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Identification and Estimation of Unconditional Policy Effects of an Endogenous Binary Treatment: An Unconditional MTE Approach
\begin{singlespace}
\title{Identification and Estimation of Unconditional Policy Effects of an
Endogenous Binary Treatment: An Unconditional MTE Approach\thanks{For their constructive feedback, we thank Xiaohong Chen (coeditor), an associate editor, and two anonymous referees. We also thank Marinho Bertanha, Michael Leung, Jessie Li, Xinwei Ma, Augusto Nieto-Barthaburu, Vitor Possebom, Christoph Rothe, Kaspar Wuthrich, and seminar participants at UC Berkeley, UC Davis, UC San Diego, University of Notre Dame, LACEA 2021, NAMES 2022, SEA 2022, the 2023 CEME Conference for Young Econometricians, and the 2\textsuperscript{nd} NorCal Junior Econometricians' Conference for their very helpful comments. This research was conducted with restricted access to
Bureau of Labor Statistics (BLS) data. The views expressed here do not
necessarily reflect the views of the BLS.}}
\author{Julian Martinez-Iriarte\thanks{
Email: [email removed]. } \\
Department of Economics\\
UC Santa Cruz \and Yixiao Sun\thanks{
Email: [email removed].} \\
Department of Economics\\
UC San Diego}
\date{\textbf{August 6, 2024}}
\maketitle
\begin{abstract}
This paper studies the identification and estimation of
policy effects when treatment status is binary and endogenous.
We introduce a new class of marginal treatment effects (MTEs)
based on the influence function of the functional underlying the policy
target. We show that an unconditional policy effect can be represented as a
weighted average of the newly defined MTEs over the
individuals who are indifferent about their treatment status. We provide
conditions for point identification of the unconditional policy effects.
When a quantile is the functional of interest, we introduce the UNconditional Instrumental Quantile
Estimator (UNIQUE) and establish its consistency and asymptotic distribution. In the
empirical application, we estimate the effect of changing college enrollment
status, induced by higher tuition subsidy, on the quantiles of the wage
distribution.
\end{abstract}
\thispagestyle{empty}
\textbf{Keywords}: marginal treatment effect, marginal policy-relevant
treatment effect, selection model, instrumental variable, unconditional
policy effect, unconditional quantile regression.
\textbf{JEL}: C14, C31, C36.
\end{singlespace}
\normalem
\clearpage\pagenumbering{arabic}
\section{Introduction}
An unconditional policy effect is the effect of a change in a target
covariate on the unconditional distribution of an outcome variable of
interest.\footnote{
There are several groups of variables in this framework. The target
covariate is the variable a policy maker aims to change. We often refer to
the target covariate as the treatment or treatment variable. The outcome
variable is the variable that a policy maker ultimately cares about. A
policy maker hopes to change the target covariate in order to achieve a
desired effect on the distribution of the outcome variable. Sometimes, a
policy maker can not change the treatment variable directly and has to
change it by\ intervening some other covariates, which may be referred to as
the policy covariates. There may also be covariates that will not be
intervened.} When the target covariate has a continuous distribution, we may
be interested in shifting its location and evaluating the effect of such a
shift on the distribution of the outcome variable. For example, we may
consider increasing the number of years of education for \emph{every} worker
in order to improve the median of the wage distribution. When the change in
the covariate distribution is small, such an effect may be referred to as
the marginal unconditional policy effect.
In this paper, we consider a binary target covariate that indicates the
treatment status. In this case, a location shift is not possible, and the
only way to change its distribution is to change the proportion of treated
individuals. We analyze the impact of such a marginal change on a general
functional of the distribution of the outcome. For example, when the
functional of interest is the mean of the outcome, this corresponds to the
marginal policy-relevant treatment effect (MPRTE) of \cite{Carneiro2010,
Carneiro2011}. For the case of quantiles, we obtain an unconditional
quantile effect (UQE). Previously, in a seminal contribution, \cite
{Firpo2009} proposed using\ an unconditional quantile regression (UQR) to
estimate the UQE.\footnote{\cite{mukhin2019} generalizes \cite{Firpo2009} to
allow for non-marginal changes in continuous covariates. Other recent
studies in the case of continuous covariates include \cite{SasakiUraZhang20}
who allow for a high-dimensional setting, \cite{InoueLiXu21} who tackle a
two-sample problem, \cite{MontesRojas} who analyze location-scale and
compensated shifts, and \cite{alejo2022} who propose an alternative
estimation method based on the slopes of (conditional) quantile regression.}
However, we show that their identification strategy can break down under
endogeneity. An extensive analysis of the resulting asymptotic bias of the
UQR estimator is provided.
The first contribution of this paper is to introduce a new class of
unconditional marginal treatment effects (MTEs) and show that the
corresponding unconditional policy effect can be represented as a weighted
average of these unconditional MTEs. The novel MTEs are derived from the
influence function of the functional of the outcome distribution that we
care about. This framework allows us to show that the MPRTE and UQE belong
to the same family of parameters. To the best of our knowledge, this was not
previously recognized in either the literature on MTEs or the literature on
unconditional policy effects.
To illustrate the usefulness of this general approach, we provide an
extensive analysis of the unconditional quantile effects. This is
empirically important since the UQR estimator proposed by \cite{Firpo2009}
is consistent for the UQE only if a certain distributional invariance
assumption holds. Such an assumption is unlikely to hold when the treatment
status is endogenous. We note that treatment endogeneity is the rule rather
than the exception in economic applications.
The second contribution of this paper is to provide a closed-form expression
for the asymptotic bias of the UQR estimator when endogeneity is overlooked.
We show that when selection into treatment follows a threshold-crossing
model, the UQR estimator can be inconsistent, even if the treatment status
is exogenous. This intriguing result underscores the need for caution in
using the UQR without careful consideration. The asymptotic bias can be
traced back to two sources. First, the subpopulation of the individuals at
the margin of indifference might have different characteristics than the
whole population. We refer to this source of bias as the \emph{marginal
heterogeneity bias. }Second, the treatment effect for a marginal individual
might be different from an apparent effect obtained by comparing the
treatment group with the control group. It is the marginal subpopulation,
not the whole population or any other subpopulation, that contributes to the
UQE. We refer to the second source of bias as the \emph{marginal selection
bias. }
The third contribution of this paper is to show that if assumptions similar
to instrument validity are imposed on the policy variables under
intervention, the resulting UQE can be point identified using the local
instrumental variable approach as in \cite{Carneiro2009}. Building on this,
we introduce the UNconditional Instrumental QUantile Estimator (UNIQUE) and
develop methods for statistical inference based on the UNIQUE when the
binary treatment is endogeneous. We take a nonparametric approach but allow
for the propensity score function to be either parametric or nonparametric.
We establish the asymptotic distribution of the UNIQUE. This is a formidable
task, as the UNIQUE is a four-step estimator, and we have to pin down
estimation errors from each step.
\textbf{Related Literature.} This paper is related to the literature on
marginal treatment effects. Introduced by \cite{Bjorklund1987}, MTEs can be
used as a building block for many different causal parameters of interest as
shown in \cite{Heckman2001}. For an excellent review, the reader is referred
to \cite{mogstad2018}. As a main departure from this literature, this paper
introduces unconditional MTEs targeted at studying unconditional policy
effects. As mentioned above, if we focus on the mean, the unconditional
marginal policy effect we study corresponds to the MPRTE of \cite
{Carneiro2010}. In this case, the unconditional MTE is the same as the usual
MTE. For quantiles, the unconditional marginal policy effect is studied by
\cite{Firpo2009}, but in a setting that does not allow for endogeneity.
\footnote{
For the case of continuous endogenous covariates, \cite{Rothe2010b} shows
that the control function approach of \cite{Imbens2009} can be used to
achieve identification. This paper considers a binary endogenous covariate.}
The unconditional MTE is novel in this case, as well as in the case with a
more general nonlinear functional of interest. In our setting that allows
some covariates to enter both the outcome equation and the selection
equation, conditioning on the propensity score is not enough, but we show
that the propensity score plays a key role in averaging the unconditional
MTEs to obtain the unconditional policy effect. This is in contrast with
\cite{Zhou2019} where the marginal treatment effect parameter is defined
based on conditioning on the propensity score. Among other contributions,
\cite{torgo2020} provide an extensive list of applications for MTEs, many of
which can also be applied to the unconditional MTEs introduced in this paper.
Our general treatment of the problem using functionals is closely related to
that of \cite{Rothe2012}. \cite{Rothe2012} analyzes the effect of an
arbitrary change in the distribution of a target covariate, either
continuous or discrete, on some feature of the distribution of the outcome
variable. By assuming a form of conditional independence, for the case of
continuous target covariates, \cite{Rothe2012} generalizes the approach of
\cite{Firpo2009}. However, for the case of a discrete treatment, instead of
point identifying the effect as we do here, bounds are obtained by assuming
that either the highest-ranked or lowest-ranked individuals enter the
program under the new policy.
We are not the first to consider unconditional quantile regressions under
endogeneity.\footnote{\cite{pereda2023} studies the unconditional quantile
treatment effect of a binary endogenous regressor. However, the effect there
differs from the unconditional quantile effect considered in this paper.
Here, unconditional refers to the marginal distribution of the observed $Y$,
while in \cite{pereda2023} unconditional means that the distribution of the
potential outcomes after the covariates are integrated out.} \cite{Kasy2016}
focuses on ranking counterfactual policies and, for the case of discrete
regressors, allows for endogeneity. However, a key difference from our
approach is that the counterfactual policies analyzed in \cite{Kasy2016} are
randomly assigned conditional on a covariate vector. In our setting,
selection into treatment follows a threshold-crossing model, where we use
the exogenous variation of an instrument to obtain different counterfactual
scenarios. \cite{martinez2020} introduces the quantile breakdown frontier in
order to perform a sensitivity analysis on departures from the
distributional invariance assumption employed by \cite{Firpo2009}.
Estimation of the MTE and parameters derived from the MTE curve is discussed
in \cite{urzua2006}, \cite{Carneiro2009}, \cite{Carneiro2010, Carneiro2011},
and \cite{ura2021}. All of these studies make a linear-in-parameters
assumption regarding the conditional means of the potential outcomes, which
yields a tractable partially linear model for the MTE curve. This strategy
is not very helpful in our case because our newly defined unconditional MTE
might involve a nonlinear function of the potential outcomes. For example,
in the case of quantiles, there is an indicator function involved. Using our
expression for the weights, we can write the UQE as a quotient of two
average derivatives. One of them, however, involves as a regressor the
estimated propensity score as in the setting of \cite{hahn2013}. We provide
conditions, different from those in \cite{hahn2013}, under which the error
from estimating the propensity score function, either parametrically or
nonparametrically, does not affect the asymptotic variance of the UNIQUE.
This may be of independent interest.
\textbf{Outline.} Section \ref{general_functional} introduces the new MTE
curve and shows how it relates to the unconditional policy effect. Section
\ref{section_uqr} presents a model for studying the UQE under endogeneity.
Section \ref{sect:IV} considers intervening an instrumental variable in
order to change the treatment status and establishes the identification of
the corresponding UQE. Section \ref{estimation} introduces and studies the
UNIQUE under a parametric specification of the propensity score. Section \ref
{simulation} provides simulation evidence. In Section \ref{empirical} we
revisit the empirical application of \cite{Carneiro2011} and focus on the
unconditional quantile effect. Section \ref{conclusion} concludes. An
appendix contains the proof of the main results. A supplementary appendix
provides the technical conditions for two lemmas, the proof of all lemmas
and a proposition, the estimation of the asymptotic variance for the UNIQUE,
and its asymptotic properties under a nonparametric specification of the
propensity score.
\textbf{Notation.} For any generic random variable $W_{1}$, we denote its
CDF and pdf by $F_{W_{1}}\left( \cdot \right) $ and $f_{W_{1}}\left( \cdot
\right) ,$ respectively. We denote its conditional CDF and pdf conditional
on a second random variable $W_{2}$ by $F_{W_{1}|W_{2}}\left( \cdot |\cdot
\right) $ and $f_{W_{1}|W_{2}}\left( \cdot |\cdot \right) ,$ respectively.
\section{Unconditional Policy Effects under Endogeneity}
\label{general_functional}
\subsection{Policy Intervention}
We employ the potential outcomes framework. For each individual, there are
two potential outcomes: $Y(0)$ and $Y(1)$, where $Y(0)$ is the outcome had
she received no treatment and $Y(1)$ is the outcome had she received
treatment. We assume that the potential outcomes are given by
\begin{equation*}
Y(0)=r_{0}(X,U_{0}),\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ and }Y(1)=r_{1}(X,U_{1}),
\end{equation*}
for a pair of unknown functions $r_{0}$ and $r_{1}$. The vector $X\in
\mathbb{R}^{d_{X}}$ consists of observables and $U:=\left( U_{0}^{\prime
},U_{1}^{\prime }\right) ^{\prime }$ consists of unobservables. Depending on
the individual's actual choice of treatment, denoted by $D,$ we observe
either $Y(0)$ or $Y(1)$, but we can never observe both. The observed outcome
is denoted by $Y$:
\begin{equation}
Y=\left( 1-D\right) Y(0)+DY(1)=\left( 1-D\right)
r_{0}(X,U_{0})+Dr_{1}(X,U_{1}). \label{model1Y}
\end{equation}
Following \cite{Heckman1999, Heckman2001, Heckman2005}, we assume that
selection into treatment is determined by a threshold-crossing equation
\begin{equation}
D=\mathds{1}\left\{ V\leq {\Greekmath 0116} \left( W\right) \right\} , \label{model1D}
\end{equation}
where $W:=(Z,X)$ and $Z\in \mathbb{R}^{d_{Z}}$ consists of covariates that
do not affect the potential outcomes directly. In the above, the unknown
function ${\Greekmath 0116} \left( W\right) $ can be regarded as the benefit from the
treatment and $V$ as the cost of the treatment. Individuals decide to take
up the treatment if and only if its benefit outweighs its cost.
While we observe $\left( D,W,Y\right) $, we observe neither $U$ nor $V.$
Also, we do not restrict the dependence among $U,W,$ and $V$. Hence, they
can be mutually dependent and $D$ could be endogenous. Moreover, the
dimension of $U$ is left unrestricted, while $V$ is a real-valued random
variable.
The propensity score is $P(w):=\Pr \left[ D=1|W=w\right] $. In view of
\eqref{model1D}, we can represent it as
\begin{equation}
P(w)=\Pr \left[ V\leq {\Greekmath 0116} \left( W\right) |W=w\right] =F_{V|W}({\Greekmath 0116} (w)|w).
\label{PS_equ}
\end{equation}
If the conditional CDF $F_{V|W}(\cdot |w)$ is a strictly increasing function
for all $w\in \mathcal{W}$, the support of $W$, we have
\begin{equation*}
D=\mathds{1}\left\{ V\leq {\Greekmath 0116} \left( W\right) \right\} =\mathds{1}\left\{
F_{V|W}(V|W)\leq F_{V|W}({\Greekmath 0116} \left( W\right) |W)\right\} =\mathds{1}\left\{
U_{D}\leq P(W)\right\} ,
\end{equation*}
where $U_{D}:=F_{V|W}(V|W)$ measures an individual's relative resistance to
the treatment, and it can be shown that $U_{D}$ is uniform on $[0,1]$ and is
independent of $W.$
To change the treatment take-up rate, we manipulate $Z,$ a subvector of $W$.
\footnote{
When we induce the covariate $Z$ to change, the distribution of this
covariate will change. However, we do not specify the new distribution \emph{
a priori}. Instead, we specify the policy rule that dictates how the value
of the covariate will change for each individual in the population. Our
intervention may then be regarded as a \emph{value} intervention. This is in
contrast to a \emph{distribution} intervention that stipulates a new
covariate distribution directly. An advantage of our policy rule is that it
is directly implementable in practice while a hypothetical distribution
intervention is not. The latter intervention may still have to be
implemented via a value intervention, which is our focus here. For a similar
comment in the continuous case, see Section 3 in \cite{MontesRojas}.} More
specifically, we consider a policy intervention that changes $Z$ into $
Z_{{\Greekmath 010E} }=\mathcal{G}\left( W,{\Greekmath 010E} \right) $ for a vector of smooth
functions $\mathcal{G}\left( \cdot ,\cdot \right) \in \mathbb{R}^{d_{Z}}$.
We assume that $\mathcal{G}(W,0)=Z$ so that the \emph{status quo} policy
corresponds to ${\Greekmath 010E} =0.$\footnote{
For notational convenience, when ${\Greekmath 010E} =0$, we drop the subscript and
denote $Y_{0}$ and $D_{0}$ as $Y$ and $D,$ respectively.} With the induced
change in $Z,$ the selection equation becomes
\begin{equation}
D_{{\Greekmath 010E} }=\mathds{1}\left\{ V\leq {\Greekmath 0116} \left( Z_{{\Greekmath 010E} },X\right) \right\}
=\mathds{1}\left\{ V\leq {\Greekmath 0116} \left( \mathcal{G}(W,{\Greekmath 010E} ),X\right) \right\}
, \label{model2D}
\end{equation}
which can be written as $D_{{\Greekmath 010E} }=\mathds{1}\left\{ U_{D}\leq P_{{\Greekmath 010E}
}(W)\right\} $, for $P_{{\Greekmath 010E} }(W)=F_{V|W}({\Greekmath 0116} \left( \mathcal{G}(W,{\Greekmath 010E}
),X\right) |W)$.\footnote{
Note that $U_{D}$ is still defined as $F_{V|W}(V|W)$, and so it does not
change under the counterfactual policy regime. This is to say that, relative
to others, an individual's resistance to the treatment is preserved across
the two policy regimes. In particular, $U_{D}$ is still uniform on $[0,1]$
and independent of $W.$} The outcome equation, in turn, is now
\begin{equation}
Y_{{\Greekmath 010E} }=\left( 1-D_{{\Greekmath 010E} }\right) Y\left( 0\right) +D_{{\Greekmath 010E} }Y\left(
1\right) =\left( 1-D_{{\Greekmath 010E} }\right) r_{0}(X,U_{0})+D_{{\Greekmath 010E}
}r_{1}(X,U_{1}). \label{model2Y}
\end{equation}
Equations (\ref{model2D}) and (\ref{model2Y}) are the same as the \emph{
status quo} equations; the only exception is that $Z$ has been replaced by $
Z_{{\Greekmath 010E} }.$ We have maintained the structural forms of the outcome
equation and the treatment selection equation. Importantly, we have also
maintained the stochastic dependence among $U,W,$ and $V,$ which is
manifested through the use of the same notation $U,W,$ and $V$ in equations (
\ref{model2D}) and (\ref{model2Y}) as in equations (\ref{model1Y}) and (\ref
{model1D}). Our policy intervention has a \emph{ceteris paribus}
interpretation at the population level: we apply the same form of
intervention on $Z$ for all individuals in the population but hold all else,
including the causal mechanism and the stochastic dependence among the \emph{
status quo} variables, constant. In particular, the conditional distribution
of $\left( U,V\right) $ given $W$ is invariant to the value of ${\Greekmath 010E} .$
We allow $\mathcal{G}(\cdot ,{\Greekmath 010E} )$ to take a general form but some
examples may be helpful. Consider the case that ${\Greekmath 0116} \left( W\right) =Z$ for
a univariate $Z.$ We may take $\mathcal{G}\left( W,{\Greekmath 010E} \right) =Z+{\Greekmath 010E} $
. Under such a policy, we change the value of $Z$ by ${\Greekmath 010E} $ for each
individual in the population so that the location of the distribution of $Z$
is shifted by ${\Greekmath 010E} .$ We may also take $\mathcal{G}\left( W,{\Greekmath 010E}
\right) =Z\left( 1+{\Greekmath 010E} \right) $, in which case, there is a proportional
or scale change in the value of $Z$ for all individuals. Both cases have
been studied by \cite{Carneiro2010}; see also table 2 in \cite{mogstad2018}.
More general location and scale changes, such as those given in \cite
{MontesRojas}, are allowed. In fact, any policy function that satisfies
Assumption \ref{Assumption_regularity} in the next subsection is permitted.
\subsection{Unconditional Policy Effects}
To define an unconditional policy effect, we first describe the space of
distributions and the functional of interest on this space. Let $\mathcal{F}
^{\ast }$ be the space of finite signed measures ${\Greekmath 0117} $ on $\mathcal{Y}
\subseteq \mathbb{R}$ with distribution function $F_{{\Greekmath 0117} }\left( y\right)
={\Greekmath 0117} (-\infty ,y]$ for $y\in \mathcal{Y}$. We endow $\mathcal{F}^{\ast }$
with the usual supremum norm: for two distribution functions $F_{{\Greekmath 0117} _{1}}$
and $F_{{\Greekmath 0117} _{2}}$ associated with the respective signed measures ${\Greekmath 0117} _{1}$
and ${\Greekmath 0117} _{2}$ on $\mathcal{Y}$, we define $\left\Vert F_{{\Greekmath 0117} _{1}}-F_{{\Greekmath 0117}
_{2}}\right\Vert _{\infty }:=\sup_{y\in \mathcal{Y}}\left\vert F_{{\Greekmath 0117}
_{1}}\left( y\right) -F_{{\Greekmath 0117} _{2}}\left( y\right) \right\vert .$ With some
abuse of notation, denote $F_{Y}$ as the distribution of $Y:$ $F_{Y}\left(
y\right) ={\Greekmath 0117} _{Y}(-\infty ,y]$ where ${\Greekmath 0117} _{Y}$ is the measure induced by
the distribution of $Y.$ Define $F_{Y_{{\Greekmath 010E} }}$ similarly. Clearly, both $
F_{Y}$ and $F_{Y_{{\Greekmath 010E} }}$ belong to $\mathcal{F}^{\ast }.$ We consider a
general functional ${\Greekmath 011A} :\mathcal{F}^{\ast }\rightarrow \mathbb{R}$ and
study the general unconditional policy effect.
\begin{definition}
\label{gue}\textbf{General Unconditional Policy Effect}
\item The general unconditional policy effect for the functional ${\Greekmath 011A} $ is
defined as
\begin{equation*}
\Pi _{{\Greekmath 011A} }:=\lim_{{\Greekmath 010E} \rightarrow 0}\frac{{\Greekmath 011A} [F_{Y_{{\Greekmath 010E} }}]- {\Greekmath 011A}[
F_{Y}]}{E[D_{{\Greekmath 010E} }]-E[D]},
\end{equation*}
whenever this limit exists.
\end{definition}
In this paper, we consider a Hadamard differentiable functional and its
associated influence function.\footnote{
An earlier working paper \cite{sun2021} considers the mean functional under
different assumptions since the mean functional is not Hadamard
differentiable.} For completeness, we provide the definitions of Hadamard
differentiability and the influence function below.
\begin{definition}
\label{Def: Hadamard}${\Greekmath 011A} :\mathcal{F}^{\ast }\rightarrow \mathbb{R}$ is
Hadamard differentiable at $F\in \mathcal{F}^{\ast }$ if there exists a
linear and continuous functional $\dot{{\Greekmath 011A}}_{F}:\mathcal{F}^{\ast
}\rightarrow \mathbb{R}$ such that for any $G\in \mathcal{F}^{\ast }$ and $
G_{{\Greekmath 010E} }\in \mathcal{F}^{\ast }$ with $\lim_{{\Greekmath 010E} \rightarrow
0}\left\Vert G_{{\Greekmath 010E} }-G\right\Vert _{\infty }=0,$ we have
\begin{equation*}
\lim_{{\Greekmath 010E} \rightarrow 0}\frac{{\Greekmath 011A} \lbrack F+{\Greekmath 010E} G_{{\Greekmath 010E} }]-{\Greekmath 011A}
\lbrack F]}{{\Greekmath 010E} }=\dot{{\Greekmath 011A}}_{F}\left[ G\right] .
\end{equation*}
\end{definition}
\begin{definition}
\label{Def: IF}The influence function of ${\Greekmath 011A} :\mathcal{F}^{\ast
}\rightarrow \mathbb{R}$ at $F\in \mathcal{F}^{\ast }$ is given by
\begin{equation*}
{\Greekmath 0120} (y,{\Greekmath 011A} ,F):=\lim_{{\Greekmath 010F} \rightarrow 0+}\frac{{\Greekmath 011A} \left[ \left(
1-{\Greekmath 010F} \right) F+{\Greekmath 010F} \Delta _{y}\right] -{\Greekmath 011A} \left[ F\right] }{
{\Greekmath 010F} },
\end{equation*}
where $\Delta _{y}$ is the distribution function that assigns all
probability mass to the single point $\left\{ y\right\} ,$ that is, $\Delta
_{y}\left( x\right) =1\left\{ x\geq y\right\} .$
\end{definition}
To see how we can use the Hadamard differentiability to obtain $\Pi _{{\Greekmath 011A} }$
, we write
\begin{equation*}
\frac{{\Greekmath 011A} \lbrack F_{Y_{{\Greekmath 010E} }}]-{\Greekmath 011A} \lbrack F_{Y}]}{{\Greekmath 010E} }=\frac{{\Greekmath 011A}
\left[ F_{Y}+{\Greekmath 010E} G_{{\Greekmath 010E} }\right] -{\Greekmath 011A} \lbrack F_{Y}]}{{\Greekmath 010E} },
\end{equation*}
for
\begin{equation}
G_{{\Greekmath 010E} }=\frac{F_{Y_{{\Greekmath 010E} }}-F_{Y}}{{\Greekmath 010E} }. \label{G_Delta}
\end{equation}
As long as we can show that $\lim_{{\Greekmath 010E} \rightarrow 0}\left\Vert G_{{\Greekmath 010E}
}-G\right\Vert _{\infty }=0$ for some $G,$ then, we obtain
\begin{equation*}
\Pi _{{\Greekmath 011A} }=\lim_{{\Greekmath 010E} \rightarrow 0}\frac{{\Greekmath 011A} \lbrack F_{Y_{{\Greekmath 010E}
}}]-{\Greekmath 011A} \lbrack F_{Y}]}{{\Greekmath 010E} }\left( \frac{E[D_{{\Greekmath 010E} }]-E[D]}{{\Greekmath 010E} }
\right) ^{-1}=\dot{{\Greekmath 011A}}_{F_{Y}}\left[ G\right] \left( \lim_{{\Greekmath 010E}
\rightarrow 0}\frac{E[D_{{\Greekmath 010E} }]-E[D]}{{\Greekmath 010E} }\right) ^{-1}.
\end{equation*}
In the proof of Theorem \ref{Theorem general} below, we show that we can use
the influence function to represent $\dot{{\Greekmath 011A}}_{F_{Y}}\left[ G\right] $ as $
\dot{{\Greekmath 011A}}_{F_{Y}}\left[ G\right] =\int_{\mathcal{Y}}{\Greekmath 0120} (y,{\Greekmath 011A}
,F_{Y})dG\left( y\right) $.
Next, we provide sufficient conditions for $\lim_{{\Greekmath 010E} \rightarrow
0}\left\Vert G_{{\Greekmath 010E} }-G\right\Vert _{\infty }=0$. We first formalize two
primary assumptions.\footnote{
In Assumptions \ref{Assumption_primary} and \ref{Assumption_regularity},
\textquotedblleft for all $w\in \mathcal{W}$\textquotedblright\ can be
replaced by \textquotedblleft for almost all $w\in \mathcal{W}$
\textquotedblright , and the supremum over $w\in \mathcal{W} $ can be
replaced by the essential supremum over $w\in \mathcal{W}$.}
\begin{assumption}
\textbf{Primary Assumptions} \label{Assumption_primary}
\begin{enumerate}[(a)]
\item \label{uniformity}$U_{D}$ is independent of $W$ and is uniformly
distributed on $\left[ 0,1\right] .$
\item \label{feasibility}For all $w=(z^{\prime },x^{\prime })^{\prime }\in
\mathcal{W}$, $\mathcal{G}(w,0)=z,$ and for sufficiently small ${\Greekmath 010E} $, $
\mathcal{G}(w,{\Greekmath 010E} )\in \mathcal{Z}$, the support of $Z.$
\end{enumerate}
\end{assumption}
As discussed earlier, Assumption \ref{Assumption_primary}(\ref{uniformity})
holds if the conditional CDF $F_{V|W}(\cdot |w)$ is a strictly increasing
function for all $w\in \mathcal{W}$. Assumption \ref{Assumption_primary}(\ref
{feasibility}) requires that the policy function $\mathcal{G}(w,{\Greekmath 010E} )$ be
feasible. This assumption, along with Assumption \ref{Assumption_regularity}(
\ref{p_delta}.ii) below, requires that the subvector of $Z$ undergoing
nontrivial intervention consists of continuous random variables. The
variables in $X$ and the part of $Z$ not subject to intervention do not need
to be continuous random variables.
\begin{assumption}
\textbf{Regularity Conditions} \label{Assumption_regularity}
\begin{enumerate}[(a)]
\item \label{regularity_x_abs}For $d=0,1,$ the conditional distribution of $
(Y(d),U_{D})$ conditional on $W=w\in \mathcal{W}$ is absolutely continuous
with conditional density function given by $f_{Y(d),U_{D}|W}(y,u|w).$
\item \label{f_y_u_x} (i) For $d=0,1,$ $u\mapsto f_{Y(d)|U_{D},W}(y|u,w)$ is
continuous for all $y\in \mathcal{Y}\left( d\right) $ and all $w\in \mathcal{
W}.$ (ii) For $d=0,1,$ $\sup_{y\in \mathcal{Y}\left( d\right) }\sup_{w\in
\mathcal{W}}\sup_{{\Greekmath 010E} \in N_{{\Greekmath 0122} }}f_{Y(d)|U_{D},W}(y|P_{{\Greekmath 010E}
}(w),w)<\infty $ where $N_{{\Greekmath 0122} }:=\left\{ {\Greekmath 010E} :\left\vert {\Greekmath 010E}
\right\vert \leq {\Greekmath 0122} \right\} $ for some ${\Greekmath 0122} >0.$
\item \label{p_delta}(i) For all $w\in \mathcal{W}$, $P(w)\in (0,1).$ (ii)
For all $w\in \mathcal{W}$, the map ${\Greekmath 010E} \mapsto P_{{\Greekmath 010E} }(w)$ is
continuously differentiable on $N_{{\Greekmath 0122} }.$ (iii) $\sup_{w\in
\mathcal{W}}\sup_{{\Greekmath 010E} \in N_{{\Greekmath 0122} }}\left\vert \frac{\partial
P_{{\Greekmath 010E} }(w)}{\partial {\Greekmath 010E} }\right\vert <\infty .$
\end{enumerate}
\end{assumption}
\begin{assumption}
\textbf{Domination Conditions} \label{Assumption_domination} For $d=0,1,$
\begin{equation*}
\int_{\mathcal{Y}(d)}\sup_{{\Greekmath 010E} \in N_{{\Greekmath 0122} }}f_{Y(d)|D_{{\Greekmath 010E}
}}(y|d)dy<\infty \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ and }\int_{\mathcal{Y}(d)}\sup_{{\Greekmath 010E} \in
N_{{\Greekmath 0122} }}\left\vert \frac{\partial f_{Y(d)|D_{{\Greekmath 010E} }}(y|d)}{
\partial {\Greekmath 010E} }\right\vert dy<\infty .
\end{equation*}
\end{assumption}
Under Assumption \ref{Assumption_regularity}(\ref{f_y_u_x}, \ref{p_delta}),
we have $E\left[ P_{{\Greekmath 010E} }(W)\right] =E\left[ P(W)\right] +E\left[ \dot{P}
\left( W\right) \right] {\Greekmath 010E} +o({\Greekmath 010E} ),$ as ${\Greekmath 010E} \rightarrow 0$,
where
\begin{equation*}
\dot{P}\left( w\right) =\left. \frac{\partial P_{{\Greekmath 010E} }\left( w\right) }{
\partial {\Greekmath 010E} }\right\vert _{{\Greekmath 010E} =0}.
\end{equation*}
So, under the new policy regime $\mathcal{G}(\cdot ,{\Greekmath 010E} )$, the
participation rate in the treatment is increased by {approximately }$E\left[
\dot{P}\left( W\right) \right] {\Greekmath 010E} .$ Locally at ${\Greekmath 010E} =0$, the policy
function $\mathcal{G}(\cdot ,{\Greekmath 010E} )$ affects the participation rate via $
\dot{P}\left( \cdot \right) $, which is the rate of change in the propensity
score at ${\Greekmath 010E} =0$. The function $\dot{P}\left( \cdot \right) $ depends on
the policy function $\mathcal{G}(\cdot ,{\Greekmath 010E} )$ used, and a more
cumbersome notation for $\dot{P}\left( \cdot \right) $ is $\dot{P}_{\mathcal{
G}}\left( \cdot \right) .$ For notational simplicity, we suppress such
dependence. Section \ref{section_dot_p} provides further analysis on $\dot{P}
\left( \cdot \right) $.
\begin{theorem}
\label{Theorem general}Let Assumptions \ref{Assumption_primary}--\ref
{Assumption_domination} hold. Assume further that ${\Greekmath 011A} :\mathcal{F}^{\ast
}\rightarrow \mathbb{R}$ is Hadamard differentiable with influence function $
{\Greekmath 0120} $, and $E\left[ \dot{P}\left( W\right) \right] \neq 0$. Then,
\begin{equation*}
\Pi _{{\Greekmath 011A} }=\int_{\mathcal{W}}\mathrm{MTE}_{{\Greekmath 011A} }\left( P(w),w\right)
\mathcal{\dot{P}}\left( w\right) dF_{W}(w),
\end{equation*}
where
\begin{equation}
\mathrm{MTE}_{{\Greekmath 011A} }\left( u_{D},w\right) :=E\left[ {\Greekmath 0120} (Y\left( 1\right)
,{\Greekmath 011A} ,F_{Y})-{\Greekmath 0120} (Y\left( 0\right) ,{\Greekmath 011A} ,F_{Y})|U_{D}=u_{D},W=w\right] ,
\label{MTE_rho}
\end{equation}
is the unconditional marginal treatment effect for the ${\Greekmath 011A} $ functional,
and
\begin{equation*}
\mathcal{\dot{P}}\left( w\right) =\frac{\dot{P}\left( w\right) }{E\left[
\dot{P}\left( W\right) \right] }.
\end{equation*}
\end{theorem}
Theorem \ref{Theorem general} reveals that the general unconditional policy
effect $\Pi _{{\Greekmath 011A} }$ is composed of two key elements: the unconditional $
\mathrm{MTE}_{{\Greekmath 011A} }\left( u_{D},w\right) $ evaluated at $u_{D}=P(w)$ and
the weighting function $\mathcal{\dot{P}}\left( w\right) .$ To understand
the first element, consider the group of individuals with the same value $w$
of $W.$ Within this group, those for whom $u_{D}=P\left( w\right) $ are
indifferent between participating and not participating. A small incentive
will induce a change in the treatment status for and only for this subgroup
of individuals. It is the change in their treatment status, and hence the
change in the composition of $Y(1)$ and $Y(0)$ in the observed outcome $Y,$
that changes its unconditional characteristics, such as the quantiles. We
refer to the individuals for whom\ $u_{D}=P\left( w\right) $ as the marginal
subpopulation. As for the second element, we defer the discussion to Section
\ref{section_dot_p}.
\subsection{Unconditional quantile effect}
Throughout the rest of this paper, we consider the case that ${\Greekmath 011A} $ is a
quantile functional at the quantile level ${\Greekmath 011C} \in \left( 0,1\right) ,$
that is, ${\Greekmath 011A} _{{\Greekmath 011C} }[F_{Y}]=F_{Y}^{-1}({\Greekmath 011C} ):=\inf_{y}\left\{ y\in
\mathcal{Y}:F_{Y}(y)\geq {\Greekmath 011C} \right\} .$ Here we have added a subscript $
{\Greekmath 011C} $ to ${\Greekmath 011A} $ to signify the quantile level under consideration. We are
interested in how an improvement in the treatment take-up rate affects the $
{\Greekmath 011C} $-quantile of the (unconditional) outcome distribution.
\begin{definition}
\textbf{Unconditional Quantile Effect}
\item The unconditional quantile effect (UQE) is defined as
\begin{equation*}
\Pi _{{\Greekmath 011C} }:=\lim_{{\Greekmath 010E} \rightarrow 0}\frac{{\Greekmath 011A} _{{\Greekmath 011C} }[F_{Y_{{\Greekmath 010E}
}}]-{\Greekmath 011A} _{{\Greekmath 011C} }[F_{Y}]}{E[D_{{\Greekmath 010E} }]-E[D]}
\end{equation*}
whenever this limit exists.
\end{definition}
Let $y_{{\Greekmath 011C} }\equiv {\Greekmath 011A} _{{\Greekmath 011C} }[F_{Y}]$ be the ${\Greekmath 011C} $-quantile of $Y.$
If $f_{Y}(y_{{\Greekmath 011C} })>0,$ then, under Assumption \ref{Assumption_regularity}(
\ref{regularity_x_abs}), the influence function of the ${\Greekmath 011C} $-quantile
functional is\footnote{
Strictly speaking, ${\Greekmath 0120} \left( y,{\Greekmath 011A} _{{\Greekmath 011C} },F_{Y}\right) =0$ for $
y=y_{{\Greekmath 011C} }$. However, redefining ${\Greekmath 0120} \left( y,{\Greekmath 011A} _{{\Greekmath 011C} },F_{Y}\right) $
at one point has no consequence on our results, as $F_{Y}\left( \cdot
\right) $ is absolutely continuous under Assumption \ref
{Assumption_regularity} (\ref{regularity_x_abs}).}
\begin{equation*}
{\Greekmath 0120} \left( y,{\Greekmath 011A} _{{\Greekmath 011C} },F_{Y}\right) =\frac{1}{f_{Y}\left( y_{{\Greekmath 011C}
}\right) }\left[ {\Greekmath 011C} -\mathds{1}\left\{ y\leq y_{{\Greekmath 011C} }\right\} \right] .
\end{equation*}
Plugging this influence function into (\ref{MTE_rho}), we obtain the
unconditional marginal treatment effect for the ${\Greekmath 011C} $-quantile.
\begin{definition}
\label{MTE_tau} The unconditional marginal treatment effect for the ${\Greekmath 011C} $
-quantile is
\begin{equation*}
\mathrm{MTE}_{{\Greekmath 011C} }\left( u,w\right) =\frac{1}{f_{Y}\left( y_{{\Greekmath 011C} }\right)
}E\left[ \mathds{1}\left\{ Y(0)\leq y_{{\Greekmath 011C} }\right\} -\mathds{1}\left\{
Y(1)\leq y_{{\Greekmath 011C} }\right\} \mid U_{D}=u,W=w\right] .
\end{equation*}
\end{definition}
The $\mathrm{MTE}_{{\Greekmath 011C} }$ defined above is a basic building block for the
unconditional quantile effect. It is different from the quantile analogue of
the marginal treatment effect of \cite{Carneiro2009} and \cite{ping2014},
which is defined as $F_{Y(1)|U_{D},W}^{-1}({\Greekmath 011C}
|u,w)-F_{Y(0)|U_{D},W}^{-1}({\Greekmath 011C} |u,w)$. An unconditional quantile effect
can not be represented as an integrated version of the latter. The next
corollary follows from applying Theorem \ref{Theorem general} to ${\Greekmath 011A}_{\Greekmath 011C}$.
\begin{corollary}
\label{Corollary_UQTE0} Let Assumptions \ref{Assumption_primary}--\ref
{Assumption_domination} hold. Assume further that $f_{Y}(y_{{\Greekmath 011C} })>0$.
Then,
\begin{eqnarray}
\Pi _{{\Greekmath 011C} } &=&\int_{\mathcal{W}}\mathrm{MTE}_{{\Greekmath 011C} }\left( P(w),w\right)
\mathcal{\dot{P}}\left( w\right) dF_{W}(w) \notag \\
&=&\frac{1}{f_{Y}(y_{{\Greekmath 011C} })}\int_{\mathcal{W}}E\left[ \mathds{1}\left\{
Y(0)\leq y_{{\Greekmath 011C} }\right\} |U_{D}=P\left( w\right) ,W=w\right] \mathcal{\dot{
P}}\left( w\right) dF_{W}\left( w\right) \notag \\
&-&\frac{1}{f_{Y}(y_{{\Greekmath 011C} })}\int_{\mathcal{W}}E\left[ \mathds{1}\left\{
Y(1)\leq y_{{\Greekmath 011C} }\right\} |U_{D}=P\left( w\right) ,W=w\right] \mathcal{\dot{
P}}\left( w\right) dF_{W}\left( w\right) . \label{UQTE_endogeneity}
\end{eqnarray}
\end{corollary}
\subsection{Understanding $\mathrm{MTE}_{\protect{\Greekmath 011C} }\left( u,w\right)$}
To understand $\mathrm{MTE}_{{\Greekmath 011C} }\left( u,w\right) ,$ we define $\Delta
(y_{{\Greekmath 011C} }):=\left( \mathds{1}\left\{ Y(0)\leq y_{{\Greekmath 011C} }\right\} -\mathds{1}
\left\{ Y(1)\leq y_{{\Greekmath 011C} }\right\} \right) /f_{Y}(y_{{\Greekmath 011C} }),$ which
underlies the above definition of $\mathrm{MTE}_{{\Greekmath 011C} }\left( u,w\right) $.
The random variable $\Delta (y_{{\Greekmath 011C} })$ can take three values:
\begin{equation*}
\Delta (y_{{\Greekmath 011C} })=
\begin{cases}
\begin{aligned} &f_Y(y_{\Greekmath 011C})^{-1}&\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ if }&Y(0)\leq y_{{\Greekmath 011C} }\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ and
}Y(1)>y_{{\Greekmath 011C} } \\ &0&\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if }&\bigg[Y(0)>y_{{\Greekmath 011C} }\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ and
}Y(1)>y_{{\Greekmath 011C} }\bigg]\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ or }\bigg[Y(0)\leq y_{{\Greekmath 011C} }\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ and }Y(1)\leq
y_{{\Greekmath 011C} }\bigg] \\ &-f_Y(y_{\Greekmath 011C})^{-1}&\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ if }&Y(0)>y_{{\Greekmath 011C} }\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ and
}Y(1)\leq y_{{\Greekmath 011C} }\end{aligned}
\end{cases}
\end{equation*}
For a given individual, $\Delta (y_{{\Greekmath 011C} })=1/f_{Y}(y_{{\Greekmath 011C} })$ when the
treatment induces the individual to \textquotedblleft
cross\textquotedblright\ the ${\Greekmath 011C} $-quantile $y_{{\Greekmath 011C} }$ of $Y$ from below,
and $\Delta (y_{{\Greekmath 011C} })=-1/f_{Y}(y_{{\Greekmath 011C} })$ when the treatment induces the
individual to \textquotedblleft cross\textquotedblright\ the ${\Greekmath 011C} $
-quantile $y_{{\Greekmath 011C} }$ of $Y$ from above. In the first case, the individual
benefits from the treatment, while in the second case, the treatment harms
her. The intermediate case, $\Delta (y_{{\Greekmath 011C} })=0,$ occurs when the
treatment induces no quantile crossing of any type. Thus, the unconditional
expected value $f_{Y}(y_{{\Greekmath 011C} })\times E[\Delta (y_{{\Greekmath 011C} })]$ equals the
difference between the proportion of individuals who benefit from the
treatment and the proportion of individuals who are harmed by it. For the
UQE, whether the treatment is beneficial or harmful is measured in terms of
quantile crossing. Among the individuals with characteristics $U_{D}=u$ and $
W=w$, $\mathrm{MTE}_{{\Greekmath 011C} }\left( u,w\right) $ is then equal to the rescaled
(by $1/f_{Y}(y_{{\Greekmath 011C} })$) difference between the proportion of individuals
who benefit from the treatment and the proportion of individuals who are
harmed by it. Thus, $\mathrm{MTE}_{{\Greekmath 011C} }\left( u,w\right) $ is positive if
the treatment leads to a greater number of individuals improving their
outcome above the threshold $y_{{\Greekmath 011C} }$, compared to those whose outcome
falls below $y_{{\Greekmath 011C} }$. Conversely, $\mathrm{MTE}_{{\Greekmath 011C} }\left( u,w\right) $
is negative if the treatment leads to a greater number of individuals
experiencing a decline in their outcome, falling below $y_{{\Greekmath 011C} }$, compared
to those experiencing an increase above $y_{{\Greekmath 011C} }.$
\subsection{Understanding $\mathcal{\dot{P}}$}
\label{section_dot_p}
To understand the weighting function $\mathcal{\dot{P}}\left( w\right) ,$
consider the case ${\Greekmath 010E} >0$, $P_{{\Greekmath 010E} }(w)\geq P(w)$ for all $w\in W$,
and $F_{W}\left( \cdot \right) $ is absolutely continuous with density $
f_{W}\left( \cdot \right) .$ Let ${\Greekmath 010F} $ be a small positive number.
Then, $f_{W}\left( w\right) {\Greekmath 010F} $ measures the proportion of
individuals for whom $W$ is in $\left[ w-{\Greekmath 010F} /2,w+{\Greekmath 010F} /2\right] .$
Note that for $W\in \left[ w-{\Greekmath 010F} /2,w+{\Greekmath 010F} /2\right] ,$ the
propensity scores under $D$ and $D_{{\Greekmath 010E} }$ are approximately $P(w)$ and $
P_{{\Greekmath 010E} }\left( w\right) $. The proportion of the individuals for whom $
W\in \left[ w-{\Greekmath 010F} /2,w+{\Greekmath 010F} /2\right] $ and who have switched their
treatment status from $0$ to $1$ is then equal to $\left[ P_{{\Greekmath 010E} }(w)-P(w)
\right] f_{W}\left( w\right) \cdot {\Greekmath 010F} .$ Scaling this by $E[D_{{\Greekmath 010E}
}]-E[D]$, which is the overall proportion of the individuals who have
switched the treatment status, we obtain
\begin{equation*}
\frac{\left[ P_{{\Greekmath 010E} }(w)-P(w)\right] f_{W}\left( w\right) }{E[D_{{\Greekmath 010E}
}]-E[D]}\cdot {\Greekmath 010F} .
\end{equation*}
Thus, we can regard $\left[ P_{{\Greekmath 010E} }(w)-P(w)\right] f_{W}\left( w\right) /
\left[ E[D_{{\Greekmath 010E} }]-E[D]\right] $ as the density function of $W$ among
those who have switched their treatment status from $0$ to $1$ as a result
of the policy intervention. On the one hand, when ${\Greekmath 010E} \rightarrow 0,$
the set of individuals who change their treatment status are the individuals
on the margin, namely, those for whom $u_{D}=P(w).$ On the other hand, when $
{\Greekmath 010E} \rightarrow 0,$ the density function approaches $\mathcal{\dot{P}}
\left( w\right) f_{W}\left( w\right) .$ So $\mathcal{\dot{P}}\left( w\right)
f_{W}\left( w\right) $ is the probability density function (with respect to
the Lebesgue measure) of the distribution of $W$ over the marginal
subpopulation. Also, note that by construction,\ $\int_{\mathcal{W}}\mathcal{
\dot{P}}\left( w\right) dF(w)=1$, and thus $\mathcal{\dot{P}}\left( w\right)
$ can be interpreted as the density of the distribution of $W$ for the
marginal subpopulation with respect to the distribution of $W$ for the
entire population.
It is now clear that the general unconditional policy effect is equal to the
average of the unconditional $\mathrm{MTE}$ over the marginal subpopulation.
Such an interpretation is still valid even if $\mathcal{\dot{P}}\left(
w\right) $ is not positive for all $w\in \mathcal{W}$. In this case, we only
need to view the distribution with density $\mathcal{\dot{P}}\left( w\right)
$ (with respect to the distribution of $W$ for the entire population) as a
signed measure.
\subsection{Understanding the Unconditional MTE}
To gain a deeper understanding of the unconditional MTE, both in the general
case and the special quantile case, we will explore another perspective
here. Note that this subsection will only provide heuristics, as the formal
developments have already been covered in the previous subsections.
\cite{heckman_prte, Heckman2005} focus on the mean functional and consider
the \emph{policy-relevant treatment effect} defined as
\begin{equation}
\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{\textrm{PRTE}}_{{\Greekmath 010E} }=\frac{E[Y_{{\Greekmath 010E} }]-E[Y]}{E[D_{{\Greekmath 010E} }]-E[D]
}. \label{prte}
\end{equation}
Taking the limit ${\Greekmath 010E} \rightarrow 0$ yields the \emph{marginal
policy-relevant treatment effect} (\textrm{MPRTE}) of \cite{Carneiro2010}: $
\mathrm{MPRTE}=\lim_{{\Greekmath 010E} \rightarrow 0}\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{\textrm{PRTE}}_{{\Greekmath 010E} }.$
\cite{Carneiro2010, Carneiro2011} show that $\mathrm{MPRTE}$ can be
represented in terms of the conventional marginal treatment effect defined
by $\mathrm{MTE}(u):=E\left[ Y(1)-Y(0)|U_{D}=u\right] $. These results are
applicable to the mean functional only.
For a general functional ${\Greekmath 011A} $ that is Hadamard differentiable, we have
\begin{equation*}
\lim_{{\Greekmath 010E} \rightarrow 0}\frac{{\Greekmath 011A} \lbrack F_{Y_{{\Greekmath 010E} }}]-{\Greekmath 011A} \lbrack
F_{Y}]}{E[D_{{\Greekmath 010E} }]-E[D]}=\dot{{\Greekmath 011A}}_{F_{Y}}\left( \lim_{{\Greekmath 010E}
\rightarrow 0}\frac{F_{Y_{{\Greekmath 010E} }}-F_{Y}}{E[D_{{\Greekmath 010E} }]-E[D]}\right) ,
\end{equation*}
where $\dot{{\Greekmath 011A}}_{F_{Y}}$ is the derivative of ${\Greekmath 011A} $ at $F_{Y}$. One way
to represent the limit on the right-hand side is to replace $E\left(
Y_{{\Greekmath 010E} }\right) $ and $E\left( Y\right) $ in (\ref{prte}) by $E[\mathds
1\left\{ Y_{{\Greekmath 010E} }\leq y\right\} ]$ and $E[\mathds1\left\{ Y\leq y\right\}
],$ respectively. So, for a given $y\in \mathcal{Y}$,
\begin{equation}
\frac{F_{Y_{{\Greekmath 010E} }}\left( y\right) -F_{Y}\left( y\right) }{E[D_{{\Greekmath 010E}
}]-E[D]}=\frac{E[\mathds1\left\{ Y_{{\Greekmath 010E} }\leq y\right\} -E[\mathds
1\left\{ Y\leq y\right\} ]}{E[D_{{\Greekmath 010E} }]-E[D]}.
\end{equation}
In the spirit of \cite{heckman_prte, Heckman2005}, we may then regard the
above as a policy-relevant treatment effect: it is the treatment effect of
the policy on the percentage of individuals whose value of $Y$ is less than $
y.$ The effect is tied to a particular value $y$, and we obtain a continuum
of policy-relevant effects indexed by $y\in \mathcal{Y}$ if $y$ is allowed
to vary over $\mathcal{Y}$. The limit
\begin{equation*}
\lim_{{\Greekmath 010E} \rightarrow 0}\frac{F_{Y_{{\Greekmath 010E} }}\left( y\right) -F_{Y}\left(
y\right) }{E[D_{{\Greekmath 010E} }]-E[D]}
\end{equation*}
can then be regarded as a continuum of marginal policy-relevant treatment
effects indexed by $y\in \mathcal{Y}$. By the results of \cite{Carneiro2009}
, for each $y\in \mathcal{Y}$, the marginal policy-relevant treatment effect
can be represented as a weighted integral of the following \emph{
policy-relevant \textquotedblleft distributional\textquotedblright }\ MTE:
\begin{eqnarray*}
\mathrm{MTE}_{d}(u,w;y) &=&E\left[ \mathds1\left\{ Y(1)\leq y\right\} -
\mathds1\left\{ Y(0)\leq y\right\} |U_{D}=u,W=w\right] \\
&=&F_{Y\left( 1\right) |U_{D},W}\left( y|u,w\right) -F_{Y\left( 0\right)
|U_{D},W}\left( y|u,w\right) .
\end{eqnarray*}
The composition $\dot{{\Greekmath 011A}}_{F_{Y}}\circ \mathrm{MTE}_{d}$, defined as $
\int_{\mathcal{Y}}{\Greekmath 0120} (y,{\Greekmath 011A} ,F_{Y})\mathrm{MTE}_{d}(u,w;dy),$ is then
\begin{eqnarray*}
\left( \dot{{\Greekmath 011A}}_{F_{Y}}\circ \mathrm{MTE}_{d}\right) \left( u,w\right)
&=&\int_{\mathcal{Y}}{\Greekmath 0120} (y,{\Greekmath 011A} ,F_{Y})\left[ F_{Y\left( 1\right)
|U_{D},W}\left( dy|u,w\right) -F_{Y\left( 0\right) |U_{D},W}\left(
dy|u,w\right) \right] \\
&=&\int_{\mathcal{Y}}{\Greekmath 0120} (y,{\Greekmath 011A} ,F_{Y})\left[ f_{Y\left( 1\right)
|U_{D},W}\left( y|u,w\right) -f_{Y\left( 0\right) |U_{D},W}\left(
y|u,w\right) \right] dy \\
&=&E\left[ {\Greekmath 0120} (Y\left( 1\right) ,{\Greekmath 011A} ,F_{Y})-{\Greekmath 0120} (Y\left( 0\right) ,{\Greekmath 011A}
,F_{Y})|U_{D}=u,W=w\right] .
\end{eqnarray*}
This shows that $\dot{{\Greekmath 011A}}_{F_{Y}}\circ \mathrm{MTE}_{d}$ is exactly the
unconditional MTE defined in (\ref{MTE_rho}). The unconditional MTE is
therefore a composition of the underlying influence function with the \emph{
policy-relevant distributional} MTE.
\section{UQR with a Threshold-crossing Model}
\label{section_uqr}
In this section, we study whether the UQR proposed by \cite{Firpo2009} can
provide a consistent estimator of UQE. When it is inconsistent, we
investigate the sources of asymptotic bias in the UQR estimator and show
that it is asymptotically biased, even when $D$ is exogenous. As a result,
the UQR may not be suitable for estimating unconditional policy effects when
the treatment variable follows a binary threshold-crossing model. In Section
\ref{estimation}, we will present a consistent estimator of the
unconditional policy effect for such a model.
\subsection{UQR with a Binary Regressor}
We provide a quick review of the UQR. As before, let $y_{{\Greekmath 011C} }$ be the $
{\Greekmath 011C} $-quantile of $Y,$ and let $y_{{\Greekmath 011C} ,{\Greekmath 010E} }$ be the ${\Greekmath 011C} $-quantile
of $Y_{{\Greekmath 010E} }$. That is, $\Pr [Y\leq y_{{\Greekmath 011C} }]=\Pr [Y_{{\Greekmath 010E} }\leq
y_{{\Greekmath 011C} ,{\Greekmath 010E} }]={\Greekmath 011C} .$ By definition, we have
\begin{equation*}
\Pr \left[ Y_{{\Greekmath 010E} }\leq y\right] =\Pr \left[ D_{{\Greekmath 010E} }=1\right] \Pr
\left[ Y_{{\Greekmath 010E} }\leq y|D_{{\Greekmath 010E} }=1\right] +\Pr \left[ D_{{\Greekmath 010E} }=0
\right] \Pr \left[ Y_{{\Greekmath 010E} }\leq y|D_{{\Greekmath 010E} }=0\right] .
\end{equation*}
When $W$ is not present, Corollary 3 in the working paper \cite{Firpo2007}
makes the following assumption to achieve identification:
\begin{equation}
\Pr \left[ Y_{{\Greekmath 010E} }\leq y|D_{{\Greekmath 010E} }=d\right] =\Pr \left[ Y\leq y|D=d
\right] \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ for }d=0\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ and }1. \label{key_assumption}
\end{equation}
We refer to this assumption as \emph{distributional invariance}, and it
readily identifies the counterfactual distribution:
\begin{equation*}
\Pr \left[ Y_{{\Greekmath 010E} }\leq y\right] =\Pr \left[ D_{{\Greekmath 010E} }=1\right] \Pr
\left[ Y\leq y|D=1\right] +\Pr \left[ D_{{\Greekmath 010E} }=0\right] \Pr \left[ Y\leq
y|D=0\right] .
\end{equation*}
Under some mild conditions, we can follow \cite{Firpo2007} to show that
\begin{eqnarray}
\Pi _{{\Greekmath 011C} } &:=&\lim_{{\Greekmath 010E} \rightarrow 0}\frac{{\Greekmath 011A} _{{\Greekmath 011C} }[F_{Y_{{\Greekmath 010E}
}}]-{\Greekmath 011A} _{{\Greekmath 011C} }[F_{Y}]}{E[D_{{\Greekmath 010E} }]-E[D]} \notag \\
&=&\frac{1}{f_{Y}(y_{{\Greekmath 011C} })}\left( \Pr [Y>y_{{\Greekmath 011C} }|D=1]-\Pr [Y>y_{{\Greekmath 011C}
}|D=0]\right) . \label{pi_no_W}
\end{eqnarray}
Hence, under the distributional invariance assumption\emph{, }the UQE can be
consistently estimated by regressing $1\left\{ Y\geq y_{{\Greekmath 011C} }\right\}
/f_{Y}(y_{{\Greekmath 011C} })$ on a constant and $D.$ Such a regression with no
additional regressor $W$ is a special case of more general unconditional
quantile regressions.
\subsection{Asymptotic Bias of the UQR Estimator}
The distributional invariance assumption given in equation
\eqref{key_assumption} is crucial for achieving\ the identification result
in \eqref{pi_no_W}. It states that the conditional distribution of the
outcome variable given the treatment status remains the same across the two
policy regimes. If treatments are randomly assigned under both policy
regimes (e.g., $D_{{\Greekmath 010E} }=\mathds{1}\left\{ U_{D}\leq P_{{\Greekmath 010E}
}(W)\right\} $ and $\left( U_{D},W\right) $ is independent of $\left(
U_{0},U_{1}\right) ),$ then $\left( U_{0},U_{1}\right) $ is clearly
independent of $D_{{\Greekmath 010E} }$. In this case, both $\Pr \left[ Y_{{\Greekmath 010E} }\leq
y|D_{{\Greekmath 010E} }=d\right] $ and $\Pr \left[ Y\leq y|D=d\right] $ are equal to $
\Pr \left[ Y\left( d\right) \leq y\right] ,$ and the distributional
invariance assumption is satisfied. However, when $D_{{\Greekmath 010E} }$ is allowed
to be correlated with $U=\left( U_{0},U_{1}\right) ^{\prime }$, the
distributional invariance assumption does not hold in general. For example,
when $d=1,$
\begin{equation*}
\Pr \left[ Y_{{\Greekmath 010E} }\leq y|D_{{\Greekmath 010E} }=1\right] =\Pr \left[ Y\left(
1\right) \leq y|D_{{\Greekmath 010E} }=1\right] =\Pr \left[ r_{1}\left( X,U_{1}\right)
\leq y|U_{D}\leq P_{{\Greekmath 010E} }\left( W\right) \right] ,
\end{equation*}
and $\Pr [Y\leq y|D=1]=\Pr \left[ r_{1}\left( X,U_{1}\right) \leq
y|U_{D}\leq P\left( W\right) \right] .$ These two conditional probabilities
are different under the general dependence of $(W,U,U_{D}).$
To allow for the endogeneity of $D$, we have dropped the distributional
invariance assumption and assumed a threshold-crossing model as in (\ref
{model1D}). The next corollary decomposes the unconditional quantile effect
given in Corollary \ref{Corollary_UQTE0} into two components. The
decomposition reveals that the UQR estimator of \cite{Firpo2009} is
asymptotically biased under a wide range of conditions, including when $D$
is exogenous.
\begin{corollary}
\label{Corollary_UQTE}Let Assumptions \ref{Assumption_primary}--\ref
{Assumption_domination} hold. Assume further that $f_{Y}(y_{{\Greekmath 011C} })>0$. Then
\footnote{
We use \textquotedblleft A\textquotedblright\ to denote the \textbf{A}
pparent component and use \textquotedblleft B\textquotedblright\ to denote
the \textbf{B}ias component.}
\begin{equation*}
\Pi _{{\Greekmath 011C} }=A_{{\Greekmath 011C} }-B_{{\Greekmath 011C} },
\end{equation*}
where
\begin{eqnarray}
A_{{\Greekmath 011C} } &=&\frac{1}{f_{Y}(y_{{\Greekmath 011C} })}\int_{\mathcal{W}}E\left[ \mathds{1}
\left\{ Y\leq y_{{\Greekmath 011C} }\right\} |D=0,W=w\right] dF_{W}\left( w\right) \notag
\\
&-&\frac{1}{f_{Y}(y_{{\Greekmath 011C} })}\int_{\mathcal{W}}E\left[ \mathds{1}\left\{
Y\leq y_{{\Greekmath 011C} }\right\} |D=1,W=w\right] dF_{W}\left( w\right) ,
\label{app_eff}
\end{eqnarray}
and $B_{{\Greekmath 011C} }=B_{1{\Greekmath 011C} }+B_{2{\Greekmath 011C} }$, for
\begin{eqnarray*}
B_{1{\Greekmath 011C} } &=&\frac{1}{f_{Y}(y_{{\Greekmath 011C} })}\int_{\mathcal{W}}\left[
F_{Y|D,W}\left( y_{{\Greekmath 011C} }|1,w\right) -F_{Y|D,W}\left( y_{{\Greekmath 011C} }|0,w\right)
\right] \mathcal{\dot{P}}\left( w\right) dF_{W}\left( w\right) \\
&-&\frac{1}{f_{Y}(y_{{\Greekmath 011C} })}\int_{\mathcal{W}}\left[ F_{Y|D,W}\left(
y_{{\Greekmath 011C} }|1,w\right) -F_{Y|D,W}\left( y_{{\Greekmath 011C} }|0,w\right) \right]
dF_{W}\left( w\right)
\end{eqnarray*}
and
\begin{eqnarray*}
B_{2{\Greekmath 011C} } &=&\frac{1}{f_{Y}(y_{{\Greekmath 011C} })}\int_{\mathcal{W}}\left[ F_{Y\left(
0\right) |D,W}\left( y_{{\Greekmath 011C} }|0,w\right) -F_{Y\left( 0\right)
|U_{D},W}\left( y_{{\Greekmath 011C} }|P\left( w\right) ,w\right) \right] \mathcal{\dot{P}
}\left( w\right) dF_{W}\left( w\right) \\
&-&\frac{1}{f_{Y}(y_{{\Greekmath 011C} })}\int_{\mathcal{W}}\left[ F_{Y\left( 1\right)
|D,W}\left( y_{{\Greekmath 011C} }|1,w\right) -F_{Y\left( 1\right) |U_{D},W}\left(
y_{{\Greekmath 011C} }|P\left( w\right) ,w\right) \right] \mathcal{\dot{P}}\left(
w\right) dF_{W}\left( w\right) . \\
&&
\end{eqnarray*}
\end{corollary}
To facilitate understanding of Corollary \ref{Corollary_UQTE}, we define and
organize the average influence functions (AIF) in a table:
\begin{equation*}
\begin{tabular}{c|c}
\hline\hline
& AIF for $Y\left( 1\right) $ $-$ AIF for $Y\left( 0\right) $ \\ \hline
$U_{D}$ & $E_{w}\left[ {\Greekmath 0120} _{{\Greekmath 011C} }\left( Y\left( 1\right) \right) -{\Greekmath 0120}
_{{\Greekmath 011C} }\left( Y\left( 0\right) \right) |U_{D}=P\left( w\right) \right] $ \\
$D$ & $E_{w}\left[ {\Greekmath 0120} _{{\Greekmath 011C} }\left( Y\left( 1\right) \right) |D=1\right]
-E_{w}\left[ {\Greekmath 0120} _{{\Greekmath 011C} }\left( Y\left( 0\right) \right) |D=0\right] $ \\
\hline
\end{tabular}
\end{equation*}
\noindent where ${\Greekmath 0120} _{{\Greekmath 011C} }\left( \cdot \right) $ is short for ${\Greekmath 0120}
(\cdot ,{\Greekmath 011A} _{{\Greekmath 011C} },F_{Y})$, the influence function of the quantile
functional. In the above, $E_{w}\left[ \cdot \right] $ stands for the
conditional mean operator given $W=w.$ For example, $E_{w}\left[ {\Greekmath 0120} _{{\Greekmath 011C}
}\left( Y\left( 0\right) \right) |D=0\right] $ stands for $E\left[ {\Greekmath 0120}
_{{\Greekmath 011C} }\left( Y\left( 0\right) \right) |D=0,W=w\right] .$ Let
\begin{eqnarray*}
{\Greekmath 0120} _{\Delta ,U_{D}}\left( w\right) := &&E_{w}\left[ {\Greekmath 0120} _{{\Greekmath 011C} }\left(
Y\left( 1\right) \right) -{\Greekmath 0120} _{{\Greekmath 011C} }\left( Y\left( 0\right) \right)
|U_{D}=P\left( w\right) \right] , \\
{\Greekmath 0120} _{\Delta ,D}\left( w\right) := &&E_{w}\left[ {\Greekmath 0120} _{{\Greekmath 011C} }\left(
Y\left( 1\right) \right) |D=1\right] -E_{w}\left[ {\Greekmath 0120} _{{\Greekmath 011C} }\left(
Y\left( 0\right) \right) |D=0\right] .
\end{eqnarray*}
The unconditional quantile effect $\Pi _{{\Greekmath 011C} }$ is the average of the
difference ${\Greekmath 0120} _{\Delta ,U_{D}}\left( w\right) $ with respect to the
distribution of $W$ over the \emph{marginal subpopulation}. The average
\emph{apparent} effect $A_{{\Greekmath 011C} }$ is the average of the difference ${\Greekmath 0120}
_{\Delta ,D}\left( w\right) $ with respect to the distribution of $W$ over
the whole population distribution. It is also equal to the limit of the UQR
estimator of \cite{Firpo2009}, where the endogeneity of the treatment
selection is ignored.\footnote{
To see why this is the case, we note that, in its simplest form, the UQR
involves regressing the \textquotedblleft influence
function\textquotedblright\ $\hat{f}_{Y}(\hat{y}_{{\Greekmath 011C} })^{-1}\left( {\Greekmath 011C} -
\mathds{1}\left\{ Y_{i}\leq \hat{y}_{{\Greekmath 011C} }\right\} \right) $ on $D_{i}$ and
$W_{i}$ by OLS and using the estimated coefficient on $D_{i}$ as the
estimator of the unconditional quantile effect. Here, $\hat{f}_{Y}(y_{{\Greekmath 011C}
}) $ is a consistent estimator of $f_{Y}(y_{{\Greekmath 011C} })$ and $\hat{y}_{{\Greekmath 011C} }$
is a consistent estimator of $y_{{\Greekmath 011C} }.$ It is now easy to see that the UQR
estimator converges in probability to $A_{{\Greekmath 011C} }$ if the conditional
expectations in (\ref{app_eff}) are linear in $W.$} Note that if our model
contains no covariate $W,$ then
\begin{eqnarray}
A_{{\Greekmath 011C} } &=&\frac{1}{f_{Y}(y_{{\Greekmath 011C} })}E\left[ \mathds{1}\left\{ Y\leq
y_{{\Greekmath 011C} }\right\} |D=0\right] -\frac{1}{f_{Y}(y_{{\Greekmath 011C} })}E\left[ \mathds{1}
\left\{ Y\leq y_{{\Greekmath 011C} }\right\} |D=1\right] \notag \\
&=&\frac{1}{f_{Y}(y_{{\Greekmath 011C} })}\left( \Pr [Y>y_{{\Greekmath 011C} }|D=1]-\Pr [Y>y_{{\Greekmath 011C}
}|D=0]\right) .
\end{eqnarray}
This is identical to the unconditional quantile effect given in (\ref
{pi_no_W}).
The discrepancy between $\Pi _{{\Greekmath 011C} }$ and $A_{{\Greekmath 011C} }$ gives rise to the
asymptotic \emph{bias} $B_{{\Greekmath 011C} }$ of the UQR estimator:
\begin{eqnarray}
B_{{\Greekmath 011C} } &=&A_{{\Greekmath 011C} }-\Pi _{{\Greekmath 011C} }=E\left[ {\Greekmath 0120} _{\Delta ,D}\left( W\right)
\right] -E\left[ {\Greekmath 0120} _{\Delta ,U_{D}}\left( W\right) \mathcal{\dot{P}}
\left( W\right) \right] \notag \\
&=&\underset{B_{1{\Greekmath 011C} }}{\underbrace{E\left\{ {\Greekmath 0120} _{\Delta ,D}\left(
W\right) \left[ 1-\mathcal{\dot{P}}\left( W\right) \right] \right\} }}+
\underset{B_{2{\Greekmath 011C} }}{\underbrace{E\left\{ \left[ {\Greekmath 0120} _{\Delta ,D}\left(
W\right) -{\Greekmath 0120} _{\Delta ,U_{D}}\left( W\right) \right] \mathcal{\dot{P}}
\left( W\right) \right\} }}. \label{Bias_decompose}
\end{eqnarray}
It is easy to see that $B_{1{\Greekmath 011C} }$ and $B_{2{\Greekmath 011C} }$ given above are
identical to those given in Corollary \ref{Corollary_UQTE}.
The decomposition in Equation (\ref{Bias_decompose}) traces the asymptotic
bias back to two sources. The first one, $B_{1{\Greekmath 011C} }$, captures\ the
heterogeneity of the averaged apparent effects averaged over two different
subpopulations. For every $w,$ ${\Greekmath 0120} _{\Delta ,D}\left( w\right) $ is the
average effect of $D$ on $\left[ {\Greekmath 011C} -\mathds{1}\left\{ Y\leq y_{{\Greekmath 011C}
}\right\} \right] /f_{Y}\left( y_{{\Greekmath 011C} }\right) $ for the individuals with $
W=w.$ These effects are averaged over two different distributions of $W$:
the distribution of $W$ for the marginal subpopulation (i.e., $\mathcal{\dot{
P}}\left( w\right) f_{W}\left( w\right) )$ and the distribution of $W$ for
the whole population (i.e., $f_{W}\left( w\right) )$. $B_{1{\Greekmath 011C} }$ is equal
to the difference between these two average effects.\emph{\ }If the effect $
{\Greekmath 0120} _{\Delta ,D}\left( w\right) $ does not depend on $w$, then $B_{1{\Greekmath 011C}
}=0 $. If $\mathcal{\dot{P}}\left( \cdot \right) $ is a constant function
that always equals 1, then the distribution of $W$ over the whole population
is the same as that over the marginal subpopulation, and hence $B_{1{\Greekmath 011C} }=0$
as well. For $B_{1{\Greekmath 011C} }\neq 0,$ it is necessary that there is an effect
heterogeneity (i.e., ${\Greekmath 0120} _{\Delta ,D}\left( w\right) $ depends on $w$) and
a distributional heterogeneity (i.e., $\mathcal{\dot{P}}\left( \cdot \right)
$ does not always equal $1$, and as a result, the distribution of $W$ over
the marginal subpopulation is different from that over the whole
population). To highlight the necessary conditions for a nonzero $B_{1{\Greekmath 011C}
}, $ we refer to $B_{1{\Greekmath 011C} }$ as the \emph{marginal heterogeneity bias. }
The second bias component, $B_{2{\Greekmath 011C} },$ embodies the second source of the
bias and has a difference-in-differences interpretation.\ Each of ${\Greekmath 0120}
_{\Delta ,D}\left( \cdot \right) $ and ${\Greekmath 0120} _{\Delta ,U_{D}}\left( \cdot
\right) $ is the difference in the average influence functions associated
with the counterfactual outcomes $Y\left( 1\right) $ and $Y\left( 0\right) .$
However, ${\Greekmath 0120} _{\Delta ,D}\left( \cdot \right) $ is the difference over the
two subpopulations who actually choose $D=1$ and $D=0$, while ${\Greekmath 0120} _{\Delta
,U_{D}}\left( \cdot \right) $ is the difference over the marginal
subpopulation. So ${\Greekmath 0120} _{\Delta ,D}\left( \cdot \right) -{\Greekmath 0120} _{\Delta
,U_{D}}\left( \cdot \right) $ is a difference in differences. $B_{2{\Greekmath 011C} }$
is simply the average of this difference in differences with respect to the
distribution of $W$ over the marginal subpopulation. This term arises
because the change in the distributions of $Y$ for those with $D=1$ and
those with $D=0$ is different from that for those whose $U_{D}$ is just
above $P\left( w\right) $ and those whose $U_{D}$ is just below $P\left(
w\right) $. Thus, we can label $B_{2{\Greekmath 011C} }$ as a \emph{marginal selection
bias}.
If ${\Greekmath 0120} _{\Delta ,D}\left( w\right) ={\Greekmath 0120} _{\Delta ,U_{D}}\left( w\right) $
for almost all $w\in \mathcal{W},$ then $B_{2{\Greekmath 011C} }=0.$ The condition ${\Greekmath 0120}
_{\Delta ,D}\left( w\right) ={\Greekmath 0120} _{\Delta ,U_{D}}\left( w\right) $ is the
same as
\begin{equation*}
E_{w}\left[ {\Greekmath 0120} _{{\Greekmath 011C} }\left( Y\left( 1\right) \right) -{\Greekmath 0120} _{{\Greekmath 011C}
}\left( Y\left( 0\right) \right) |U_{D}=P\left( w\right) \right] =E_{w}\left[
{\Greekmath 0120} _{{\Greekmath 011C} }\left( Y\left( 1\right) \right) |D=1\right] -E_{w}\left[ {\Greekmath 0120}
_{{\Greekmath 011C} }\left( Y\left( 0\right) \right) |D=0\right] .
\end{equation*}
Equivalently,
\begin{eqnarray*}
&&E_{w}\left[ {\Greekmath 0120} _{{\Greekmath 011C} }\left( Y\left( 1\right) \right) |U_{D}=P\left(
w\right) \right] -E_{w}\left[ {\Greekmath 0120} _{{\Greekmath 011C} }\left( Y\left( 1\right) \right)
|D=1\right] \\
&=&E_{w}\left[ {\Greekmath 0120} _{{\Greekmath 011C} }\left( Y\left( 0\right) \right) |U_{D}=P\left(
w\right) \right] -E_{w}\left[ {\Greekmath 0120} _{{\Greekmath 011C} }\left( Y\left( 0\right) \right)
|D=0\right] .
\end{eqnarray*}
The condition resembles the parallel-paths assumption or the constant-bias
assumption in a difference-in-differences analysis. If $U_{D}$ is
independent of $\left( U_{0},U_{1}\right) $ given $W,$ then this condition
holds, and $B_{2{\Greekmath 011C} }=0.$
In general, when $U_{D}$ is not independent of $\left( U_{0},U_{1}\right) $
given $W$, and $W$ enters the selection equation, we have $B_{1{\Greekmath 011C} }\neq 0$
and $B_{2{\Greekmath 011C} }\neq 0,$ hence $\Pi _{{\Greekmath 011C} }\neq A_{{\Greekmath 011C} }.$ If $\mathcal{
\dot{P}}\left( w\right) $ is not identified, then $B_{1{\Greekmath 011C} }$ is not
identified. In general, $B_{2{\Greekmath 011C} }$ is not identified without additional
assumptions. Therefore, in the absence of additional assumptions, the
asymptotic bias can not be eliminated, and $\Pi _{{\Greekmath 011C} }$ is not identified.
It is not surprising that in the presence of endogeneity, the UQR estimator
is asymptotically biased. The virtue of Corollary \ref{Corollary_UQTE} is
that it provides a closed-form characterization and clear interpretations of
the asymptotic bias. To the best of our knowledge, this bias formula is new
in the literature. If point identification can not be achieved, then the
bias formula can be used in a bound analysis or sensitivity analysis. From a
broad perspective, the asymptotic bias $B_{{\Greekmath 011C} }$ is the unconditional
quantile counterpart of the endogenous bias of the OLS estimator in a linear
regression framework.\footnote{
The bias decomposition is not unique. Corollary \ref{Corollary_UQTE} gives
only one possibility. We can also write
\begin{equation}
B_{{\Greekmath 011C} }=\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ }\underset{\tilde{B}_{1{\Greekmath 011C} }}{\underbrace{E\left\{ {\Greekmath 0120}
_{\Delta ,U_{D}}\left( W\right) \left[ 1-\dot{P}\left( W\right) \right]
\right\} }}+\underset{\tilde{B}_{2{\Greekmath 011C} }}{\underbrace{E\left[ {\Greekmath 0120} _{\Delta
,D}\left( W\right) -{\Greekmath 0120} _{\Delta ,U_{D}}\left( W\right) \right] }}. \notag
\end{equation}
The interpretations of $\tilde{B}_{1{\Greekmath 011C} }$ and $\tilde{B}_{2{\Greekmath 011C} }$ are
similar to those of $B_{1{\Greekmath 011C} }$ and $B_{2{\Greekmath 011C} }$ with obvious and minor
modifications. In this case, it may be more revealing to call $\tilde{B}
_{1{\Greekmath 011C} }$ the marginal heterogeneity bias, because a necessary condition
for a nonzero $\tilde{B}_{1{\Greekmath 011C} }$ is that there is an effect heterogeneity
among the marginal subpopulation.}
\section{Unconditional Quantile Effect under Instrumental Intervention\label
{sect:IV}}
In this section, we consider two types of interventions: one on a valid
instrumental variable and the other on an invalid instrumental variable. We
show how we may recover the effect of the latter by using a valid instrument.
\subsection{Instrumental Intervention with a valid instrument}
To introduce the instrumental intervention, we partition $Z$ into two parts
and write $Z=\left( Z_{1},Z_{-1}\right) $\textbf{\ }where $Z_{1}$\ is a
univariate continuous policy variable and $Z_{-1}$ consists of other
variables. If there is only one variable in $Z,$ then $Z=Z_{1},$ and $Z_{-1}$
is not present. We consider the intervention
\begin{equation}
\mathcal{G}\left( W,{\Greekmath 010E} \right) =\left( Z_{1}+g\left( W\right) s\left(
{\Greekmath 010E} \right) ,Z_{2},\cdots ,Z_{d_{Z}}\right) \label{loc_inst}
\end{equation}
where $g\left( \cdot \right) $ is a measurable function and $s\left( {\Greekmath 010E}
\right) $ is a smooth function satisfying $s\left( 0\right) =0$. That is, we
intervene to change the first component of $Z.$ Note that while $s({\Greekmath 010E} )$
is the same for all individuals, $g\left( W\right) $ depends on the value of
$W,$ and hence it is individual-specific. Thus, we allow the intervention to
be heterogeneous. In empirical applications, $Z$ may consist of a few
variables, and $Z_{1}$ is the \emph{continuous} target variable that we
attempt to change. There are no continuity requirements for the other
elements of $Z.$
To present our identification assumptions, we write $W=(W_{1},W_{-1})$ where
$W_{1}$ is the first element of $W$ and $W_{-1}$ consists of other elements
of $W.$ Since $W=(Z,X),$ the first element of $W$ is also the first element
of $Z$, and hence $W_{1}=Z_{1}$. By definition, we have $W_{-1}=\left(
Z_{-1},X\right) .$ We maintain the following assumptions, which are similar
to the corresponding assumptions in \cite{Heckman1999, Heckman2001,
Heckman2005}.
\begin{assumption}
\textbf{Relevance and Exogeneity} \label{Assumption_heckman}
\begin{enumerate}[(a)]
\item \label{relevance}Conditional on $W_{-1}=\left( Z_{-1},X\right) ,$ ${\Greekmath 0116}
(Z,X)$ is a non-degenerate random variable.
\item \label{exogeneity}Conditional on $W_{-1}=\left( Z_{-1},X\right) ,$ $
Z_{1}$ is independent of $(U_{0},U_{1},V).$
\end{enumerate}
\end{assumption}
Assumption \ref{Assumption_heckman}(\ref{relevance}) is a relevance
assumption: for any given level of $W_{-1}$, $Z_{1}$ can induce some
variation in $D$. Assumption \ref{Assumption_heckman}(\ref{exogeneity}) is a
conditional exogeneity assumption: for any given level of $W_{-1}$, $Z_{1}$
is independent of the unobservables. These two assumptions are essentially
the conditions for $Z_{1}$ to be a valid instrumental variable, hence we
will refer to $Z_{1}$ as the instrumental variable, and the intervention in
\eqref{loc_inst} as the instrumental intervention.\footnote{
If all variables in $Z$ satisfy the conditional exogeneity assumption, we
may replace Assumption \ref{Assumption_heckman} by the following: \textit{
(a) Conditional on }$X,$ ${\Greekmath 0116} \left( Z,X\right) $\textit{\ is a
non-degenerate random variable,} and \textit{(b) Conditional on }$X,$ $Z$
\textit{\ is independent of }$(U_{0},U_{1},V)$\textit{.} With minor
modifications, our results will remain valid. More specifically, in the
statement of each result, we only need to replace \textquotedblleft
conditioning on $W_{-1}$\textquotedblright\ by \textquotedblleft
conditioning on $X$\textquotedblright . The working paper \cite{sun2021} is
based on this alternative assumption. Here we will work with Assumption \ref
{Assumption_heckman}, which appears to be more plausible.}
Let $w:=\left( w_{1},w_{-1}\right) =(z_{1},w_{-1}).$ Under Assumption \ref
{Assumption_heckman}(\ref{exogeneity}), the unconditional MTE for the
quantile functional ${\Greekmath 011A} _{{\Greekmath 011C} }$ becomes
\begin{eqnarray}
\mathrm{MTE}_{{\Greekmath 011C} }\left( u,w\right) &=&\frac{1}{f_{Y}\left( y_{{\Greekmath 011C}
}\right) }E\left[ \mathds{1}\left\{ Y(0)\leq y_{{\Greekmath 011C} }\right\} -\mathds{1}
\left\{ Y(1)\leq y_{{\Greekmath 011C} }\right\} \mid U_{D}=u,W=w\right] \notag \\
&=&\frac{1}{f_{Y}\left( y_{{\Greekmath 011C} }\right) }E\left[ \mathds{1}\left\{ Y(0)\leq
y_{{\Greekmath 011C} }\right\} -\mathds{1}\left\{ Y(1)\leq y_{{\Greekmath 011C} }\right\} \mid
U_{D}=u,W_{-1}=w_{-1}\right] \notag \\
&:=&\widetilde{\mathrm{MTE}}_{{\Greekmath 011C} }\left( u,w_{-1}\right) ,
\label{MTE_tilde}
\end{eqnarray}
where in the second line above, conditioning on $Z_{1}$ is not necessary and
has been dropped.
Next, we present a lemma that characterizes the weighting function for the
instrumental intervention in \eqref{loc_inst}.
\begin{lemma}
\label{Lemma_SZ}Assume that (i) for almost all $w\in \mathcal{W}$, the
conditional distribution of$\ V$ conditional on $W=w$ is absolutely
continuous conditional density $f_{V|W}\left( v|w\right) $;\ (ii) ${\Greekmath 0116}
\left( w\right) $\ is differentiable in $z_{1}$\ for almost all $w\in
\mathcal{W}$ with derivative ${\Greekmath 0116} _{z_{1}}^{\prime }\left( \cdot \right) $
such that $E\left[ f_{V|W}\left( {\Greekmath 0116} (W)|W\right) {\Greekmath 0116} _{z_{1}}^{\prime
}\left( W\right) g\left( W\right) \right] $ is well defined and is not equal
to zero; (iii) $s\left( {\Greekmath 010E} \right) $\ is a differentiable function in a
neighborhood of zero and $\left. \partial s\left( {\Greekmath 010E} \right) /\partial
{\Greekmath 010E} \right\vert _{{\Greekmath 010E} =0}\neq 0$. Then
\begin{equation}
\mathcal{\dot{P}}\left( w\right) =\frac{f_{V|W}\left( {\Greekmath 0116} (w)|w\right) {\Greekmath 0116}
_{z_{1}}^{\prime }\left( w\right) g(w)}{E\left[ f_{V|W}\left( {\Greekmath 0116} \left(
W\right) |W\right) {\Greekmath 0116} _{z_{1}}^{\prime }\left( W\right) g(W)\right] }.
\label{P_dot_w}
\end{equation}
\end{lemma}
The lemma shows that the weighting function $\mathcal{\dot{P}}\left(
w\right) $ does not depend on the function form of $s\left( \cdot \right) .$
Hence, the unconditional policy effect does not depend on $s\left( \cdot
\right) .$
\begin{corollary}
\label{Corollary_UQTE_w} Let Assumptions \ref{Assumption_primary}--\ref
{Assumption_domination} and \ref{Assumption_heckman}, and the assumptions of
Lemma \ref{Lemma_SZ} hold. Assume further that $f_{Y}(y_{{\Greekmath 011C} })>0$. Then,
the unconditional quantile effect of the instrumental intervention given in
\eqref{loc_inst} is
\begin{equation}
\Pi _{{\Greekmath 011C} }=\int_{\mathcal{W}}\widetilde{\mathrm{MTE}}_{{\Greekmath 011C} }\left(
P(w),w_{-1}\right) \mathcal{\dot{P}}\left( w\right) dF_{W}\left( w\right).
\notag
\end{equation}
\end{corollary}
The proposition below establishes the identifiability of $\widetilde{\mathrm{
MTE}}_{{\Greekmath 011C} }$ and of the weighting function $\mathcal{\dot{P}}\left(
w\right) $ given in Corollary \ref{Corollary_UQTE_w}.
\begin{proposition}
\label{Prop_MTE_identification}Let Assumptions \ref{Assumption_regularity}(
\ref{regularity_x_abs}), \ref{Assumption_regularity}(\ref{f_y_u_x}), and \ref
{Assumption_heckman}(\ref{exogeneity}), and the assumptions in Lemma \ref
{Lemma_SZ} hold. Then, for every $u=P(w)$ for some $w=(w_{1},w_{-1})\in
\mathcal{W}$, we have\footnote{
For a more general functional ${\Greekmath 011A} ,$ we can show that
\begin{equation*}
\widetilde{\mathrm{MTE}}_{{\Greekmath 011A} }(u,w_{-1})=\frac{\partial E\left[ {\Greekmath 0120}
(Y,{\Greekmath 011A} ,F_{Y})|P(W)=u,W_{-1}=w_{-1}\right] }{\partial u}
\end{equation*}
for every $u$ equal to $P(w)$ for some $w=(w_{1},w_{-1})\in \mathcal{W}$.}
\begin{equation*}
\widetilde{\mathrm{MTE}}_{{\Greekmath 011C} }\left( u,w_{-1}\right) =-\frac{1}{
f_{Y}\left( y_{{\Greekmath 011C} }\right) }\frac{\partial E\left[ \mathds{1}\left\{ Y\leq
y_{{\Greekmath 011C} }\right\} |P(W)=u,W_{-1}=w_{-1}\right] }{\partial u},
\end{equation*}
and
\begin{equation}
\mathcal{\dot{P}}\left( w\right) =\frac{\frac{\partial P(w)}{\partial z_{1}}
g(w)}{E\left[ \frac{\partial P(W)}{\partial z_{1}}g\left( W\right) \right] }
\label{id_prop_weight}
\end{equation}
where $\frac{\partial P(W)}{\partial z_{1}}$ is short for $\left. \frac{
\partial P(w)}{\partial z_{1}}\right\vert _{w=W}.$
\end{proposition}
Using Proposition \ref{Prop_MTE_identification}, and the fact that $g(w)$ is
known, we can represent $\Pi _{{\Greekmath 011C} }$ as
\begin{equation}
\Pi _{{\Greekmath 011C} }=-\frac{1}{f_{Y}(y_{{\Greekmath 011C} })}\int_{\mathcal{W}}\frac{\partial E
\left[ \mathds{1}\left\{ Y\leq y_{{\Greekmath 011C} }\right\} |P(W)=P(w),W_{-1}=w_{-1}
\right] }{\partial P(w)}\frac{\frac{\partial P(w)}{\partial z_{1}}g(w)}{E
\left[ \frac{\partial P(W)}{\partial z_{1}}g\left( W\right) \right] }
dF_{W}\left( w\right) . \label{UQE_representation}
\end{equation}
All objects in the above are point identified, hence $\Pi _{{\Greekmath 011C} }$ is point
identified.\footnote{
We note that Assumption \ref{Assumption_heckman}(\ref{exogeneity}) plays a
key role in identifying $\mathcal{\dot{P}}\left( w\right) .$ Without the
assumption that $V$ is independent of $Z_{1}$ conditional on $\left(
Z_{-1},X\right) ,$ we can have only that
\begin{equation*}
\frac{\partial P(w)}{\partial z_{1}}=f_{V|Z,X}({\Greekmath 0116} (w)|w){\Greekmath 0116}
_{z_{1}}^{\prime }\left( w\right) +\frac{\partial F_{V|Z,X}({\Greekmath 0116} (w)|\tilde{z}
,x)}{\partial \tilde{z}_{1}}\bigg |_{\tilde{z}=z}.
\end{equation*}
The presence of the second term in the above equation invalidates the
identification result in (\ref{id_prop_weight}).}
\subsection{Instrumental Intervention with an invalid instrument}
While identifying the unconditional effect of an instrumental intervention
is of interest in its own right, the identification result can be further
leveraged to identify the unconditional policy effect of another
intervention. Theorem \ref{Theorem general} has shown that the unconditional
effects of two interventions will be the same if their weighting functions
coincide. Consider a counterfactual policy with a target weighting function $
\mathcal{\dot{P}}^{\circ }\left( \cdot \right) .$ By strategically choosing
the instrument function $g\left( \cdot \right) $, we can ensure that the
weighting function under the policy intervention in \eqref{loc_inst} is the
same as $\mathcal{\dot{P}}^{\circ }\left( \cdot \right) .$ In other words,
with appropriate choices of $g\left( \cdot \right) ,$ the unconditional
effect of intervening $Z_{1}$ is the same as the unconditional effect of
another counterfactual policy. If the former is identified, then the latter
is also identified.
As an example, suppose we intervene on the second element $Z_{2}$ of $Z$
with
\begin{equation}
\mathcal{G}^{\circ }\left( W,{\Greekmath 010E} \right) =\left( Z_{1},Z_{2}+g^{\circ
}\left( W\right) s^{\circ }\left( {\Greekmath 010E} \right) ,\cdots ,Z_{d_{Z}}\right)
\label{loc_intervention}
\end{equation}
for some $g^{\circ }\left( \cdot \right) $ and $s^{\circ }\left( {\Greekmath 010E}
\right) .$ Under conditions similar to those in Lemma \ref{Lemma_SZ}, the
weighting function for this intervention is
\begin{equation*}
\mathcal{\dot{P}}^{\circ }\left( w\right) =\frac{f_{V|W}\left( {\Greekmath 0116}
(w)|w\right) {\Greekmath 0116} _{z_{2}}^{\prime }\left( w\right) g^{\circ }(w)}{E\left[
f_{V|W}\left( {\Greekmath 0116} \left( W\right) |W\right) {\Greekmath 0116} _{z_{2}}^{\prime }\left(
W\right) g^{\circ }(W)\right] }.
\end{equation*}
Letting $\mathcal{\dot{P}}^{\circ }\left( w\right) =\mathcal{\dot{P}}\left(
w\right) $ for $\mathcal{\dot{P}}\left( w\right) $ given in Lemma \ref
{Lemma_SZ} and solving for $g(\cdot )$ yields
\begin{equation*}
g(w)=\frac{{\Greekmath 0116} _{z_{2}}^{\prime }\left( w\right) }{{\Greekmath 0116} _{z_{1}}^{\prime
}\left( w\right) }g^{\circ }(w).
\end{equation*}
So, the unconditional effect of the intervention given in (\ref
{loc_intervention}) is the same as that of the intervention given in
\eqref{loc_inst} when $g(w)$ is chosen appropriately. It is important to
point out that $Z_{2}$ may not be a valid instrument, and its unconditional
effect is identified via \textquotedblleft intervention
matching.\textquotedblright
In general, identifying an unconditional effect of an invalid instrument via
intervention matching\ is feasible only if we have prior knowledge of the
ratio ${\Greekmath 0116} _{z_{2}}^{\prime }\left( w\right) /{\Greekmath 0116} _{z_{1}}^{\prime }\left(
w\right) .$ This information may be available from economic theory. When
such knowledge is unavailable, an alternative approach can be employed.
The alternative approach hinges on the assumption that $\left(
Z_{1},Z_{2}\right) $ is independent of $V$ conditional on the remaining
elements in $W$, denoted as $W_{-(1,2)}$. In this case, for $
W=(Z_{1},Z_{2},W_{-(1,2)})$ and $w=(z_{1},z_{2},w_{-\left( 1,2\right) }),$
we have
\begin{eqnarray*}
P(w) &=&\Pr \left[ V\leq {\Greekmath 0116} \left( z_{1},z_{2},w_{-\left( 12\right)
}\right) |W=w\right] \\
&=&\Pr \left[ V\leq {\Greekmath 0116} \left( z_{1},z_{2},w_{-\left( 12\right) }\right)
|W_{-(1,2)}=w_{-\left( 1,2\right) }\right] \\
&=&F_{V|W_{-\left( 1,2\right) }}\left( {\Greekmath 0116} \left( z_{1},z_{2},w_{-\left(
12\right) }\right) |w_{-\left( 1,2\right) }\right) ,
\end{eqnarray*}
and
\begin{equation*}
\left( \frac{\partial P(w)}{\partial z_{2}}\right) /\left( \frac{\partial
P(w)}{\partial z_{1}}\right) =\frac{f_{V|W_{-\left( 1,2\right) }}\left( {\Greekmath 0116}
\left( z_{1},z_{2},w_{-12}\right) |w_{-\left( 1,2\right) }\right) {\Greekmath 0116}
_{z_{2}}^{\prime }\left( w\right) }{f_{V|W_{-\left( 1,2\right) }}\left( {\Greekmath 0116}
\left( z_{1},z_{2},w_{-12}\right) |w_{-\left( 1,2\right) }\right) {\Greekmath 0116}
_{z_{1}}^{\prime }\left( w\right) }=\frac{{\Greekmath 0116} _{z_{2}}^{\prime }\left(
w\right) }{{\Greekmath 0116} _{z_{1}}^{\prime }\left( w\right) }.
\end{equation*}
Thus, the ratio ${\Greekmath 0116} _{z_{2}}^{\prime }\left( w\right) /{\Greekmath 0116} _{z_{1}}^{\prime
}\left( w\right) $ can be identified via the ratio of two partial
derivatives of the propensity score function.
It is important to note that the conditional independence of $Z_{2}$ from $V$
given $W_{-(1,2)}$ does not rule out the possibility that $Z_{2}$ may still
be dependent on $(U_{0},U_{1})$, and consequently, $Z_{2}$ could still be an
invalid instrument. This shows that intervention matching may be used to
identify the effect of intervening on an invalid instrument.
\section{Unconditional Instrumental Quantile Estimation}
\label{estimation}
This section is devoted to the estimation and inference of the UQE under the
instrumental intervention in (\ref{loc_inst}). We assume that the propensity
score function is parametric, and we leave the case with a nonparametric
propensity score to Section \ref{non_parametric_ps_appendix} of the
supplementary appendix. In order to simplify the notation, we set $g(\cdot
)\equiv 1$ for the remainder of this paper.\footnote{
The presence of $g\left( \cdot \right) $ amounts to a change of measure:
from a measure with density $f_{W}(w)$ to a measure with density $
g(w)f_{W}\left( w\right) $. When $g\left( \cdot \right) $ is not equal to a
constant function, we only need to change the population expectation
operator $E\left[ h(W)\right] $ that involves the distribution of $W$ into $E
\left[ h(W)g(W)\right] $ and the empirical average operator $\mathbb{P}_{n}
\left[ h\left( W\right) \right] $ into $\mathbb{P}_{n}\left[ h(W)g\left(
W\right) \right] .$ All of our results will remain valid.}
Letting
\begin{equation*}
m_{0}(y_{{\Greekmath 011C} },P(w),w_{-1}):=E\left[ \mathds{1}\left\{ Y\leq y_{{\Greekmath 011C}
}\right\} |P(W)=P(w),W_{-1}=w_{-1}\right]
\end{equation*}
and using (\ref{UQE_representation}), we have
\begin{equation}
\Pi _{{\Greekmath 011C} }=-\frac{1}{f_{Y}(y_{{\Greekmath 011C} })}E\left[ \frac{\partial P(W)}{
\partial Z_{1}}\right] ^{-1}E\left[ \frac{\partial m_{0}(y_{{\Greekmath 011C}
},P(W),W_{-1})}{\partial Z_{1}}\right] . \label{equ_pi_tau}
\end{equation}
$\Pi _{{\Greekmath 011C} }$ consists of two average derivatives and a density evaluated
at a point, some of which depend on the unconditional ${\Greekmath 011C} $-quantile $
y_{{\Greekmath 011C} }$. Altogether $\Pi _{{\Greekmath 011C} }$ depends on four unknown quantities.
The method of unconditional instrumental quantile estimation involves first
estimating the four quantities separately and then plugging these estimates
into $\Pi _{{\Greekmath 011C} }$ to obtain the estimator $\hat{\Pi}_{{\Greekmath 011C} }.$ See (\ref
{UNIQUE_Expression}) in Subsection \ref{parap_uqr} for the formula of $\hat{
\Pi}_{{\Greekmath 011C} }.$
We consider estimating the four quantities in the next few subsections. For
a given sample $\left\{ O_{i}=(Y_{i},Z_{i},X_{i},D_{i})\right\} _{i=1}^{n}$,
we will use $\mathbb{P}_{n}$ to denote the empirical measure. The
expectation of a function ${\Greekmath 011F} \left( O\right) $ with respect to $\mathbb{P}
_{n}$ is then $\mathbb{P}_{n}{\Greekmath 011F} =n^{-1}\sum_{i=1}^{n}{\Greekmath 011F} (O_{i})$.
Similarly, we use $\mathbb{P}$ to denote the population measure, and so $
\mathbb{P}{\Greekmath 011F} \left( O\right) =E{\Greekmath 011F} \left( O\right) .$
\subsection{Estimating the Quantile and Density}
\label{two_step_den_section}
For a given ${\Greekmath 011C} $, we estimate $y_{{\Greekmath 011C} }$ using the (generalized) inverse
of the empirical distribution function of $Y$: $\hat{y}_{{\Greekmath 011C} }=\inf \left\{
y:\mathbb{F}_{n}(y)\geq {\Greekmath 011C} \right\} $, where
\begin{equation*}
\mathbb{F}_{n}(y):=\frac{1}{n}\sum_{i=1}^{n}\mathds{1}\left\{ Y_{i}\leq
y\right\} .
\end{equation*}
By Lemma \ref{quantile_an} in the appendix, we can write $\hat{y}_{{\Greekmath 011C}
}-y_{{\Greekmath 011C} }=\mathbb{P}_{n}{\Greekmath 0120} _{Q}(Y,y_{{\Greekmath 011C} })+o_{p}(n^{-1/2})$
where
\begin{equation*}
{\Greekmath 0120} _{Q}(Y,y_{{\Greekmath 011C} }):=\frac{{\Greekmath 011C} -\mathds{1}\left\{ Y\leq y_{{\Greekmath 011C}
}\right\} }{f_{Y}(y_{{\Greekmath 011C} })}.
\end{equation*}
Here the subscript \textquotedblleft $Q$\textquotedblright\ on ${\Greekmath 0120} _{Q}$
signifies that it is the influence function for a \textbf{Q}uantile
functional.
We use a kernel density estimator to estimate $f_{Y}(y)$. We maintain the
following assumptions on the kernel function and the bandwidth.
\begin{assumption}
\textbf{Kernel Assumption} \label{Assumption_Kernel}
\item The kernel function $K(\cdot )$ satisfies (i) $\int_{-\infty }^{\infty
}K(u)du=1$, (ii) $\int_{-\infty }^{\infty }u^{2}K(u)du<\infty $, and (iii) $
K(u)=K(-u)$, and it is twice differentiable with Lipschitz continuous
second-order derivative $K^{\prime \prime }\left( u\right) $ satisfying (i) $
\int_{-\infty }^{\infty }K^{\prime \prime }(u)udu<\infty $ and $\left(
ii\right) $ there exist positive constants $C_{1}$ and $C_{2}$ such that $
\left\vert K^{\prime \prime }\left( u_{1}\right) -K^{\prime \prime }\left(
u_{2}\right) \right\vert \leq C_{2}\left\vert u_{1}-u_{2}\right\vert ^{2}$
for $\left\vert u_{1}-u_{2}\right\vert \geq C_{1}.$
\end{assumption}
\begin{assumption}
\textbf{Rate Assumption }\label{Assumption_rate}: $n\uparrow \infty $ and $
h\downarrow 0$ such that $nh^{3}\uparrow \infty $ but $nh^{5}=O(1)$.
\end{assumption}
The non-standard condition $nh^{3}\uparrow \infty $ is due to the estimation
of $y_{{\Greekmath 011C} }$. Since we need to expand $\hat{f}_{Y}(\hat{y}_{{\Greekmath 011C} })-\hat{f}
_{Y}(y_{{\Greekmath 011C} })$, which involves the derivative of $\hat{f}_{Y}(y),$ we have
to impose a slower rate of decay for $h$ to control the remainder. The
details can be found in the proof of Lemma \ref{two_step_density}. We note,
however, that $nh^{3}\uparrow \infty $ implies the usual rate condition $
nh\uparrow \infty $.
The estimator of $f_{Y}(y)$ is then given by
\begin{equation*}
\hat{f}_{Y}(y)=\frac{1}{n}\sum_{i=1}^{n}K_{h}\left( Y_{i}-y\right) ,\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ }
\end{equation*}
where $K_{h}\left( u\right) :=K(u/h)/h.$
By Lemma \ref{two_step_density} in the appendix, we can isolate the
estimation errors from estimating the density and quantile and write:
\begin{align}
\hat{f}_{Y}(\hat{y}_{{\Greekmath 011C} })-{f}_{Y}(y_{{\Greekmath 011C} })& =\hat{f}_{Y}({y}_{{\Greekmath 011C} })-{f
}_{Y}(y_{{\Greekmath 011C} })+f_{Y}(\hat{y}_{{\Greekmath 011C} })-{f}_{Y}(y_{{\Greekmath 011C}
})+o_{p}(n^{-1/2}h^{-1/2}) \notag \\
& =\mathbb{P}_{n}{\Greekmath 0120} _{f_{Y}}\left( Y,y_{{\Greekmath 011C} }\right) +B_{f_{Y}}(y_{{\Greekmath 011C}
})+f_{Y}^{\prime }(y_{{\Greekmath 011C} })\mathbb{P}_{n}{\Greekmath 0120} _{Q}(Y,y_{{\Greekmath 011C}
})+o_{p}(n^{-1/2}h^{-1/2}). \label{decom_f_y}
\end{align}
Here, ${\Greekmath 0120} _{f_{Y}}\left( Y,y_{{\Greekmath 011C} }\right) :=K_{h}\left( Y-y_{{\Greekmath 011C}
}\right) -E\left[ K_{h}\left( Y-y_{{\Greekmath 011C} }\right) \right] $ and $
B_{f_{Y}}(y_{{\Greekmath 011C} })$ is the bias term, which is $O(h^{2})$. The first two
terms on the right-hand side of (\ref{decom_f_y}) represent the dominating
terms in the error from estimating $f_{Y}$. The third term reflects the
error from estimating $y_{{\Greekmath 011C} }$.
\subsection{Estimating the Average Derivatives}
\label{param_ps_section}
To estimate the two average derivatives, we make a parametric assumption on
the propensity score, leaving the nonparametric specification to Section \ref
{non_parametric_ps_appendix} of the supplementary appendix.
\begin{assumption}
\label{Assumption_parametric_ps} The propensity score $P(Z,X,{\Greekmath 010B} _{0})$
is known up to a finite-dimensional vector ${\Greekmath 010B} _{0}\in \mathbb{R}
^{d_{{\Greekmath 010B} }}$.
\end{assumption}
Under Assumption \ref{Assumption_parametric_ps}, the UQE $\Pi _{{\Greekmath 011C} }$ can
be written as
\begin{equation}
\Pi _{{\Greekmath 011C} }=-\frac{1}{f_{Y}(y_{{\Greekmath 011C} })}\cdot \frac{1}{T_{1}}\cdot T_{2},
\label{param_interest_3}
\end{equation}
where
\begin{equation*}
T_{1}=E\left[ \frac{\partial P(Z,X,{\Greekmath 010B} _{0})}{\partial Z_{1}}\right]
\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ and }T_{2}=E\left[ \frac{\partial m_{0}(y_{{\Greekmath 011C} },P(Z,X,{\Greekmath 010B}
_{0}),W_{-1})}{\partial Z_{1}}\right] \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{.}
\end{equation*}
First, we estimate $T_{1},$ which is the mean of the derivative of the
propensity score, by
\begin{equation*}
T_{1n}(\hat{{\Greekmath 010B}}):=\frac{1}{n}\sum_{i=1}^{n}\frac{\partial P(z,x,\hat{
{\Greekmath 010B}})}{\partial z_{1}}\bigg |_{\left( z,x\right) =\left(
Z_{i},X_{i}\right) }
\end{equation*}
where $\hat{{\Greekmath 010B}}$ is an estimator of ${\Greekmath 010B} _{0}$ satisfying $\hat{{\Greekmath 010B}
}-{\Greekmath 010B} _{0}=\mathbb{P}_{n}{\Greekmath 0120} _{{\Greekmath 010B} _{0}}\left( D,W\right)
+o_{p}(n^{-1/2})$ for some measurable function ${\Greekmath 0120} _{{\Greekmath 010B} _{0}}\left(
\cdot ,\cdot \right) .$ To save space, we slightly abuse notation and write
\begin{equation*}
\frac{\partial P(Z,X,{\Greekmath 010B} )}{\partial z_{1}}=\frac{\partial P(z,x,{\Greekmath 010B} )
}{\partial z_{1}}\bigg |_{\left( z,x\right) =\left( Z,X\right) }.
\end{equation*}
We adopt this convention in the rest of the paper. Under Lemma \ref
{ps_estimation_param} in the appendix, we have
\begin{eqnarray}
T_{1n}(\hat{{\Greekmath 010B}})-T_{1} &=&\mathbb{P}_{n}{\Greekmath 0120} _{\partial P}\left(
W\right) +\left\{ E\left[ \frac{\partial ^{2}P(Z,X,{\Greekmath 010B} _{0})}{\partial
z_{1}\partial {\Greekmath 010B} }\right] \right\} ^{\prime }\mathbb{P}_{n}{\Greekmath 0120} _{{\Greekmath 010B}
_{0}}\left( D,W\right) +o_{p}(n^{-1/2}), \notag \\
&& \label{decom_ps_non}
\end{eqnarray}
where
\begin{equation*}
{\Greekmath 0120} _{{\partial P}}\left( W\right) :=\frac{\partial P(Z,X,{\Greekmath 010B} _{0})}{
\partial z_{1}}-E\left[ \frac{\partial P(Z,X,{\Greekmath 010B} _{0})}{\partial z_{1}}
\right] .
\end{equation*}
Equation (\ref{decom_ps_non}) has a similar interpretation to equation (\ref
{decom_f_y}). It consists of a term that ignores the estimation uncertainty
in $\hat{{\Greekmath 010B}}$ but accounts for the variability of the sample mean, and
another term that accounts for the uncertainty in $\hat{{\Greekmath 010B}}$ but ignores
the variability of the sample mean.
We estimate the second average derivative $T_{2}$ by
\begin{equation}
T_{2n}(\hat{y}_{{\Greekmath 011C} },\hat{m},\hat{{\Greekmath 010B}}):=\frac{1}{n}\sum_{i=1}^{n}\frac{
\partial \hat{m}(\hat{y}_{{\Greekmath 011C} },P(Z_{i},X_{i},\hat{{\Greekmath 010B}}),W_{-1,i})}{
\partial z_{1}}; \label{t2n}
\end{equation}
See (\ref{t2n_2}) for an explicit construction. We can regard $T_{2n}(\hat{y}
_{{\Greekmath 011C} },\hat{m},\hat{{\Greekmath 010B}})$ as a four-step estimator. The first step
estimates $y_{{\Greekmath 011C} }$, the second step estimates ${\Greekmath 010B} _{0}$, the third
step estimates the conditional expectation $m_{0}(y,P(Z,X,{\Greekmath 010B}
_{0}),W_{-1})$ using the generated regressor $P(Z,X,\hat{{\Greekmath 010B}})$, and the
fourth step averages the derivative (with respect to $Z_{1})$ over the
generated regressor $P(Z,X,\hat{{\Greekmath 010B}})$ and $W_{-1}$.
We use the series method to estimate $m_{0}$. To alleviate notation, define
the vector $\tilde{w}({\Greekmath 010B} ):=(P(z,x,{\Greekmath 010B} ),w_{-1})^{\prime }\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ and }
\tilde{W}_{i}({\Greekmath 010B} ):=(P(Z_{i},X_{i},{\Greekmath 010B} ),W_{-1,i})^{\prime }.$ We
write $\tilde{w}=\tilde{w}\left( {\Greekmath 010B} _{0}\right) $ and $\tilde{W}_{i}=
\tilde{W}_{i}({\Greekmath 010B} _{0})$ to suppress their dependence on the true
parameter value ${\Greekmath 010B} _{0}.$
Both $\tilde{w}({\Greekmath 010B} )$ and $\tilde{W}_{i}({\Greekmath 010B} )$ are in $\mathbb{R}
^{d_{W}}.$ Let ${\Greekmath 011E} ^{J}(\tilde{w}({\Greekmath 010B} ))=({\Greekmath 011E} _{1J}(\tilde{w}({\Greekmath 010B}
)),\ldots ,{\Greekmath 011E} _{JJ}(\tilde{w}({\Greekmath 010B} )))^{\prime }$
be a vector of $J$ basis functions of $\tilde{w}({\Greekmath 010B} )$ with finite
second moments\footnote{
If any variable in $W_{-1}$ is discrete with a small number of possible
values, we can exclude it in the basis functions. Instead, we can construct
the basis functions using only the remaining variables and apply the series
method to each subsample defined by the values of the discrete variable.}.
Here, each ${\Greekmath 011E} _{jJ}\left( \cdot \right) $ is a differentiable basis
function. Then, the series estimator of $m_{0}(y_{{\Greekmath 011C} },\tilde{w}({\Greekmath 010B} ))$
is $\hat{m}(\hat{y}_{{\Greekmath 011C} },\tilde{w}(\hat{{\Greekmath 010B}}))={\Greekmath 011E} ^{J}(\tilde{w}(
\hat{{\Greekmath 010B}}))^{\prime }\hat{b}(\hat{{\Greekmath 010B}},\hat{y}_{{\Greekmath 011C} }),$
where $\hat{b}(\hat{{\Greekmath 010B}},\hat{y}_{{\Greekmath 011C} })$ is:
\begin{equation*}
\hat{b}(\hat{{\Greekmath 010B}},\hat{y}_{{\Greekmath 011C} })=\left( \sum_{i=1}^{n}{\Greekmath 011E} ^{J}(\tilde{W
}_{i}(\hat{{\Greekmath 010B}})){\Greekmath 011E} ^{J}(\tilde{W}_{i}(\hat{{\Greekmath 010B}}))^{\prime }\right)
^{-1}\sum_{i=1}^{n}{\Greekmath 011E} ^{J}(\tilde{W}_{i}(\hat{{\Greekmath 010B}}))\mathds{1}\left\{
Y_{i}\leq \hat{y}_{{\Greekmath 011C} }\right\} .
\end{equation*}
The estimator of the average derivative $T_{2}$ is then
\begin{equation}
T_{2n}(\hat{y}_{{\Greekmath 011C} },\hat{m},\hat{{\Greekmath 010B}})=\frac{1}{n}\sum_{i=1}^{n}\frac{
\partial {\Greekmath 011E} ^{J}(\tilde{W}_{i}(\hat{{\Greekmath 010B}}))}{\partial z_{1}}^{\prime }
\hat{b}(\hat{{\Greekmath 010B}},\hat{y}_{{\Greekmath 011C} }). \label{t2n_2}
\end{equation}
We use the path derivative approach of \cite{newey1994} to obtain a
decomposition of $T_{2n}(\hat{y}_{{\Greekmath 011C} },\hat{m},\hat{{\Greekmath 010B}})-T_{2},$ which
is similar to that in Section 2.1 of \cite{hahn2013}. To describe the idea,
let $\left\{ F_{{\Greekmath 0112} }\right\} $ be a path of distributions indexed by $
{\Greekmath 0112} \in \mathbb{R}$ such that $F_{{\Greekmath 0112} _{0}}$ is the true distribution
of $O:=(Y,Z,X,D)$. The parametric assumption on the propensity score does
not need to be imposed on the path.\footnote{
As we show later, the error from estimating the propensity score does not
affect the asymptotic variance of $T_{2n}(\hat{y}_{{\Greekmath 011C} },\hat{m},\hat{{\Greekmath 010B}
}).$} The score of the parametric submodel is $S(O)=\frac{\partial \log
dF_{{\Greekmath 0112} }(O)}{\partial {\Greekmath 0112} }\big|_{{\Greekmath 0112} ={\Greekmath 0112} _{0}}.$
For any ${\Greekmath 0112} ,$ we define
\begin{equation*}
T_{2,{\Greekmath 0112} }=E_{{\Greekmath 0112} }\left[ \frac{\partial m_{{\Greekmath 0112} }(y_{{\Greekmath 011C} ,{\Greekmath 0112} },
\tilde{W}\left( {\Greekmath 010B} _{{\Greekmath 0112} }\right) )}{\partial z_{1}}\right]
\end{equation*}
where $m_{{\Greekmath 0112} },$ $y_{{\Greekmath 011C} ,{\Greekmath 0112} },$ and ${\Greekmath 010B} _{{\Greekmath 0112} }$ are the
probability limits of $\hat{m},$ $\hat{y}_{{\Greekmath 011C} },$ and $\hat{{\Greekmath 010B}},$
respectively, when the distribution of $O$ is $F_{{\Greekmath 0112} }$. Note that when $
{\Greekmath 0112} ={\Greekmath 0112} _{0},$ we have ${\Greekmath 010B} _{{\Greekmath 0112} _{0}}={\Greekmath 010B} _{0},m_{{\Greekmath 0112}
_{0}}=m_{0}$ and $T_{2,{\Greekmath 0112} _{0}}=T_{2}.$ Suppose the set of scores $
\left\{ S(O)\right\} $ for all parametric submodels $\left\{ F_{{\Greekmath 0112}
}\right\} $ can approximate any zero-mean, finite-variance function of $O$
in the mean square sense.\footnote{
This is the \textquotedblleft generality\textquotedblright\ requirement of
the family of distributions in \cite{newey1994}.} If the function ${\Greekmath 0112}
\rightarrow T_{2,{\Greekmath 0112} }$ is differentiable at ${\Greekmath 0112} _{0}$ and we can
write
\begin{equation}
\frac{\partial T_{2,{\Greekmath 0112} }}{\partial {\Greekmath 0112} }\bigg|_{{\Greekmath 0112} ={\Greekmath 0112} _{0}}=E
\left[ \Gamma (O)S(O)\right] \label{path_derivative}
\end{equation}
for some mean-zero and finite second-moment function $\Gamma (\cdot )$ and
any path $F_{{\Greekmath 0112} },$ then, by Theorem 2.1 of \cite{newey1994}, the
asymptotic variance of $T_{2n}(\hat{y}_{{\Greekmath 011C} },\hat{m},\hat{{\Greekmath 010B}})$ is\ $
E[\Gamma (O)^{2}]$.
In Lemma \ref{param_ave_estimation} in the appendix, we show that ${\Greekmath 0112}
\rightarrow T_{2,{\Greekmath 0112} }$ is differentiable at ${\Greekmath 0112} _{0}$. Then, by the
chain rule, we can write
\begin{eqnarray*}
\left. \frac{\partial T_{2,{\Greekmath 0112} }}{\partial {\Greekmath 0112} }\right\vert _{{\Greekmath 0112}
={\Greekmath 0112} _{0}} &=&\left. \frac{\partial }{\partial {\Greekmath 0112} }E_{{\Greekmath 0112} }\left[
\frac{\partial m_{{\Greekmath 0112} _{0}}(y_{{\Greekmath 011C} },\tilde{W}\left( {\Greekmath 010B} _{0}\right) )
}{\partial z_{1}}\right] \right\vert _{{\Greekmath 0112} ={\Greekmath 0112} _{0}}+\left. \frac{
\partial }{\partial {\Greekmath 0112} }E\left[ \frac{\partial m_{{\Greekmath 0112} }(y_{{\Greekmath 011C} },
\tilde{W}\left( {\Greekmath 010B} _{0}\right) )}{\partial z_{1}}\right] \right\vert
_{{\Greekmath 0112} ={\Greekmath 0112} _{0}} \\
&+&\left. \frac{\partial }{\partial {\Greekmath 0112} }E\left[ \frac{\partial m_{{\Greekmath 0112}
_{0}}(y_{{\Greekmath 011C} ,{\Greekmath 0112} },\tilde{W}\left( {\Greekmath 010B} _{0}\right) )}{\partial z_{1}}
\right] \right\vert _{{\Greekmath 0112} ={\Greekmath 0112} _{0}}+\left. \frac{\partial }{\partial
{\Greekmath 0112} }E\left[ \frac{\partial m_{{\Greekmath 0112} _{0}}(y_{{\Greekmath 011C} },\tilde{W}\left(
{\Greekmath 010B} _{{\Greekmath 0112} }\right) )}{\partial z_{1}}\right] \right\vert _{{\Greekmath 0112}
={\Greekmath 0112} _{0}}.
\end{eqnarray*}
To use Theorem 2.1 of \cite{newey1994}, we need to write all these terms in
an outer-product form, namely the form of the right-hand side of (\ref
{path_derivative}). To search for the required function $\Gamma (\cdot )$,
we follow \cite{newey1994} and examine the components of $T_{2,{\Greekmath 0112} }$ one
at a time, treating the remaining components as known.
Lemma \ref{lemma_hahn_ridder} in the appendix shows that under some
conditions
\begin{equation*}
\left. \frac{\partial }{\partial {\Greekmath 0112} }E\left[ \frac{\partial m_{{\Greekmath 0112}
_{0}}(y_{{\Greekmath 011C} },\tilde{W}\left( {\Greekmath 010B} _{{\Greekmath 0112} }\right) )}{\partial z_{1}}
\right] \right\vert _{{\Greekmath 0112} ={\Greekmath 0112} _{0}}=0,
\end{equation*}
that is, we can ignore the error from estimating the propensity score in our
asymptotic analysis. Lemma \ref{param_ave_estimation} in the appendix
characterizes the influence function of $T_{2n}(\hat{y}_{{\Greekmath 011C} },\hat{m},\hat{
{\Greekmath 010B}})$ and establishes a stochastic approximation of $T_{2n}(\hat{y}
_{{\Greekmath 011C} },\hat{m},\hat{{\Greekmath 010B}})-T_{2}$ as follows:
\begin{equation}
T_{2n}(\hat{y}_{{\Greekmath 011C} },\hat{m},\hat{{\Greekmath 010B}})-T_{2}=\mathbb{P}_{n}{\Greekmath 0120}
_{\partial m_{0}}\left( W,y_{{\Greekmath 011C} }\right) +\mathbb{P}_{n}{\Greekmath 0120}
_{m_{0}}\left( Y,W,y_{{\Greekmath 011C} }\right) +\mathbb{P}_{n}\tilde{{\Greekmath 0120}}
_{Q}(Y,y_{{\Greekmath 011C} })+o_{p}(n^{-1/2}), \label{decom_derps_non0}
\end{equation}
where
\begin{align*}
{\Greekmath 0120} _{\partial m_{0}}\left( W,y_{{\Greekmath 011C} }\right) & :=\frac{\partial
m_{0}(y_{{\Greekmath 011C} },\tilde{W})}{\partial z_{1}}-T_{2}, \\
{\Greekmath 0120} _{m_{0}}\left( Y,W,y_{{\Greekmath 011C} }\right) & :=-\left[ \mathds{1}\left\{ Y\leq
y_{{\Greekmath 011C} }\right\} -m_{0}(y_{{\Greekmath 011C} },\tilde{W})\right] \times E\left[ \frac{
\partial \log f_{W}(W)}{\partial z_{1}}\bigg|\tilde{W}\right] ,
\end{align*}
and
\begin{equation*}
\tilde{{\Greekmath 0120}}_{Q}(Y,y_{{\Greekmath 011C} }):=\left[ \frac{{\Greekmath 011C} -\mathds{1}\left\{ Y\leq
y_{{\Greekmath 011C} }\right\} }{f_{Y}(y_{{\Greekmath 011C} })}\right] \times E\left[ \frac{\partial
f_{Y|\tilde{W}}(y_{{\Greekmath 011C} }|\tilde{W})}{\partial z_{1}}\right] .
\end{equation*}
This characterizes the contribution of each stage to the influence function
of $T_{2n}(\hat{y}_{{\Greekmath 011C} },\hat{m},\hat{{\Greekmath 010B}})$. The contribution from
estimating $m_{0}$, given by $\mathbb{P}_{n}{\Greekmath 0120} _{m_{0}}$, corresponds to
the one in Proposition 5 of \cite{newey1994} (p. 1362).
\subsection{Estimating the UQE}
\label{parap_uqr}
With $\hat{y}_{{\Greekmath 011C} },\hat{f}_{Y},\hat{m},\hat{{\Greekmath 010B}}$ given in the
previous subsections, we estimate the UQE by
\begin{equation}
\hat{\Pi}_{{\Greekmath 011C} }(\hat{y}_{{\Greekmath 011C} },\hat{f}_{Y},\hat{m},\hat{{\Greekmath 010B}})=-\frac{1
}{\hat{f}_{Y}(\hat{y}_{{\Greekmath 011C} })}\frac{T_{2n}(\hat{y}_{{\Greekmath 011C} },\hat{m},\hat{
{\Greekmath 010B}})}{T_{1n}(\hat{{\Greekmath 010B}})}. \label{UNIQUE_Expression}
\end{equation}
This is our unconditional instrumental quantile estimator (UNIQUE). With the
asymptotic linear representations of all three components $\hat{f}_{Y}(\hat{y
}_{{\Greekmath 011C} }),$ $T_{1n}(\hat{{\Greekmath 010B}}),$ and $T_{2n}(\hat{y}_{{\Greekmath 011C} },\hat{m},
\hat{{\Greekmath 010B}}),$ we can obtain the asymptotic linear representation of $\hat{
\Pi}_{{\Greekmath 011C} }(\hat{y}_{{\Greekmath 011C} },\hat{f}_{Y},\hat{m},\hat{{\Greekmath 010B}}).$ The next
theorem follows from combining Lemmas \ref{two_step_density}, \ref
{ps_estimation_param}, \ref{lemma_hahn_ridder}, and \ref
{param_ave_estimation}.
\begin{theorem}
\label{uqr_if_param} Under the assumptions of Lemmas \ref{two_step_density},
\ref{ps_estimation_param}, \ref{lemma_hahn_ridder}, and \ref
{param_ave_estimation}, we have
\begin{eqnarray}
\hat{\Pi}_{{\Greekmath 011C} }-\Pi _{{\Greekmath 011C} } &=&\frac{T_{2}}{f_{Y}(y_{{\Greekmath 011C} })^{2}T_{1}}
\left[ \mathbb{P}_{n}{\Greekmath 0120} _{f_{Y}}(Y,y_{{\Greekmath 011C} })+B_{f_{Y}}(y_{{\Greekmath 011C} })\right] +
\frac{T_{2}}{f_{Y}(y_{{\Greekmath 011C} })^{2}T_{1}}f_{Y}^{\prime }(y_{{\Greekmath 011C} })\mathbb{P}
_{n}{\Greekmath 0120} _{Q}(Y,y_{{\Greekmath 011C} }) \notag \\
&+&\frac{T_{2}}{f_{Y}(y_{{\Greekmath 011C} })T_{1}^{2}}\mathbb{P}_{n}{\Greekmath 0120} _{\partial
P}\left( W\right) +\frac{T_{2}}{f_{Y}(y_{{\Greekmath 011C} })T_{1}^{2}}E\left[ \frac{
\partial ^{2}P(Z,X,{\Greekmath 010B} _{0})}{\partial z_{1}\partial {\Greekmath 010B} _{0}^{\prime }
}\right] \mathbb{P}_{n}{\Greekmath 0120} _{{\Greekmath 010B} _{0}}\left( D,W\right)
\label{est_param_pi_decomp_5} \\
&-&\frac{1}{f_{Y}(y_{{\Greekmath 011C} })T_{1}}\mathbb{P}_{n}{\Greekmath 0120} _{\partial m_{0}}\left(
W,y_{{\Greekmath 011C} }\right) -\frac{1}{f_{Y}(y_{{\Greekmath 011C} })T_{1}}\mathbb{P}_{n}{\Greekmath 0120}
_{m_{0}}\left( Y,W,y_{{\Greekmath 011C} }\right) \notag \\
&&-\frac{1}{f_{Y}(y_{{\Greekmath 011C} })T_{1}}\mathbb{P}_{n}\tilde{{\Greekmath 0120}}_{Q}(Y,y_{{\Greekmath 011C}
})+R_{\Pi }, \notag
\end{eqnarray}
where
\begin{eqnarray*}
R_{\Pi } &=&O_{p}\left( |\hat{f}_{Y}(\hat{y}_{{\Greekmath 011C} })-f_{Y}(y_{{\Greekmath 011C}
})|^{2}\right) +O_{p}\left( n^{-1}\right) +O_{p}\left( n^{-1/2}|\hat{f}_{Y}(
\hat{y}_{{\Greekmath 011C} })-f_{Y}(y_{{\Greekmath 011C} })|\right) \\
&+&o_{p}\left( n^{-1/2}h^{-1/2}\right) +o_{p}(h^{2}).
\end{eqnarray*}
Furthermore, under Assumption \ref{Assumption_rate}, $\sqrt {nh}
R_\Pi=o_p(1).$
\end{theorem}
Equation (\ref{est_param_pi_decomp_5}) consists of six influence functions
and a bias term. The bias term $B_{f_{Y}}(y_{{\Greekmath 011C} })$ arises from estimating
the density and is of order $O(h^{2})$. The six influence functions reflect
the impact of each estimation stage. The rate of convergence of $\hat{\Pi}
_{{\Greekmath 011C} }$ is slowed down through $\mathbb{P}_{n}{\Greekmath 0120} _{f_{Y}}(Y)$, which is
of order $O_{p}(n^{-1/2}h^{-1/2})$. We can summarize the results of Theorem
\ref{uqr_if_param} in a single equation:
\begin{equation*}
\hat{\Pi}_{{\Greekmath 011C} }-\Pi _{{\Greekmath 011C} }=\mathbb{P}_{n}{\Greekmath 0120} _{\Pi _{{\Greekmath 011C} }}\left(
O\right) +\tilde{B}_{f_{Y}}(y_{{\Greekmath 011C} })+o_{p}(n^{-1/2}h^{-1/2}),
\end{equation*}
where ${\Greekmath 0120} _{\Pi _{{\Greekmath 011C} }}$ collects all the influence functions in (\ref
{est_param_pi_decomp_5}) except for the bias, and
\begin{equation*}
\tilde{B}_{f_{Y}}(y_{{\Greekmath 011C} }):=\frac{T_{2}}{f_{Y}(y_{{\Greekmath 011C} })^{2}T_{1}}
B_{f_{Y}}(y_{{\Greekmath 011C} }).
\end{equation*}
If $nh^{5}\rightarrow 0,$ then the bias term is $o(n^{-1/2}h^{-1/2})$. The
following corollary provides the asymptotic distribution of $\hat{\Pi}_{{\Greekmath 011C}
}$.
\begin{corollary}
\label{corollary_param}Under the assumptions of Theorem \ref{uqr_if_param}
and the assumption that $nh^{5}\rightarrow 0,$
\begin{equation*}
\sqrt{nh}\left( \hat{\Pi}_{{\Greekmath 011C} }-\Pi _{{\Greekmath 011C} }\right) =\sqrt{n}\mathbb{P}_{n}
\sqrt{h}{\Greekmath 0120} _{\Pi _{{\Greekmath 011C} }}\left( O\right) +o_{p}(1)\Rightarrow \mathcal{N}
(0,V_{{\Greekmath 011C} }),
\end{equation*}
where
\begin{equation}
V_{{\Greekmath 011C} }=\lim_{h\downarrow 0}E\left\{ h\left[ {\Greekmath 0120} _{\Pi _{{\Greekmath 011C} }}\left(
O\right) \right] ^{2}\right\} . \label{variance_param}
\end{equation}
\end{corollary}
From the perspective of asymptotic theory, all of the following terms are
all of order $O_{p}\left( h\right) =o_{p}\left( 1\right) $ and hence can be
ignored in large samples: $\sqrt{nh}\mathbb{P}_{n}{\Greekmath 0120} _{Q},$ $\sqrt{nh}
\mathbb{P}_{n}{\Greekmath 0120} _{\partial P},$ $\sqrt{nh}\mathbb{P}_{n}{\Greekmath 0120} _{{\Greekmath 010B}
_{0}},$ $\sqrt{nh}\mathbb{P}_{n}{\Greekmath 0120} _{\partial m_{0}},$ $\sqrt{nh}\mathbb{P}
_{n}{\Greekmath 0120} _{m_{0}},$ and $\sqrt{nh}\mathbb{P}_{n}\tilde{{\Greekmath 0120}}_{Q}.$ The
asymptotic variance is then given by
\begin{equation*}
V_{{\Greekmath 011C} }=\frac{T_{2}^{2}}{f_{Y}(y_{{\Greekmath 011C} })^{4}T_{1}^{2}}\lim_{h\downarrow
0}E\left[ h{\Greekmath 0120} _{f_{Y}}^{2}(y_{{\Greekmath 011C} })\right] =\frac{T_{2}^{2}}{
f_{Y}(y_{{\Greekmath 011C} })^{3}T_{1}^{2}}\int_{-\infty }^{\infty }K^{2}\left( u\right)
du.
\end{equation*}
However, $V_{{\Greekmath 011C} }$ ignores all estimation uncertainties except that in $
\hat{f}_{Y}(y_{{\Greekmath 011C} })$, and we do not expect it to reflect the
finite-sample variability of $\sqrt{nh}(\hat{\Pi}_{{\Greekmath 011C} }-\Pi _{{\Greekmath 011C} })$
well. To improve the finite-sample performances, we keep the dominating term
from each source of estimation errors and employ a sample counterpart of $
E[h{\Greekmath 0120} _{\Pi _{{\Greekmath 011C} }}^{2}]$ to estimate $V_{{\Greekmath 011C} }.$ The details can be
found in Section \ref{estimation_variance_appendix} of the supplementary
appendix.
\subsection{Testing the Null of No Effect}
\label{param_hypothesis}
We can apply Corollary \ref{corollary_param} to conduct hypothesis testing
on $\Pi _{{\Greekmath 011C} }$. Since $\hat{\Pi}_{{\Greekmath 011C} }$ converges to $\Pi _{{\Greekmath 011C} }$ at
a nonparametric rate, in general, the test will have power only against a
local departure of a nonparametric rate. However, if we are interested in
testing the null of a zero effect, that is, $H_{0}:\Pi _{{\Greekmath 011C} }=0$ vs. $
H_{1}:\Pi _{{\Greekmath 011C} }\neq 0,$ we can detect a parametric rate of departure from
the null. The reason is that, by (\ref{param_interest_3}), $\Pi _{{\Greekmath 011C} }=0$
if and only if $T_{2}=0,$ and $T_{2}$ can be estimated at the usual
parametric rate. Hence, instead of testing $H_{0}:\Pi _{{\Greekmath 011C} }=0$ vs. $
H_{1}:\Pi _{{\Greekmath 011C} }\neq 0,$ we can test the equivalent hypotheses $
H_{0}:T_{2}=0$ vs. $H_{1}:T_{2}\neq 0.$
Our test is based on the estimator $T_{2n}(\hat{y}_{{\Greekmath 011C} },\hat{m},\hat{
{\Greekmath 010B}})$ of $T_{2}$. In view of its influence function given in Lemma \ref
{param_ave_estimation}, we can estimate the asymptotic variance of $T_{2n}(
\hat{y}_{{\Greekmath 011C} },\hat{m},\hat{{\Greekmath 010B}})$ by
\begin{equation*}
\hat{V}_{2}=\frac{1}{n}\sum_{i=1}^{n}\left( \hat{{\Greekmath 0120}}_{\partial m,i}+\hat{
{\Greekmath 0120}}_{m,i}+\hat{E}\left[ \frac{\partial f_{Y|\tilde{W}\left( {\Greekmath 010B}
_{0}\right) }(\hat{y}_{{\Greekmath 011C} }|\tilde{W}\left( \hat{{\Greekmath 010B}}\right) )}{
\partial z_{1}}\right] \hat{{\Greekmath 0120}}_{Q,i}\right) ^{2}
\end{equation*}
where $\hat{{\Greekmath 0120}}_{\partial m,i},$ $\hat{{\Greekmath 0120}}_{m,i}$ and $\hat{{\Greekmath 0120}}_{Q,i}$
are plug-in estimates of ${\Greekmath 0120} _{\partial m_{0}}\left( W_{i},y_{{\Greekmath 011C}
}\right) ,{\Greekmath 0120} _{m_{0}}\left( Y_{i},W_{i},y_{{\Greekmath 011C} }\right) ,$ and $\hat{{\Greekmath 0120}}
_{Q}\left( Y_{i},y_{{\Greekmath 011C} }\right) ,$ respectively.
We can then form the test statistic:
\begin{equation}
T_{2n}^{o}:=\sqrt{n}\frac{T_{2n}(\hat{y}_{{\Greekmath 011C} },\hat{m},\hat{{\Greekmath 010B}})}{
\sqrt{\hat{V}_{2}}}. \label{test_statistic_t2}
\end{equation}
By Lemma \ref{param_ave_estimation} and using standard arguments, we can
show that $T_{2n}^{o}\Rightarrow \mathcal{N}(0,1).$ To save space, we omit
the details here.
\section{Simulation Evidence}
\label{simulation}
In the simulation study, we consider the following structural equations:
\begin{align*}
Y(0)& =X_{1}+X_{2}+U_{0}, \\
Y(1)& ={\Greekmath 010C} +X_{1}+X_{2}+U_{1}, \\
D& =\mathds{1}\left\{ (Z_{1}+Z_{2}+Z_{3}+X_{1}+X_{2})/\sqrt{5}>V\right\} .
\end{align*}
Here, $(U_{0},U_{1},Z_{1},Z_{2},Z_{3},X_{1},X_{2})$ are jointly normal,
independent variables, with mean $0$ and unit variances, and
\begin{equation*}
V=\frac{{\Greekmath 0125}}{\sqrt{2}} U_{0}+\frac{{\Greekmath 0125}}{\sqrt{2}} U_{1}+\sqrt{
1-{\Greekmath 0125} ^{2}}e,
\end{equation*}
where $e$ is standard normal, independent of $
(U_{0},U_{1},Z_{1},Z_{2},Z_{3},X_{1},X_{2})$.\footnote{
Note that $V$ has also unit variance.} The parameter ${\Greekmath 0125} $ governs the
endogeneity of $D$.
The target of interest is the UQE $\Pi _{{\Greekmath 011C} }$, as defined in (\ref
{equ_pi_tau}), involving $y_{{\Greekmath 011C} },f_{Y},$ $T_{1},$ and $T_{2}$.\footnote{
Detailed calculations of $\Pi _{{\Greekmath 011C} }$ are available from Appendix B of an
earlier working paper \cite{sun2021},} We estimate $\Pi _{{\Greekmath 011C} }$ using the
UNIQUE outlined in Section \ref{estimation}. More specifically, for
estimating the (population) quantile $y_{{\Greekmath 011C} },$ we utilize the sample
quantile. To estimate $f_{Y}$, we use a Gaussian kernel with bandwidth $
h=1.06\times \hat{{\Greekmath 011B}}_{Y}\times n^{-1/4}$, where $\hat{{\Greekmath 011B}}_{Y}$ is
the sample standard deviation of $Y$. For estimating $T_{1}$, we employ a
probit model, and for $T_{2}$, we run a cubic series regression.
\subsection{Testing the Null Hypothesis of No Effect}
If we set ${\Greekmath 010C} =0$, $\Pi _{{\Greekmath 011C} }=0$ and so the null hypothesis of a zero
effect holds. The test statistic $T_{2n}^{o}$ is constructed following
equation \eqref{test_statistic_t2}. Because the test statistic does not
involve estimating the density (or $T_{1}$), the test has nontrivial power
again $1/\sqrt{n}$-departures (i.e., ${\Greekmath 010C} =c/\sqrt{n}$ for some $c\neq 0)$
from the null.
To simulate the power function of the nominal 5\% test, we consider a range
of 25 values of ${\Greekmath 010C} $ between $-1$ and $1$. The endogeneity, governed by
the parameter ${\Greekmath 0125} $, takes five values: 0, 0.25, 0.5, 0.75, and 0.9. We
perform 1,000 simulations with 1,000 observations. For different values of $
{\Greekmath 011C} $, the power functions are shown below in Figure \ref{power_1}.
\ The test has the desired null rejection probability, except for the
extreme quantile ${\Greekmath 011C} =0.1$, where under high endogeneity, the rejection
probability does not increase fast enough. The power function for the
median, ${\Greekmath 011C} =0.5$, is not shown here as it is almost identical to that of $
{\Greekmath 011C} =0.4$. Furthermore, simulation results not reported here show that the
power functions for ${\Greekmath 011C} =0.6,0.7,0.8,0.9$ are very similar to those of $
{\Greekmath 011C} =0.4,0.3,0.2,0.1,$ respectively.
\begin{figure}[h]
\centering
\includegraphics[scale =0.7]{figures_power.png}
\caption{ Power functions for different values of $\protect{\Greekmath 0125} $ and $
\protect{\Greekmath 011C} $.}
\label{power_1}
\end{figure}
\subsection{Empirical Coverage of Confidence Intervals}
In this subsection, we investigate the empirical coverage of confidence
intervals built using $\hat{V}_{{\Greekmath 011C} }$, the variance estimator given in
\eqref{var_param_if}. Since $\sqrt{nh}\left( \hat{\Pi}_{{\Greekmath 011C} }-\Pi _{{\Greekmath 011C}
}\right) \approx \mathcal{N}(0,\hat{V}_{{\Greekmath 011C} }),$ a 95\% confidence interval
for $\Pi _{{\Greekmath 011C} }$ can be constructed using
\begin{equation*}
\hat{\Pi}_{{\Greekmath 011C} }\pm 1.96\times \frac{\sqrt{\hat{V}_{{\Greekmath 011C} }}}{\sqrt{nh}}.
\end{equation*}
We use a grid of ${\Greekmath 010C} $ that takes values $-1,-0.5,-0.25,0,0.25,0.5,$ and $
1$. For the endogeneity parameter, ${\Greekmath 0125} $, we take the values $
0,0.25,0.5,0.75,$ and $0.9$. Finally, ${\Greekmath 011C} $ takes the values from $0.1$ to
$0.9$ with an increment of 0.1. We note that, for values of ${\Greekmath 010C} \neq 0$,
where the effect is not 0, we need to numerically compute the value of $\Pi
_{{\Greekmath 011C} }$. We perform 10,000 simulations with 1,000 observations each. The
results are reported in the tables below for ${\Greekmath 011C} =0.1$ and ${\Greekmath 011C} =0.5$. It
is clear that the confidence intervals have reasonable coverage accuracy in
almost all cases.
\begin{table}[h]
\centering
\begin{tabular}{llllll}
${\Greekmath 010C} \setminus {\Greekmath 0125} $ & 0 & 0.25 & 0.5 & 0.75 & 0.9 \\ \hline\hline
-1 & 0.9701 & 0.9739 & 0.9717 & 0.9715 & 0.9740 \\
-0.5 & 0.9696 & 0.9706 & 0.9754 & 0.9691 & 0.9745 \\
-0.25 & 0.9732 & 0.9724 & 0.9753 & 0.9743 & 0.9731 \\
0 & 0.9754 & 0.9727 & 0.9746 & 0.9750 & 0.9782 \\
0.25 & 0.9793 & 0.9780 & 0.9758 & 0.9775 & 0.9768 \\
0.5 & 0.9807 & 0.9824 & 0.9800 & 0.9817 & 0.9772 \\
1 & 0.9808 & 0.9829 & 0.9821 & 0.9789 & 0.9808 \\ \hline
\end{tabular}
\caption{Empirical coverage of 95\% confidence intervals for $\protect{\Greekmath 011C} =
0.1$.}
\end{table}
\bigskip
\bigskip
\begin{table}[h]
\centering
\begin{tabular}{llllll}
${\Greekmath 010C} \setminus {\Greekmath 0125} $ & 0 & 0.25 & 0.5 & 0.75 & 0.9 \\ \hline\hline
-1 & 0.9679 & 0.9741 & 0.9737 & 0.9769 & 0.9748 \\
-0.5 & 0.9608 & 0.9622 & 0.9623 & 0.9705 & 0.9697 \\
-0.25 & 0.9587 & 0.9605 & 0.9630 & 0.9607 & 0.9602 \\
0 & 0.9616 & 0.9565 & 0.9569 & 0.9591 & 0.9524 \\
0.25 & 0.9579 & 0.9598 & 0.9594 & 0.9571 & 0.9579 \\
0.5 & 0.9640 & 0.9587 & 0.9551 & 0.9575 & 0.9569 \\
1 & 0.9621 & 0.9667 & 0.9666 & 0.9672 & 0.9668 \\ \hline
\end{tabular}
\caption{Empirical coverage of 95\% confidence intervals for $\protect{\Greekmath 011C} =
0.5$.}
\end{table}
\section{Empirical Application}
\label{empirical}
We estimate the unconditional quantile effect of expanding college
enrollment on (log) wages. The outcome variable $Y$ is the log wage, and the
binary treatment is the college enrollment status. Thus, $p=\Pr [D=1]$ is
the proportion of individuals who ever enrolled in a college. Arguably, the
cost of tuition $(Z_{1})$, assumed to be continuous, is an important factor
that affects the college enrollment status but not the wage directly. In
order to alter the proportion of enrolled individuals, we consider a policy
that subsidizes tuition by a certain amount. The UQE is the effect of this
policy on the different quantiles of the unconditional distribution of wages
when the subsidy is small.
This policy shifts $Z_{1}$, the tuition, to $Z_{1{\Greekmath 010E} }=Z_{1}+s({\Greekmath 010E} )$
for some $s({\Greekmath 010E} )$, which is the same for all individuals, and induces a
small change in college enrollment. Note that we do not need to specify $
s({\Greekmath 010E} )$ because we look at the limiting version as ${\Greekmath 010E} \rightarrow 0$
. In practice, we may set $s({\Greekmath 010E} )$ equal to a number that is relatively
small compared to the total tuition.
We use the same data as in \cite{Carneiro2010} and \cite{Carneiro2011}: a
sample of white males from the 1979 National Longitudinal Survey of Youth
(NLSY1979). The web appendix to \cite{Carneiro2011} contains a detailed
description of the variables. The outcome variable $Y$ is the log wage in
1991. The treatment indicator $D$ is equal to $1$ if the individual ever
enrolled in college by 1991, and $0$ otherwise. The other covariates are
AFQT score, mother's education, number of siblings, average log earnings
1979--2000 in the county of residence at age 17, average unemployment
1979--2000 in the state of residence at age 17, urban residence dummy at age
14, cohort dummies, years of experience in 1991, average local log earnings
in 1991, and local unemployment in 1991. We collect these variables into a
vector and denote it by $X$.
We assume that the following four variables (denoted by $
Z_{1},Z_{2},Z_{3},Z_{4}$) enter the selection equation but not the outcome
equation: tuition at local public four-year colleges at age 17, presence of
a four-year college in the county of residence at age 14, local earnings at
age 17, and local unemployment at age 17. The total sample size is 1747, of
which 882 individuals had never enrolled in a college ($D=0$) by 1991, and
865 individuals had enrolled in a college by 1991 ($D=1)$. We estimate the
UQE of a marginal shift in the tuition at local public four-year colleges at
age 17 ($Z_{1})$ using the UNIQUE.
Here are some details of the UNIQUE. To estimate the propensity score, we
use a parametric logistic specification. To estimate the conditional
expectation function $m_{0}$,$\ $we run a series regression using the
estimated propensity score and the covariates $Z_{2},Z_{3},Z_{4},$ and $X$\
as the regressors. Due to the large number of variables involved, a
penalization of ${\Greekmath 0115} =10^{-4}$ was imposed on the $L_{2}$-norm of the
coefficients, excluding the constant term as in ridge regressions. We
estimate the UQE at the quantile level ${\Greekmath 011C} =0.1,0.15,\ldots ,0.9$. For
each ${\Greekmath 011C} ,$ we also construct the 95\% (pointwise) confidence interval.
Figure \ref{emp_app_1} presents the results. The estimated UQE ranges
between 0.22 and 0.47 across the quantiles with an average of 0.37. When we
estimate the unconditional mean effect, we obtain an estimate of 0.21, which
is somewhat consistent with the quantile cases. We interpret these estimates
in the following way: the effect of a ${\Greekmath 010E} $ (small) increase in college
enrollment induced by an additive change in tuition increases (log) wages
between $0.22\times {\Greekmath 010E} $ and $0.47\times {\Greekmath 010E} $ across quantiles. For
example, for ${\Greekmath 010E} =0.01$, we obtain an increase in the quantiles of the
wage distribution between $0.22\%$ and $0.47\%$.
\begin{figure}[h]
\centering
\includegraphics[scale =0.7]{app_emp.eps}
\caption{ \emph{Solid line: point estimates of the UQEs; Dashed line: 95\%
confidence intervals.}}
\label{emp_app_1}
\end{figure}
\section{Conclusion}
\label{conclusion}
In this paper we study the unconditional policy effect with an endogenous
binary treatment. Framing the selection equation as a threshold-crossing
model allows us to introduce a novel class of unconditional marginal
treatment effects and represent the unconditional effect as a weighted
average of these unconditional marginal treatment effects. When the policy
variable used to change the participation rate satisfies a conditional
exogeneity condition, it is possible to recover the unconditional policy
effect using the proposed UNIQUE method.
To illustrate the usefulness of unconditional MTEs, we focus on the
unconditional quantile effect. We find that the unconditional quantile
regression estimator that neglects endogeneity can be severely biased.
Moreover, the bias may not be uniform across quantiles. Any attempt to sign
the bias \emph{a priori} requires very strong assumptions on the
data-generating process. Intriguingly, the unconditional quantile regression
estimator can be inconsistent even if the treatment status is exogenously
determined. This happens when the treatment selection is partly determined
by some covariates that also influence the outcome variable.
Our findings reveal that the unconditional quantile effect (UQE) and the
marginal policy-relevant treatment effect (MPRTE) can be seen as part of the
same family of policy effects. From a purely robustness perspective, the
unconditional median effect---a special UQE---can be considered a more
robust version of the MPRTE, in the same way that the median is the robust
counterpart of the mean. Both represent specific examples of a general
unconditional policy effect. To the best of our knowledge, this connection
has not been established in the literature on either UQE or MPRTE.