is short hand for the $dim(\mathcal{X})$-vector of CDFs
$F_{Y|D=1,X=x}$ for each value of $x\in\mathcal{X}$. Same comment applies to
$p_{X}$: it is a $dim(\mathcal{X})$-vector that collects the (joint) pmf of
$X$. This map is linear, so the Hadamard derivative161616We do not derivate
with respect to $\delta$. tangentially to
$\ell^{\infty}(\mathcal{Y})^{2+dim(X)}\times(0,1)^{1+dim(\mathcal{X})}$ at

$(F_{Y|D=1},F_{Y|D_{\delta}=0},F_{Y|D=1,X},p,p_{X})$

is the map

$\displaystyle\psi^{\prime}_{F_{Y|D=1},F_{Y|D_{\delta}=0},F_{Y|D=1,X},p,p_{X}}(h_{1},h_{2},h_{X},h_{3},\tilde{h}_{X})$
$\displaystyle=ph_{1}+(1-p-\delta)h_{2}$
$\displaystyle+(F_{Y|D=1}(\cdot)-F_{Y|D_{\delta}=0}(\cdot))h_{3}$
$\displaystyle+\delta\sum_{x\in\mathcal{X}}(h_{x}p_{x}+F_{Y|D=1,X=x}\tilde{h}_{x})$

Here $h_{X}$ is $dim(X)$-vector of functions in $\ell^{\infty}(\mathcal{Y})$,
and $\tilde{h}_{X}$ is a $dim(X)$-vector that belongs to
$(0,1)^{dim(\mathcal{X})}$. By the functional Delta method (see Theorem 20.8
in van der Vaart (1998)) and Assumption 8, we have

$\displaystyle\sqrt{n}(\hat{F}_{A}-F_{A})$
$\displaystyle=\sqrt{n}(\psi(\hat{F}_{Y|D=1},\hat{F}_{Y|D_{\delta}=0},\hat{F}_{Y|D=1,X},\hat{p},\hat{p}_{X};\delta)-\psi(F_{Y|D=1},F_{Y|D_{\delta}=0},F_{Y|D=1,X},p,p_{X};\delta))$
$\displaystyle\rightsquigarrow
p\mathbb{G}_{1}+(1-p-\delta)\mathbb{G}_{0,\delta}+(F_{Y|D=1}(\cdot)-F_{Y|D_{\delta}=0}(\cdot))\mathbb{Z}_{p}$
$\displaystyle+\delta\sum_{x\in\mathcal{X}}(\mathbb{G}_{1,x}p_{x}+F_{Y|D=1,X=x}\mathbb{Z}_{x}):=\mathbb{G}_{A},$

and convergence takes place in $\ell^{\infty}(\mathcal{Y})$. The random
element $\mathbb{G}_{A}$ is Gaussian.

Now we deal with

$\displaystyle\sqrt{n}(\hat{c}_{L}-c_{L})=-\frac{1}{\delta}\sqrt{n}\left(\hat{F}_{A}(\hat{F}_{Y}^{-1}+g)-F_{A}(F_{Y}^{-1}+g)\right)$

We introduce some new notation related to Assumption 9. Let
$\mathbb{D}_{\delta}\subset\ell^{\infty}(\mathcal{Y})$ denote the set of all
restrictions of distribution functions on $\mathbb{R}$ to
$[F_{Y}^{-1}(\delta)-\varepsilon,F_{Y}^{-1}(1-\delta)+\varepsilon]$.
Additionally, $\mathbb{C}_{\delta}$ is the set of continuous functions on
$[F_{Y}^{-1}(\delta)-\varepsilon,F_{Y}^{-1}(1-\delta)+\varepsilon]$. Also,
$\mathbb{U}\mathbb{C}(\mathcal{Y})$ is the set of uniformly continuous
functions defined on $\mathcal{Y}$.

Going back to $c_{L}$, it can be written as the composition of two maps. The
first one is
$\phi:\mathbb{D}(\mathcal{Y})\times\mathbb{D}_{\delta}\mapsto\mathbb{D}(\mathcal{Y})\times\ell^{\infty}(\delta,1-\delta)$
given by $\phi(H_{1},H_{2})\mapsto(H_{1},H_{2}^{-1})$. The second one is
$\psi:\mathbb{D}(\mathcal{Y})\times\ell^{\infty}(\delta,1-\delta)\mapsto\mathcal{\ell}^{\infty}(\delta,1-\delta)$
given by $\psi(H_{1},H_{2})\mapsto H_{1}\circ(H_{2}+g)$. Thus

$\displaystyle\psi\circ\phi(F_{A},F_{Y})=F_{A}(F_{Y}^{-1}+g).$

By Assumption 9.1 and Lemma 21.4(i) in van der Vaart (1998), $\phi$ has
Hadamard derivative at $(F_{A},F_{Y})$ tangentially to
$\ell^{\infty}(\mathcal{Y})\times\mathbb{C}_{\delta}$ given by the map

$\displaystyle\phi^{\prime}_{(F_{A},F_{Y})}(h_{1},h_{2})=\left(h_{1},-\frac{h_{2}\circ
F_{Y}^{-1}}{f_{Y}\circ F_{Y}^{-1}}\right).$

The second map
$\psi:\mathbb{D}(\mathcal{Y})\times\ell^{\infty}(\delta,1-\delta)\mapsto\mathcal{\ell}^{\infty}(\delta,1-\delta)$
is given by $\psi(H_{1},H_{2})\mapsto H_{1}\circ(H_{2}+g)$. It has Hadamard
derivative tangentially to
$\mathbb{U}\mathbb{C}(\mathcal{Y})\times\ell^{\infty}(\delta,1-\delta)$ at any
$H_{1}\in\mathbb{U}\mathbb{C}(\mathcal{Y})$ such that its derivative is
bounded and uniformly continuous on $\mathcal{Y}$, and any
$H_{2}\in\ell^{\infty}(\delta,1-\delta)$. To see, this we combine Lemmas
3.9.25 and 3.9.27 in van der Vaart and Wellner (1996). Let
$\alpha_{t}\to\alpha$ and $\beta_{t}\to\beta$ in $\mathbb{D}(\mathcal{Y})$ and
$\ell^{\infty}(\delta,1-\delta)$ respectively, as $t\to 0$.

$\displaystyle\frac{\psi(H_{1}+t\alpha_{t},H_{2}+t\beta_{t})-\psi(H_{1},H_{2})}{t}-\alpha\circ(H_{2}+g)-h_{1}\circ(H_{2}+g)\cdot\beta$
$\displaystyle=\frac{H_{1}\circ(H_{2}+g+t\beta_{t})+t\alpha_{t}\circ(H_{2}+g+t\beta_{t})-H_{1}\circ(H_{2}+g)}{t}-\alpha\circ(H_{2}+g)-h_{1}\circ(H_{2}+g)\cdot\beta$
$\displaystyle=(\alpha_{t}-\alpha)\circ(H_{2}+g+t\beta_{t})+\alpha\circ(H_{2}+g+t\beta_{t})-\alpha\circ(H_{2}+g)$
$\displaystyle+\frac{H_{1}\circ(H_{2}+g+t\beta_{t})-H_{1}\circ(H_{2}+g)}{t}-h_{1}\circ(H_{2}+g)\cdot\beta$

The first term, $(\alpha_{t}-\alpha)\circ(H_{2}+g+t\beta_{t})$, converges to
$0$ in $\mathbb{D}(\mathcal{Y})$ (that is, uniformly) because convergence of
$\alpha_{t}\to\alpha$ is uniform. The second term,
$\alpha\circ(H_{2}+g+t\beta_{t})-\alpha\circ(H_{2}+g)$, converges to $0$ in
$\mathbb{D}(\mathcal{Y})$ because $\alpha$ is uniformly continuous on
$\mathcal{Y}$. For the last term, fix a $\tau\in(\delta,1-\delta)$. By the
mean-value theorem

$\displaystyle\frac{H_{1}(H_{2}(\tau)+g+t\beta_{t}(\tau))-H_{1}(H_{2}(\tau)+g)}{t}-h_{1}(H_{2}(\tau)+g)\cdot\beta(\tau)$
$\displaystyle=h_{1}(\varepsilon_{\tau,t})\beta_{t}(\tau)-h_{1}(H_{2}(\tau)+g)\cdot\beta(\tau)$
$\displaystyle=h_{1}(\varepsilon_{\tau,t})(\beta_{t}(\tau)-\beta(\tau))+(h_{1}(\varepsilon_{\tau,t})-h_{1}(H_{2}(\tau)+g))\cdot\beta(\tau)$

The first term, $h_{1}(\varepsilon_{\tau,t})(\beta_{t}(\tau)-\beta(\tau))$,
converges uniformly to $0$ because $h_{1}$ is bounded on $\mathcal{Y}$, and
$\beta_{t}$ converges uniformly to $\beta$. The second term converges to $0$
uniformly because $h_{1}$ is uniformly continuous on $\mathcal{Y}$.

Hence, by Assumption 9.2, $\psi$ has Hadamard derivative at
$(F_{A},F_{Y}^{-1})$ tangentially to
$\mathbb{U}\mathbb{C}(\mathcal{Y})\times\ell^{\infty}(\delta,1-\delta)$ given
by the map

$\displaystyle\psi^{\prime}_{(F_{A},F_{Y}^{-1})}(h_{1},h_{2})=h_{1}\circ(F_{Y}^{-1}+g)+f_{A}\circ(F_{Y}^{-1}+g)\cdot
h_{2}.$

We use the chain rule (see Theorem 20.9 in van der Vaart (1998)) to conclude
that $\psi\circ\phi$ has Hadamard derivative at $(F_{A},F_{Y})$ tangentially
to $\mathbb{U}\mathbb{C}(\mathcal{Y})\times\mathbb{C}_{\delta}$ given by the
map

$\displaystyle(\psi\circ\phi)^{\prime}_{(F_{A},F_{Y})}(h_{1},h_{2})$
$\displaystyle=\psi^{\prime}_{\phi(F_{A},F_{Y})}\circ\phi^{\prime}_{(F_{A},F_{Y})}(h_{1},h_{2})$
$\displaystyle=\psi^{\prime}_{(F_{A},F_{Y}^{-1})}\circ(h_{1},-h_{2}(F_{Y}^{-1})/f_{Y}(F_{Y}^{-1}))$
$\displaystyle=h_{1}\circ(F_{Y}^{-1}+g)-f_{A}\circ(F_{Y}^{-1}+g)\frac{h_{2}\circ
F_{Y}^{-1}}{f_{Y}\circ F_{Y}^{-1}}.$

By the functional Delta method (see Theorem 20.8 in van der Vaart (1998)) and
the continuous mapping theorem (because of the $-1/\delta$ factor), we have
that

$\displaystyle\sqrt{n}(\hat{c}_{L}-c_{L})$
$\displaystyle=-\frac{1}{\delta}\sqrt{n}\left(\hat{F}_{A}\circ(\hat{F}_{Y}^{-1}+g)-F_{A}\circ(F_{Y}^{-1}+g)\right)$
$\displaystyle\rightsquigarrow-\frac{1}{\delta}(\psi\circ\phi)^{\prime}_{(F_{A},F_{Y})}(\mathbb{G}_{A},\mathbb{G}_{Y}))$
$\displaystyle=-\frac{1}{\delta}\mathbb{G}_{A}\circ(F_{Y}^{-1}+g)+\frac{1}{\delta}f_{A}\circ(F_{Y}^{-1}+g)\frac{\mathbb{G}_{Y}\circ
F_{Y}^{-1}}{f_{Y}\circ F_{Y}^{-1}}.$ (A.10)

Convergence takes place in $\ell^{\infty}(\delta,1-\delta)$. The limiting
element is indeed Gaussian, as it is the linear combination of two Gaussian
elements.

∎

###### Proof of Theorem 5.

For $G(y)=E[F_{Y|D=1,X}(y)|\partial D]$ or $G(y)=F_{Y|D=0}(y)$, we find the
asymptotic distribution of $\sqrt{n}(\hat{G}\circ\hat{F}_{Y}^{-1}-G\circ
F_{Y}^{-1})$. Consider first the map
$\psi:\mathbb{D}(\mathcal{Y})^{2}\to\mathbb{D}(\mathcal{Y})\times\ell^{\infty}(0,1)$,
given by $\psi(H_{1},H_{2})=(H_{1},H_{2}^{-1})$. Here,
$\mathbb{D}(\mathcal{Y})$ is the set of all restrictions of distribution
functions on $\mathbb{R}$ to $\mathcal{Y}=[y_{l},y_{u}]$, such that they give
mass 1 to $(y_{l},y_{u}]$. Also, $\mathbb{C}(\mathcal{Y})$ is the set of all
(uniformly) continuous functions defined on $\mathcal{Y}$.

By Lemma 21.4.(ii) in van der Vaart (1998), and Assumption 11, $\psi$ is
Hadamard differentiable tangentially to
$\ell^{\infty}(\mathcal{Y})\times\mathbb{C}(\mathcal{Y})$ at $(G,F_{Y})$, with
derivative given by the map

$\displaystyle\psi^{\prime}_{(G,F_{Y})}(h_{1},h_{2})=\left(h_{1},-\frac{h_{2}\circ
F_{Y}^{-1}}{f_{y}\circ F_{Y}^{-1}}\right).$

Now, consider the map
$\phi:\mathbb{D}(\mathcal{Y})\times\ell^{\infty}(0,1)\to\ell^{\infty}(0,1)$
given by $\phi(H_{1},H_{2})=H_{1}\circ H_{2}^{-1}$. By Lemmas 3.9.25 and
3.9.27 in van der Vaart and Wellner (1996), and Assumption 11, $\phi$ has
Hadamard derivative at $(G,F_{Y}^{-1})$ tangentially to
$\mathbb{U}\mathbb{C}(\mathcal{Y})\times\ell^{\infty}(0,1)$ given by the map

$\displaystyle\phi^{\prime}_{(G,F_{Y}^{-1})}(h_{1},h_{2})=h_{1}\circ
F_{Y}^{-1}+g\circ F_{Y}^{-1}\cdot h_{2},$

where $g$ is the density of $G$.

We use the chain rule (see Theorem 20.9 in van der Vaart (1998)) to conclude
that $\phi\circ\psi$ has Hadamard derivative at $(G,F_{Y})$ tangentially to
$\mathbb{U}\mathbb{C}(\mathcal{Y})\times\mathbb{C}(\mathcal{Y})$ given by the
map

$\displaystyle(\phi\circ\psi)^{\prime}_{(G,F_{Y})}(h_{1},h_{2})$
$\displaystyle=\phi^{\prime}_{\phi(G,F_{Y})}\circ\psi^{\prime}_{(G,F_{Y})}(h_{1},h_{2})$
$\displaystyle=\phi^{\prime}_{(G,F_{Y}^{-1})}\circ(h_{1},-h_{2}\circ
F_{Y}^{-1}/f_{Y}\circ F_{Y}^{-1})$ $\displaystyle=h_{1}\circ F_{Y}^{-1}-g\circ
F_{Y}^{-1}\cdot\frac{h_{2}\circ F_{Y}^{-1}}{f_{Y}\circ F_{Y}^{-1}}.$

By the functional Delta method (see Theorem 20.8 in van der Vaart (1998)) we
have that

$\displaystyle\sqrt{n}(\hat{G}\circ\hat{F}_{Y}^{-1}-G\circ F_{Y}^{-1})$
$\displaystyle\rightsquigarrow(\phi\circ\psi)^{\prime}_{(G,F_{Y})}(\mathbb{G}_{G},\mathbb{G}_{Y}))$
$\displaystyle=\mathbb{G}_{G}\circ F_{Y}^{-1}-g\circ
F_{Y}^{-1}\cdot\frac{\mathbb{G}_{Y}\circ F_{Y}^{-1}}{f_{Y}\circ F_{Y}^{-1}}$
$\displaystyle=:\mathbb{G}_{G,Y},$

where $\mathbb{G}_{G}$ is either $\mathbb{G}_{\partial D}$ or
$\mathbb{G}_{0}$. We have

$\displaystyle\hat{c}_{\tau,U}^{M}$
$\displaystyle:=\sum_{x\in\mathcal{X}}\hat{F}_{Y|D=1,X=x}(\hat{F}^{-1}_{Y}(\tau))\hat{p}_{x|\partial
D}-\hat{F}_{Y|D=0}(\hat{F}^{-1}_{Y}(\tau)),$ $\displaystyle c_{\tau,U}^{M}$
$\displaystyle:=E[F_{Y|D=1,X}(F^{-1}_{Y}(\tau))|\partial
D]-F_{Y|D=0}(F^{-1}_{Y}(\tau)).$

By the continuous mapping theorem

$\displaystyle\sqrt{n}(\hat{c}_{\tau,U}^{M}-c_{\tau,U}^{M})\rightsquigarrow\mathbb{G}_{\theta_{M}}:=\mathbb{G}_{\partial
D,Y}-\mathbb{G}_{0,Y}.$

∎

###  C Bounds derived from the QBF

The goal is to find the joint distribution of

$\displaystyle\hat{L}_{\tau,\tau^{*}}$
$\displaystyle:=\max\left\\{\hat{\tilde{F}}_{A}^{-1}(\tau-\delta),\hat{F}_{A}^{-1}(\tau-\delta\hat{\tilde{c}}_{\tau^{*},L})\right\\}-\hat{F}_{Y}^{-1}(\tau),$
$\displaystyle\hat{U}_{\tau,\tau^{*}}$
$\displaystyle:=\min\left\\{\hat{\tilde{F}}_{A}^{-1}(\tau),\hat{F}_{A}^{-1}(\tau+\delta\hat{\tilde{c}}_{\tau^{*},L})\right\\}-\hat{F}_{Y}^{-1}(\tau).$

where
$\hat{\tilde{c}}_{\tau^{*},L}=\max\left\\{\min\left\\{\hat{c}_{\tau^{*},L},1\right\\},0\right\\}$,
and by (9),

$\displaystyle\hat{c}_{\tau^{*},L}=\frac{\tau^{*}-\hat{F}_{A}\left(\hat{F}_{Y}^{-1}(\tau^{*})+g\right)}{\delta}.$

as a process in $\ell^{\infty}(\delta,1-\delta)^{2}$. The main assumption we
need is the following.

######  Assumption 12.

The following multivariate functional central limit theorem holds

$\displaystyle\sqrt{n}\begin{pmatrix}\hat{F}_{Y}-F_{Y}\\\ \hat{F}_{A}-F_{A}\\\
\hat{\tilde{F}}_{A}-\tilde{F}_{A}\\\
\end{pmatrix}\rightsquigarrow\begin{pmatrix}\mathbb{G}_{Y}\\\
\mathbb{G}_{A}\\\ \tilde{\mathbb{G}}_{A}\end{pmatrix},$

where $\mathbb{G}_{Y}$ , $\mathbb{G}_{A}$, and $\tilde{\mathbb{G}}_{A}$ are
Brownian bridges in $\ell^{\infty}(\mathcal{Y})$.

First consider the map
$\Gamma_{1}:\mathbb{D}_{\delta}\times\mathbb{D}_{A}\times\tilde{\mathbb{D}}_{A}\to\ell^{\infty}(\delta,1-\delta)^{3}\times[0,1]$,
given by

$\displaystyle\Gamma_{1}(H_{1},H_{2},H_{3})=\left(H_{1}^{-1},H_{2}^{-1},H_{3}^{-1},\tilde{\Gamma}_{1}(H_{1},H_{2})\right).$

where

$\displaystyle\tilde{\Gamma}_{1}(H_{1},H_{2})=\max\left\\{\min\left\\{\frac{\tau^{*}-H_{2}(H_{1}^{-1}(\tau^{*})+g)}{\delta},1\right\\},0\right\\}.$

Here, $\mathbb{D}_{\delta}\subset\ell^{\infty}(\mathcal{Y})$ denote the set of
all restrictions of distribution functions on $\mathbb{R}$ to
$[F_{Y}^{-1}(\delta)-\varepsilon,F_{Y}^{-1}(1-\delta)+\varepsilon]$.
$\mathbb{D}_{A}$ and $\tilde{\mathbb{D}}_{A}$ are defined analogously but with
respect to $F_{A}$ and $\tilde{F}_{A}$ Let $\mathbb{C}_{\delta}$ bet the set
of continuous functions on
$[F_{Y}^{-1}(\delta)-\varepsilon,F_{Y}^{-1}(1-\delta)+\varepsilon]$. Define
$\mathbb{C}_{A}$ and $\tilde{\mathbb{C}}_{A}$ similarly but with respect to
$F_{A}$ and $\tilde{F}_{A}$.

The next Assumption, similar to Assumption 9 is needed for Hadamard
Differentiability.

######  Assumption 13 (Conditions for Hadamard Differentiability).

  1. 1.

Let $G$ be either $F_{Y}$, $F_{A}$, and $\tilde{F}_{A}$. For some
$\varepsilon>0$, $G$ is continuously differentiable in
$[G^{-1}(\delta)-\varepsilon,G^{-1}(1-\delta)+\varepsilon]\subset\mathcal{Y}$
with strictly positive derivative $g$.

By Assumption 13 and Lemma 21.4(i) in van der Vaart (1998), $\Gamma_{1}$ has
Hadamard derivative at $(F_{Y},F_{A},\tilde{F}_{A})$ tangentially to
$\mathbb{C}_{\delta}\times\mathbb{C}_{A}\times\tilde{\mathbb{C}}_{A}$ given by
the map

$\displaystyle\Gamma^{\prime}_{1,(F_{Y},F_{A},\tilde{F}_{A})}(h_{1},h_{2},h_{3})=\left(-\frac{h_{1}\circ
F_{Y}^{-1}}{f_{Y}\circ F_{Y}^{-1}},-\frac{h_{2}\circ F_{A}^{-1}}{f_{A}\circ
F_{A}^{-1}},-\frac{h_{3}\circ\tilde{F}_{A}^{-1}}{\tilde{f}_{A}\circ\tilde{F}_{A}^{-1}},\tilde{\Gamma}^{\prime}_{1,(F_{Y},F_{A})}(h_{1},h_{2})\right).$

where the derivative $\tilde{\Gamma}^{\prime}_{1,(F_{Y},F_{A})}(h_{1},h_{2})$
is shown below in an auxiliary Lemma 3.

Now consider the map
$\Gamma_{2}:\ell^{\infty}(\delta,1-\delta)^{3}\times[0,1]\to\ell^{\infty}(\delta,1-\delta)^{5}$
given by

$\displaystyle\Gamma_{2}(H_{1},H_{2},H_{3},H_{4})=(H_{1},H_{2}(\cdot+\delta
H_{4}),H_{2}(\cdot-\delta H_{4}),H_{3},H_{3}(\cdot-\delta))$

which is a combination of identity and composition maps, so it is Hadamard
differentiable at $(F_{Y},F_{A},\tilde{F}_{A},\hat{\tilde{c}}_{\tau^{*},L})$
by assumption to 9.2 tangentially to
$\ell^{\infty}(\delta,1-\delta)\times\mathbb{U}\mathbb{C}(\mathcal{Y})^{2}\times[0,1]$,
with derivative given by

$\displaystyle\Gamma^{\prime}_{2,(F_{Y},F_{A},\tilde{F}_{A},\hat{\tilde{c}}_{\tau^{*},L})}(h_{1},h_{2},h_{3},h_{4})$
$\displaystyle=(h_{1},h_{2}(\cdot+\delta\hat{\tilde{c}}_{\tau^{*},L})+f_{A}(\cdot+\delta\hat{\tilde{c}}_{\tau^{*},L})\delta
h_{4},h_{2}(\cdot-\delta\hat{\tilde{c}}_{\tau^{*},L})+f_{A}(\cdot-\delta\hat{\tilde{c}}_{\tau^{*},L})\delta
h_{4},h_{3},h_{3}(\cdot-\delta))$

Finally, consider the map
$\Gamma_{3}:\ell^{\infty}(\delta,1-\delta)^{5}\to\ell^{\infty}(\delta,1-\delta)^{2}$,
given by

$\displaystyle\Gamma_{3}(H_{1},H_{2},H_{3},H_{4},H_{5})=\begin{pmatrix}\max\left\\{H_{5},H_{3}\right\\}-H_{1}\\\
\min\left\\{H_{4},H_{2}\right\\}-H_{1}\end{pmatrix}$

In particular

$\displaystyle\Gamma_{3}(F_{Y}^{-1},F_{A}^{-1}(\cdot+\delta\hat{\tilde{c}}_{\tau^{*},L}),F_{A}^{-1}(\cdot-\delta\hat{\tilde{c}}_{\tau^{*},L}),{\tilde{F}}_{A}^{-1},{\tilde{F}}_{A}^{-1}(\cdot-\delta))=\begin{pmatrix}L_{\cdot,\tau^{*}}\\\
U_{\cdot,\tau^{*}}\end{pmatrix}$

This map is Hadamard directional differentiable at
$(F_{Y}^{-1},F_{A}^{-1}(\cdot+\delta{\tilde{c}}_{\tau^{*},L}),F_{A}^{-1}(\cdot-\delta{\tilde{c}}_{\tau^{*},L}),{\tilde{F}}_{A}^{-1},{\tilde{F}}_{A}^{-1}(\cdot-\delta))$
tangentially to $\ell^{\infty}(\delta,1-\delta)^{5}$ with derivative given by

$\displaystyle\Gamma^{\prime}_{3,(F_{Y}^{-1},F_{A}^{-1}(\cdot+\delta{\tilde{c}}_{\tau^{*},L}),F_{A}^{-1}(\cdot-\delta{\tilde{c}}_{\tau^{*},L}),{\tilde{F}}_{A}^{-1},{\tilde{F}}_{A}^{-1}(\cdot-\delta))}(h_{1},h_{2},h_{3},h_{4},h_{5})$
$\displaystyle=\begin{pmatrix}h_{5}\mathds{1}_{\\{{\tilde{F}}_{A}^{-1}(\cdot-\delta)>F_{A}^{-1}(\cdot-\delta{\tilde{c}}_{\tau^{*},L})\\}}+h_{3}\mathds{1}_{\\{{\tilde{F}}_{A}^{-1}(\cdot-\delta)<F_{A}^{-1}(\cdot-\delta{\tilde{c}}_{\tau^{*},L})\\}}+\max\left\\{h_{5},h_{3}\right\\}\mathds{1}_{\\{{\tilde{F}}_{A}^{-1}(\cdot-\delta)=F_{A}^{-1}(\cdot-\delta{\tilde{c}}_{\tau^{*},L})\\}}-h_{1}\\\
h_{4}\mathds{1}_{\\{{\tilde{F}}_{A}^{-1}<F_{A}^{-1}(\cdot+\delta{\tilde{c}}_{\tau^{*},L})\\}}+h_{2}\mathds{1}_{\\{{\tilde{F}}_{A}^{-1}>F_{A}^{-1}(\cdot+\delta{\tilde{c}}_{\tau^{*},L})\\}}+\min\left\\{h_{4},h_{2}\right\\}\mathds{1}_{\\{{\tilde{F}}_{A}^{-1}=F_{A}^{-1}(\cdot+\delta{\tilde{c}}_{\tau^{*},L})\\}}-h_{1}\end{pmatrix}$

Therefore, the map

$\displaystyle\begin{pmatrix}L_{\cdot,\tau^{*}}\\\
U_{\cdot,\tau^{*}}\end{pmatrix}=\Gamma_{3}\circ\Gamma_{2}\circ\Gamma_{1}(F_{Y},F_{A},\tilde{F}_{A})$

is Hadamard directional differentiable at $F_{Y},F_{A},\tilde{F}_{A}$
tangentially to
$\mathbb{C}_{\delta}\times\mathbb{C}_{A}\times\tilde{\mathbb{C}}_{A}$ by the
chain rule for Hadamard directional differentiable maps (see Proposition 3.6
in Shapiro (1990); Lemma C2 of Masten and Poirier (2020)). By Theorem 2.1 in
Fang and Santos (2019) and Assumption 12, we get

$\displaystyle\sqrt{n}\begin{pmatrix}L_{\cdot,\tau^{*}}-\hat{L}_{\cdot,\tau^{*}}\\\
U_{\cdot,\tau^{*}}-\hat{U}_{\cdot,\tau^{*}}\end{pmatrix}\rightsquigarrow\left[\Gamma_{3}\circ\Gamma_{2}\circ\Gamma_{1}\right]^{\prime}_{(F_{Y},F_{A},\tilde{F}_{A})}(\mathbb{G}_{Y},\mathbb{G}_{A},\tilde{\mathbb{G}}_{A})$

and convergence takes place in $\ell^{\infty}(\delta,1-\delta)^{2}$.

######  Lemma 3.

Under the assumptions of Theorem 4, the map
$\tilde{\Gamma}_{1}:\mathbb{D}_{\delta}\times\mathbb{D}_{A}\mapsto\mathbb{R}$
given by

$\displaystyle\tilde{\Gamma}_{1}(H_{1},H_{2})\mapsto\max\left\\{\min\left\\{\frac{\tau^{*}-H_{2}(H_{1}^{-1}(\tau^{*})+g)}{\delta},1\right\\},0\right\\}$

is Hadamard directionally differentiable at $(F_{Y},F_{A})$ tangentially to
$\mathbb{C}_{\delta}\times\ell^{\infty}(\mathcal{Y})$.

###### Proof of Lemma 3.

We analyze the map

$\displaystyle\tilde{\Gamma}_{1}(H_{1},H_{2})\mapsto\max\left\\{\min\left\\{\frac{\tau^{*}-H_{2}(H_{1}^{-1}(\tau^{*})+g)}{\delta},1\right\\},0\right\\}$

separately. We write it as the composition of several maps. The first one is
$\phi:\mathbb{D}_{\delta}\times\mathbb{D}_{A}\mapsto\mathbb{R}\times\mathbb{D}_{A}$
given by $\phi(H_{1},H_{2})\mapsto(H_{1}^{-1}(\tau^{*}),H_{2})$. The second
one is $\psi:\mathbb{R}\times\mathbb{D}_{A}\mapsto\mathbb{R}$ given by
$\psi(H_{1},H_{2})\mapsto(\tau^{*}-H_{2}(H_{1}+g))/\delta$. Thus

$\displaystyle\psi\circ\phi(F_{Y},F_{A})=\frac{\tau^{*}-F_{A}(F_{Y}^{-1}(\tau^{*})+g)}{\delta}.$

The map $\phi$ essentially the same map $\phi$ in Theorem 4. By the results
there, it has Hadamard derivative at $(F_{Y},F_{A})$ tangentially to
$\mathbb{C}_{\delta}\times\ell^{\infty}(\mathcal{Y})$ given by the map

$\displaystyle\phi^{\prime}_{(F_{Y},F_{A})}(h_{1},h_{2})=\left(-\frac{h_{1}(F_{Y}^{-1}(\tau^{*}))}{f_{Y}(F_{Y}^{-1}(\tau^{*}))},h_{2}\right).$

Likewise, the map $\psi$ is essentially the map in Theorem 4. Therefore, it
has Hadamard derivative at $(F_{Y}^{-1}(\tau^{*}),F_{A})$ tangentially to
$\mathbb{R}\times\mathbb{U}\mathbb{C}(\mathcal{Y})$ given by the map

$\displaystyle\psi^{\prime}_{(F_{Y}^{-1}(\tau^{*}),F_{A})}(h_{1},h_{2})=-\frac{h_{2}(F_{Y}^{-1}(\tau^{*})+g)}{\delta}-\frac{f_{A}(F_{Y}^{-1}(\tau^{*})+g)\cdot
h_{1}}{\delta}.$

Now define

$\displaystyle\tilde{\phi}(H)=\max\\{\min\\{H,1\\},0\\}.$

and it is the composition of an evaluation map $H\in\mathbb{R}\mapsto H$ and
of the max/min composition. The evaluation map is linear, hence fully Hadamard
differentiable. The composition of max/min is Hadamard directional
differentiable by the chain rule for Hadamard directional differentiable maps
(see Proposition 3.6 in Shapiro (1990); Lemma C2 of Masten and Poirier
(2020)). Hence, another application of the chain rule yields that $\phi(H)$ is
Hadamard directional differentiable at any $H\in\mathbb{R}$ tangentially to
$\mathbb{R}$. By direct computation, the derivative, for any $h\in\mathbb{R}$,
is given by the map

$\displaystyle\tilde{\phi}^{\prime}_{H}(h)=h\mathds{1}_{\left\\{0<H<1\right\\}}+\max(0,h)\mathds{1}_{\left\\{H=0\right\\}}+\min(0,h)\mathds{1}_{\left\\{H=1\right\\}}.$
(A.11)

Therefore,

$\displaystyle\tilde{\phi}\circ\psi\circ\phi(F_{Y},F_{A})=\max\left\\{\min\left\\{\frac{\tau^{*}-F_{A}(F_{Y}^{-1}(\tau^{*})+g)}{\delta},1\right\\},0\right\\}$

By the chain rule, $\tilde{\phi}\circ\psi\circ\phi$ has Hadamard directional
derivative at $(F_{Y},F_{A})$ tangentially to
$\mathbb{C}_{\delta}\times\ell^{\infty}(\mathcal{Y})$, given by

$\displaystyle[\tilde{\phi}\circ\psi\circ\phi]^{\prime}_{(F_{Y},F_{A})}(h_{1},h_{2})=\tilde{\phi}^{\prime}_{\psi\circ\phi(F_{Y},F_{A})}\circ\psi^{\prime}_{\phi(F_{Y},F_{A})}\circ\phi^{\prime}_{(F_{Y},F_{A})}(h_{1},h_{2}).$

∎