# Regression Discontinuity Design under Self-selection ††thanks: We would like
to thank Zhuan Pei, Yanqing Fan, Peter Hull and participants in seminars and
conferences at which this paper was presented. All remaining errors are ours.

Sida Peng and Yang Ning Microsoft Research<EMAIL_ADDRESS>of
Statistics and Data Science, Cornell University<EMAIL_ADDRESS>
###### Abstract

In Regression Discontinuity (RD) design, self-selection leads to different
distributions of covariates on two sides of the policy intervention, which
essentially violates the continuity of potential outcome assumption. The
standard RD estimand becomes difficult to interpret due to the existence of
some indirect effect, i.e. the effect due to self selection. We show that the
direct causal effect of interest can still be recovered under a class of
estimands. Specifically, we consider a class of weighted average treatment
effects tailored for potentially different target populations. We show that a
special case of our estimands can recover the average treatment effect under
the conditional independence assumption per Angrist and Rokkanen (2015), and
another example is the estimand recently proposed in Frölich and Huber (2018).
We propose a set of estimators through a weighted local linear regression
framework and prove the consistency and asymptotic normality of the
estimators. Our approach can be further extended to the fuzzy RD case. In
simulation exercises, we compare the performance of our estimator with the
standard RD estimator. Finally, we apply our method to two empirical data
sets: the U.S. House elections data in Lee (2008) and a novel data set from
Microsoft Bing on Generalized Second Price (GSP) auction.
Keywords: causal inference, regression discontinuity, weighted average
treatment effect, inverse probability weighted estimator

##  1 Introduction

Regression discontinuity (RD) design is an important policy evaluation tool
that has been widely used in empirical studies. Under the continuity
assumptions (Lee, 2008), the RD design gives rise to many testable restriction
similar to a randomized control trial, and it allows for the identification of
causal effects (Hahn et al., 2001). Standard nonparametric tools like series
expansion method or kernel regression method can be applied to estimate this
quantity under this minimal assumption, see Imbens and Lemieux (2008) and
Cattaneo and Escanciano (2017).

An implication of the continuity assumption is that the distribution of the
covariates conditional on the assignment variable is continuous at the cutoff.
To valid this assumption, a variety of statistical tests and nonparametric
inference procedures have been proposed, including Cattaneo et al. (2015) and
Canay and Kamat (2018). However, this assumption may not hold in many
applications. In particular, we consider the following two motivating
examples.

######  Example 1 (College scholarship).

Consider the classical scholarship example per Thistlethwaite and Campbell
(1960). Students receive scholarships if their test score is higher than a
threshold. The authors applied the RD framework to estimate the causal effect
of the scholarship on student’s future education outcomes. In this example,
the continuity assumption implies that the distributions of the covariate
information for students whose score is just above and below the threshold are
similar. However, if students are aware of this policy before they take the
SAT exam, some of them may study harder to pass this threshold. As a result,
students whose score is on two sides of the threshold may have distinct
characteristics (e.g., gender).

######  Example 2 (Generalized Second Price (GSP) auction).

GSP is widely used by internet search engines like Google or Bing to allocate
sponsored search advertisements. In reality, it is implemented through a
reservation score system. Each bidder is assigned a score as a function of
their bids and characteristics. The bidder’s advertisement is displayed if her
score passes the reservation. However, due to design of the score system,
bidders with low quality may be selected below the reservation score while
bidders with high quality may be selected above the reservation score. Given
that low quality bidders may have a high willingness to pay, we often observe
the distribution of bids to be different on two sides of the reservation
score. We will further elaborate this example in the real data analysis.

Due to different distributions of the covariates at the cutoff in these
examples, the standard RD estimand is no longer valid. We show in those cases
the standard RD estimand can be decomposed into a direct treatment effect and
an indirect treatment effect. The indirect effect is due to the unbalanced
covariates near the cut-off. For example, the policy intervention may result
more boys than girls to receive scholarship. If in general boys perform
different from girls in SAT test, the difference in SAT score due to gender
will also be accounted into the standard RD estimand as if the policy
intervention is selecting on genders. However as no direct causal mechanism is
assumed between the unbalanced covariates and running variable, the selection
could be generated due to an unknown equilibrium, completely reversed or
purely spurious. For example, in a GSP auction, the reservation score is
designed to separate the bidders by quality and in the meanwhile bidders may
self-select such that low quality bidders may have incentive to bid higher. As
a result, policy changes to move reservation score can be risky if the
equilibrium between reservation score and quality of bidders are not
disentangled.

In this paper, we propose a new framework to address this problem by adjusting
unbalanced covariates due to self-selection. Consider the following sharp RD
setup: $T$ is a binary treatment variable, $Y(1)$ and $Y(0)$ are the potential
outcomes under $T=1$ and $T=0$ and $X$ is the running variable so that the
treatment is fully determined by $T=\mathbb{1}(X>c)$ for a known threshold
$c$. In the classical RD framework, it is assumed that $\mathbb{E}(Y(t)|X=x)$
is continuous at $x=c$ for $t=0,1$; see Assumption 3. This essentially assumes
away self-selection based on both observed and unobserved covariates. To
account for the self-selection effect, we assume that further covariate
information can be collected. In particular, we define $Z(1)$ and $Z(0)$ as
the potential covariates with or without treatment. By using the potential
“outcome” formulation, we allow the distribution of the covariates on two
sides of the threshold to be different, i.e., discontinuity of the covariate
distribution. By controlling all unbalanced covariates, we assume that
$\mathbb{E}(Y(t)|X=x,Z(t)=z)$ is continuous in $x$ at the threshold; see
Assumption 5. This is the main assumption made in this paper.

In the classical RD framework, the standard causal parameter of interest is
the marginal treatment effect $\mathbb{E}(Y(1)-Y(0)|X=c)$ at the cutoff. In
the presence of self-selection, the continuity assumption on
$\mathbb{E}(Y(t)|X=x)$ may fail. Thus, the marginal treatment effect can be
confounded by the discontinuity of the conditional mean of covariates at the
cutoff, as seen in equation (2.4). In this paper, we propose a class of
estimands for RD design in the framework of weighted average treatment effect
(WATE). We show that our estimands can tease out the effect of discontinuity
of the conditional distribution of covariates through re-weighting the
marginal treatment effects. For instance, under the constant treatment effect
model with unbalanced covariates, our estimands reduce to the direct treatment
effect. One special case of our estimands is Frölich and Huber (2018). Another
special case can be interpreted as the “global” average treatment effect (ATE)
$\mathbb{E}(Y(1)-Y(0))$ under the conditional independence assumption (CIA) as
in Angrist and Rokkanen (2015). The CIA assumes that the treatment is mean
independent of the running variable near the cutoff conditional on the
covariates. Intuitively, after projecting the outcome variable onto a rich set
of covariates (excluding running variable), the residual should not depend on
the running variable and can be viewed as an experiment with randomly assigned
treatment. We show that our method identifies ATE under CIA assumption,
whereas the standard RD estimand remains a “local” treatment effect at the
cutoff (Lee, 2008).

We further provide the nonparametric identification for our estimands and
propose nonparametric estimators based on the inverse propensity score
weighted (IPW) approach, see Horvitz and Thompson (1952) and Abadie and Imbens
(2016). However, notice that the treatment assignment is degenerate with
respect to the running variable. We get around this problem by considering the
marginal effect of the treatment and re-weighting on the other covariates
first. The kernel method is applied to estimate the conditional mean function.
The consistency and asymptotic normality of the proposed estimator are
established. We further extend our method to the fuzzy RD design where the
treatment compliance is imperfect. Similarly, we provide the nonparametric
identification of the causal effect under the fuzzy RD design, and propose a
nonparametric estimator. The proposed estimator is similar to that of the
local average treatment effect (LATE) in a fractional format but with
numerator being adjusted to incorporate additional selections.

This work is connected to the growing literature on the RD design with
covariates. In particular, two recent papers provide insightful guidance on
the subject. Calonicoy et al. (2017) estimated the marginal treatment effect
by a local linear regression with the linear-in-parameters specification for
the covariates. The main advantage of their method is that the nonparametric
estimation of $\mathbb{E}(Y(t)|X=x,Z(t)=z)$ is avoided. In another paper,
Frölich and Huber (2018) proposed a fully nonparametric estimator of the
marginal treatment effect by estimating $\mathbb{E}(Y(t)|X=x,Z(t)=z)$
nonparametrically. They allowed the conditional density of $Z(t)$ given $X$ to
be discontinuous. Our work differs from the above papers by considering a
different estimand that is less local under CIA and identifies the direct
treatment effect under the constant treatment effect model. Unlike Calonicoy
et al. (2017), we do not require the continuity of the conditional mean of
$Z(t)$ given $X$. Instead, our Assumption 5 is similar to assumption 1 (iv) in
Frölich and Huber (2018) under the sharp RD design. The proposed IPW estimator
is also different from the above regression based estimators.

The rest of the paper is organized as follows. In Section 2, we propose our
new estimands in the form of weighted average treatment effects and establish
its nonparametric identification. In Section 3, we propose a set of weighted
local linear estimators. The theoretical properties are established in Section
4. In Section 5, we conduct simulation studies and also apply the method to
two empirical data sets: the U.S. House elections data in Lee (2008) and a
novel data set from Microsoft Bing on Generalized Second Price (GSP) auction.
Finally, we consider the extension to the fuzzy RD case in Section 6. The
proofs are deferred to the appendix.

##  2 Sharp RD Design

###  2.1 Problem Setup and Continuity Assumption

In the standard RD design setting, we observe $n$ i.i.d. random samples
$\\{Y_{i},X_{i},Z_{i},T_{i}\\}_{i=1}^{n}$, where $Y_{i}$ is the outcome
variable of interest for the $i$th sample, $T_{i}\in\\{0,1\\}$ is the binary
treatment variable, $X_{i}\in\mathbb{R}$ is the running variable and
$Z_{i}\in\mathbb{R}^{p}$ is the covariate. In the sharp RD design, the
treatment $T_{i}$ is perfectly assigned through the running variable $X_{i}$
relative to a known cutoff $c$. For example, we have

$T_{i}=\mathbb{1}(X_{i}>c).$

Adopting a potential outcome framework, we can write the observed outcome
variable $Y_{i}$ as

$Y_{i}=Y_{i}(0)\cdot(1-T_{i})+Y_{i}(1)\cdot T_{i},$

where $Y_{i}(0)$ and $Y_{i}(1)$ represent the potential outcomes without or
with treatment. The average treatment effect is defined as
$\mathbb{E}(Y_{i}(1)-Y_{i}(0))$. However, this estimand is not identifiable
under the RD design as the treatment assignment $T_{i}$ is a deterministic
function of $X_{i}$. In this framework, Hahn et al. (2001), Lee (2008) and
Cattaneo et al. (2015) showed that one can still identify the treatment effect
at the cutoff

$\tau_{SRD}=\mathbb{E}(Y_{i}(1)-Y_{i}(0)|X_{i}=c),$

under the following continuity assumption.

######  Assumption 3 (Continuity Assumption of
$\mathbb{E}(Y_{i}(t)|X_{i}=x)$).

$\mathbb{E}(Y_{i}(1)|X_{i}=x)\quad\text{ and
}\quad\mathbb{E}(Y_{i}(0)|X_{i}=x)$

are continuous in $x$ at $x=c$.

This assumption implies that the conditional mean of the potential outcomes
near the cutoff $x=c$ are similar. There is no discontinuity of the
conditional mean functions at the cutoff. This assumption enables us to
identify $\tau_{SRD}$ in the RD design. We refer to Hahn et al. (2001), Lee
(2008) and Cattaneo et al. (2015) for further discussion on this assumption.

Now let us consider the case that additional covariates $Z_{i}$ are observed.
Denote

$Z_{i}=Z_{i}(0)\cdot(1-T_{i})+Z_{i}(1)\cdot T_{i},$

where $Z_{i}(0)$ and $Z_{i}(1)$ represent the potential covariates without or
with treatment. In the presence of covariates $Z_{i}$, the causal parameter
$\tau_{SRD}$ can be rewritten as

$\tau_{SRD}=\mathbb{E}\Big{(}\mathbb{E}[Y_{i}(1)|X_{i}=c,Z_{i}(1)]-\mathbb{E}[Y_{i}(0)|X_{i}=c,Z_{i}(0)]\Big{|}X_{i}=c\Big{)}.$
(2.1)

In a recent work, Calonicoy et al. (2017) proposed a kernel based estimator of
$\tau_{SRD}$ by accounting for the additional covariates $Z_{i}$. In addition
to the continuity Assumption 3, it is also assumed that
$\mathbb{E}(Z_{i}(1)|X_{i}=c)=\mathbb{E}(Z_{i}(0)|X_{i}=c)$ for the
consistency of the resulting kernel estimator, that is the potential
covariates $Z_{i}(1)$ and $Z_{i}(0)$ have the same conditional mean at the
cutoff $X_{i}=c$.

However, in some applications we may observe
$\mathbb{E}(Z_{i}(1)|X_{i}=c)\neq\mathbb{E}(Z_{i}(0)|X_{i}=c)$, when self-
selection based on the covariates exists. For example, consider the classical
scholarship example. Students with SAT score higher than a threshold will
receive scholarship. The treatment effect of interest is the effect of
scholarship on the students’ first semester GPAs. If the cutoff is pre-
released, it might be possible that students with some common characteristics
(i.e. gender) may study harder to pass the bar. This leads to an ex-ante
selection based on the covariates. Thus, one may observe that the conditional
mean functions of $Z_{i}$ given $X_{i}$ right below or above the threshold are
different, i.e.,

$\lim_{x\rightarrow c^{+}}\mathbb{E}(Z_{i}|X_{i}=x)\neq\lim_{x\rightarrow
c^{-}}\mathbb{E}(Z_{i}|X_{i}=x).$ (2.2)

The following simple lemma essentially says that the self-selection based on
the covariates (i.e., eq 2.2) implies
$\mathbb{E}(Z_{i}(1)|X_{i}=c)\neq\mathbb{E}(Z_{i}(0)|X_{i}=c)$.

######  Lemma 4.

If $\mathbb{E}(Z_{i}(t)|X_{i}=x)$ is continuous at $x=c$ for $t\in\\{0,1\\}$,
then

$\mathbb{E}(Z_{i}(1)|X_{i}=c)=\mathbb{E}(Z_{i}(0)|X_{i}=c)\Longleftrightarrow\lim_{x\rightarrow
c^{+}}\mathbb{E}(Z_{i}|X_{i}=x)=\lim_{x\rightarrow
c^{-}}\mathbb{E}(Z_{i}|X_{i}=x).$

###### Proof.

By the continuity assumption on $\mathbb{E}(Z_{i}(t)|X_{i}=x)$,

$\mathbb{E}(Z_{i}(1)|X_{i}=c)=\lim_{x\rightarrow
c^{+}}\mathbb{E}(Z_{i}(1)|X_{i}=x)=\lim_{x\rightarrow
c^{+}}\mathbb{E}(Z_{i}|X_{i}=x),$

and similarly $\mathbb{E}(Z_{i}(0)|X_{i}=c)=\lim_{x\rightarrow
c^{-}}\mathbb{E}(Z_{i}|X_{i}=x)$. The lemma holds. ∎

The above lemma provides a convenient way to check whether
$\mathbb{E}(Z_{i}(1)|X_{i}=c)=\mathbb{E}(Z_{i}(0)|X_{i}=c)$ holds in empirical
studies. One may simply plot the observed covariates $Z_{i}$ against $X_{i}$
and examine whether there is a discontinuity of the trend around $x=c$. The
method is applied in the real data analysis.

In the following, we investigate the consequence of
$\mathbb{E}(Z_{i}(1)|X_{i}=c)\neq\mathbb{E}(Z_{i}(0)|X_{i}=c)$. To be
specific, we consider the following constant treatment effect model

$\mathbb{E}(Y_{i}(t)|X_{i},Z_{i}(t))=\alpha+\tau\mathbb{1}(t=1)+g(X_{i})+Z_{i}(t)\gamma,$
(2.3)

for $t\in\\{0,1\\}$, where $g(\cdot)$ is an arbitrary continuous function. By
(2.1), one can show that

$\tau_{SRD}=\tau+\Big{(}\mathbb{E}(Z_{i}(1)|X_{i}=c)-\mathbb{E}(Z_{i}(0)|X_{i}=c)\Big{)}\gamma.$
(2.4)

The estimand $\tau_{SRD}$ can be decomposed into two terms. The first term
$\tau$ represents the direct treatment effect after controlling the running
variable $X_{i}$ and the covariates $Z_{i}(t)$. The second term in the right
hand side of (2.4) can be interpreted as the indirect effect of the policy due
to the unbalanced covariates near the cutoff or self-selection, which is
nonzero if $\gamma\neq 0$ and
$\mathbb{E}(Z_{i}(1)|X_{i}=c)\neq\mathbb{E}(Z_{i}(0)|X_{i}=c)$. In many
applications, the direct treatment effect $\tau$ is usually more meaningful
and interpretable than $\tau_{SRD}$, as $\tau_{SRD}$ is confounded by the
self-selection effect.

In this example, when the indirect effects in $\tau_{SRD}$ is assumed away by
requiring the continuity of the conditional density of the covariates $Z_{i}$
at the cutoff $X_{i}=c$, the data around the cutoff can be viewed as a natural
experiment and continuity on the covariates implies a balanced design for this
experiment so that we can estimate a local average treatment effect. However
when the self-selection exits, the experiment is no longer balanced and the
average treatment effect $\tau_{SRD}$ will typically differ from the direct
effect $\tau$.

###  2.2 Weighted Average Treatment Effect

As seen in (2.4), if
$\mathbb{E}(Z_{i}(1)|X_{i}=c)\neq\mathbb{E}(Z_{i}(0)|X_{i}=c)$ (i.e., the
covariates are unbalanced at the cutoff) and $\gamma\neq 0$, $\tau_{SRD}$ can
be different from the causal parameter of interest. To overcome this
difficulty, we propose a new class of causal parameters, called the weighted
average treatment effect (WATE), which are defined as

$\displaystyle\tau^{w}_{SRD}$
$\displaystyle=\mathbb{E}\\{Y(1)w_{1}(Z(1))|X=c\\}-\mathbb{E}\\{Y(0)w_{0}(Z(0))|X=c\\}$
$\displaystyle=\int\Big{[}\mathbb{E}(Y(1)|X=c,Z(1)=z)w_{1}(z)f_{Z(1)|X}(z|c)$
$\displaystyle~{}~{}~{}~{}~{}~{}-\mathbb{E}(Y(0)|X=c,Z(1)=z)w_{0}(z)f_{Z(0)|X}(z|c)\Big{]}dz,$
(2.5)

where $w_{1}(\cdot)$ and $w_{0}(\cdot)$ denote different choices of weights to
form the estimand, and $f_{Z(1)|X}(\cdot|\cdot)$ and $f_{Z(0)|X}(\cdot|\cdot)$
are the conditional density of $Z(1)$ and $Z(0)$ given $X$. In order to
interpret (2.5) as the WATE, we require the following normalization condition
for $w_{1}(\cdot)$ and $w_{0}(\cdot)$:

$\int w_{1}(z)f_{Z(1)|X}(z|c)dz=\int w_{0}(z)f_{Z(0)|X}(z|c)dz=1.$

In particular, by choosing appropriate $w_{1}(\cdot)$ and $w_{0}(\cdot)$,
(2.5) can be interpreted as the average of the difference of the conditional
mean functions corresponding to a target population. To see this, we consider
the following examples. Denote

$\Delta(c,z)=\mathbb{E}(Y(1)|X=c,Z(1)=z)-\mathbb{E}(Y(0)|X=c,Z(0)=z).$

  * •

Average treatment effect over entire population:

$\tau^{w1}_{SRD}=\int\Delta(c,z)f_{Z}(z)dz,$

where $f_{Z}(\cdot)$ is the p.d.f of the covariates $Z$. In $\tau^{w1}_{SRD}$,
we average the conditional mean difference over the entire population whose
covariates follow from the marginal distribution $f_{Z}(z)$. It is easy to see
that WATE defined in (2.5) recovers $\tau^{w1}_{SRD}$ by taking

$w_{1}(z)=\frac{f_{Z}(z)}{f_{Z(1)|X}(z|c)},~{}~{}\textrm{and}~{}~{}w_{0}(z)=\frac{f_{Z}(z)}{f_{Z(0)|X}(z|c)}.$

  * •

Average treatment effect over locally untreated population:

$\tau^{w2}_{SRD}=\int\Delta(c,z)f_{Z(0)|X}(z|c)dz.$

In this causal parameter, we average the conditional mean difference over the
untreated population right below the threshold whose covariates follow from
the conditional distribution $f_{Z(0)|X}(z|c)$. Similarly, we obtain
$\tau^{w2}_{SRD}$ by taking

$w_{1}(z)=\frac{f_{Z(0)|X}(z|c)}{f_{Z(1)|X}(z|c)}~{}~{}\textrm{and}~{}~{}w_{0}(z)=1.$

  * •

Average treatment effect over locally randomized population:

$\tau^{w3}_{SRD}=\int\Delta(c,z)\frac{f_{Z(0)|X}(z|c)+f_{Z(1)|X}(z|c)}{2}dz.$

This is the estimand studied by Frölich and Huber (2018) in the sharp RD case.
Under the proposed WATE framework, $\tau^{w3}_{SRD}$ can be viewed as the
average treatment effect over the population around the threshold which is
randomized so that their covariates follow from $f_{Z(0)|X}(z|c)$ and
$f_{Z(1)|X}(z|c)$ with equal probability. Similarly, we obtain
$\tau^{w3}_{SRD}$ by taking

$w_{1}(z)=\frac{f_{Z(1)|X}(z|c)+f_{Z(0)|X}(z|c)}{2f_{Z(1)|X}(z|c)}~{}~{}\textrm{and}~{}~{}w_{0}(z)=\frac{f_{Z(1)|X}(z|c)+f_{Z(0)|X}(z|c)}{2f_{Z(0)|X}(z|c)}.$

  * •

Average treatment effect via classical RD estimand: We note that the proposed
WATE reduces to the classical RD estimand $\tau_{SRD}$ (2.1) by taking
$w_{1}(z)=w_{0}(z)=1$. However, we note that unlike the previous three
examples, $\tau_{SRD}$ may not be written as the average treatment effect over
one well defined population. To see this, recall that when there exists self-
selection, the conditional distributions $f_{Z(1)|X}(z|c)$ and
$f_{Z(0)|X}(z|c)$ usually differ from each other. Then $\tau_{SRD}$ in (2.1)
can be written as the difference of the average of
$\mathbb{E}(Y_{i}(t)|X_{i}=c,Z_{i}(t)=z)$ over two populations (i.e., the
population right below and above the threshold) with covariate distributions
$f_{Z(1)|X}(z|c)$ and $f_{Z(0)|X}(z|c)$ respectively. This is the reason for
which $\tau_{SRD}$ is confounded by the unbalanced covariates.

A summary of the first three estimands is provided in Table 1. In practice,
which causal estimand in above examples to use should depend on the target
population of interest and is often determined on a case-by-case basis.
Indeed, our framework opens a door towards designing new causal parameters
tailored to specific applications. For instance, similar to $\tau^{w2}_{SRD}$,
one can define the average treatment effect over locally treated population
i.e., $\int\Delta(c,z)f_{Z(1)|X}(z|c)dz$. Since the goal of the paper is to
deal with unbalanced covariates, to fix the idea we will mainly focus on the
first three examples.

In the following, we comment on two properties of our estimands
$\tau^{w1}_{SRD}$, $\tau^{w2}_{SRD}$ and $\tau^{w3}_{SRD}$. First, under the
constant treatment effect model (2.3), direct calculation shows that
$\Delta(c,z)=\tau$ and thus $\tau^{w1}_{SRD}=\tau^{w2}_{SRD}=\tau^{w3}_{SRD}$
equals to the direct treatment effect $\tau$ without any further assumption.
In contrast, the classical RD estimand $\tau_{SRD}$ reduces to $\tau$ under
the extra assumption that $\gamma=0$ or
$\mathbb{E}(Z_{i}(1)|X_{i}=c)=\mathbb{E}(Z_{i}(0)|X_{i}=c)$.

Second, our estimand $\tau^{w1}_{SRD}$ generalizes to the overall average
treatment effect (ATE) under the conditional independence assumption (CIA)
proposed by Angrist and Rokkanen (2015). Assume that $Z_{i}$ are the pre-
treatment covariates, i.e, $Z_{i}(1)=Z_{i}(0)=Z_{i}$. The CIA is defined as

$\mathbb{E}(Y_{i}(1)|X_{i},Z_{i})=\mathbb{E}(Y_{i}(1)|Z_{i}),\qquad\mathbb{E}(Y_{i}(0)|X_{i},Z_{i})=\mathbb{E}(Y_{i}(0)|Z_{i}),$
(2.6)

which says that the potential outcomes are mean independent of the running
variable conditional on the covariates. By controlling a rich set of
covariates, CIA seems to be a reasonable assumption as the link between the
running variable and outcomes can be blocked (Angrist and Rokkanen, 2015).
Since the CIA (2.6) implies $\Delta(c,z)=\Delta(z)$, our estimand
$\tau^{w1}_{SRD}$ reduces to

$\tau^{w1}_{SRD}=\int\Delta(z)f_{Z}(z)dz=\mathbb{E}(Y_{i}(1)-Y_{i}(0)),$

which is the overall ATE. In contrast,

$\tau^{w2}_{SRD}=\tau^{w3}_{SRD}=\tau_{SRD}=\mathbb{E}(Y_{i}(1)-Y_{i}(0)|X_{i}=c)$

remains a “local” treatment effect at the cutoff. It requires further
conditions to generalize to the overall ATE. For instance, if the constant
treatment assumption $\mathbb{E}(Y_{i}(1)|Z_{i})-\mathbb{E}(Y_{i}(0)|Z_{i})=a$
holds for some constant $a$, then $\tau_{SRD}$ reduces to
$\mathbb{E}(Y_{i}(1)-Y_{i}(0))$. Thus, the new estimand $\tau^{w1}_{SRD}$ can
represent a causal effect that is less local than the standard RD estimand
$\tau_{SRD}$.

Estimand | $w_{1}(z)$ | $w_{0}(z)$ | $\pi_{1}(z)$ | $\pi_{0}(z)$
---|---|---|---|---
$\int\Delta(c,z)f_{Z}(z)dz$ | $\frac{f_{Z}(z)}{f_{Z(1)|X}(z|c)}$ | $\frac{f_{Z}(z)}{f_{Z(0)|X}(z|c)}$ | $\frac{f_{X,Z(1)}(c,z)}{2f_{Z}(z)}$ | $\frac{f_{X,Z(0)}(c,z)}{2f_{Z}(z)}$
$\int\Delta(c,z)f_{Z(0)|X}(z|c)dz$ | $\frac{f_{Z(0)|X}(z|c)}{f_{Z(1)|X}(z|c)}$ | $1$ | $\frac{f_{X,Z(1)}(c,z)}{2f_{Z(0)|X}(z|c)}$ | $\frac{f_{X}(c)}{2}$
$\int\Delta(c,z)\frac{f_{Z(0)|X}(z|c)+f_{Z(1)|X}(z|c)}{2}dz$ | $\frac{f_{Z(1)|X}(z|c)+f_{Z(0)|X}(z|c)}{2f_{Z(1)|X}(z|c)}$ | $\frac{f_{Z(1)|X}(z|c)+f_{Z(0)|X}(z|c)}{2f_{Z(0)|X}(z|c)}$ | $\frac{f_{X,Z(1)}(c,z)}{f_{Z(1)|X}(z|c)+f_{Z(0)|X}(z|c)}$ | $\frac{f_{X,Z(0)}(c,z)}{f_{Z(1)|X}(z|c)+f_{Z(0)|X}(z|c)}$
Table 1: Three examples of the weighted average treatment effect
$\tau^{w}_{SRD}$, where
$\Delta(c,z)=\mathbb{E}(Y_{i}(1)|X_{i}=c,Z_{i}(1)=z)-\mathbb{E}(Y_{i}(0)|X_{i}=c,Z_{i}(0)=z)$.

###  2.3 Nonparametric Identification

In this subsection, we study the nonparametric identification of
$\tau^{w1}_{SRD}$, $\tau^{w2}_{SRD}$ and $\tau^{w3}_{SRD}$. Instead of
Assumption 3, we impose the following continuity assumption.

######  Assumption 5 (Continuity Assumption of
$\mathbb{E}(Y_{i}(t)|X_{i}=x,Z_{i}(t)=z)$).

$\mathbb{E}(Y_{i}(1)|X_{i}=x,Z_{i}(1)=z)\quad\text{ and
}\quad\mathbb{E}(Y_{i}(0)|X_{i}=x,Z_{i}(0)=z)$

are right and left continuous in $x$ at $x=c$ for any $z\in\mathcal{Z}$
respectively.

Intuitively, this assumption says, once all unbalanced covariates are
controlled, there is no further discontinuity between the running variable and
outcomes at the threshold. This assumption is similar to assumption 1 (iv) in
Frölich and Huber (2018) under the sharp RD design. In addition, this
assumption is weaker than CIA, as (2.6) implies our Assumption 5 when the
covariates are pre-determined.

The following theorem shows that $\tau^{w1}_{SRD}$ is identifiable based on
the distribution of the observed data under Assumption 5. In addition,
$\tau^{w2}_{SRD}$ and $\tau^{w3}_{SRD}$ are identifiable under some extra
continuity assumptions.

######  Theorem 6 (Nonparametric Identification).

Under Assumption 5, $\tau^{w1}_{SRD}$ is identifiable:

$\tau^{w1}_{SRD}=\int\big{[}\mathbb{E}(Y|X=c^{+},Z=z)-\mathbb{E}(Y|X=c^{-},Z=z)\big{]}f_{Z}(z)dz,$

where $\mathbb{E}(Y|X=c^{+},Z)=\lim_{x\rightarrow c^{+}}\mathbb{E}(Y|X=x,Z)$
and $\mathbb{E}(Y|X=c^{-},Z)=\lim_{x\rightarrow c^{-}}\mathbb{E}(Y|X=x,Z)$. In
addition, if $f_{Z(0)|X}(z|x)$ is left continuous in $x$ at $x=c$ for any
$z\in\mathcal{Z}$, then $\tau^{w2}_{SRD}$ is identifiable:

$\tau^{w2}_{SRD}=\int\big{[}\mathbb{E}(Y|X=c^{+},Z=z)-\mathbb{E}(Y|X=c^{-},Z=z)\big{]}f_{Z|X}(z|c^{-})dz,$

where $f_{Z|X}(z|c^{-})=\lim_{x\rightarrow c^{-}}f_{Z|X}(z|x)$. Furthermore,
if $f_{Z(1)|X}(z|x)$ is right continuous in $x$ at $x=c$ for any
$z\in\mathcal{Z}$, then $\tau^{w3}_{SRD}$ is identifiable:

$\tau^{w3}_{SRD}=\int\big{[}\mathbb{E}(Y|X=c^{+},Z=z)-\mathbb{E}(Y|X=c^{-},Z=z)\big{]}\frac{f_{Z|X}(z|c^{-})+f_{Z|X}(z|c^{+})}{2}dz.$

###### Proof.

To show the identifiability of $\tau^{w1}_{SRD}$, we note that Assumption 5
implies

$\displaystyle\int\mathbb{E}(Y(1)|X=c,Z(1)=z)f_{Z}(z)dz$
$\displaystyle=\int\lim_{\delta\rightarrow
0^{+}}\mathbb{E}(Y(1)|X=c+\delta,Z(1)=z)f_{Z}(z)dz,$

By the definition of the potential outcome, we have
$\mathbb{E}(Y(1)|X=c+\delta,Z(1)=z)=\mathbb{E}(Y|X=c+\delta,Z=z)$ for any
$\delta>0$. Thus,

$\int\mathbb{E}(Y(1)|X=c,Z(1)=z)f_{Z}(z)dz=\int\lim_{\delta\rightarrow
0^{+}}\mathbb{E}(Y|X=c+\delta,Z=z)f_{Z}(z)dz.$

Following the same step, we can show that

$\int\mathbb{E}(Y(0)|X=c,Z(0)=z)f_{Z}(z)dz=\int\lim_{\delta\rightarrow
0^{-}}\mathbb{E}(Y|X=c+\delta,Z=z)f_{Z}(z)dz.$

This implies $\tau^{w1}_{SRD}$ is identifiable. To show $\tau^{w2}_{SRD}$ is
identifiable, similarly we have
$\mathbb{E}(Y(1)|X=c,Z(1)=z)=\mathbb{E}(Y|X=c^{+},Z=z)$. In addition, by the
left continuity of $f_{Z(0)|X}(z|x)$, we further have
$f_{Z(0)|X}(z|c)=\lim_{\delta\rightarrow
0^{-}}f_{Z(0)|X}(z|c+\delta)=f_{Z|X}(z|c^{-})$. Thus, we have

$\displaystyle\int\mathbb{E}(Y(1)|X=c,Z(1)=z)f_{Z(0)|X}(z|c)dz$
$\displaystyle=\int\mathbb{E}(Y|X=c^{+},Z=z)f_{Z|X}(z|c^{-})dz.$

This implies $\tau^{w2}_{SRD}$ is identifiable. The identifiability of
$\tau^{w3}_{SRD}$ follows from the same argument. This completes the proof. ∎

##  3 Nonparametric Estimation

In the causal inference literature, inverse propensity score weighting (IPW)
is one of the most widely used tools to handle the unbalanced covariates in
the treatment and control groups (Horvitz and Thompson, 1952). However, the
standard IPW method is not directly applicable because in the sharp RD design
the treatment assignment is a deterministic function of the running variable
and thus the propensity score is degenerate. In this section, we propose a
class of nonparametric estimators of $\tau^{w1}_{SRD}$, $\tau^{w2}_{SRD}$ and
$\tau^{w3}_{SRD}$ by modifying the inverse propensity score weighting
approach.

To motivate our nonparametric estimator, we consider the following notation.
Denote by $\pi_{1}(Z_{i})$ a function of the covariate $Z_{i}$ to be chosen
later, $K(\cdot)$ a symmetric kernel function and $h$ a bandwidth that shrinks
to 0. The detailed conditions on the kernel function and bandwidth are
deferred to the next section. Consider the following inverse weighted kernel
estimator

$\frac{1}{n}\sum_{i=1}^{n}\frac{Y_{i}T_{i}}{\pi_{1}(Z_{i})}\cdot
h^{-1}K\Big{(}\frac{X_{i}-c}{h}\Big{)},$ (3.1)

for the estimand $\mathbb{E}\\{Y(1)w_{1}(Z(1))|X=c\\}$, where $\pi_{1}(Z)$
plays the same role as the propensity score in the IPW method. Since in the RD
design the propensity score function is degenerate, in the following we will
show that the choice of $\pi_{1}(Z_{i})$ differs from the standard propensity
score model. The rationale is to choose $\pi_{1}(Z_{i})$ so that the estimator
(3.1) is asymptotically unbiased,

$\mathbb{E}\left(\frac{Y_{i}T_{i}}{\pi_{1}(Z_{i})}\cdot
h^{-1}K\Big{(}\frac{X_{i}-c}{h}\Big{)}\right)\approx\mathbb{E}\\{Y_{i}(1)w_{1}(Z_{i}(1))|X_{i}=c\\}.$
(3.2)

Thus, it suffices to calculate the expectation of the kernel estimator as
follows

$\displaystyle\mathbb{E}\left(\frac{Y_{i}T_{i}}{\pi_{1}(Z_{i})}\cdot
h^{-1}K\Big{(}\frac{X_{i}-c}{h}\Big{)}\right)$
$\displaystyle=\mathbb{E}\left(\frac{T_{i}}{\pi_{1}(Z_{i}(1))}\cdot
h^{-1}K\Big{(}\frac{X_{i}-c}{h}\Big{)}\cdot\mathbb{E}(Y_{i}(1)|X_{i},Z_{i}(1))\right)$
$\displaystyle=\int\int_{x>c}\mathbb{E}(Y_{i}(1)|X_{i}=x,Z_{i}(1)=z)\frac{h^{-1}}{\pi_{1}(z)}K\Big{(}\frac{x-c}{h}\Big{)}f_{Z(1),X}(z,x)dxdz,$
(3.3)

where the first step follows from $Z_{i}=Z_{i}(1)$ under $T=1$ and the last
step follows from the definition $T_{i}=\mathbb{1}(X_{i}>c)$. Thus, (3.3)
implies

$\displaystyle\mathbb{E}\left(\frac{Y_{i}T_{i}}{\pi_{1}(Z_{i})}\cdot
h^{-1}K\Big{(}\frac{X_{i}-c}{h}\Big{)}\right)$
$\displaystyle=\int\int_{u>0}\mathbb{E}(Y_{i}(1)|X_{i}=uh+c,Z_{i}(1)=z)\frac{K(u)}{\pi_{1}(z)}f_{Z(1),X}(z,uh+c)dudz$
$\displaystyle\approx\int\mathbb{E}(Y_{i}(1)|X_{i}=c,Z_{i}(1)=z)\frac{1}{\pi_{1}(z)}\frac{f_{Z(1),X}(z,c)}{2}dz,$
(3.4)

where the last step follows from the symmetry of the kernel function
$\int_{u>0}K(u)du=1/2$ and can be made rigorous given the regularity
conditions specified in the next section. Comparing with

$\mathbb{E}\\{Y_{i}(1)w_{1}(Z_{i}(1))|X_{i}=c\\}=\int\mathbb{E}(Y_{i}(1)|X_{i}=c,Z_{i}(1)=z)w_{1}(z)f_{Z(1)|X}(z|c)dz,$

we can see that (3.2) holds provided

$\pi_{1}(z)=\frac{f_{X}(c)}{2w_{1}(z)},$ (3.5)

where $f_{X}(c)$ is the p.d.f of $X$ at $x=c$. Following from the same
argument, one can show that $\pi_{0}(z)=\frac{f_{X}(c)}{2w_{0}(z)}$. In Table
1, we give the detailed expression of $\pi_{1}(z)$ and $\pi_{0}(z)$ for the
proposed three estimands $\tau^{w1}_{SRD}$, $\tau^{w2}_{SRD}$ and
$\tau^{w3}_{SRD}$. Since the weight $\pi_{1}(z)$ depends on the unknown
density functions, we propose to estimate those densities by the following
kernel estimators

$\begin{split}\widehat{f}_{X,Z(1)}(c,z)=2\cdot(nh_{1}^{2})^{-1}\sum_{x_{i}>c}K_{1}(\frac{c-x_{i}}{h_{1}},\frac{z-z_{i}}{h_{1}}),\end{split}$
(3.6)
$\begin{split}\widehat{f}_{X,Z(0)}(c,z)=2\cdot(nh_{1}^{2})^{-1}\sum_{x_{i}<c}K_{1}(\frac{c-x_{i}}{h_{1}},\frac{z-z_{i}}{h_{1}}),\end{split}$
(3.7)
$\widehat{f}_{Z}(z)=(nh_{2})^{-1}\sum_{i=1}^{n}K(\frac{z-z_{i}}{h_{2}}),~{}~{}\textrm{and}~{}~{}\widehat{f}_{X}(c)=(nh_{2})^{-1}\sum_{i=1}^{n}K(\frac{c-x_{i}}{h_{2}}),$
(3.8)

where $h_{1}$ and $h_{2}$ are bandwidth parameters. Note that the kernel
estimator is known to suffer from the curse of dimensionality and is only
applicable when the dimension of $Z_{i}$ is small. For the applications in
which a large number of covariates can be collected, one may consider
alternative parametric or semiparametric approaches for density estimation. In
this work, we only focus on the above kernel estimators and leave the
alternatives for future investigation.

Replacing the unknown density functions in $\pi_{1}(z)$ and $\pi_{0}(z)$ as
shown in Table 1 with the corresponding kernel estimators, we can obtain
$\widehat{\pi}_{1}(z)$ and $\widehat{\pi}_{0}(z)$. While we can construct the
final estimator by plugging $\widehat{\pi}_{1}(z)$ into (3.1), for practical
use and theoretical analysis we recommend the local linear estimator, since it
has smaller asymptotic bias and better finite sample behavior near the
boundary (Fan and Gijbels, 1996). Motivated by the formulation of the kernel
estimator (3.1), we propose the following weighted local linear (WLL)
estimator

$\begin{split}\Big{(}\widehat{\alpha}_{0},\widehat{\beta}_{0}\Big{)}&=\arg\min_{\alpha,\beta}\sum_{\\{i:X_{i}<c\\}}\frac{1-T_{i}}{\widehat{\pi}_{0}(Z_{i})}\Big{(}Y_{i}-\alpha-(X_{i}-c)\beta\Big{)}^{2}K\Big{(}\frac{X_{i}-c}{h}\Big{)},\\\
\Big{(}\widehat{\alpha}_{1},\widehat{\beta}_{1}\Big{)}&=\arg\min_{\alpha,\beta}\sum_{\\{i:X_{i}>c\\}}\frac{T_{i}}{\widehat{\pi}_{1}(Z_{i})}\Big{(}Y_{i}-\alpha-(X_{i}-c)\beta\Big{)}^{2}K\Big{(}\frac{X_{i}-c}{h}\Big{)},\end{split}$
(3.9)

Thus, we can estimate the WATE ${\tau}_{SDR}^{w}$ by

$\widehat{\tau}_{SDR}^{w}=\widehat{\alpha}_{1}-\widehat{\alpha}_{0}.$ (3.10)

##  4 Theoretical Results

In this section, we study the asymptotic properties of the proposed estimator.
We focus on the local linear estimator (3.9) for the estimand
$\tau^{w1}_{SRD}$, due to the nice properties of $\tau^{w1}_{SRD}$ as
explained in Section 2.2. The analysis of the estimators for the other two
estimands $\tau^{w2}_{SRD}$ and $\tau^{w3}_{SRD}$ is similar, but may require
different technical conditions.

Let $\delta$ denote a small positive constant. For notational simplicity,
define $\mathcal{F}^{-}$ as the class of functions of $x\in(c-\delta,c]$ and
$z\in\mathcal{Z}$ such that for any $f\in\mathcal{F}^{-}$,
$\frac{\partial^{3}}{\partial a\partial b\partial c}f(x,z)$ is continuous,
where $a,b,c=\\{x,z\\}$. Here, the derivatives of $f(x,z)$ with respect to $x$
at $x=c$ (say $\frac{\partial^{3}}{\partial x^{3}}f(c,z)$) are interpreted as
left derivatives. Similarly, we define $\mathcal{F}^{+}$ as the class of
functions of $x\in[c,c+\delta)$ and $z\in\mathcal{Z}$ such that for any
$f\in\mathcal{F}^{+}$, $\frac{\partial^{3}}{\partial a\partial b\partial
c}f(x,z)$ is continuous, where $a,b,c=\\{x,z\\}$. The derivatives at $x=c$
correspond to the right derivatives. For simplicity, we only consider the case
that $\textrm{dim}(Z)=1$. The generalization to multivariate covariates
follows from the similar argument.

######  Assumption 7 (Smoothness condition).

Assume $f_{X,Z(0)}(x,z)\in\mathcal{F}^{-}$ and
$f_{X,Z(1)}(x,z)\in\mathcal{F}^{+}$. Denote
$m_{t}(x,z)=\mathbb{E}(Y_{i}(t)|X_{i}=x,Z_{i}(t)=z)$ for $t=\\{0,1\\}$. Assume
$m_{0}(x,z)\in\mathcal{F}^{-}$ and $m_{1}(x,z)\in\mathcal{F}^{+}$.

Since the proposed method requires to estimate the unknown density functions
$f_{X,Z(0)}(x,z)$ and $f_{X,Z(1)}(x,z)$, we assume they are sufficiently
smooth. In addition, we also need the smoothness of $m_{t}(x,z)$ in order to
study the local linear estimator (3.9). This assumption implies Assumption 5,
which is required for identification purpose.

######  Assumption 8.

Assume the following conditions hold:

  1. 1.

$X\in[x_{l},x_{u}]$ such that $x_{l}<c<x_{u}$ and $Z$ also has bounded
support.

  2. 2.

The kernel $K(u)$ is a non-negative, symmetric and bounded function with
compact support which satisfies

$\int K(z)\,dz=1,~{}~{}~{}\textrm{and}~{}~{}~{}K(u)=K(-u).$

The bivariate kernel $K_{1}(u,v)$ is a product kernel $K_{1}(u,v)=K(u)K(v)$,
where $K(\cdot)$ satisfies the above conditions.

  3. 3.

$f_{X,Z(1)}(x,z)$ is bounded way from 0 by a constant for $x\in[c,c+\delta)$
and $z\in\mathcal{Z}$ and $f_{X,Z(0)}(x,z)$ is bounded way from 0 by a
constant for $x\in(c-\delta,c]$ and $z\in\mathcal{Z}$.

  4. 4.

$\sigma_{t}^{2}=\mathbb{E}[(Y(t)-m_{t}(X,Z(t)))^{2}|X=x,Z(t)=z]<\infty$ for
$t\in\\{0,1\\}$.

Assumption 8 has four parts. The first and second parts are standard
assumptions in kernel density estimation problems. Since our RD estimator
applies local linear estimators at the boundary, the uniform kernel or
triangular kernel has been shown to have good performance under such scenario
(Calonicoy et al., 2017). The third part requires $f_{X,Z(1)}(x,z)$ and
$f_{X,Z(0)}(x,z)$ to be bounded away from 0 so that the inverse weights in the
local linear estimator can be well controlled. This is similar to IPW
estimator which requires the propensity score to be bounded away from 0. The
last part assumes that the (homoscedastic) noise has finite variance.

Denote $\alpha_{t}=\int\mathbb{E}(Y(t)|X=c,Z(t)=z)f_{Z}(z)dz$, and recall that
$\tau^{w1}_{SRD}=\alpha_{1}-\alpha_{0}$. The main theorem in this section
shows the rate of convergence of the local linear estimators
$\widehat{\alpha}_{1}$ and $\widehat{\alpha}_{0}$ and their limiting
distributions.

######  Theorem 1.

Assume that assumptions 7 and 8 hold, and we choose $h\asymp\sqrt{h_{1}}\asymp
h_{2}$. Then

$|\widehat{\alpha}_{t}-\alpha_{t}|=O_{p}(h^{2}+(nh^{2})^{-1/2}).$ (4.1)

Denote $d_{t}(x_{i},z_{i})=m_{t}(x_{i},z_{i})-\alpha_{t}$. Furthermore, if
$h^{3}n^{1/2}=o(1)$ holds, we have

$\begin{split}\sqrt{nh^{2}}(\widehat{\alpha}_{1}-\alpha_{1})&\sim
N\left(0,C_{v}\cdot\mathbb{E}_{Z}\left(\frac{f_{Z}(z_{i})}{f_{X,Z(1)}(c^{+},z_{i})}d_{1}(c^{+},z_{i})^{2}\right)\right),\end{split}$

and

$\begin{split}\sqrt{nh^{2}}(\widehat{\alpha}_{0}-\alpha_{0})&\sim
N\left(0,C_{v}\cdot\mathbb{E}_{Z}\left(\frac{f_{Z}(z_{i})}{f_{X,Z(0)}(c^{-},z_{i})}d_{0}(c^{-},z_{i})^{2}\right)\right),\end{split}$

where $\mathbb{E}_{Z}$ denotes the expectation under the marginal distribution
of $Z$, and

$C_{v}=\frac{\kappa_{2}^{2}\kappa_{20}+\kappa_{1}^{2}\kappa_{22}-2\kappa_{1}\kappa_{2}\kappa_{21}}{\left(\frac{1}{2}\kappa_{2}-\kappa_{1}^{2}\right)^{2}},$

with

$\kappa_{q}=\int_{u>0}K(u)u^{q}du\quad\text{ and
}\quad\kappa_{2q}=\int_{u>0}K(u)^{2}u^{q}du,\quad\text{ for }q=0,1,2.$

We note that from (4.1) the optimal choice of $h$ is of order $O(n^{-1/6})$
and the corresponding convergence rate is
$|\widehat{\alpha}_{t}-\alpha_{t}|=O_{p}(n^{-1/3})$ due to the boundary
effect. Specifically, the estimated weights $\widehat{\pi}_{1}(z)$ and
$\widehat{\pi}_{0}(z)$ depend on the density estimators at the boundary. Since
we only require $f_{X,Z(0)}(x,z)\in\mathcal{F}^{-}$ and
$f_{X,Z(1)}(x,z)\in\mathcal{F}^{+}$ to be smooth from one side, the
corresponding density estimators have a slower rate. So, the plug-in error
becomes the dominant term when establishing the rate of
$\widehat{\alpha}_{t}$. However, if $Z_{i}$ are the pre-treatment covariates,
i.e, $Z_{i}(1)=Z_{i}(0)=Z_{i}$, we can estimate the density $f_{X,Z}(c,z)$ by

$\begin{split}\widehat{f}_{X,Z}(c,z)=(nh_{1}^{2})^{-1}\sum_{i=1}^{n}K_{1}(\frac{c-x_{i}}{h_{1}},\frac{z-z_{i}}{h_{1}}).\end{split}$
(4.2)

The following corollary shows that in this case $\widehat{\alpha}_{t}$ has an
improved rate. With an optimal choice of the bandwidth parameters, we prove
that $|\widehat{\alpha}_{t}-\alpha_{t}|=O_{p}(n^{-2/5})$, that is the boundary
effect for estimating $\alpha_{t}$ is automatically removed without applying
any additional bias correction procedures.

######  Corollary 1.

Assume that assumptions 7 and 8 hold and $Z_{i}$ are the pre-treatment
covariates. Choosing $h\asymp h_{1}\asymp h_{2}$, we have

$\begin{split}|\widehat{\alpha}_{t}-\alpha_{t}|=O_{p}(h^{2}+(nh)^{-1/2}).\end{split}$

If $h^{5}n=o(1)$, we have

$\begin{split}\sqrt{nh}(\widehat{\alpha}_{t}-\alpha_{t})&\sim
N\left(0,C_{v}\cdot\sigma^{2}\int\frac{f_{Z}(z)^{2}}{f_{X,Z}(c,z)}dz\right).\end{split}$

##  5 Simulation and Empirical Examples

###  5.1 Simulation Study

In this first setting, consider the following data generating process:

$y_{i}(1)=2+x_{i}+\beta z_{i}+\epsilon_{i},$ $y_{i}(0)=1+x_{i}+\beta
z_{i}+\epsilon_{i},$

where $x_{i}$, $z_{i}$, and $\epsilon_{i}$ are generated independently from
$N(0,1)$ distribution. The treatment $T_{i}$ is assigned at the cutoff 0:
$T_{i}=\mathbb{1}(x_{i}>0)$. In this case, there is no discontinuity of the
conditional distribution of $z_{i}$ given $x_{i}=0$. Thus, our estimaind
$\tau^{w1}_{SRD}$ equals the standard RD estimand $\tau_{SRD}$ (both are equal
to 1). We vary $\beta$ from 0 to 5 and compare weighted local linear (WLL)
estimator with the standard RD estimator (Imbens and Lemieux (2008)) in terms
of bias, variance, root-mean-squared error (MSE), coverage probability of 95%
confidence intervals (Coverage) and its length (CI length). When implementing
both methods, we set the bandwidth parameter for standard RD estimator using
cross-validation and then use the same bandwidth for WLL. The results based on
500 simulations are shown in Table 2. When $\beta=0$, there is no covariates
involved in the outcome function. Standard RD estimator performs neck to neck
with our estimator. When $\beta\neq 0$, the standard RD estimator performs
slightly better in terms of bias, however, our estimator consistently has
smaller variance and MSE.

|  | bias | variance | MSE | Coverage | CI length
---|---|---|---|---|---|---
n | $\beta$ | RD | WLL | RD | WLL | RD | WLL | RD | WLL | RD | WLL
500 | 0 | -0.0106 | -0.0102 | 0.3731 | 0.3767 | 0.3733 | 0.3768 | 0.9650 | 0.9750 | 1.6666 | 1.6853
1 | 0.0236 | 0.0226 | 0.6090 | 0.5423 | 0.6094, | 0.5427 | 0.9250 | 0.9400 | 2.2335 | 2.1753
2 | -0.0583 | -0.0548 | 0.9004 | 0.7886 | 0.9023 | 0.7905 | 0.9600 | 0.9600 | 3.6103 | 3.2369
5 | -0.0522 | -0.0423 | 2.1757 | 1.8264 | 2.1763 | 1.8268 | 0.8800 | 0.9000 | 7.1005 | 6.1439
1000 | 0 | 0.0021 | 0.0034 | 0.2853 | 0.2841 | 0.2854 | 0.2841 | 0.9900 | 0.9900 | 1.3447 | 1.4083
1 | -0.0088 | -0.0118 | 0.4379 | 0.4387 | 0.4740 | 0.4388 | 0.9200 | 0.9450 | 1.6833 | 1.6661
2 | 0.0380 | 0.0431 | 0.6046 | 0.5118 | 0.6058 | 0.5136 | 0.9400 | 0.9550 | 2.2516 | 2.2635
5 | 0.1106 | 0.1129 | 1.4676 | 1.2770 | 1.4718 | 1.2820 | 0.9600 | 0.9850 | 6.3293 | 6.9139
2000 | 0 | -0.0107 | -0.0111 | 0.2098 | 0.2096 | 0.2101 | 0.2099 | 0.9300 | 0.9200 | 0.7400 | 0.7229
1 | -0.0320 | -0.0225 | 0.3151 | 0.2920 | 0.3167 | 0.2929 | 0.9850 | 0.9850 | 1.3572 | 1.3237
2 | -0.0177 | -0.0151 | 0.5234 | 0.4595 | 0.5237 | 0.4598 | 0.9500 | 0.9650 | 2.0911 | 1.9680
5 | 0.1331 | 0.1360 | 1.1707 | 1.0019 | 1.1783 | 1.0111 | 0.9150 | 0.9200 | 4.2082 | 3.7723
5000 | 0 | -0.0149 | -0.0136 | 0.1606 | 0.1616 | 0.1613 | 0.1622 | 0.9200 | 0.9250 | 0.5699 | 0.5725
1 | 0.0034 | 0.0049 | 0.2451 | 0.2275 | 0.2451 | 0.2275 | 0.9350 | 0.9400 | 0.9012 | 0.8793
2 | -0.0130 | -0.0137 | 0.3301 | 0.2940 | 0.3304 | 0.2943 | 0.9450 | 0.9600 | 1.3654 | 1.2931
5 | 0.0053 | 0.0139 | 0.8355 | 0.7016 | 0.8355 | 0.7018 | 0.9250 | 0.9750 | 3.1578 | 3.1535

Table 2: Comparison of the standard RD estimator and the proposed weighted
local linear estimator (WLL) in the first setting.

Figure 1 compares the MSE of the standard RD estimator with our WLL estimator
across different bandwidth choices. The figure is consistent with corollary 1
as our estimator is asymptotically more efficient through including additional
covariates into the estimation. Moreover, the advantage of WLL estimator is
greater when perform under-smoothing and the difference of the two estimators
becomes smaller as the bandwidth increases.

Figure 1: Comparing MSE of WLL and RD estimator under the same bandwidth
choices

In the second setting, we consider the following data generating process:

$y_{i}(1)=3+x_{i}+z_{i}+\epsilon_{1i},$
$y_{i}(0)=1+x_{i}+z_{i}+\epsilon_{0i},$

where $x_{i}$ and $\epsilon_{i}$ are generated independently from $N(0,1)$
distribution, however, $z_{i}$ is generated from another independent $N(0,1)$
process with a discontinuity at $X>0$, i.e.
$z_{i}=\gamma\cdot\mathbb{1}(x_{i}>0)+z_{i}^{*}$, where $z_{i}^{*}\sim
N(0,1)$. The treatment $T_{i}$ is assigned at the cutoff 0:
$T_{i}=\mathbb{1}(x_{i}>0)$. When $\gamma=0$, again there is no discontinuity
of the conditional distribution of $z_{i}$ given $x_{i}=0$. Both our estimaind
$\tau^{w1}_{SRD}$ and the standard RD estimand $\tau_{SRD}$ are equal to 2.
However, as $\gamma$ differs from 0, the conditional distribution of $z_{i}$
given $x_{i}$ is discontinuous at $x_{i}=0$. Our estimand $\tau^{w1}_{SRD}$ is
still equal to 2, which is the direct causal effect of interest. If we adopt
the standard RD framework and ignore the discontinuity of the conditional
distribution of $z_{i}$ given $x_{i}$, we would expect that the standard RD
estimator is biased for estimating the direct causal effect of interest (which
is 2 in this example). In the data generating process, we vary $\gamma$ from 0
to 1 and compare our estimator with the standard RD estimator. The results are
shown in Table 3. The standard RD estimator has large bias and very poor
coverage probability when $\gamma$ is close to 1, which agrees with our
expectation. In contrast, the proposed estimator has relatively small MSE and
accurate coverage probabilities across different choices of the sample size
$n$ and the parameter $\gamma$. In summary, our simulation studies confirm
that one should apply the proposed framework to the RD study if there exists
some potential discontinuity of the conditional distribution of the covariates
given the running variable.

|  | bias | variance | MSE | Coverage | CI length
---|---|---|---|---|---|---
n | $\gamma$ | RD | WLL | RD | WLL | RD | WLL | RD | WLL | RD | WLL
500 | 0.2 | 0.1858 | 0.0762 | 0.6181 | 0.4318 | 0.4166 | 0.1922 | 0.9350 | 0.9550 | 2.4229 | 1.6925
0.4 | 0.4457 | 0.1150 | 0.5519 | 0.4327 | 0.5032 | 0.2005 | 0.8700 | 0.9450 | 2.1634 | 1.6963
0.6 | 0.5721 | 0.1817 | 0.5898 | 0.4913 | 0.6751 | 0.2744 | 0.8500 | 0.9400 | 2.3119 | 1.9260
0.8 | 0.7620 | 0.2445 | 0.5686 | 0.4698 | 0.9040 | 0.2805 | 0.7100 | 0.9250 | 2.2290 | 1.8416
1 | 0.9753 | 0.3628 | 0.5346 | 0.4404 | 1.2369 | 0.3256 | 0.5750 | 0.8800 | 2.0956 | 1.7264
1000 | 0.2 | 0.1851 | 0.0869 | 0.4124 | 0.4897 | 0.2044 | 0.2473 | 0.9200 | 0.9650 | 1.6168 | 1.9196
0.4 | 0.4082 | 0.1271 | 0.4101 | 0.3360 | 0.3349 | 0.1291 | 0.8450 | 0.9350 | 1.6077 | 1.3171
0.6 | 0.5534 | 0.1795 | 0.4034 | 0.3428 | 0.4690 | 0.1497 | 0.7200 | 0.9200 | 1.5814 | 1.3437
0.8 | 0.8240 | 0.2458 | 0.4625 | 0.3859 | 0.8930 | 0.2093 | 0.5750 | 0.900 | 1.8131 | 1.5125
1 | 1.0130 | 0.3193 | 0.4445 | 0.5142 | 1.2237 | 0.3664 | 0.3950 | 0.9400 | 1.7423 | 2.0157
2000 | 0.2 | 0.2052 | 0.0592 | 0.3162 | 0.2705 | 0.1421 | 0.0767 | 0.9050 | 0.9700 | 1.2393 | 1.0602
0.4 | 0.3755 | 0.0887 | 0.3072 | 0.2500 | 0.2354 | 0.0704 | 0.7550 | 0.9400 | 1.2042 | 0.9802
0.6 | 0.5787 | 0.1697 | 0.3196 | 0.2568 | 0.4370 | 0.0947 | 0.5500 | 0.9000 | 1.2529 | 1.0067
0.8 | 0.8448 | 0.2208 | 0.3173 | 0.2979 | 0.8143 | 0.1375 | 0.2450 | 0.9100 | 1.2438 | 1.1676
1 | 1.0107 | 0.2234 | 0.2876 | 0.3556 | 1.1043 | 0.1763 | 0.0450 | 0.9500 | 1.1272 | 1.3938
5000 | 0.2 | 0.2110 | 0.0418 | 0.2249 | 0.1748 | 0.0951 | 0.0323 | 0.8300 | 0.9450 | 0.8817 | 0.6852
0.4 | 0.4138 | 0.0650 | 0.2397 | 0.1947 | 0.2287 | 0.04212 | 0.6200 | 0.9450 | 0.9396 | 0.7631
0.6 | 0.6099 | 0.1301 | 0.2319 | 0.2230 | 0.4257 | 0.06666 | 0.3000 | 0.9350 | 0.9091 | 0.8741
0.8 | 0.8327 | 0.1826 | 0.2305 | 0.2741 | 0.7465 | 0.1085 | 0.0500 | 0.9150 | 0.9035 | 1.0744
1 | 1.0259 | 0.2331 | 0.2250 | 0.2130 | 1.1032 | 0.0997 | 0.0050 | 0.9750 | 0.8821 | 0.8350

Table 3: Comparison of the standard RD estimator and the proposed weighted
local linear estimator (WLL) in the second setting.

###  5.2 Incumbency Advantage

We further apply our method to study the “incumbency advantage” in the U.S.
House elections as in Lee (2008). The “incumbency advantage” states that
current incumbent party in a district are more likely to win the next. The
dataset contains 6560 observations on elections to the United States House of
Representatives (1946-1998). We evaluate the probability a Democrat both
running in and winning election $t+1$ as a function of the Democratic vote
share margin of victory in election $t$. The other covariates we are
considering includes the Democratic vote share at time $t-1$, Democrat winning
at $t-1$, Democrat’s political experience, opponent’s political experience,
Democrat’s electoral experience, opponent’s electoral experience. This is the
same setting as studied in Lee (2008) where he uses local 4th order polynomial
RD estimates and includes these additional covariates as robustness check. We
apply our method in the same five settings and results are reported in Table 4
below:

Vote share $t+1$ |  | (1) | (2) | (3) | (4) | (5)
---|---|---|---|---|---|---
|  | Polyn | WLL | Polyn | WLL | Polyn | WLL | Polyn | WLL | Polyn | WLL
Victory, election $t$ |  | 0.077 | 0.076 | 0.078 | 0.076 | 0.077 | 0.076 | 0.077 | 0.076 | 0.078 | 0.076
|  | (0.011) | (0.010) | (0.011) | (0.011) | (0.011) | (0.010) | (0.011) | (0.010) | (0.011) | (0.010)
Dem. vote share |  | - | - | Y | Y | - | - | - | - | Y | Y
$t-1$ |  |  |  |  |  |  |  |  |  |  |
Dem. win, $t-1$ |  | - | - | Y | Y | - | - | - | - | Y | Y
Dem. political |  | - | - | - | - | Y | Y | - | - | Y | Y
experience |  |  |  |  |  |  |  |  |  |  |
Opp. political |  | - | - | - | - | Y | Y | - | - | Y | Y
experience |  |  |  |  |  |  |  |  |  |  |
Dem. electoral |  | - | - | - | - | - | - | Y | Y | Y | Y
experience |  |  |  |  |  |  |  |  |  |  |
Opp. electoral |  | - | - | - | - | - | - | Y | Y | Y | Y
experience |  |  |  |  |  |  |  |  |  |  |

Table 4: Effect of winning an election on subsequent party electoral success.
Polyn method is replicated as in Lee (2008) using 4th order polynomial. WLL
represents the results using our method. $Y$ indicates a covariates is been
included in the calculation. Standard error is provided in the bracket using
bootstrap for both method.

As shown in Table 4, there is no significant change on the estimated effect
when using our method. The main reason is that the conditional densities of
covariates are truly continuous at the cutoff in this data set. We expect that
both standard RD estimator and our method are valid and therefore the results
are very similar.

###  5.3 Generalized Second Price Auction (GSP)

Next we apply our method to study the generalized second price auction (GSP)
problem. GSP is an auction mechanism for multiple items and it has been used
widely for the assignment of advertisement positions by internet search engine
like google and bing. Let $n$ be the number of bidders, and let $b^{1}\geq
b^{2}\geq\cdots\geq b^{n}$ be the bids from high to low. Denote by
$v_{(1)},v_{(2)},\cdots,v_{(n)}$ the bidders’ valuation associated with the
rank of bids and $r^{k}$ the click through rate for the $k$th position. The
$k$th bidder’s payoff in a GSP is given as $(v_{(k)}-b^{k+1})r^{k}$.

An important metrics is the click through rate $r^{k}$ for the $k$th position.
Bidders are interested in the potential growth in their search traffic by
winning the auction. And furthermore, in a Vickrey-Clarke-Groves (VCG)
auction, $r^{k}$ will determine the total cost for placing each bidder in the
sponsored advertisement region. In real world, GSP is usually implemented
through a reservation score. A search score is formed for every bidder based
on their bid and other quality measures. When the search score is bigger than
a pre-set reservation score, the bidder’s link will be displayed in the
sponsored area. Otherwise, they will be displayed after all the sponsored
advertisements. The reservation score cut-off creates a natural regression
discontinuity setting to evaluate $r^{k}$.

We study the Microsoft Bing search data from Oct 2nd to Oct 22nd in 2015 and
estimate the effect of advertisement positions. We focus on a set of searches
with first advertisement positions displayed but without third advertisement
position displayed.111Microsoft Bing allows a maximum of 4 advertisement to be
displayed at the time of study. This allows us to analyze the effect of second
advertisement positions by comparing click-abilities for bidders near the
search score cutoff. Figure 2 plots the connection between search score and
click-ability for the second position in the sponsored advertisement area.
Once the score passed the cut-off at 15.17, the customers’ links will be
placed at the sponsored advertisement area and a significant increase in the
click traffic can be observed.

Figure 2: Plot of search score and click-ability. If the search score is above
15.17, the bidder’s advertisement will be displayed at the second position.
Otherwise, the advertisement will be displayed based on its regular search
orders.

Consider the bid as a covariate. Figure 3 plots the mean bidding price before
and after the search score cut-off. The mean bids before the cut-off is higher
than the mean bids after the cut-off, implying the discontinuity of the
conditional distribution of the covariate. See also Figure 4 for the
conditional density of the covariate before and after the cut-off. Although a
local envy free equilibrium exists when bidders are all bidding their true
valuation (Edelman et al., 2007), information asymmetry or bidder inertia may
still lead to bidder selections. For example, active bidders may have the
incentive to bid more aggressively to take advantage of the bidders with high
inertia.

Figure 3: Search Score v.s. Bids Figure 4: Density plot for bids before and
after the cut-off

Table 5 presents the results of our estimator and a polynomial RD estimator.
The classic RD estimator may not be valid in this case due to the
discontinuity in the covariates. It estimates that placing the advertisement
on the second position can increase the click-ability by 1.91% and it is
statistically significant. On the other hand, the proposed estimator delivers
only 1.20%, 57% less than the RD estimator and it is not statistically
significant at 5% level. The difference is mainly because low quality bidders
with high willingness to pay have the incentive to bid higher to move their
search scores pass the threshold. But when we match bidders with similar bids
before and after the cut-off, the effect goes away.

| RD | WLL
---|---|---
Estimates | 1.91%∗∗∗ | 1.20%
Standard Error | 0.0033 | 0.0090

Table 5: Estimated effect of adversing at second position using the standard
RD estimator and the proposed method. The bandwidth parameters are selected
using cross-validation and standard errors are obtained via bootstrap. ∗∗∗
represents significance at $p<0.05$

##  6 Extension to Fuzzy RD Design

Fuzzy RD design refers to the case of a RD design when the treatment
compliance is imperfect. Although $T_{i}$ is no longer a deterministic
function of $X_{i}$, there is still a discontinuity in
$\mathbb{P}(T_{i}|X_{i},Z_{i})$. Hahn et al. (2001) showed the equivalence
between the fuzzy RD design and local average treatment effect (LATE). Under
independence and monotonicity assumption, the cutoff can be used as an
instrument for the treatment status and the LATE can be interpreted as an
intention-to-treat effect on the compliers: subjects take treatment as their
assigned one.

Introducing self-selection under the Fuzzy RD design is the same as adding an
additional endogenous variable. As a result, additional instrumental variable
is required for identification. We consider a scenario when the self-selection
is based on the cut-off instead of the treatment assignment mechanism. This
setting allows us to deal with unobservables that affect the covariates. Let
$\tau$ denote the type of individual: complier ($co$), always-taker ($at$) and
never taker ($nt$). A post-selection may exist and we can write

$Z_{i}=\mathbb{1}(X_{i}>c)Z_{i}(1)+\mathbb{1}(X_{i}\leq c)Z_{i}(0).$

Notice that in sharp RD design, the self-selection based on the cut-off is the
same as the selection on treatment assignment mechanism since the assignment
mechanism is a deterministic function of $X_{i}$. We study the selection based
on the cut-off in fuzzy RD design as it is a more practical design. For
example, students may notice that with SAT score higher than a threshold may
increase their chance to receive scholarship, but the true assignment
mechanism (gender, family income, etc.) may not be disclosed to them.

To consider self-selection under this setting, we first need to modify the
independence and monotonicity assumption to incorporate additional covariates.

######  Assumption 9 (Fuzzy RD design).

.

  1. 1.

Independence:
$\mathbb{P}(\tau_{\epsilon}|X_{i}=c^{+},Z_{i}(1)=z)=\mathbb{P}(\tau_{\epsilon}|X_{i}=c^{-},Z_{i}(0)=z)$
for $\tau_{\epsilon}\in\\{co,at,nt\\}$.

  2. 2.

Monotonicity:
$\mathbb{P}(\tau_{\epsilon}=co)+\mathbb{P}(\tau=at)+\mathbb{P}(\tau_{\epsilon}=nt)=1$

  3. 3.

Exclusion:

$\displaystyle\mathbb{E}(Y_{i}(1)|Z_{i}(1)=z,\tau_{\epsilon}=at)=\mathbb{E}(Y_{i}(1)|Z_{i}(0)=z,\tau_{\epsilon}=at)$
$\displaystyle\mathbb{E}(Y_{i}(1)|Z_{i}(1)=z,\tau_{\epsilon}=nt)=\mathbb{E}(Y_{i}(1)|Z_{i}(0)=z,\tau_{\epsilon}=nt)$

Due to self-selection, $Z_{i}$ is discontinuous at $X_{i}=c$, thus the
independence assumption needs to be modified to insure individual types are
remain constant under self-selection. Monotonicity assumption is essential to
the identification by restricting the direction of selection. We may only have
compliers, always-takers, never-takers in the population. As a result, we can
view the treatment as an instrument as in Imbens and Angrist (1994).

By allowing types of individuals to depend on $Z_{i}$, our framework allows
selection on treatment compliance to vary across $Z_{i}$. For example,
individuals with family incomes less than a threshold are more likely to be
always takers for the scholarship. An extreme scenario when $Z_{i}$ is binary
would be that always takers all have $Z_{i}=1$. In a similar spirit to the
sharp RD design, we consider the following three estimands under our WATE
framework:

$\begin{split}\mathbb{E}_{Z(0)}\Big{(}\mathbb{E}(Y_{i}(1)|Z_{i}(1)=z,\tau_{\epsilon}=co)-\mathbb{E}(Y_{i}(0)|Z_{i}(0)=z,\tau_{\epsilon}=co)\Big{|}X_{i}=c^{-},\tau_{\epsilon}=co\Big{)},\end{split}$
(6.1) $\begin{split}\lim_{\epsilon\rightarrow
0}\mathbb{E}\Big{(}\mathbb{E}(Y_{i}(1)|Z_{i}(1)=z,\tau_{\epsilon}=co)-\mathbb{E}(Y_{i}(0)|Z_{i}(0)=z,\tau_{\epsilon}=co)\Big{|}X\in\mathcal{N}_{\epsilon},\tau_{\epsilon}=co\Big{)},\end{split}$
(6.2)
$\begin{split}\mathbb{E}_{Z}\Big{(}\mathbb{E}(Y_{i}(1)|Z_{i}(1)=z,\tau_{\epsilon}=co)-\mathbb{E}(Y_{i}(0)|Z_{i}(0)=z,\tau_{\epsilon}=co)\Big{)},\end{split}$
(6.3)

where $\mathcal{N}_{\epsilon}$ is a symmetric $\epsilon$ neighborhood around
$c$.

######  Theorem 2 (Identification under Fuzzy RD Design).

Under assumption 9, we have

$\displaystyle\mathbb{E}_{Z(0)}\Big{(}\mathbb{E}(Y_{i}(1)|Z_{i}(1)=z,\tau_{\epsilon}=co)-\mathbb{E}(Y_{i}(0)|Z_{i}(0)=z,\tau_{\epsilon}=co)\Big{|}X_{i}=c^{-},\tau_{\epsilon}=co\Big{)}$
$\displaystyle=\frac{\mathbb{E}_{Z(0)}\left(\mathbb{E}(Y_{i}|X_{i}=c^{+},Z_{i}=z)-\mathbb{E}(Y_{i}|X_{i}=c^{-},Z_{i}=z)\Big{|}X_{i}=c^{-}\right)}{\mathbb{E}_{Z(0)}\left(\mathbb{E}(T_{i}|X_{i}=c^{+},Z_{i}=z)-\mathbb{E}(T_{i}|X_{i}=c^{-},Z_{i}=z)\Big{|}X_{i}=c^{-}\right)},$
$\displaystyle\lim_{\epsilon\rightarrow
0}\mathbb{E}\Big{(}\mathbb{E}(Y_{i}(1)|Z_{i}(1)=z,\tau_{\epsilon}=co)-\mathbb{E}(Y_{i}(0)|Z_{i}(0)=z,\tau_{\epsilon}=co)\Big{|}X\in\mathcal{N}_{\epsilon},\tau_{\epsilon}=co\Big{)}$
$\displaystyle=\frac{\int\left(\mathbb{E}(Y_{i}|X_{i}=c^{+},Z_{i}=z)-\mathbb{E}(Y_{i}|X_{i}=c^{-},Z_{i}=z)\right)(f_{Z(0)|X}(z_{i}|c)+f_{Z(1)|X}(z_{i}|c))dz}{\int\left(\mathbb{E}(T_{i}|X_{i}=c^{+},Z_{i}=z)-\mathbb{E}(T_{i}|X_{i}=c^{-},Z_{i}=z)\right)(f_{Z(0)|X}(z_{i}|c)+f_{Z(1)|X}(z_{i}|c))dz},$
If we further have the conditional independence assumption (CIA),
$\displaystyle\mathbb{E}_{Z}\Big{(}\mathbb{E}(Y_{i}(1)|Z_{i}(1)=z,\tau_{\epsilon}=co)-\mathbb{E}(Y_{i}(0)|Z_{i}(0)=z,\tau_{\epsilon}=co)\Big{)}$
$\displaystyle=\frac{\mathbb{E}_{Z}\left(\mathbb{E}(Y_{i}|X_{i}=c^{+},Z_{i}=z)-\mathbb{E}(Y_{i}|X_{i}=c^{-},Z_{i}=z)\right)}{\mathbb{E}_{Z}\left(\mathbb{E}(T_{i}|X_{i}=c^{+},Z_{i}=z)-\mathbb{E}(T_{i}|X_{i}=c^{-},Z_{i}=z)\right)}.$

Theorem 2 proves the identification results for the three WATEs under the
fuzzy RD design. The proof for the first two results are straightforward. The
third result relies on the CIA in addition. This is because the sharp RD
estimand is to ignore the conditioning on $X_{i}$, however, the fuzzy RD has
to condition on $X_{i}$ to cancel with the denominator. The CIA is to directly
remove the conditioning on $X_{i}$ and thus is necessary for the results to
hold. The local linear estimators of these three estimands can be developed
following the argument in Section 3. However, the technical details and the
theoretical results are much more involved, and are beyond the scope of this
work. Given the practical importance of the fuzzy RD design, we think this is
an important future research problem to explore.

##  7 Discussion

In this paper we study the RD problem when the conditional distribution of the
covariates given running variables is discontinuous at the cutoff, for
example, due to self-selection. The standard RD design is no longer valid as
the continuity of potential outcomes assumption is violated. We show that
casual effect can still be recovered when the covariates related to self-
selection are observed. We thus propose a set of estimands under the framework
of WATE and show that these estimands can be estimated using a class of
weighted local linear estimators. We derive the theory for our estimators
include consistency and asymptotic normality. We further compare our estimator
with the standard RD estimator in simulation exercises to demonstrate its
finite sample performance.

We apply our estimator to two empirical examples. First, we study the U.S.
House elections data in Lee (2008), which is the classical data for RD
estimator. Our estimator and local polynomial RD estimator have similar
performance. Second, we apply our method to evaluate the effect of a GSP
auction. We show that the result from our estimator is different from standard
RD estimator due to self-selection. In particular, the obtained effect of
advertising by using our estimator is smaller and statistically insignificant
when self-selection is taking into account.

##  Appendix A Proofs

In the proof, we use $C$ to denote a generic constant which may change from
line to line. Define the following notations:

$\kappa_{\iota}=\int_{u>0}K(u)u^{\iota}du\quad\text{ and
}\quad\kappa_{2\iota}=\int_{u>0}K(u)^{2}u^{\iota}du,\quad\text{ for
}\iota=0,1,2,\cdots$

For the ease of presentation, we introduce the following notations. Let
$f_{X|Z(1)}^{\prime}(c^{+}|z_{i})$, $f_{X|Z(1)}^{\prime\prime}(c^{+}|z_{i})$
and $f_{X|Z(1)}^{\prime\prime\prime}(c^{+}|z_{i})$ denote the right
derivatives $\frac{\partial f_{X|Z(1)}(c^{+}|z_{i})}{\partial x}$,
$\frac{\partial^{2}f_{X|Z(1)}(c^{+}|z_{i})}{\partial x^{2}}$ and
$\frac{\partial^{3}f_{X|Z(1)}(c^{+}|z_{i})}{\partial x^{3}}$. Denote

$m_{1}(x,z)=\mathbb{E}(Y(1)|X=x,Z(1)=z),$
$\alpha_{1}=\mathbb{E}\\{Y(1)w_{1}(Z(1))|X=c^{+}\\}=\int\frac{f_{X|Z(1)}(c^{+}|z)}{\pi_{1}(z)}m_{1}(c^{+},z)f_{Z(1)}(z)d{z},$

and

$\beta_{1}=\int\frac{f^{\prime}_{X|Z(1)}(c^{+}|z)}{\pi_{1}(z)}m_{1}(c^{+},z)f_{Z(1)}(z)d{z}+\int\frac{f_{X|Z(1)}(c^{+}|z)}{\pi_{1}(z)}m^{\prime}_{1}(c^{+},z)f_{Z(1)}(z)d{z},$

where $m^{\prime}_{1}(c^{+},z)=\frac{\partial m_{1}(c^{+},z)}{\partial x}$.
Define $R_{i}=y_{i}-\alpha_{1}-\beta_{1}(x_{i}-c)$,

$\displaystyle A_{n}$
$\displaystyle=\sum_{i=1}^{n}\frac{T_{i}}{\pi_{1}(z_{i})}K\Big{(}\frac{x_{i}-c}{h}\Big{)}(x_{i}-c)^{2},$
$\displaystyle B_{n}$
$\displaystyle=-\sum_{i=1}^{n}\frac{T_{i}}{\pi_{1}(z_{i})}K\Big{(}\frac{x_{i}-c}{h}\Big{)}(x_{i}-c),$
$\displaystyle C_{n}$
$\displaystyle=\sum_{i=1}^{n}\frac{T_{i}}{\pi_{1}(z_{i})}K\Big{(}\frac{x_{i}-c}{h}\Big{)},$
$\displaystyle D_{n}$ $\displaystyle=A_{n}C_{n}-B_{n}^{2}.$

Since $A_{n},B_{n},C_{n},D_{n}$ all depend on $\pi_{t}(z)$, we denote the
corresponding version with $\widehat{\pi}_{1}(z)$ as
$\widehat{A}_{n},\widehat{B}_{n},\widehat{C}_{n},\widehat{D}_{n}$.

######  Lemma 10.

Under assumption 5, 7 and 8, we have

$\Big{|}\frac{\widehat{A}_{n}}{n}-h^{3}\kappa_{2}-h^{4}\mathbb{E}\left(\frac{f_{X|Z(1)}^{\prime}(c^{+}|z_{i})}{\pi_{1}(z_{i})}\right)\kappa_{3}\Big{|}=O_{p}(h^{5}+rh^{3}+\frac{h^{5/2}}{n^{1/2}}),$
$\Big{|}\frac{\widehat{B}_{n}}{n}+h^{2}\kappa_{1}+h^{3}\mathbb{E}\left(\frac{f_{X|Z(1)}^{\prime}(c^{+}|z_{i})}{\pi_{1}(z_{i})}\right)\kappa_{2}\Big{|}=O(h^{4}+rh^{2}+\frac{h^{3/2}}{n^{1/2}}),$
$\Big{|}\frac{\widehat{C}_{n}}{n}-h/2\Big{|}=O_{p}(h^{2}+rh+\frac{h^{1/2}}{n^{1/2}}),$
(A.1)
$\Big{|}\frac{\widehat{D}_{n}}{n^{2}}-h^{4}\left(\frac{1}{2}\kappa_{2}-\kappa_{1}^{2}\right)\Big{|}=O_{p}(h^{5}+rh^{4}+\frac{h^{7/2}}{n^{1/2}}),$

where $r$ satisfies
$\sup_{z\in\mathcal{Z}}|\widehat{\pi}_{1}(z)-\pi_{1}(z)|=O_{p}(r)$.

###### Proof.

We will focus on the proof of (A.1). The remain results can be shown following
the similar steps. By triangle inequality,

$\displaystyle\Big{|}\frac{\widehat{C}_{n}}{n}-\frac{h}{2}\Big{|}$
$\displaystyle\leq\Big{|}\frac{1}{n}\sum_{i=1}^{n}\frac{T_{i}}{\pi_{1}(z_{i})}K\Big{(}\frac{x_{i}-c}{h}\Big{)}-\frac{h}{2}\Big{|}+\Big{|}\frac{1}{n}\sum_{i=1}^{n}\frac{T_{i}(\pi_{1}(z_{i})-\widehat{\pi}_{1}(z_{i}))}{\pi_{1}(z_{i})\widehat{\pi}_{1}(z_{i})}K\Big{(}\frac{x_{i}-c}{h}\Big{)}\Big{|}:=I_{1}+I_{2}.$

For term $I_{1}$, we further decompose it into two terms,

$I_{1}\leq\Big{|}\frac{1}{n}\sum_{i=1}^{n}\frac{T_{i}}{\pi_{1}(z_{i})}K\Big{(}\frac{x_{i}-c}{h}\Big{)}-\mathbb{E}\Big{(}\frac{T_{i}}{\pi_{1}(z_{i})}K\Big{(}\frac{x_{i}-c}{h}\Big{)}\Big{)}\Big{|}+\Big{|}\mathbb{E}\Big{(}\frac{T_{i}}{\pi_{1}(z_{i})}K\Big{(}\frac{x_{i}-c}{h}\Big{)}\Big{)}-h/2\Big{|}:=I_{11}+I_{12}.$
(A.2)

The bias term $I_{12}$ is computed as

$\displaystyle I_{12}$
$\displaystyle=\Big{|}\int\int_{x>c}\frac{1}{\pi_{1}(z)}K\Big{(}\frac{x_{i}-c}{h}\Big{)}f_{X,Z(1)}(x,z)dxdz-\frac{h}{2}\Big{|}$
$\displaystyle=h\Big{|}\int\int_{u>0}\frac{1}{\pi_{1}(z)}K(u)f_{X,Z(1)}(uh+c,z)dudz-\frac{1}{2}\Big{|}$
$\displaystyle\leq
h\Big{|}\int\int_{u>0}\frac{1}{\pi_{1}(z)}K(u)f_{X,Z(1)}(c^{+},z)dudz-\frac{1}{2}\Big{|}$
$\displaystyle~{}~{}~{}~{}+h\Big{|}\int\int_{u>0}\frac{1}{\pi_{1}(z)}K(u)f_{X|Z(1)}^{\prime}(\widetilde{u}|z)uhf_{Z(1)}(z)dudz\Big{|},$

where the last step follows from the mean value theorem for some intermediate
value $\widetilde{u}$. Our assumption implies that
$|f_{X|Z(1)}^{\prime}(\widetilde{u}|z)|$ is bounded. In addition, by
assumption $\pi_{1}(z)$ is bounded away from 0 by a constant, thus the second
term is of order $h^{2}$. For the first term, by the choice of $\pi_{1}(z)$ we
get

$\int\frac{1}{\pi_{1}(z)}f_{X|Z(1)}(c^{+}|z)f_{Z(1)}(z)dz=1,$ (A.3)

which implies

$\int\int_{u>0}\frac{1}{\pi_{1}(z)}K(u)f_{X|Z(1)}(c^{+}|z)f_{Z(1)}(z)dudz=\int\frac{1}{\pi_{1}(z)}f_{X|Z(1)}(c^{+}|z)f_{Z(1)}(z)dz\int_{u>0}K(u)du=\frac{1}{2}.$

Thus, we have $I_{12}=O(h^{2})$. Now we consider $I_{11}$. By the Markov
inequality, $I_{11}\lesssim(\mathbb{E}I_{11}^{2})^{1/2}$. Thus, it suffices to
compute $\mathbb{E}I_{11}^{2}$,

$\displaystyle\mathbb{E}I_{11}^{2}$
$\displaystyle=\frac{1}{n}\mathbb{E}\Big{(}\frac{T_{i}}{\pi_{1}^{2}(z_{i})}K^{2}\Big{(}\frac{x_{i}-c}{h}\Big{)}\Big{)}-\frac{1}{n}\Big{[}\mathbb{E}\Big{(}\frac{T_{i}}{\pi_{1}(z_{i})}K\Big{(}\frac{x_{i}-c}{h}\Big{)}\Big{)}\Big{]}^{2}$
$\displaystyle\leq\frac{1}{n}\int\int_{x>c}\frac{1}{\pi_{1}^{2}(z)}K^{2}\Big{(}\frac{x-c}{h}\Big{)}f_{X,Z(1)}(x,z)dxdz$
$\displaystyle=\frac{h}{n}\int\int_{u>0}\frac{1}{\pi_{1}^{2}(z)}K^{2}(u)f_{X,Z(1)}(uh+c,z)dudz$
$\displaystyle=O(\frac{h}{n}),$

where the last step follows from the same argument in $I_{21}$. Putting them
together into (A.2), we have $I_{1}=O_{p}(h^{2}+\frac{h^{1/2}}{n^{1/2}})$. For
the last term $I_{2}$, we have

$\displaystyle I_{2}$
$\displaystyle\leq\sup_{z\in\mathcal{Z}}|\widehat{\pi}_{1}(z)-\pi_{1}(z)|\frac{1}{n}\sum_{i=1}^{n}\frac{T_{i}}{\pi_{1}(z_{i})\widehat{\pi}_{1}(z_{i})}K\Big{(}\frac{x_{i}-c}{h}\Big{)}$
$\displaystyle\leq
O_{p}(r)\frac{1}{n}\sum_{i=1}^{n}\frac{T_{i}}{\pi_{1}(z_{i})}K\Big{(}\frac{x_{i}-c}{h}\Big{)}=O_{p}(rh),$

where the last step holds by the bound for the $I_{1}$ term. This implies
(A.1). ∎

######  Lemma 11.

Under assumption 5, 7 and 8, we have

$\displaystyle\mathbb{E}\left((h_{2})^{-1}K(\frac{z_{i}-z_{j}}{h_{2}})\Big{|}z_{i}\right)=f_{Z}(z_{i})+O_{p}(h_{2}^{2}),$
$\displaystyle\mathbb{E}\left(2(h_{1}^{2})^{-1}T_{j}K_{1}(\frac{c-x_{j}}{h_{1}},\frac{z_{i}-z_{j}}{h_{1}})\Big{|}z_{i},x_{i}\right)=f_{X,Z(1)}(c^{+},z_{i})+2h_{1}\kappa_{1}\frac{\partial
f_{X,Z(1)}(c^{+},z_{i})}{\partial x_{i}}+O_{p}(h_{1}^{2}),$
$\displaystyle\sup_{z_{i}}\Big{|}(nh_{2})^{-1}\sum_{j=1}^{n}K(\frac{z_{i}-z_{j}}{h_{2}})-f_{Z}(z_{i})\Big{|}=O_{p}\left(\sqrt{\frac{\log
n}{nh_{2}}}+h_{2}^{2}\right),$
$\displaystyle\sup_{z_{i}}\Big{|}2(nh_{1}^{2})^{-1}\sum_{j=1}^{n}T_{j}K(\frac{c-x_{j}}{h_{1}},\frac{z_{i}-z_{j}}{h_{1}})-f_{X,Z(1)}(c^{+},z_{i})\Big{|}=O_{p}\left(\sqrt{\frac{\log
n}{nh_{1}^{2}}}+h_{1}\right).$

In case when $Z=Z(1)=Z(0)$,

$\displaystyle\mathbb{E}\left((h_{1}^{2})^{-1}T_{j}K_{1}(\frac{c-x_{j}}{h_{1}},\frac{z_{i}-z_{j}}{h_{1}})\Big{|}z_{i},x_{i}\right)=f_{X,Z}(c,z_{i})+O_{p}(h_{1}^{2}),$
$\displaystyle\sup_{z_{i}}\Big{|}(nh_{1}^{2})^{-1}\sum_{j=1}^{n}K(\frac{c-x_{j}}{h_{1}},\frac{z_{i}-z_{j}}{h_{1}})-f_{X,Z}(c,z_{i})\Big{|}=O_{p}\left(\sqrt{\frac{\log
n}{nh_{1}^{2}}}+h_{1}^{2}\right).$

###### Proof.

First, note that

$\mathbb{P}(Z=z)=\mathbb{P}(Z=z,T=1)+\mathbb{P}(Z=z,T=0)=\mathbb{P}(Z(1)=z,X>c)+\mathbb{P}(Z(0)=z,X<c).$

Then, we have

$f_{Z}(z)=\int_{x>c}f_{X,Z(1)}(x,z)dx+\int_{x<c}f_{X,Z(0)}(x,z)dx,$

which implies that $f_{Z}(z)$ is second order continuously differentiable by
the continuously differentiable property of $f_{X,Z(1)}(x,z)$ and
$f_{X,Z(0)}(x,z)$ in assumption 7. Thus, the standard calculation in
nonparametric density estimation yields

$\begin{split}\mathbb{E}\left((h_{2})^{-1}K(\frac{z_{i}-z_{j}}{h_{2}})\Big{|}z_{i}\right)&=\frac{1}{h_{2}}\int
K\left(\frac{z_{i}-z_{j}}{h_{2}}\right)f_{Z}(z_{j})dz_{j}=\int
K\left(u\right)f_{Z}(z_{i}+uh_{2})du\\\
&=f_{Z}(z_{i})+O_{p}(h_{2}^{2}).\end{split}$

To show the second result, following the similar argument, we get

$\begin{split}&\mathbb{E}\left(2(h_{11}h_{12})^{-1}T_{j}K_{1}(\frac{c-x_{j}}{h_{11}},\frac{z_{i}-z_{j}}{h_{12}})\Big{|}z_{i},x_{i}\right)=\frac{2}{h_{11}h_{12}}\int\int
T_{j}K_{1}(\frac{c-x_{j}}{h_{11}},\frac{z_{i}-z_{j}}{h_{12}})f_{X,Z(1)}(x_{j},z_{j})dz_{j}dx_{j}\\\
&=\frac{2}{h_{11}}\int T_{j}K(\frac{c-x_{j}}{h_{11}})\int
K(v)f_{X,Z(1)}(x_{j},z_{i}+vh_{12})dvdx_{j}\\\ &=\frac{2}{h_{11}}\int
T_{j}K(\frac{c-x_{j}}{h_{11}})\left(f_{X,Z(1)}(x_{j},z_{i})+Ch_{12}^{2}\right)dx_{j}\\\
&=2\int_{u>0}K(u)\left(f_{X,Z(1)}(c+uh_{11},z_{i})+Ch_{12}^{2}\right)du\\\
&=f_{X,Z(1)}(c^{+},z_{i})+2h_{11}\kappa_{1}\frac{\partial
f_{X,Z(1)}(c^{+},z_{i})}{\partial x_{i}}+O(h_{12}^{2}+h_{11}^{2}),\end{split}$

where $C$ is a generic constant. The last two results follow from Fan (1993)
together with the above bias calculation. ∎

###  A.1 Proof of Theorem 1

Since the local linear estimator is invariant to the scale of $\pi_{1}(z)$, we
can simply take $\frac{1}{\pi_{1}(z)}=\frac{f_{Z}(z)}{f_{X,Z(1)}(c,z)}$ in the
rest of the proof. It can be estimated by the following kernel estimator:

$\begin{split}\frac{1}{\widehat{\pi}_{1}(z)}=\frac{\widehat{f}_{Z}(z)}{\widehat{f}_{X,Z(1)}(c,z)}=\frac{(nh_{2})^{-1}\sum_{i=1}^{n}K(\frac{z-z_{i}}{h_{2}})}{2\cdot(nh_{1}^{2})^{-1}\sum_{x_{i}>c}K_{1}(\frac{c-x_{i}}{h_{1}},\frac{z-z_{i}}{h_{1}})}.\end{split}$

Start with the following minimization problem:

$\begin{split}\Big{(}\widehat{\alpha}_{1},\widehat{\beta}_{1}\Big{)}&=\arg\min_{\alpha,\beta}\sum_{i}\frac{T_{i}}{\widehat{\pi}_{1}{(z_{i})}}\Big{(}y_{i}-\alpha-(x_{i}-c)\beta\Big{)}^{2}K\Big{(}\frac{x_{i}-c}{h}\Big{)}.\end{split}$
(A.4)

Recall that for any kernel estimates $\widehat{f}=\widehat{f}(x)$ and
$\widehat{g}=\widehat{g}(x)$, if $f$ is bounded away from 0, then

$\begin{split}\frac{1}{\widehat{f}}=\frac{1}{f}-\frac{1}{f^{2}}(\widehat{f}-f)+O_{p}(r^{2}),\end{split}$
(A.5)

where $\|\widehat{f}-f\|_{\infty}=O_{p}(r)$ and
$\|\widehat{g}-g\|_{\infty}=O_{p}(s)$. Thus, if $g$ is bounded from above,
then

$\begin{split}\frac{\widehat{g}}{\widehat{f}}&=\frac{g+(\widehat{g}-g)}{f}-\frac{g+(\widehat{g}-g)}{f^{2}}(\widehat{f}-f)+O_{p}(r^{2})\\\
&=\frac{g}{f}-\frac{g}{f^{2}}(\widehat{f}-f)+\frac{\widehat{g}-g}{f}+O_{p}(r^{2}+rs)\\\
&=\frac{g}{f}-\frac{g\widehat{f}}{f^{2}}+\frac{\widehat{g}}{f}+O_{p}(r^{2}+rs).\end{split}$

Following the above discussion, we can show that

$\begin{split}&\frac{1}{2\cdot(nh_{1}^{2})^{-1}\sum_{j=1}^{n}T_{j}K_{1}(\frac{c-x_{j}}{h_{1}},\frac{z_{i}-z_{j}}{h_{1}})}\\\
&=-\frac{1}{f_{X,Z(1)}(c^{+},z_{i})^{2}}\left(2(nh_{1}^{2})^{-1}\sum_{j=1}^{n}T_{j}K_{1}(\frac{c-x_{j}}{h_{1}},\frac{z_{i}-z_{j}}{h_{1}})-f_{X,Z(1)}(c^{+},z_{i})\right)\\\
&+\frac{1}{f_{X,Z(1)}(c^{+},z_{i})}+O_{p}\left(r^{2}\right),\end{split}$

where Lemma 11 implies

$r=\sqrt{\frac{\log n}{nh_{1}^{2}}}+h_{1}.$

The gradient of (A.4) can be written as the following $U$-statistic:

$\begin{split}&\frac{1}{n}\sum_{i=1}^{n}\frac{T_{i}\widehat{f}_{Z}(z_{i})}{\widehat{f}_{X,Z(1)}(c,z_{i})}K\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}=\frac{1}{n}\sum_{i=1}^{n}\frac{T_{i}(nh_{2})^{-1}\sum_{j=1}^{n}K(\frac{z_{i}-z_{j}}{h_{2}})}{2\cdot(nh_{1}^{2})^{-1}\sum_{j=1}^{n}T_{j}K_{1}(\frac{c-x_{j}}{h_{1}},\frac{z_{i}-z_{j}}{h_{1}})}K\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}\\\
&=\frac{1}{n}\sum_{i=1}^{n}\frac{T_{i}f_{Z}(z_{i})}{f_{X,Z(1)}(c^{+},z_{i})}K\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}+\frac{T_{i}}{f_{X,Z(1)}(c^{+},z_{i})}\left((nh_{2})^{-1}\sum_{j=1}^{n}K(\frac{z_{i}-z_{j}}{h_{2}})\right)K\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}\\\
&-\frac{T_{i}f_{Z}(z_{i})}{f_{X,Z(1)}(c^{+},z_{i})^{2}}\left(2\cdot(nh_{1}^{2})^{-1}\sum_{j=1}^{n}T_{j}K_{1}(\frac{c-x_{j}}{h_{1}},\frac{z_{i}-z_{j}}{h_{1}})\right)K\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}+O_{p}\left(r^{2}+rs\right)\\\
&=\frac{1}{n^{2}}\sum_{i=1}^{n}\sum_{j=1}^{n}\Big{\\{}\frac{T_{i}}{f_{X,Z(1)}(c^{+},z_{i})}(h_{2})^{-1}K(\frac{z_{i}-z_{j}}{h_{2}})K\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}\\\
&-2\cdot\frac{T_{i}T_{j}f_{Z}(z_{i})}{f_{X,Z(1)}(c^{+},z_{i})^{2}}(h_{1}^{2})^{-1}K_{1}(\frac{c-x_{j}}{h_{1}},\frac{z_{i}-z_{j}}{h_{1}})K\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}+\frac{T_{i}f_{Z}(z_{i})}{f_{X,Z(1)}(c^{+},z_{i})}K\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}\Big{\\}}+O_{p}\left(r^{2}+rs\right),\end{split}$

where

$s=\sqrt{\frac{\log n}{nh_{2}}}+h_{2}^{2},$

implied by Lemma 11. Define

$\begin{split}\phi_{i,j}&=\frac{T_{i}}{f_{X,Z(1)}(c^{+},z_{i})}(h_{2})^{-1}K(\frac{z_{i}-z_{j}}{h_{2}})K\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}\\\
&-2\cdot\frac{T_{i}T_{j}f_{Z}(z_{i})}{f_{X,Z(1)}(c^{+},z_{i})^{2}}(h_{1}^{2})^{-1}K_{1}(\frac{c-x_{j}}{h_{1}},\frac{z_{i}-z_{j}}{h_{1}})K\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}+\frac{T_{i}f_{Z}(z_{i})}{f_{X,Z(1)}(c^{+},z_{i})}K\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}.\end{split}$

Thus,

$\begin{split}\frac{1}{n}\sum_{i=1}^{n}\frac{T_{i}\widehat{f}_{Z}(z)}{\widehat{f}_{X|Z(1)}(c,z)}K\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}&=\frac{1}{n^{2}}\sum_{i}\sum_{j}\phi_{i,j}+O_{p}\left(r^{2}+rs\right)\\\
&=\frac{1}{2n^{2}}\sum_{i}\sum_{j}(\phi_{i,j}+\phi_{j,i})+O_{p}\left(r^{2}+rs\right)\\\
&=\frac{1}{n^{2}}\sum_{i<j}(\phi_{i,j}+\phi_{j,i})+\frac{1}{n^{2}}\sum_{i}\phi_{i,i}+O_{p}\left(r^{2}+rs\right),\end{split}$
(A.6)

where the first (leading) term is a $U$-statistic after rescaling. By lemma 15
and Theorem 12.3 in Van der Vaart (2000), we have

$\frac{n^{1/2}}{h^{1/2}}\Big{(}\frac{1}{n(n-1)}\sum_{i<j}(\phi_{i,j}+\phi_{j,i})-\delta/2-\frac{1}{n}\sum_{i=1}^{n}\\{\mathbb{E}(\phi_{i,j}+\phi_{j,i}|i)-\delta\\}\Big{)}=o_{p}(1),$
(A.7)

where $\delta=\mathbb{E}(\phi_{i,j}+\phi_{j,i})$, and we use
$\mathbb{E}(\cdot|i)$ to denote the conditional expectation given the $i$th
sample. In the following, we approximate
$\mathbb{E}(\phi_{i,j}+\phi_{j,i}|i)$. Define
$d(x_{i},z_{i})=m(x_{i},z_{i})-\alpha_{1}$. By Lemma 13, we have

$\mathbb{E}(\phi_{i,j})=O_{p}(h^{3}).$

The central limit theorem implies

$\frac{1}{(nh)^{1/2}}\sum_{i=1}^{n}\\{\mathbb{E}(\phi_{i,j}+\phi_{j,i}|i)-\delta\\}+O_{p}(\frac{n^{1/2}}{h^{1/2}}(h^{3}))\rightarrow_{d}N(0,\xi^{2}/h),$
(A.8)

where $\delta=O_{p}(h^{3})$ and
$\xi^{2}=\mathbb{E}\\{(\mathbb{E}(\phi_{i,j}+\phi_{j,i}|i)-\delta)^{2}\\}=\mathbb{E}\left\\{\mathbb{E}(\phi_{ij}|i)^{2}+\mathbb{E}(\phi_{ji}|i)^{2}+2\mathbb{E}(\phi_{ij}|i)\mathbb{E}(\phi_{ji}|i)\right\\}-\delta^{2}$.
We now calculate the asymptotic variance as follows. Since
$\mathbb{E}(\phi_{j,i}|i)=\mathbb{E}(\phi_{i,j}|j)$, from lemma 13, we have

$\begin{split}\mathbb{E}(\phi_{ij}|j)&=\frac{h}{2}d_{1}(c^{+},z_{j})-T_{j}\frac{h}{h_{1}}K(\frac{c-x_{j}}{h_{1}})\frac{f_{Z}(z_{j})}{f_{X,Z(1)}(c^{+},z_{j})}d_{1}(c^{+},z_{j})+O_{p}(h^{2}+hh_{1}+hh_{2}).\end{split}$

Similarly, we can show that

$\mathbb{E}(\phi_{i,j}|i)=\frac{T_{i}f_{Z}(z_{i})}{f_{X,Z(1)}(c^{+},z_{i})}K\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}+O_{p}(h_{2}^{2}+h_{1}).$

Recall that $\sigma^{2}=\mathbb{E}(Y(1)-m_{1}(X,Z(1)))^{2}$. Since
$h\asymp\sqrt{h_{1}}\asymp h_{2}$, after some tedious calculation we can show
that

$\begin{split}\frac{1}{n}\mathbb{E}\left(\mathbb{E}(\phi_{ji}|i)^{2}\right)&=\frac{1}{n}\frac{h^{2}}{h_{1}^{2}}\int_{z_{i}}\int_{x_{i}}T_{i}K(\frac{c-x_{i}}{h_{1}})^{2}\frac{f_{Z}(z_{i})^{2}}{f_{X,Z(1)}(c^{+},z_{i})^{2}}d(c^{+},z_{i})^{2}f_{X,Z(1)}(x_{i},z_{i})dx_{i}dz_{i}+O\left(\frac{h^{2}}{n}\right)\\\
&=\frac{h^{2}}{h_{1}}\frac{\kappa_{20}}{n}\int_{z_{i}}\frac{f_{Z}(z_{i})^{2}}{f_{X,Z(1)}(c^{+},z_{i})}d(c^{+},z_{i})^{2}dz_{i}+O\left(\frac{h^{2}}{n}\right)\\\
&=\frac{h^{2}}{h_{1}}\frac{\kappa_{20}}{n}\mathbb{E}_{Z}\left(\frac{f_{Z}(z_{i})}{f_{X,Z(1)}(c^{+},z_{i})}d(c^{+},z_{i})^{2}\right)+O\left(\frac{h^{2}}{n}\right),\end{split}$

And

$\xi^{2}=\kappa_{20}\underbrace{\mathbb{E}_{Z}\left(\frac{f_{Z}(z_{i})}{f_{X,Z(1)}(c^{+},z_{i})}d_{1}(c^{+},z_{i})^{2}\right)}_{\omega}+O(h).$

Combining (A.7) and (A.9),

$n^{1/2}\Big{(}\frac{1}{n(n-1)}\sum_{i<j}(\phi_{i,j}+\phi_{j,i})-\delta/2\Big{)}+O\Big{(}n^{1/2}h^{3}\Big{)}\rightarrow_{d}N(0,\xi^{2}).$

Finally, note that in (A.6),

$\frac{1}{n^{2}}\sum_{i=1}^{n}\phi_{i,i}\lesssim\frac{1}{n}\mathbb{E}(\phi_{i,j})=O_{p}\left(\frac{h^{2}}{n}\right),$

and therefore we obtain that

$\frac{1}{n^{1/2}}\sum_{i=1}^{n}\frac{T_{i}\widehat{f}_{Z}(z)}{\widehat{f}_{X,Z(1)}(c,z)}K\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}+\phi\rightarrow_{d}N(n^{1/2}\delta/2,\xi^{2}),$

where

$\phi=O_{p}\Big{(}n^{1/2}(h^{3}+r^{2}+rs)\Big{)}.$

Following the similar argument, we can show the joint convergence

$\frac{1}{n^{1/2}}\sum_{i=1}^{n}\frac{T_{i}\widehat{f}_{Z}(z)}{\widehat{f}_{X,Z(1)}(c,z)}K\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}[1,(x_{i}-c)]^{T}\rightarrow_{d}N\left(n^{1/2}\left(\begin{matrix}O_{p}(h^{3})\\\
O_{p}(h^{4})\end{matrix}\right),\omega\left(\begin{matrix}\kappa_{20}&h\kappa_{21}\\\
h\kappa_{21}&h^{2}\kappa_{22}\\\ \end{matrix}\right)\right).$

By the least squared formulation, the estimator $\widehat{\alpha}_{1}$
satisfies

$\sqrt{nh^{2}}(\widehat{\alpha}_{1}-\alpha_{1})=-e_{1}^{T}\left(\begin{matrix}\widehat{C}_{n}/(nh)&-\widehat{B}_{n}/(nh)\\\
-\widehat{B}_{n}/(nh)&\widehat{A}_{n}/(nh)\end{matrix}\right)^{-1}\frac{1}{n^{1/2}}\sum_{i=1}^{n}\frac{T_{i}\widehat{f}_{Z}(z)}{\widehat{f}_{X,Z(1)}(c,z)}K\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}[1,(x_{i}-c)]^{T},$

where $e_{1}^{T}=(1,0)$. From lemma 10 and the matrix inversion formula,

$\left(\begin{matrix}\widehat{C}_{n}/(nh)&-\widehat{B}_{n}/(nh)\\\
-\widehat{B}_{n}/(nh)&\widehat{A}_{n}/(nh)\end{matrix}\right)^{-1}=\frac{1}{\widehat{D}_{n}/(nh)^{2}}\left(\begin{matrix}\widehat{A}_{n}/(nh)&\widehat{B}_{n}/(nh)\\\
\widehat{B}_{n}/(nh)&\widehat{C}_{n}/(nh)\end{matrix}\right)\rightarrow_{p}\frac{1}{h^{2}(\kappa_{2}/2-\kappa_{1}^{2})}\left(\begin{matrix}h^{2}\kappa_{2}&-h\kappa_{1}\\\
-h\kappa_{1}&\frac{1}{2}\end{matrix}\right).$

Thus, the asymptotic bias of $\sqrt{nh^{2}}(\widehat{\alpha}_{1}-\alpha_{1})$
is

$\frac{-e_{1}^{T}}{h^{2}(\kappa_{2}/2-\kappa_{1}^{2})}\left(\begin{matrix}h^{2}\kappa_{2}&-h\kappa_{1}\\\
-h\kappa_{1}&\frac{1}{2}\end{matrix}\right)n^{1/2}\left(\begin{matrix}O_{p}(h^{3})\\\
O_{p}(h^{4})\end{matrix}\right)=O(n^{1/2}h^{3})=o(1).$

Similarly, the asymptotic variance of
$\sqrt{nh^{2}}(\widehat{\alpha}_{1}-\alpha_{1})$ is

$\displaystyle\frac{\omega}{h^{4}(\kappa_{2}/2-\kappa_{1}^{2})^{2}}e_{1}^{T}\left(\begin{matrix}h^{2}\kappa_{2}&-h\kappa_{1}\\\
-h\kappa_{1}&\frac{1}{2}&\end{matrix}\right)\left(\begin{matrix}\kappa_{20}&h\kappa_{21}\\\
h\kappa_{21}&h^{2}\kappa_{22}\\\
\end{matrix}\right)\left(\begin{matrix}h^{2}\kappa_{2}&-h\kappa_{1}\\\
-h\kappa_{1}&\frac{1}{2}\end{matrix}\right)e_{1}$
$\displaystyle=\mathbb{E}_{Z}\left(\frac{f_{Z}(z_{i})}{f_{X,Z(1)}(c^{+},z_{i})}d(c^{+},z_{i})^{2}\right)\cdot
C_{v},$

where

$C_{v}=\frac{\kappa_{2}^{2}\kappa_{20}+\kappa_{1}^{2}\kappa_{22}-2\kappa_{1}\kappa_{2}\kappa_{21}}{\left(\frac{1}{2}\kappa_{2}-\kappa_{1}^{2}\right)^{2}}.$

This completes the proof.

######  Lemma 12.

Under the same condition as in Theorem 1,

$\begin{split}M(c,z_{i})&:=\frac{1}{h}\int
T_{i}K\left(\frac{x_{i}-c}{h}\right)\mathbb{E}(R_{i}|x_{i},z_{i})f_{X,Z(1)}(x_{i},z_{i})dx_{i}\\\
&=\frac{1}{2}m_{1}(c^{+},z_{i})f_{X,Z(1)}(c^{+},z_{i})+h\kappa_{1}\frac{\partial
m_{1}(c^{+},z_{i})}{\partial
x_{i}}f_{X,Z(1)}(c^{+},z_{i})+h\kappa_{1}m_{1}(c^{+},z_{i})\frac{\partial
f_{X,Z(1)}(c^{+},z_{i})}{\partial x_{i}}\\\
&-\frac{1}{2}\alpha_{1}f_{X,Z(1)}(c^{+},z_{i})-h\kappa_{1}\beta_{1}f_{X,Z(1)}(c^{+},z_{i})-h\kappa_{1}\alpha_{1}\frac{\partial
f_{X,Z(1)}(c^{+},z_{i})}{\partial x_{i}}+O_{p}(h^{2}),\end{split}$

where $O_{p}$ terms are valid uniformly over $i$.

###### Proof.

Following the standard Taylor expansion, we can show that

$\displaystyle M(c,z_{i})=\frac{1}{h}\int
T_{i}K\left(\frac{x_{i}-c}{h}\right)\mathbb{E}(R_{i}|x_{i},z_{i})f_{X,Z(1)}(x_{i},z_{i})dx_{i}$
$\displaystyle=\int_{u>0}K(u)\left(m_{1}(c+uh,z_{i})-\alpha_{1}-uh\beta_{1}\right)f_{X,Z(1)}(c+uh,z_{i})du$
$\displaystyle=\int_{u>0}K(u)\left(m_{1}(c^{+},z_{i})+\frac{\partial
m_{1}(c^{+},z_{i})}{\partial
x_{i}}uh+\frac{\partial^{2}m_{1}(c^{+},z_{i})}{\partial
x_{i}^{2}}\frac{u^{2}h^{2}}{2}+\frac{\partial^{3}m_{1}(\widetilde{c}^{(2)},z_{i})}{\partial
x_{i}^{3}}\frac{u^{3}h^{3}}{3!}-\alpha_{1}-uh\beta_{1}\right)$
$\displaystyle\left(f_{X,Z(1)}(c^{+},z_{i})+\frac{\partial
f_{X,Z(1)}(c^{+},z_{i})}{\partial
x_{i}}uh+\frac{\partial^{2}f_{X,Z(1)}(c^{+},z_{i})}{\partial
x_{i}^{2}}\frac{u^{2}h^{2}}{2}+\frac{\partial^{3}f_{X,Z(1)}(\widetilde{c}^{(3)},z_{i})}{\partial
x_{i}^{3}}\frac{u^{3}h^{3}}{3!}\right)du$
$\displaystyle=\frac{1}{2}(m_{1}(c^{+},z_{i})-\alpha_{1})f_{X,Z(1)}(c^{+},z_{i})+h\kappa_{1}\left(\frac{\partial
m_{1}(c^{+},z_{i})}{\partial x_{i}}-\beta_{1}\right)f_{X,Z(1)}(c^{+},z_{i})$
$\displaystyle+h\kappa_{1}(m_{1}(c^{+},z_{i})-\alpha_{1})\frac{\partial
f_{X,Z(1)}(c^{+},z_{i})}{\partial x_{i}}+O_{p}(h^{2}),$

where $O_{p}$ terms are valid uniformly over $i$ as the (mixed) third
derivatives of $f_{X,Z(1)}(x_{i},z_{i})$ are all bounded. ∎

######  Lemma 13.

Recall that

$\begin{split}\phi_{i,j}&=\frac{T_{i}}{f_{X,Z(1)}(c^{+},z_{i})}(h_{2})^{-1}K(\frac{z_{i}-z_{j}}{h_{2}})K\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}\\\
&-2\cdot\frac{T_{i}T_{j}f_{Z}(z_{i})}{f_{X,Z(1)}(c^{+},z_{i})^{2}}(h_{1}^{2})^{-1}K_{1}(\frac{c-x_{j}}{h_{1}},\frac{z_{i}-z_{j}}{h_{1}})K\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}+\frac{T_{i}f_{Z}(z_{i})}{f_{X,Z(1)}(c^{+},z_{i})}K\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}.\end{split}$

Under the same condition as in Theorem 1, and when $h_{1}=h^{2}$

$\displaystyle\mathbb{E}(\phi_{i,j}|j)=\frac{h}{2}d(c^{+},z_{j})-T_{j}\frac{h}{h_{1}}K(\frac{c-x_{j}}{h_{1}})\frac{f_{Z}(z_{j})}{f_{X,Z(1)}(c^{+},z_{j})}d_{1}(c^{+},z_{j})+O_{p}(h^{2}+hh_{1}+hh_{2}),$
$\displaystyle\mathbb{E}(\phi_{i,j}|i)=\frac{T_{i}f_{Z}(z_{i})}{f_{X,Z(1)}(c^{+},z_{i})}K\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}+O_{p}(h_{2}^{2}+h_{1}),$
$\displaystyle\mathbb{E}(\phi_{i,j})=O_{p}(h^{3})$

where $O_{p}$ terms are valid uniformly over $i$ or $j$.

###### Proof.

$\begin{split}\mathbb{E}(\phi_{i,j}|j)&=\underbrace{\mathbb{E}\left(\frac{T_{i}}{f_{X,Z(1)}(c^{+},z_{i})}(h_{2})^{-1}K(\frac{z_{i}-z_{j}}{h_{2}})K\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}\Big{|}z_{j}\right)}_{\text{(Part.BI)}}\\\
&-2T_{j}\cdot\underbrace{\mathbb{E}\left(\frac{T_{i}f_{Z}(z_{i})}{f_{X,Z(1)}(c^{+},z_{i})^{2}}(h_{1}^{2})^{-1}K_{1}(\frac{c-x_{j}}{h_{1}},\frac{z_{i}-z_{j}}{h_{1}})K\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}\Big{|}z_{j},x_{j}\right)}_{\text{(Part.BII)}}\\\
&+\underbrace{\mathbb{E}\left(\frac{T_{i}f_{Z}(z_{i})}{f_{X,Z(1)}(c^{+},z_{i})}K\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}\Big{|}z_{j},x_{j}\right)}_{\text{(Part.BIII)}}.\end{split}$

From lemma 12,

$\begin{split}\text{(Part.BI)}&=\mathbb{E}\left(\frac{T_{i}}{f_{X,Z(1)}(c^{+},z_{i})}(h_{2})^{-1}K(\frac{z_{i}-z_{j}}{h_{2}})K\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}\Big{|}z_{j}\right)\\\
&=h\int_{z_{i}}\frac{1}{f_{X,Z(1)}(c^{+},z_{i})}(h_{2})^{-1}K(\frac{z_{i}-z_{j}}{h_{2}})M(c,z_{i})dz_{i}\\\
&=h\int_{z_{i}}(h_{2})^{-1}K(\frac{z_{i}-z_{j}}{h_{2}})\frac{1}{2}(m_{1}(c^{+},z_{i})-\alpha_{1})dz_{i}+O_{p}(h^{2})\\\
&=\frac{h}{2}(m_{1}(c^{+},z_{j})-\alpha_{1})+O_{p}(h^{2}+h_{2}h),\end{split}$

where $O_{p}$ terms are valid uniformly over $j$. Similarly, for Part.BII and
Part.BIII, we can show that

$\begin{split}&\text{(Part.BII)}=\mathbb{E}\left(\frac{T_{i}f_{Z}(z_{i})}{f_{X,Z(1)}(c^{+},z_{i})^{2}}(h_{1}^{2})^{-1}K_{1}(\frac{c-x_{j}}{h_{1}},\frac{z_{i}-z_{j}}{h_{1}})K\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}\Big{|}z_{j},x_{j}\right)\\\
&=h\int_{z_{i}}\frac{f_{Z}(z_{i})}{f_{X,Z(1)}(c^{+},z_{i})^{2}}(h_{1}^{2})^{-1}K_{1}(\frac{c-x_{j}}{h_{1}},\frac{z_{i}-z_{j}}{h_{1}})M(c,z_{i})dz_{i}\\\
&=\frac{h}{2}\int_{z_{i}}\frac{f_{Z}(z_{i})}{f_{X,Z(1)}(c^{+},z_{i})}(h_{1}^{2})^{-1}K_{1}(\frac{c-x_{j}}{h_{1}},\frac{z_{i}-z_{j}}{h_{1}})(m_{1}(c^{+},z_{i})-\alpha_{1})dz_{i}+O_{p}(h^{2})\\\
&=\frac{h}{2h_{1}}K(\frac{c-x_{j}}{h_{1}})\frac{f_{Z}(z_{j})}{f_{X,Z(1)}(c^{+},z_{j})}(m_{1}(c^{+},z_{j})-\alpha_{1})+O_{p}\left(hh_{1}\right)+O_{p}(h^{2}),\end{split}$
$\begin{split}\text{(Part.BIII)}&=\mathbb{E}\left(\frac{T_{i}f_{Z}(z_{i})}{f_{X,Z(1)}(c^{+},z_{i})}K\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}\Big{|}z_{j},x_{j}\right)=\frac{h}{2}\int_{z_{i}}(m_{1}(c^{+},z_{i})-\alpha_{1})f_{Z}(z_{i})dz_{i}+O_{p}(h^{2})=O_{p}(h^{2}),\end{split}$

where the last step follows from the definition of $\alpha_{1}$. Define
$d_{1}(x_{i},z_{i})=m_{1}(x_{i},z_{i})-\alpha_{1}$. Combining the Part BI, BII
and BIII, we obtain

$\begin{split}\mathbb{E}(\phi_{ij}|j)&=\frac{h}{2}d_{1}(c^{+},z_{j})-T_{j}\frac{h}{h_{1}}K(\frac{c-x_{j}}{h_{1}})\frac{f_{Z}(z_{j})}{f_{X,Z(1)}(c^{+},z_{j})}d_{1}(c^{+},z_{j})+O_{p}(h^{2}+hh_{1}+hh_{2}).\end{split}$

Following the similar calculation, by lemma 11 we have

$\begin{split}\mathbb{E}(\phi_{i,j}|i)&=\underbrace{\frac{T_{i}}{f_{X,Z(1)}(c^{+},z_{i})}\mathbb{E}\left((h_{2})^{-1}K(\frac{z_{i}-z_{j}}{h_{2}})\Big{|}z_{i}\right)K\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}}_{\text{Part.
AI}}\\\
&-\underbrace{2\cdot\frac{T_{i}f_{Z}(z_{i})}{f_{X,Z(1)}(c^{+},z_{i})^{2}}\mathbb{E}\left((h_{1}^{2})^{-1}T_{j}K_{1}(\frac{c-x_{j}}{h_{1}},\frac{z_{i}-z_{j}}{h_{1}})\Big{|}z_{i},x_{i}\right)K\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}}_{\text{Part.
AII}}\\\
&+\underbrace{\frac{T_{i}f_{Z}(z_{i})}{f_{X,Z(1)}(c^{+},z_{i})}K\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}}_{\text{Part.
AIII}}\\\
&=\frac{T_{i}f_{Z}(z_{i})}{f_{X,Z(1)}(c^{+},z_{i})}K\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}+O_{p}(h_{1}+h_{2}^{2}).\end{split}$

Finally, we calculate $\mathbb{E}(\phi_{i,j})$ using Lemma 12,

$\begin{split}\mathbb{E}(\text{
Part.AIII})&=h\int\frac{f_{Z}(z_{i})}{f_{X,Z(1)}(c^{+},z_{i})}M(c,z_{i})dz_{i}\\\
&=\frac{h}{2}\int(m_{1}(c^{+},z_{i})-\alpha_{1})f_{Z}(z_{i})dz_{i}-h^{2}\kappa_{1}\beta_{1}-h^{2}\kappa_{1}\alpha_{1}\int\frac{\partial
f_{X,Z(1)}(c^{+},z_{i})}{\partial
x_{i}}\frac{f_{Z}(z_{i})}{f_{X,Z(1)}(c^{+},z_{i})}dz_{i}\\\
&+h^{2}\kappa_{1}\int\left(\frac{\partial m_{1}(c^{+},z_{i})}{\partial
x_{i}}f_{Z}(z_{i})+m_{1}(c^{+},z_{i})\frac{\partial
f_{X,Z(1)}(c^{+},z_{i})}{\partial
x_{i}}\frac{f_{Z}(z_{i})}{f_{X,Z(1)}(c^{+},z_{i})}\right)dz_{i}+O_{p}(h^{3})\\\
&=-h^{2}\kappa_{1}\alpha_{1}\int\frac{\partial
f_{X,Z(1)}(c^{+},z_{i})}{\partial
x_{i}}\frac{f_{Z}(z_{i})}{f_{X,Z(1)}(c^{+},z_{i})}dz_{i}+O_{p}(h^{3})\end{split}$

where the $O_{p}$ terms are valid uniformly over $i$ and the last equality
follows as

$\alpha_{1}=\int m_{1}(c^{+},z)f_{Z}(z)d{z},$

and

$\beta_{1}=\int\frac{\partial m_{1}(c^{+},z_{i})}{\partial
x_{i}}f_{Z}(z_{i})dz_{i}+\int m_{1}(c^{+},z_{i})\frac{\partial
f_{X,Z(1)}(c^{+},z_{i})}{\partial
x_{i}}\frac{f_{Z}(z_{i})}{f_{X,Z(1)}(c^{+},z_{i})}dz_{i}.$

From Lemma 11, some tedious calculation implies

$\begin{split}\mathbb{E}(\text{Part.AI})&=\mathbb{E}(\text{Part.AIII})+O_{p}(hh_{2}^{2}),\end{split}$

and

$\begin{split}\mathbb{E}(\text{Part.AII})&=\mathbb{E}(\text{Part.AIII})+2h_{1}\kappa_{1}\mathbb{E}\left(\frac{T_{i}f_{Z}(z_{i})}{f_{X,Z(1)}(c^{+},z_{i})^{2}}\frac{\partial
f_{X,Z(1)}(c^{+},z_{i})}{\partial
x_{i}}K\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}\right)+O_{p}(hh_{1}^{2}).\end{split}$

From Lemma 11 and Lemma 12,

$\begin{split}&h_{1}\kappa_{1}\mathbb{E}\left(\frac{T_{i}f_{Z}(z_{i})}{f_{X,Z(1)}(c^{+},z_{i})^{2}}\frac{\partial
f_{X,Z(1)}(c^{+},z_{i})}{\partial
x_{i}}K\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}\right)\\\
&=h_{1}\kappa_{1}\int\int\frac{T_{i}f_{Z}(z_{i})}{f_{X,Z(1)}(c^{+},z_{i})^{2}}\frac{\partial
f_{X,Z(1)}(c^{+},z_{i})}{\partial
x_{i}}K\Big{(}\frac{x_{i}-c}{h}\Big{)}\mathbb{E}(R_{i}|x_{i},z_{i})f_{X,Z(1)}(x_{i},z_{i})dx_{i}dz_{i}\\\
&=\frac{hh_{1}\kappa_{1}}{2}\int\frac{f_{Z}(z_{i})}{f_{X,Z(1)}(c^{+},z_{i})}\frac{\partial
f_{X,Z(1)}(c^{+},z_{i})}{\partial
x_{i}}(m_{1}(c^{+},z_{i})-\alpha_{1})dz_{i}+O_{p}(h_{1}h^{2}).\end{split}$

Thus when $h_{1}\asymp h^{2}$

$\mathbb{E}(\phi_{i,j})=O_{p}(h^{3}).$

This completes the proof. ∎

###  A.2 Proof of Corollary 1

In the case when $Z=Z(1)=Z(0)$, we can estimate
$\frac{1}{\pi_{1}(z)}=\frac{f_{Z}(z)}{f_{X,Z}(c,z)}$ by the following kernel
estimator:

$\begin{split}\frac{1}{\widehat{\pi}_{1}(z)}=\frac{\widehat{f}_{Z}(z)}{\widehat{f}_{X,Z}(c,z)}=\frac{(nh_{2})^{-1}\sum_{i=1}^{n}K(\frac{z-z_{i}}{h_{2}})}{(nh_{1}^{2})^{-1}\sum_{i=1}^{n}K_{1}(\frac{c-x_{i}}{h_{1}},\frac{z-z_{i}}{h_{1}})}.\end{split}$

Similar as in theorem 1,

$\begin{split}&\frac{1}{(nh_{1}^{2})^{-1}\sum_{j=1}^{n}K_{1}(\frac{c-x_{j}}{h_{1}},\frac{z_{i}-z_{j}}{h_{1}})}\\\
&=-\frac{1}{f_{X,Z}(c,z_{i})^{2}}\left((nh_{1}^{2})^{-1}\sum_{j=1}^{n}K_{1}(\frac{c-x_{j}}{h_{1}},\frac{z_{i}-z_{j}}{h_{1}})-f_{X,Z}(c,z_{i})\right)\\\
&+\frac{1}{f_{X,Z}(c,z_{i})}+O_{p}\left(r^{2}\right),\end{split}$

where Lemma 11 implies

$r=\sqrt{\frac{\log n}{nh_{1}^{2}}}+h_{1}^{2}.$

The gradient in (A.4) can be written as the following $U$-statistic:

$\begin{split}&\frac{1}{n}\sum_{i=1}^{n}\frac{T_{i}\widehat{f}_{Z}(z_{i})}{\widehat{f}_{X,Z}(c,z_{i})}K\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}=\frac{1}{n}\sum_{i=1}^{n}\frac{T_{i}(nh_{2})^{-1}\sum_{j=1}^{n}K(\frac{z_{i}-z_{j}}{h_{2}})}{(nh_{1}^{2})^{-1}\sum_{j=1}^{n}K_{1}(\frac{c-x_{j}}{h_{1}},\frac{z_{i}-z_{j}}{h_{1}})}K\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}\\\
&=\frac{1}{n}\sum_{i=1}^{n}\frac{T_{i}f_{Z}(z_{i})}{f_{X,Z}(c,z_{i})}K\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}+\frac{T_{i}}{f_{X,Z}(c,z_{i})}\left((nh_{2})^{-1}\sum_{j=1}^{n}K(\frac{z_{i}-z_{j}}{h_{2}})\right)K\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}\\\
&-\frac{T_{i}f_{Z}(z_{i})}{f_{X,Z}(c,z_{i})^{2}}\left((nh_{1}^{2})^{-1}\sum_{j=1}^{n}K_{1}(\frac{c-x_{j}}{h_{1}},\frac{z_{i}-z_{j}}{h_{1}})\right)K\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}+O_{p}\left(r^{2}+rs\right)\\\
&=\frac{1}{n^{2}}\sum_{i=1}^{n}\sum_{j=1}^{n}\Big{\\{}\frac{T_{i}}{f_{X,Z}(c,z_{i})}(h_{2})^{-1}K(\frac{z_{i}-z_{j}}{h_{2}})K\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}\\\
&-\frac{T_{i}f_{Z}(z_{i})}{f_{X,Z}(c,z_{i})^{2}}(h_{1}^{2})^{-1}K_{1}(\frac{c-x_{j}}{h_{1}},\frac{z_{i}-z_{j}}{h_{1}})K\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}+\frac{T_{i}f_{Z}(z_{i})}{f_{X,Z}(c,z_{i})}K\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}\Big{\\}}+O_{p}\left(r^{2}+rs\right),\end{split}$

where

$s=\sqrt{\frac{\log n}{nh_{2}}}+h_{2}^{2},$

Define

$\begin{split}\phi_{i,j}^{rd}&=\frac{T_{i}}{f_{X,Z}(c,z_{i})}(h_{2})^{-1}K(\frac{z_{i}-z_{j}}{h_{2}})K\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}\\\
&-\frac{T_{i}f_{Z}(z_{i})}{f_{X,Z}(c,z_{i})^{2}}(h_{1}^{2})^{-1}K_{1}(\frac{c-x_{j}}{h_{1}},\frac{z_{i}-z_{j}}{h_{1}})K\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}+\frac{T_{i}f_{Z}(z_{i})}{f_{X,Z}(c,z_{i})}K\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}.\end{split}$

By Lemma 14, we have

$\mathbb{E}(\phi_{i,j}^{rd})=O_{p}(h^{3}).$

The central limit theorem implies

$\frac{1}{(nh)^{1/2}}\sum_{i=1}^{n}\\{\mathbb{E}(\phi_{i,j}^{rd}+\phi_{j,i}^{rd}|i)-\delta^{rd}\\}+O_{p}(\frac{n^{1/2}}{h^{1/2}}(h^{3}+h^{2}h_{1}))\rightarrow_{d}N(0,\xi^{2}/h),$
(A.9)

where $\delta^{rd}=O_{p}(h^{3})$ and
$\xi^{2}=\mathbb{E}\\{(\mathbb{E}(\phi_{i,j}^{rd}+\phi_{j,i}^{rd}|i)-\delta)^{2}\\}=\mathbb{E}\left\\{\mathbb{E}(\phi_{ij}|i)^{2}+\mathbb{E}(\phi_{ji}|i)^{2}+2\mathbb{E}(\phi_{ij}|i)\mathbb{E}(\phi_{ji}|i)\right\\}-\delta^{2}$.
We now calculate the asymptotic variance as follows. Since
$\mathbb{E}(\phi_{j,i}|i)=\mathbb{E}(\phi_{i,j}|j)$, from lemma 13, we have

$\begin{split}\mathbb{E}(\phi_{ij}|j)&=\frac{h}{2}d_{1}(c^{+},z_{j})-T_{j}\frac{h}{h_{1}}K(\frac{c-x_{j}}{h_{1}})\frac{f_{Z}(z_{j})}{f_{X,Z}(c,z_{j})}d_{1}(c^{+},z_{j})+O_{p}(h^{2}+hh_{1}+hh_{2}).\end{split}$

Similarly, we can show that

$\mathbb{E}(\phi_{i,j}|i)=\frac{T_{i}f_{Z}(z_{i})}{f_{X,Z}(c,z_{i})}K\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}+O_{p}(h_{2}^{2}+h_{1}).$

Recall that $\sigma^{2}=\mathbb{E}(Y(1)-m_{1}(X,Z))^{2}$. Since $h\asymp
h_{1}\asymp h_{2}$, after some tedious calculation we can show that

$\begin{split}\frac{1}{n}\mathbb{E}\left(\mathbb{E}(\phi_{ij}|i)^{2}\right)&=\frac{\sigma^{2}}{n}\int_{z_{i}}\int_{x_{i}}\frac{T_{i}f_{Z}(z_{i})^{2}}{f_{X,Z}(c,z_{i})^{2}}K\Big{(}\frac{x_{i}-c}{h}\Big{)}^{2}f_{X,Z}(x_{i},z_{i})dx_{i}dz_{i}\\\
&+\frac{1}{n}\int_{z_{i}}\int_{x_{i}}\frac{T_{i}f_{Z}(z_{i})^{2}}{f_{X,Z}(c,z_{i})^{2}}K\Big{(}\frac{x_{i}-c}{h}\Big{)}^{2}f_{X,Z}(x_{i},z_{i})(m_{1}(x_{i},z_{i})-\alpha_{1})^{2}dx_{i}dz_{i}+O\left(\frac{h^{2}}{n}\right)\\\
&=\frac{\sigma^{2}}{n}h\kappa_{20}\int_{z_{i}}\frac{f_{Z}(z_{i})^{2}}{f_{X,Z}(c,z_{i})}dz_{i}+\frac{h}{n}\kappa_{20}\int_{z_{i}}\frac{f_{Z}(z_{i})^{2}}{f_{X,Z}(c,z_{i})}d_{1}(c^{+},z_{i})^{2}dz_{i}+O\left(\frac{h^{2}}{n}\right)\\\
&=\kappa_{20}\frac{h}{n}\sigma^{2}\mathbb{E}_{Z}\left(\frac{f_{Z}(z_{i})}{f_{X,Z}(c,z_{i})}\right)+\kappa_{20}\frac{h}{n}\mathbb{E}_{Z}\left(\frac{f_{Z}(z_{i})}{f_{X,Z}(c,z_{i})}d_{1}(c^{+},z_{i})^{2}\right)+O\left(\frac{h^{2}}{n}\right),\end{split}$
$\begin{split}\frac{1}{n}\mathbb{E}\left(\mathbb{E}(\phi_{ji}|i)^{2}\right)&=\frac{1}{n}\frac{h^{2}}{h_{1}^{2}}\int_{z_{i}}\int_{x_{i}}T_{i}K(\frac{c-x_{i}}{h_{1}})^{2}\frac{f_{Z}(z_{i})^{2}}{f_{X,Z}(c^{+},z_{i})^{2}}d_{1}(c^{+},z_{i})^{2}f_{X,Z}(x_{i},z_{i})dx_{i}dz_{i}+O\left(\frac{h^{2}}{n}\right)\\\
&=\frac{h^{2}}{h_{1}}\frac{\kappa_{20}}{n}\int_{z_{i}}\frac{f_{Z}(z_{i})^{2}}{f_{X,Z}(c,z_{i})}d_{1}(c^{+},z_{i})^{2}dz_{i}+O\left(\frac{h^{2}}{n}\right)\\\
&=\frac{h^{2}}{h_{1}}\frac{\kappa_{20}}{n}\mathbb{E}_{Z}\left(\frac{f_{Z}(z_{i})}{f_{X,Z}(c,z_{i})}d_{1}(c^{+},z_{i})^{2}\right)+O\left(\frac{h^{2}}{n}\right),\end{split}$
$\begin{split}\frac{2}{n}\mathbb{E}\left(\mathbb{E}(\phi_{ij}|i)\mathbb{E}(\phi_{ji}|i)\right)&=-2\cdot\frac{h}{h_{1}n}\int_{z_{i}}\int_{x_{i}}T_{i}K(\frac{c-x_{j}}{h_{1}})^{2}\frac{f_{Z}(z_{j})^{2}}{f_{X,Z}(c,z_{j})^{2}}d_{1}(c^{+},z_{j})^{2}f_{X,Z}(x_{i},z_{i})dx_{i}dz_{i}+O\left(\frac{h^{2}}{n}\right)\\\
&=-2\cdot\kappa_{20}\frac{h}{n}\int_{z_{i}}\frac{f_{Z}(z_{i})^{2}}{f_{X,Z}(c,z_{i})}d_{1}(c^{+},z_{j})^{2}dz_{i}+O\left(\frac{h^{2}}{n}\right)\\\
&=-2\cdot\kappa_{20}\frac{h}{n}\mathbb{E}_{Z}\left(\frac{f_{Z}(z_{i})}{f_{X,Z}(c|z_{i})}d_{1}(c^{+},z_{i})^{2}\right)+O\left(\frac{h^{2}}{n}\right).\end{split}$

Thus, choosing $h_{1}=h$, we have

$\xi^{2}=h\kappa_{20}\underbrace{\sigma^{2}\mathbb{E}_{Z}\left(\frac{f_{Z}(z_{i})}{f_{X,Z}(c,z_{i})}\right)}_{\omega}+O(h^{2}).$

Combining (A.7) and (A.9),

$\frac{n^{1/2}}{h^{1/2}}\Big{(}\frac{1}{n(n-1)}\sum_{i<j}(\phi_{i,j}+\phi_{j,i})-\delta/2\Big{)}+O\Big{(}\frac{n^{1/2}}{h^{1/2}}(h^{3}+h^{2}h_{1})\Big{)}\rightarrow_{d}N(0,\xi^{2}/h).$

Finally, note that in (A.6),

$\frac{1}{n^{2}}\sum_{i=1}^{n}\phi_{i,i}\lesssim\frac{1}{n}\mathbb{E}(\phi_{i,j})=O_{p}\left(\frac{h^{2}}{n}\right),$

and therefore we obtain that

$\frac{1}{(nh)^{1/2}}\sum_{i=1}^{n}\frac{T_{i}\widehat{f}_{Z}(z)}{\widehat{f}_{X,Z}(c,z)}K\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}+\phi\rightarrow_{d}N(\frac{n^{1/2}\delta}{2h^{1/2}},\xi^{2}/h),$

where

$\phi=O_{p}\Big{(}\frac{n^{1/2}}{h^{1/2}}(h^{3}+r^{2}+rs)\Big{)}.$

Following the similar argument, we can show the joint convergence

$\frac{1}{(nh)^{1/2}}\sum_{i=1}^{n}\frac{T_{i}\widehat{f}_{Z}(z)}{\widehat{f}_{X,Z}(c,z)}K\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}[1,(x_{i}-c)]^{T}\rightarrow_{d}N\left(\frac{n^{1/2}}{h^{1/2}}\left(\begin{matrix}O_{p}(h^{3})\\\
O_{p}(h^{4})\end{matrix}\right),\omega\left(\begin{matrix}\kappa_{20}&h\kappa_{21}\\\
h\kappa_{21}&h^{2}\kappa_{22}\\\ \end{matrix}\right)\right).$

By the least squared formulation, the estimator $\widehat{\alpha}_{1}$
satisfies

$\sqrt{nh}(\widehat{\alpha}_{1}-\alpha_{1})=-e_{1}^{T}\left(\begin{matrix}\widehat{C}_{n}/(nh)&-\widehat{B}_{n}/(nh)\\\
-\widehat{B}_{n}/(nh)&\widehat{A}_{n}/(nh)\end{matrix}\right)^{-1}\frac{1}{(nh)^{1/2}}\sum_{i=1}^{n}\frac{T_{i}\widehat{f}_{Z}(z)}{\widehat{f}_{X,Z(1)}(c,z)}K\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}[1,(x_{i}-c)]^{T},$

where $e_{1}^{T}=(1,0)$. From lemma 10 and the matrix inversion formula,

$\left(\begin{matrix}\widehat{C}_{n}/(nh)&-\widehat{B}_{n}/(nh)\\\
-\widehat{B}_{n}/(nh)&\widehat{A}_{n}/(nh)\end{matrix}\right)^{-1}=\frac{1}{\widehat{D}_{n}/(nh)^{2}}\left(\begin{matrix}\widehat{A}_{n}/(nh)&\widehat{B}_{n}/(nh)\\\
\widehat{B}_{n}/(nh)&\widehat{C}_{n}/(nh)\end{matrix}\right)\rightarrow_{p}\frac{1}{h^{2}(\kappa_{2}/2-\kappa_{1}^{2})}\left(\begin{matrix}h^{2}\kappa_{2}&-h\kappa_{1}\\\
-h\kappa_{1}&\frac{1}{2}\end{matrix}\right).$

Thus, the asymptotic bias of $\sqrt{nh}(\widehat{\alpha}_{1}-\alpha_{1})$ is

$\frac{-e_{1}^{T}}{h^{2}(\kappa_{2}/2-\kappa_{1}^{2})}\left(\begin{matrix}h^{2}\kappa_{2}&-h\kappa_{1}\\\
-h\kappa_{1}&\frac{1}{2}\end{matrix}\right)\frac{n^{1/2}}{h^{1/2}}\left(\begin{matrix}O_{p}(h^{3})\\\
O_{p}(h^{4})\end{matrix}\right)=O(n^{1/2}h^{5/2})=o(1).$

Similarly, the asymptotic variance of
$\sqrt{nh}(\widehat{\alpha}_{1}-\alpha_{1})$ is

$\displaystyle\frac{\omega}{h^{4}(\kappa_{2}/2-\kappa_{1}^{2})^{2}}e_{1}^{T}\left(\begin{matrix}h^{2}\kappa_{2}&-h\kappa_{1}\\\
-h\kappa_{1}&\frac{1}{2}&\end{matrix}\right)\left(\begin{matrix}\kappa_{20}&h\kappa_{21}\\\
h\kappa_{21}&h^{2}\kappa_{22}\\\
\end{matrix}\right)\left(\begin{matrix}h^{2}\kappa_{2}&-h\kappa_{1}\\\
-h\kappa_{1}&\frac{1}{2}\end{matrix}\right)e_{1}$
$\displaystyle=\sigma^{2}\int\frac{f_{Z}(z_{i})^{2}}{f_{X,Z}(c,z_{i})}dz_{i}\cdot
C_{v},$

where

$C_{v}=\frac{\kappa_{2}^{2}\kappa_{20}+\kappa_{1}^{2}\kappa_{22}-2\kappa_{1}\kappa_{2}\kappa_{21}}{\left(\frac{1}{2}\kappa_{2}-\kappa_{1}^{2}\right)^{2}}.$

This completes the proof.

######  Lemma 14.

Recall that

$\begin{split}\phi_{i,j}^{rd}&=\frac{T_{i}}{f_{X,Z}(c,z_{i})}(h_{2})^{-1}K(\frac{z_{i}-z_{j}}{h_{2}})K\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}\\\
&-\frac{T_{i}f_{Z}(z_{i})}{f_{X,Z}(c,z_{i})^{2}}(h_{1}^{2})^{-1}K_{1}(\frac{c-x_{j}}{h_{1}},\frac{z_{i}-z_{j}}{h_{1}})K\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}+\frac{T_{i}f_{Z}(z_{i})}{f_{X,Z}(c,z_{i})}K\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}.\end{split}$

Under the same condition as in Theorem 1, and choose $h_{1}\asymp h$

$\displaystyle\mathbb{E}(\phi_{i,j}^{rd}|j)=\frac{h}{2}d_{1}(c^{+},z_{j})-\frac{h}{2h_{1}}K(\frac{c-x_{j}}{h_{1}})\frac{f_{Z}(z_{j})}{f_{X,Z}(c,z_{j})}d_{1}(c^{+},z_{j})+O_{p}(h^{2}+hh_{1}+hh_{2}),$
$\displaystyle\mathbb{E}(\phi_{i,j}^{rd}|i)=\frac{T_{i}f_{Z}(z_{i})}{f_{X,Z}(c,z_{i})}K\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}+O_{p}(h_{2}^{2}+h_{1}^{2}),$
$\displaystyle\mathbb{E}(\phi_{i,j}^{rd})=O_{p}(h^{3}),$

where $O_{p}$ terms are valid uniformly over $i$ or $j$.

###### Proof.

$\begin{split}\mathbb{E}(\phi_{i,j}^{rd}|j)&=\underbrace{\mathbb{E}\left(\frac{T_{i}}{f_{X,Z}(c,z_{i})}(h_{2})^{-1}K(\frac{z_{i}-z_{j}}{h_{2}})K\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}\Big{|}z_{j}\right)}_{\text{(Part.EI)}}\\\
&-\underbrace{\mathbb{E}\left(\frac{T_{i}f_{Z}(z_{i})}{f_{X,Z}(c,z_{i})^{2}}(h_{1}^{2})^{-1}K_{1}(\frac{c-x_{j}}{h_{1}},\frac{z_{i}-z_{j}}{h_{1}})K\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}\Big{|}z_{j},x_{j}\right)}_{\text{(Part.EII)}}\\\
&+\underbrace{\mathbb{E}\left(\frac{T_{i}f_{Z}(z_{i})}{f_{X,Z}(c,z_{i})}K\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}\Big{|}z_{j},x_{j}\right)}_{\text{(Part.EIII)}}.\end{split}$

From lemma 12,

$\begin{split}\text{(Part.EI)}&=\mathbb{E}\left(\frac{T_{i}}{f_{X,Z}(c,z_{i})}(h_{2})^{-1}K(\frac{z_{i}-z_{j}}{h_{2}})K\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}\Big{|}z_{j}\right)\\\
&=h\int_{z_{i}}\frac{1}{f_{X,Z}(c,z_{i})}(h_{2})^{-1}K(\frac{z_{i}-z_{j}}{h_{2}})M(c,z_{i})dz_{i}\\\
&=h\int_{z_{i}}(h_{2})^{-1}K(\frac{z_{i}-z_{j}}{h_{2}})\frac{1}{2}(m_{1}(c^{+},z_{i})-\alpha_{1})dz_{i}+O_{p}(h^{2})\\\
&=\frac{h}{2}(m_{1}(c^{+},z_{j})-\alpha_{1})+O_{p}(h^{2}+h_{2}h),\end{split}$

where $O_{p}$ terms are valid uniformly over $j$. Similarly, for Part.EII and
Part.EIII, we can show that

$\begin{split}&\text{(Part.EII)}=\mathbb{E}\left(\frac{T_{i}f_{Z}(z_{i})}{f_{X,Z}(c,z_{i})^{2}}(h_{1}^{2})^{-1}K_{1}(\frac{c-x_{j}}{h_{1}},\frac{z_{i}-z_{j}}{h_{1}})K\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}\Big{|}z_{j},x_{j}\right)\\\
&=h\int_{z_{i}}\frac{f_{Z}(z_{i})}{f_{X,Z}(c,z_{i})^{2}}(h_{1}^{2})^{-1}K_{1}(\frac{c-x_{j}}{h_{1}},\frac{z_{i}-z_{j}}{h_{1}})M(c,z_{i})dz_{i}\\\
&=\frac{h}{2}\int_{z_{i}}\frac{f_{Z}(z_{i})}{f_{X,Z}(c,z_{i})}(h_{1}^{2})^{-1}K_{1}(\frac{c-x_{j}}{h_{1}},\frac{z_{i}-z_{j}}{h_{1}})(m_{1}(c^{+},z_{i})-\alpha_{1})dz_{i}+O_{p}(h^{2})\\\
&=\frac{h}{2h_{1}}K(\frac{c-x_{j}}{h_{1}})\frac{f_{Z}(z_{j})}{f_{X,Z}(c,z_{j})}(m_{1}(c^{+},z_{j})-\alpha_{1})+O_{p}\left(hh_{1}\right)+O_{p}(h^{2}),\end{split}$
$\begin{split}\text{(Part.EIII)}&=\mathbb{E}\left(\frac{T_{i}f_{Z}(z_{i})}{f_{X,Z}(c,z_{i})}K\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}\Big{|}z_{j},x_{j}\right)=\frac{h}{2}\int_{z_{i}}(m_{1}(c^{+},z_{i})-\alpha_{1})f_{Z}(z_{i})dz_{i}+O_{p}(h^{2})=O_{p}(h^{2}),\end{split}$

where the last step follows from the definition of $\alpha_{1}$. Define
$d_{1}(x_{i},z_{i})=m_{1}(x_{i},z_{i})-\alpha_{1}$. Combining the Part EI, EII
and EIII, we obtain

$\begin{split}\mathbb{E}(\phi_{ij}^{rd}|j)&=\frac{h}{2}d_{1}(c^{+},z_{j})-\frac{h}{2h_{1}}K(\frac{c-x_{j}}{h_{1}})\frac{f_{Z}(z_{j})}{f_{X,Z}(c,z_{j})}d_{1}(c^{+},z_{j})+O_{p}(h^{2}+hh_{1}+hh_{2}).\end{split}$

Following the similar calculation, by lemma 11 we have

$\begin{split}\mathbb{E}(\phi_{i,j}^{rd}|i)&=\underbrace{\frac{T_{i}}{f_{X,Z}(c,z_{i})}\mathbb{E}\left((h_{2})^{-1}K(\frac{z_{i}-z_{j}}{h_{2}})\Big{|}z_{i}\right)K\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}}_{\text{Part.
FI}}\\\
&-\underbrace{\frac{T_{i}f_{Z}(z_{i})}{f_{X,Z}(c,z_{i})^{2}}\mathbb{E}\left((h_{1}^{2})^{-1}K_{1}(\frac{c-x_{j}}{h_{1}},\frac{z_{i}-z_{j}}{h_{1}})\Big{|}z_{i},x_{i}\right)K\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}}_{\text{Part.
FII}}\\\
&+\underbrace{\frac{T_{i}f_{Z}(z_{i})}{f_{X,Z}(c,z_{i})}K\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}}_{\text{Part.
FIII}}\\\
&=\frac{T_{i}f_{Z}(z_{i})}{f_{X,Z}(c,z_{i})}K\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}+O_{p}(h_{1}^{2}+h_{2}^{2}).\end{split}$

Finally, we calculate $\mathbb{E}(\phi_{i,j}^{rd})$ using Lemma 12,

$\begin{split}\mathbb{E}(\text{
Part.FIII})&=h\int\frac{f_{Z}(z_{i})}{f_{X,Z}(c,z_{i})}M(c,z_{i})dz_{i}\\\
&=\frac{h}{2}\int(m_{1}(c^{+},z_{i})-\alpha_{1})f_{Z}(z_{i})dz_{i}-h^{2}\kappa_{1}\beta_{1}-h^{2}\kappa_{1}\alpha_{1}\int\frac{\partial
f_{X,Z}(c,z_{i})}{\partial
x_{i}}\frac{f_{Z}(z_{i})}{f_{X,Z}(c,z_{i})}dz_{i}\\\
&+h^{2}\kappa_{1}\int\left(\frac{\partial m_{1}(c^{+},z_{i})}{\partial
x_{i}}f_{Z}(z_{i})+m_{1}(c^{+},z_{i})\frac{\partial f_{X,Z}(c,z_{i})}{\partial
x_{i}}\frac{f_{Z}(z_{i})}{f_{X,Z}(c,z_{i})}\right)dz_{i}+O_{p}(h^{3})\\\
&=-h^{2}\kappa_{1}\alpha_{1}\int\frac{\partial f_{X,Z}(c,z_{i})}{\partial
x_{i}}\frac{f_{Z}(z_{i})}{f_{X,Z}(c,z_{i})}dz_{i}+O_{p}(h^{3})\end{split}$

where the $O_{p}$ terms are valid uniformly over $i$ and the last equality
follows as

$\alpha_{1}=\int m_{1}(c^{+},z)f_{Z}(z)d{z},$

and

$\beta_{1}=\int\frac{\partial m_{1}(c^{+},z_{i})}{\partial
x_{i}}f_{Z}(z_{i})dz_{i}+\int m_{1}(c^{+},z_{i})\frac{\partial
f_{X,Z}(c,z_{i})}{\partial x_{i}}\frac{f_{Z}(z_{i})}{f_{X,Z}(c,z_{i})}dz_{i}.$

From Lemma 11, some tedious calculation implies

$\begin{split}\mathbb{E}(\text{Part.FI})&=\mathbb{E}(\text{Part.FIII})+O_{p}(hh_{2}^{2}),\end{split}$

and

$\begin{split}\mathbb{E}(\text{Part.FII})&=\mathbb{E}(\text{Part.FIII})+O_{p}(hh_{1}^{2}).\end{split}$

Thus when $h_{1}\asymp h$

$\mathbb{E}(\phi_{i,j})=O_{p}(h^{3}).$

This completes the proof. ∎

###  A.3 Proof of Theorem 2

###### Proof.

Assume the treatment assignment function for individual $i$ is $T_{i}(X_{i})$
and only depends on $X_{i}$ directly. The assignment function is not known to
the individual but they are aware of a jump in $T_{i}(X_{i})$ if $X_{i}>c$.
Thus a post selection may exist and we can write

$Z_{i}=(X_{i}>c)Z_{i}(1)+(X_{i}\leq c)Z_{i}(0)$

$\begin{split}\mathbb{E}(T_{i}Y_{i}|X_{i}=c^{+},Z_{i}=z)&=\mathbb{E}(T_{i}Y_{i}|X_{i}=c^{+},Z_{i}=z,\tau_{\epsilon}=co)\mathbb{P}(\tau_{\epsilon}=co|X_{i}=c^{+},Z_{i}=z)\\\
&+\mathbb{E}(T_{i}Y_{i}|X_{i}=c^{+},Z_{i}=z,\tau_{\epsilon}=at)\mathbb{P}(\tau_{\epsilon}=at|X_{i}=c^{+},Z_{i}=z)\\\
&+\mathbb{E}(T_{i}Y_{i}|X_{i}=c^{+},Z_{i}=z,\tau_{\epsilon}=nt)\mathbb{P}(\tau_{\epsilon}=nt|X_{i}=c^{+},Z_{i}=z)\\\
\end{split}$

$\begin{split}\mathbb{E}(T_{i}Y_{i}|X_{i}=c^{-},Z_{i}=z)&=\mathbb{E}(T_{i}Y_{i}|X_{i}=c^{-},Z_{i}=z,\tau_{\epsilon}=co)\mathbb{P}(\tau_{\epsilon}=co|X_{i}=c^{-},Z_{i}=z)\\\
&+\mathbb{E}(T_{i}Y_{i}|X_{i}=c^{-},Z_{i}=z,\tau_{\epsilon}=at)\mathbb{P}(\tau_{\epsilon}=at|X_{i}=c^{-},Z_{i}=z)\\\
&+\mathbb{E}(T_{i}Y_{i}|X_{i}=c^{-},Z_{i}=z,\tau_{\epsilon}=nt)\mathbb{P}(\tau_{\epsilon}=nt|X_{i}=c^{-},Z_{i}=z)\\\
\end{split}$

Assume
$\mathbb{P}(\tau_{\epsilon}|X_{i}=c^{+},Z_{i}(1)=z)=\mathbb{P}(\tau_{\epsilon}|X_{i}=c^{-},Z_{i}(0)=z)$

$\begin{split}&\mathbb{E}(T_{i}Y_{i}|X_{i}=c^{+},Z_{i}=z)-\mathbb{E}(T_{i}Y_{i}|X_{i}=c^{-},Z_{i}=z)\\\
&=\Big{(}\mathbb{E}(T_{i}Y_{i}|X_{i}=c^{+},Z_{i}=z,\tau_{\epsilon}=co)-\mathbb{E}(T_{i}Y_{i}|X_{i}=c^{-},Z_{i}=z,\tau_{\epsilon}=co)\Big{)}\mathbb{P}(\tau_{\epsilon}=co|X_{i}=c,Z_{i}=z)\\\
&+\Big{(}\mathbb{E}(T_{i}Y_{i}|X_{i}=c^{+},Z_{i}=z,\tau_{\epsilon}=at)-\mathbb{E}(T_{i}Y_{i}|X_{i}=c^{-},Z_{i}=z,\tau_{\epsilon}=at)\Big{)}\mathbb{P}(\tau_{\epsilon}=at|X_{i}=c,Z_{i}=z)\\\
&+\Big{(}\mathbb{E}(T_{i}Y_{i}|X_{i}=c^{+},Z_{i}=z,\tau_{\epsilon}=nt)-\mathbb{E}(T_{i}Y_{i}|X_{i}=c^{-},Z_{i}=z,\tau_{\epsilon}=nt)\Big{)}\mathbb{P}(\tau_{\epsilon}=nt|X_{i}=c,Z_{i}=z)\\\
&=\mathbb{E}(Y_{i}(1)|Z_{i}(1)=z,\tau_{\epsilon}=co)\cdot\mathbb{P}(\tau_{\epsilon}=co|X_{i}=c,Z_{i}=z)\\\
&+\Big{(}\mathbb{E}(Y_{i}(1)|Z_{i}(1)=z,\tau_{\epsilon}=at)-\mathbb{E}(Y_{i}(1)|Z_{i}(0)=z,\tau_{\epsilon}=at)\Big{)}\mathbb{P}(\tau_{\epsilon}=at|X_{i}=c,Z_{i}=z)\end{split}$

Similarly

$\begin{split}&\mathbb{E}((1-T_{i})Y_{i}|X_{i}=c^{+},Z_{i}=z)-\mathbb{E}((1-T_{i})Y_{i}|X_{i}=c^{-},Z_{i}=z)\\\
&=\Big{(}\mathbb{E}((1-T_{i})Y_{i}|X_{i}=c^{+},Z_{i}=z,\tau_{\epsilon}=co)-\mathbb{E}((1-T_{i})Y_{i}|X_{i}=c^{-},Z_{i}=z,\tau_{\epsilon}=co)\Big{)}\mathbb{P}(\tau_{\epsilon}=co|X_{i}=c,Z_{i}=z)\\\
&+\Big{(}\mathbb{E}((1-T_{i})Y_{i}|X_{i}=c^{+},Z_{i}=z,\tau_{\epsilon}=at)-\mathbb{E}((1-T_{i})Y_{i}|X_{i}=c^{-},Z_{i}=z,\tau_{\epsilon}=at)\Big{)}\mathbb{P}(\tau_{\epsilon}=at|X_{i}=c,Z_{i}=z)\\\
&+\Big{(}\mathbb{E}((1-T_{i})Y_{i}|X_{i}=c^{+},Z_{i}=z,\tau_{\epsilon}=nt)-\mathbb{E}((1-T_{i})Y_{i}|X_{i}=c^{-},Z_{i}=z,\tau_{\epsilon}=nt)\Big{)}\mathbb{P}(\tau_{\epsilon}=nt|X_{i}=c,Z_{i}=z)\\\
&=-\mathbb{E}(Y_{i}(0)|Z_{i}(0)=z,\tau_{\epsilon}=co)\cdot\mathbb{P}(\tau_{\epsilon}=co|X_{i}=c,Z_{i}=z)\\\
&+\Big{(}\mathbb{E}(Y_{i}(0)|Z_{i}(1)=z,\tau_{\epsilon}=nt)-\mathbb{E}(Y_{i}(0)|Z_{i}(0)=z,\tau_{\epsilon}=nt)\Big{)}\mathbb{P}(\tau_{\epsilon}=nt|X_{i}=c,Z_{i}=z)\\\
\end{split}$

Assume
$\mathbb{E}(Y_{i}(1)|Z_{i}(c^{+})=z,\tau_{\epsilon}=at)=\mathbb{E}(Y_{i}(1)|Z_{i}(c^{-})=z,\tau_{\epsilon}=at)$
and
$\mathbb{E}(Y_{i}(0)|Z_{i}(c^{+})=z,\tau_{\epsilon}=nt)=\mathbb{E}(Y_{i}(0)|Z_{i}(c^{-})=z,\tau_{\epsilon}=nt)$,
then

$\begin{split}&\mathbb{E}(Y_{i}|X_{i}=c^{+},Z_{i}=z)-\mathbb{E}(Y_{i}|X_{i}=c^{-},Z_{i}=z)\\\
&=\underbrace{\Big{(}\mathbb{E}(Y_{i}(1)|Z_{i}(1)=z,\tau_{\epsilon}=co)-\mathbb{E}(Y_{i}(0)|Z_{i}(0)=z,\tau_{\epsilon}=co)\Big{)}}_{:=\Delta}\cdot\mathbb{P}(\tau_{\epsilon}=co|X_{i}=c,Z_{i}=z)\end{split}$
(A.10)

Apply the same decomposition to $T_{i}$

$\begin{split}&\mathbb{E}(T_{i}|X_{i}=c^{+},Z_{i}=z)-\mathbb{E}(T_{i}|X_{i}=c^{-},Z_{i}=z)\\\
&=\Big{(}\mathbb{E}(T_{i}|X_{i}=c^{+},Z_{i}=z,\tau_{\epsilon}=co)-\mathbb{E}(T_{i}|X_{i}=c^{-},Z_{i}=z,\tau_{\epsilon}=co)\Big{)}\mathbb{P}(\tau_{\epsilon}=co|X_{i}=c,Z_{i}=z)\\\
&+\Big{(}\mathbb{E}(T_{i}|X_{i}=c^{+},Z_{i}=z,\tau_{\epsilon}=at)-\mathbb{E}(T_{i}|X_{i}=c^{-},Z_{i}=z,\tau_{\epsilon}=at)\Big{)}\mathbb{P}(\tau_{\epsilon}=at|X_{i}=c,Z_{i}=z)\\\
&+\Big{(}\mathbb{E}(T_{i}|X_{i}=c^{+},Z_{i}=z,\tau_{\epsilon}=nt)-\mathbb{E}(T_{i}|X_{i}=c^{-},Z_{i}=z,\tau_{\epsilon}=nt)\Big{)}\mathbb{P}(\tau_{\epsilon}=nt|X_{i}=c,Z_{i}=z)\\\
&=\mathbb{P}(\tau_{\epsilon}=co|X_{i}=c,Z_{i}=z)\\\ \end{split}$ (A.11)

First notice that when integrating both sides with $f_{Z(0)|X}(z|c)$,

$\displaystyle\mathbb{E}_{Z(0)}\left(\mathbb{E}(Y_{i}|X_{i}=c^{+},Z_{i}=z)-\mathbb{E}(Y_{i}|X_{i}=c^{-},Z_{i}=z)\Big{|}X_{i}=c^{-}\right)$
$\displaystyle=\int\Delta\cdot\mathbb{P}(\tau_{\epsilon}=co|X_{i}=c,Z_{i}=z)f_{Z(0)|X}(z|c^{-})dz$
$\displaystyle=\mathbb{P}(\tau_{\epsilon}=co|X_{i}=c)\int\Delta\cdot
f_{Z(0)|X,\tau_{\epsilon}=co}(z|c^{-})dz$
$\displaystyle=\mathbb{P}(\tau_{\epsilon}=co|X_{i}=c)\cdot\mathbb{E}_{Z(0)}\Big{(}\mathbb{E}(Y_{i}(1)|Z_{i}(c^{+})=z,\tau_{\epsilon}=co)-\mathbb{E}(Y_{i}(0)|Z_{i}(c^{-})=z,\tau_{\epsilon}=co)\Big{|}X_{i}=c^{-},\tau_{\epsilon}=co\Big{)}$

And devide (A.10) with (A.11), we obtain

$\displaystyle\mathbb{E}_{Z(0)}\Big{(}\mathbb{E}(Y_{i}(1)|Z_{i}(c^{+})=z,\tau_{\epsilon}=co)-\mathbb{E}(Y_{i}(0)|Z_{i}(c^{-})=z,\tau_{\epsilon}=co)\Big{|}X_{i}=c^{-},\tau_{\epsilon}=co\Big{)}$
$\displaystyle=\frac{\mathbb{E}_{Z(0)}\left(\mathbb{E}(Y_{i}|X_{i}=c^{+},Z_{i}=z)-\mathbb{E}(Y_{i}|X_{i}=c^{-},Z_{i}=z)\Big{|}X_{i}=c^{-}\right)}{\mathbb{E}_{Z(0)}\left(\mathbb{E}(T_{i}|X_{i}=c^{+},Z_{i}=z)-\mathbb{E}(T_{i}|X_{i}=c^{-},Z_{i}=z)\Big{|}X_{i}=c^{-}\right)}$

Similarly, when integrating both sides with
$\frac{(f_{Z(0)|X}(z_{i},c^{-})+f_{Z(1)|X}(z_{i},c^{+}))}{2}$

$\displaystyle\int\big{(}\mathbb{E}(Y_{i}|X_{i}=c^{+},Z_{i}=z)-\mathbb{E}(Y_{i}|X_{i}=c^{-},Z_{i}=z)\Big{)}\frac{f_{Z(0)|X}(z_{i}|c^{-})+f_{Z(1)|X}(z_{i}|c^{+})}{2}dz$
$\displaystyle=\int\Delta\cdot\mathbb{P}(\tau_{\epsilon}=co|X_{i}=c,Z_{i}=z)\frac{f_{Z(0)|X}(z_{i}|c^{-})+f_{Z(1)|X}(z_{i}|c^{+})}{2}dz$
$\displaystyle=\mathbb{P}(\tau_{\epsilon}=co|X_{i}=c)\int\Delta\cdot\frac{f_{Z(0)|X,\tau_{\epsilon}=co}(z_{i}|c^{-})+f_{Z(1)|X,\tau_{\epsilon}=co}(z_{i}|c^{+})}{2}dz$

And we obtain the estimand

$\displaystyle\lim_{\epsilon\rightarrow
0}\mathbb{E}\Big{(}\mathbb{E}(Y_{i}(1)|Z_{i}(1)=z,\tau_{\epsilon}=co)-\mathbb{E}(Y_{i}(0)|Z_{i}(0)=z,\tau_{\epsilon}=co)\Big{|}X\in\mathcal{N}_{\epsilon},\tau_{\epsilon}=co\Big{)}$
$\displaystyle=\frac{\int\left(\mathbb{E}(Y_{i}|X_{i}=c^{+},Z_{i}=z)-\mathbb{E}(Y_{i}|X_{i}=c^{-},Z_{i}=z)\right)(f_{Z(0)|X}(z_{i}|c^{-})+f_{Z(1)|X}(z_{i}|c^{+}))dz}{\int\left(\mathbb{E}(T_{i}|X_{i}=c^{+},Z_{i}=z)-\mathbb{E}(T_{i}|X_{i}=c^{-},Z_{i}=z)\right)(f_{Z(0)|X}(z_{i}|c^{-})+f_{Z(1)|X}(z_{i}|c^{+}))dz}$

Lastly under CIA, one can show that

$\begin{split}&\mathbb{E}(Y_{i}|X_{i}=c^{+},Z_{i}=z)-\mathbb{E}(Y_{i}|X_{i}=c^{-},Z_{i}=z)\\\
&=\underbrace{\Big{(}\mathbb{E}(Y_{i}(1)|Z_{i}(1)=z,\tau_{\epsilon}=co)-\mathbb{E}(Y_{i}(0)|Z_{i}(0)=z,\tau_{\epsilon}=co)\Big{)}}_{:=\Delta}\cdot\mathbb{P}(\tau_{\epsilon}=co|Z_{i}=z)\end{split}$
(A.12)

and

$\begin{split}\mathbb{E}(T_{i}|X_{i}=c^{+},Z_{i}=z)-\mathbb{E}(T_{i}|X_{i}=c^{-},Z_{i}=z)=\mathbb{P}(\tau_{\epsilon}=co|Z_{i}=z)\\\
\end{split}$ (A.13)

when we integrating with both sides with $f_{Z}(z_{i})$,

$\displaystyle\int\big{(}\mathbb{E}(Y_{i}|X_{i}=c^{+},Z_{i}=z)-\mathbb{E}(Y_{i}|X_{i}=c^{-},Z_{i}=z)\Big{)}f_{Z}(z_{i})dz$
$\displaystyle=\int\Delta\cdot\mathbb{P}(\tau_{\epsilon}=co|Z_{i}=z)f_{Z}(z_{i})dz$
$\displaystyle=\mathbb{P}(\tau_{\epsilon}=co)\int\Delta\cdot f_{Z}(z_{i})dz$

Thus

$\displaystyle\mathbb{E}_{Z}\Big{(}\mathbb{E}(Y_{i}(1)|Z_{i}(1)=z,\tau_{\epsilon}=co)-\mathbb{E}(Y_{i}(0)|Z_{i}(0)=z,\tau_{\epsilon}=co)\Big{)}$
$\displaystyle=\frac{\mathbb{E}_{Z}\left(\mathbb{E}(Y_{i}|X_{i}=c^{+},Z_{i}=z)-\mathbb{E}(Y_{i}|X_{i}=c^{-},Z_{i}=z)\right)}{\mathbb{E}_{Z}\left(\mathbb{E}(T_{i}|X_{i}=c^{+},Z_{i}=z)-\mathbb{E}(T_{i}|X_{i}=c^{-},Z_{i}=z)\right)}$

∎

######  Lemma 15.

When either $h\asymp h_{1}\asymp h_{2}$ or $h\asymp\sqrt{h_{1}}\asymp h_{2}$,
the condition in Theorem 12.3 in Van der Vaart (2000) holds, such that
$\mathbb{E}(\phi_{ij}^{2})<\infty$

###### Proof.

$\mathbb{E}(\phi_{ij}^{2})$ is a linear combination of the following
quantities:

$\displaystyle\mathbb{E}\left(\frac{1}{f_{X,Z(1)}(c^{+},z_{i})^{2}}(h_{2})^{-2}K^{2}(\frac{z_{i}-z_{j}}{h_{2}})K^{2}\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}^{2}\right)$
(A.14)
$\displaystyle\mathbb{E}\left(\frac{f_{Z}(z_{i})^{2}}{f_{X,Z(1)}(c^{+},z_{i})^{4}}(h_{1})^{-4}K_{1}^{2}(\frac{c-x_{j}}{h_{1}},\frac{z_{i}-z_{j}}{h_{1}})K^{2}\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}^{2}\right)$
(A.15)
$\displaystyle\mathbb{E}\left(\frac{f_{Z}^{2}(z_{i})}{f_{X,Z(1)}^{2}(c^{+},z_{i})}K^{2}\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}^{2}\right)$
(A.16)
$\displaystyle\mathbb{E}\left(\frac{f_{Z}(z_{i})}{f_{X,Z(1)}^{3}(c^{+},z_{i})}(h_{2})^{-1}(h_{1})^{-2}K_{1}(\frac{c-x_{j}}{h_{1}},\frac{z_{i}-z_{j}}{h_{1}})K(\frac{z_{i}-z_{j}}{h_{2}})K^{2}\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}^{2}\right)$
(A.17)
$\displaystyle\mathbb{E}\left(\frac{f_{Z}(z_{i})}{f_{X,Z(1)}^{2}(c^{+},z_{i})}(h_{2})^{-1}K(\frac{z_{i}-z_{j}}{h_{2}})K^{2}\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}^{2}\right)$
(A.18)
$\displaystyle\mathbb{E}\left(\frac{f_{Z}^{2}(z_{i})}{f_{X,Z(1)}(c^{+},z_{i})^{3}}(h_{1}^{2})^{-1}K_{1}(\frac{c-x_{j}}{h_{1}},\frac{z_{i}-z_{j}}{h_{1}})K^{2}\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}^{2}\right)$
(A.19)

  * •

From Lemma 16, equation A.14 can be written as

$\displaystyle\mathbb{E}\left(\frac{1}{f_{X,Z(1)}(c^{+},z_{i})^{2}}(h_{2})^{-2}K^{2}(\frac{z_{i}-z_{j}}{h_{2}})K^{2}\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}^{2}\Big{|}z_{j}\right)$
$\displaystyle=\int_{z_{i}}\frac{1}{f_{X,Z(1)}(c^{+},z_{i})^{2}}(h_{2})^{-1}K^{2}(\frac{z_{i}-z_{j}}{h_{2}})J(x_{i},z_{i})dz_{i}\qquad\text{($z_{j}\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}(x_{i},z_{i})$)}$
$\displaystyle=\kappa_{20}\int_{z_{i}}\frac{1}{f_{X,Z(1)}(c^{+},z_{i})}(h_{2})^{-1}K^{2}(\frac{z_{i}-z_{j}}{h_{2}})(\sigma^{2}+d_{1}(c^{+},z_{i})^{2})dz_{i}+O_{p}(h)$
$\displaystyle=\frac{\kappa_{20}^{2}}{f_{X,Z(1)}(c^{+},z_{j})}(\sigma^{2}+d_{1}(c^{+},z_{j})^{2})+O_{p}(h)$

Thus

$\displaystyle\mathbb{E}\left(\frac{1}{f_{X,Z(1)}(c^{+},z_{i})^{2}}(h_{2})^{-2}K^{2}(\frac{z_{i}-z_{j}}{h_{2}})K^{2}\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}^{2}\right)$
$\displaystyle=\kappa_{20}^{2}\left(\sigma^{2}\int\frac{f_{Z}(z_{j})}{f_{X,Z(1)}(c^{+},z_{j})}dz_{j}+\int
d_{1}(c^{+},z_{j})^{2}\frac{f_{Z}(z_{j})}{f_{X,Z(1)}(c^{+},z_{j})}dz_{j}\right)+O_{p}(h)<\infty$

  * •

When $h\asymp h_{1}\asymp h_{2}$, by law of iterated expectation and Lemma 16,
equation A.15 can be written as

$\displaystyle\mathbb{E}\left(\frac{f_{Z}(z_{i})^{2}}{f_{X,Z(1)}(c^{+},z_{i})^{4}}(h_{1})^{-4}K_{1}^{2}(\frac{c-x_{j}}{h_{1}},\frac{z_{i}-z_{j}}{h_{1}})K^{2}\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}^{2}\Big{|}x_{j},z_{j}\right)$
$\displaystyle=\int_{z_{i}}\frac{f_{Z}(z_{i})^{2}}{f_{X,Z(1)}(c^{+},z_{i})^{4}}(h_{1})^{-3}K_{1}^{2}(\frac{c-x_{j}}{h_{1}},\frac{z_{i}-z_{j}}{h_{1}})J(c,z_{i})dz_{i}$
$\displaystyle=\kappa_{20}\int_{z_{i}}\frac{f_{Z}(z_{i})^{2}}{f_{X,Z(1)}(c^{+},z_{i})^{3}}(h_{1})^{-3}K_{1}^{2}(\frac{c-x_{j}}{h_{1}},\frac{z_{i}-z_{j}}{h_{1}})(\sigma^{2}+d_{1}(c^{+},z_{i})^{2})dz_{i}+O_{p}(h)$
$\displaystyle=\kappa_{20}^{2}\frac{f_{Z}(z_{j})^{2}}{f_{X,Z(1)}(c^{+},z_{j})^{3}}(\sigma^{2}+d_{1}(c^{+},z_{j})^{2})(h_{1})^{-1}K^{2}(\frac{c-x_{j}}{h_{1}})+O_{p}(h)$

Thus

$\displaystyle\mathbb{E}\left(\frac{f_{Z}(z_{i})^{2}}{f_{X,Z(1)}(c^{+},z_{i})^{4}}(h_{1})^{-4}K_{1}^{2}(\frac{c-x_{j}}{h_{1}},\frac{z_{i}-z_{j}}{h_{1}})K^{2}\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}^{2}\right)$
$\displaystyle=\kappa_{20}^{2}\int\int\frac{f_{Z}(z_{j})^{2}}{f_{X,Z(1)}(c^{+},z_{j})^{3}}(\sigma^{2}+d_{1}(c^{+},z_{j})^{2})(h_{1})^{-1}K^{2}(\frac{c-x_{j}}{h_{1}})f_{X,Z(1)}(x_{j},z_{j})dz_{j}dx_{j}+O_{p}(h)$
$\displaystyle=\kappa_{20}^{3}\int\frac{f_{Z}(z_{j})^{2}}{f_{X,Z(1)}(c^{+},z_{j})^{2}}(\sigma^{2}+d_{1}(c^{+},z_{j})^{2})dz_{j}+O_{p}(h)$
$\displaystyle=\kappa_{20}^{3}\left(\sigma^{2}\int\frac{f_{Z}(z_{j})^{2}}{f_{X,Z(1)}(c^{+},z_{j})^{2}}dz_{j}+\int
d_{1}(c^{+},z_{j})^{2}\frac{f_{Z}(z_{j})^{2}}{f_{X,Z(1)}(c^{+},z_{j})^{2}}dz_{j}\right)+O_{p}(h)<\infty$

When $h\asymp\sqrt{h_{1}}\asymp h_{2}$

$\displaystyle\mathbb{E}\left(\frac{f_{Z}(z_{i})^{2}}{f_{X,Z(1)}(c^{+},z_{i})^{4}}(h_{1})^{-4}K_{1}^{2}(\frac{c-x_{j}}{h_{1}},\frac{z_{i}-z_{j}}{h_{1}})K^{2}\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}^{2}\Big{|}x_{j},z_{j}\right)$
$\displaystyle=\int_{z_{i}}\frac{f_{Z}(z_{i})^{2}}{f_{X,Z(1)}(c^{+},z_{i})^{4}}(h_{1})^{-2}K_{1}^{2}(\frac{c-x_{j}}{h_{1}},\frac{z_{i}-z_{j}}{h_{1}})J(c,z_{i})dz_{i}$
$\displaystyle=\kappa_{20}\int_{z_{i}}\frac{f_{Z}(z_{i})^{2}}{f_{X,Z(1)}(c^{+},z_{i})^{3}}(h_{1})^{-2}K_{1}^{2}(\frac{c-x_{j}}{h_{1}},\frac{z_{i}-z_{j}}{h_{1}})(\sigma^{2}+d_{1}(c^{+},z_{i})^{2})dz_{i}+O_{p}(h)$
$\displaystyle=\kappa_{20}^{2}\frac{f_{Z}(z_{j})^{2}}{f_{X,Z(1)}(c^{+},z_{j})^{3}}(\sigma^{2}+d_{1}(c^{+},z_{j})^{2})K^{2}(\frac{c-x_{j}}{h_{1}})+O_{p}(h)$

Thus

$\displaystyle\mathbb{E}\left(\frac{f_{Z}(z_{i})^{2}}{f_{X,Z(1)}(c^{+},z_{i})^{4}}(h_{1})^{-4}K_{1}^{2}(\frac{c-x_{j}}{h_{1}},\frac{z_{i}-z_{j}}{h_{1}})K^{2}\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}^{2}\right)$
$\displaystyle=\kappa_{20}^{2}\int\int\frac{f_{Z}(z_{j})^{2}}{f_{X,Z(1)}(c^{+},z_{j})^{3}}(\sigma^{2}+d_{1}(c^{+},z_{j})^{2})K^{2}(\frac{c-x_{j}}{h_{1}})f_{X,Z(1)}(x_{j},z_{j})+O_{p}(h)$
$\displaystyle=O_{p}(h)<\infty$

  * •

From Lemma 16, equation A.16 can be written as

$\displaystyle\mathbb{E}\left(\frac{f_{Z}(z_{i})^{2}}{f_{X,Z(1)^{2}}(c^{+},z_{i})}K^{2}\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}^{2}\right)$
$\displaystyle=h\int_{z_{i}}\frac{f_{Z}(z_{i})^{2}}{f_{X,Z(1)}(c,z_{i})^{2}}J(c,z_{i})dz_{i}$
$\displaystyle=h\kappa_{20}\int_{z_{i}}\frac{f_{Z}(z_{i})^{2}}{f_{X,Z(1)}(c,z_{i})}(\sigma^{2}+d_{1}(c^{+},z_{j})^{2})dz_{i}+O_{p}(h^{2})$
$\displaystyle=O_{p}(h)<\infty$

  * •

From Lemma 16, and using the fact that the kernel is bounded by a constant
$\mathcal{K}$, equation A.17 can be written as

$\displaystyle\mathbb{E}\left(\frac{f_{Z}(z_{i})}{f_{X,Z(1)}(c^{+},z_{i})^{3}}(h_{2})^{-1}(h_{1})^{-2}K_{1}(\frac{c-x_{j}}{h_{1}},\frac{z_{i}-z_{j}}{h_{1}})K(\frac{z_{i}-z_{j}}{h_{2}})K^{2}\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}^{2}\Big{|}x_{j},z_{j}\right)$
$\displaystyle=\int_{z_{i}}\frac{f_{Z}(z_{i})}{f_{X,Z(1)}(c^{+},z_{i})^{3}}(h_{1})^{-2}K_{1}(\frac{c-x_{j}}{h_{1}},\frac{z_{i}-z_{j}}{h_{1}})K(\frac{z_{i}-z_{j}}{h_{2}})J(c,z_{j})dz_{i}$
$\displaystyle=\kappa_{20}K(\frac{c-x_{j}}{h_{1}})\int_{z_{i}}\frac{f_{Z}(z_{i})}{f_{X,Z(1)}(c^{+},z_{i})^{2}}(h_{1})^{-2}K(\frac{z_{i}-z_{j}}{h_{1}})K(\frac{z_{i}-z_{j}}{h_{2}})(\sigma^{2}+d_{1}(c^{+},z_{i})^{2})dz_{i}+O_{p}(h)$
$\displaystyle\leq\mathcal{K}\kappa_{20}K(\frac{c-x_{j}}{h_{1}})(h_{1})^{-1}\frac{f_{Z}(z_{j})}{f_{X,Z(1)}(c^{+},z_{j})^{2}}(\sigma^{2}+d_{1}(c^{+},z_{j})^{2})+O_{p}(h)$

Thus

$\displaystyle\mathbb{E}\left(\frac{f_{Z}(z_{i})}{f_{X,Z(1)}(c^{+},z_{i})^{3}}(h_{2})^{-1}(h_{1})^{-2}K_{1}(\frac{c-x_{j}}{h_{1}},\frac{z_{i}-z_{j}}{h_{1}})K(\frac{z_{i}-z_{j}}{h_{2}})K^{2}\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}^{2}\right)$
$\displaystyle\leq\mathcal{K}\kappa_{20}\int\int
K(\frac{c-x_{j}}{h_{1}})(h_{1})^{-1}\frac{f_{Z}(z_{j})}{f_{X,Z(1)}(c^{+},z_{j})^{2}}(\sigma^{2}+d_{1}(c^{+},z_{j})^{2})f_{X,Z(1)}(x_{j},z_{j})dz_{j}dx_{j}+O_{p}(h)$
$\displaystyle=\mathcal{K}\kappa_{20}\int\frac{f_{Z}(z_{j})}{f_{X,Z(1)}(c^{+},z_{j})}(\sigma^{2}+d_{1}(c^{+},z_{j})^{2})dz_{j}+O_{p}(h)$
$\displaystyle=\mathcal{K}\kappa_{20}\left(\sigma^{2}\int\frac{f_{Z}(z_{j})}{f_{X,Z(1)}(c^{+},z_{j})}dz_{j}+\int
d_{1}(c^{+},z_{j})^{2}\frac{f_{Z}(z_{j})}{f_{X,Z(1)}(c^{+},z_{j})}dz_{j}\right)+O_{p}(h)<\infty$

  * •

From Lemma 16, equation A.18 can be written as

$\displaystyle\mathbb{E}\left(\frac{f_{Z}(z_{i})}{f_{X,Z(1)}(c^{+},z_{i})^{2}}(h_{2})^{-1}K(\frac{z_{i}-z_{j}}{h_{2}})K^{2}\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}^{2}\Big{|}x_{j},z_{j}\right)$
$\displaystyle=\int\frac{f_{Z}(z_{i})}{f_{X,Z(1)}(c^{+},z_{i})^{2}}K(\frac{z_{i}-z_{j}}{h_{2}})J(c,z_{i})dz_{i}$
$\displaystyle=\kappa_{20}\int\frac{f_{Z}(z_{i})}{f_{X,Z(1)}(c^{+},z_{i})}K(\frac{z_{i}-z_{j}}{h_{2}})(\sigma^{2}+d_{1}(c^{+},z_{i})^{2})dz_{i}+O_{p}(h)$
$\displaystyle=O_{p}(h)$

Thus

$\displaystyle\mathbb{E}\left(\frac{f_{Z}(z_{i})}{f_{X,Z(1)}(c^{+},z_{i})^{2}}(h_{2})^{-1}K(\frac{z_{i}-z_{j}}{h_{2}})K^{2}\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}^{2}\right)<\infty$

  * •

From Lemma 16, equation A.19 can be written as

$\displaystyle\mathbb{E}\left(\frac{f_{Z}(z_{i})^{2}}{f_{X,Z(1)}(c^{+},z_{i})^{3}}(h_{1}^{2})^{-1}K_{1}(\frac{c-x_{j}}{h_{1}},\frac{z_{i}-z_{j}}{h_{1}})K^{2}\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}^{2}\Big{|}x_{j},z_{j}\right)$
$\displaystyle=h\int\frac{f_{Z}(z_{i})^{2}}{f_{X,Z(1)}(c^{+},z_{i})^{3}}(h_{1}^{2})^{-1}K_{1}(\frac{c-x_{j}}{h_{1}},\frac{z_{i}-z_{j}}{h_{1}})J(c,z_{i})dz_{i}$
$\displaystyle=h\kappa_{20}K(\frac{c-x_{j}}{h_{1}})\int\frac{f_{Z}(z_{i})^{2}}{f_{X,Z(1)}(c^{+},z_{i})^{2}}(h_{1}^{2})^{-1}K(\frac{z_{i}-z_{j}}{h_{1}})(\sigma^{2}+d_{1}(c^{+},z_{i})^{2})dz_{i}+O_{p}(h)$
$\displaystyle=hh_{1}^{-1}\kappa_{20}K(\frac{c-x_{j}}{h_{1}})\frac{f_{Z}(z_{j})^{2}}{f_{X,Z(1)}(c^{+},z_{j})^{2}}(\sigma^{2}+d_{1}(c^{+},z_{j})^{2})dz_{i}+O_{p}(h)$

Thus when $h\asymp h_{1}\asymp h_{2}$

$\displaystyle\mathbb{E}\left(\frac{f_{Z}(z_{i})^{2}}{f_{X,Z(1)}(c^{+},z_{i})^{2}}K_{1}^{2}(\frac{c-x_{j}}{h_{1}},\frac{z_{i}-z_{j}}{h_{1}})K^{2}\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}^{2}\right)$
$\displaystyle=\kappa_{20}^{2}\int\int\frac{f_{Z}(z_{j})^{2}}{f_{X,Z(1)}(c^{+},z_{j})^{3}}(\sigma^{2}+d_{1}(c^{+},z_{j})^{2})K(\frac{c-x_{j}}{h_{1}})f_{X,Z(1)}(x_{j},z_{j})dz_{j}dx_{j}+O_{p}(h)$
$\displaystyle=O_{p}(h)<\infty$

and when $h\asymp\sqrt{h_{1}}\asymp h_{2}$

$\displaystyle\mathbb{E}\left(\frac{f_{Z}(z_{i})^{2}}{f_{X,Z(1)}(c^{+},z_{i})^{2}}K_{1}^{2}(\frac{c-x_{j}}{h_{1}},\frac{z_{i}-z_{j}}{h_{1}})K^{2}\Big{(}\frac{x_{i}-c}{h}\Big{)}R_{i}^{2}\right)=O_{p}(\sqrt{h})<\infty$

∎

######  Lemma 16.

Under the same condition as in Theorem 1,

$\begin{split}J(c,z_{i})&:=\frac{1}{h}\int
K^{2}\left(\frac{x_{i}-c}{h}\right)\mathbb{E}(R_{i}^{2}|x_{i},z_{i})f_{X,Z(1)}(x_{i},z_{i})dx_{i}\\\
&=(\sigma^{2}+d_{1}(c^{+},z_{i})^{2})\cdot
f_{X,Z(1)}(c^{+},z_{i})\kappa_{20}+O_{p}(h),,\end{split}$

where $O_{p}$ terms are valid uniformly over $i$.

###### Proof.

Recall
$R_{i}=\epsilon_{i}+m_{1}(x_{i},z_{i})-\alpha_{1}-(x_{i}-c)\beta_{1}=\epsilon_{i}+d_{1}(x_{i},z_{i})-(x_{i}-c)\beta_{1}$,
thus

$\displaystyle\mathbb{E}(R_{i}^{2}|x_{i},z_{i})$
$\displaystyle=d_{1}(x_{i},z_{i})^{2}+\mathbb{E}(\epsilon_{i}^{2}|x_{i},z_{i})+(x_{i}-c)^{2}\beta_{1}^{2}$
$\displaystyle+2d_{1}(x_{i},z_{i})\mathbb{E}(\epsilon_{i}|x_{i},z_{i})+2(x_{i}-c)\beta_{1}\mathbb{E}(\epsilon_{i}|x_{i},z_{i})+2d_{1}(x_{i},z_{i})(x_{i}-c)\beta_{1}$
$\displaystyle=\sigma^{2}+d_{1}(x_{i},z_{i})^{2}+(x_{i}-c)^{2}\beta_{1}^{2}+2d_{1}(x_{i},z_{i})\cdot(x_{i}-c)\beta_{1}$

Following the standard Taylor expansion, we can show that

$\displaystyle J(c,z_{i})=\frac{1}{h}\int
K^{2}\left(\frac{x_{i}-c}{h}\right)\mathbb{E}(R_{i}^{2}|x_{i},z_{i})f_{X,Z(1)}(x_{i},z_{i})dx_{i}$
$\displaystyle=\sigma^{2}\int_{u}K^{2}(u)f_{X,Z(1)}(c+uh,z_{i})du+\int_{u}K^{2}(u)d_{1}(c+uh,z_{i})^{2}f_{X,Z(1)}(c+uh,z_{i})du$
$\displaystyle+\beta_{1}^{2}h^{2}\int_{u}K^{2}(u)u^{2}f_{X,Z(1)}(c+uh,z_{i})du+2h\beta_{1}\cdot\int_{u}K^{2}(u)ud_{1}(c+uh,z_{i})f_{X,Z(1)}(c+uh,z_{i})du$
$\displaystyle=(\sigma^{2}+d_{1}(c^{+},z_{i})^{2})\cdot
f_{X,Z(1)}(c^{+},z_{i})\int_{u}K^{2}(u)du+O_{p}(h),$

where $O_{p}$ terms are valid uniformly over $i$ as the (mixed) third
derivatives of $f_{X,Z(1)}(x_{i},z_{i})$ are all bounded. ∎

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