# On semiparametric estimation of the intercept of the sample selection model:
a kernel approach

Zhewen Pan
School of Economics, Zhejiang University of Finance & Economics
18 Xueyuan Street, Xiasha, Hangzhou 310018, China. Email<EMAIL_ADDRESS>
###### Abstract

This paper presents a new perspective on the identification at infinity for
the intercept of the sample selection model as identification at the boundary
via a transformation of the selection index. This perspective suggests
generalizations of estimation at infinity to kernel regression estimation at
the boundary and further to local linear estimation at the boundary. The
proposed kernel-type estimators with an estimated transformation are proven to
be nonparametric-rate consistent and asymptotically normal under mild
regularity conditions. A fully data-driven method of selecting the optimal
bandwidths for the estimators is developed. The Monte Carlo simulation shows
the desirable finite sample properties of the proposed estimators and
bandwidth selection procedures.

_Keywords_ : identification at infinity, kernel regression estimation,
boundary effect, local linear estimation, bandwidth selection

_JEL codes_ : C01, C13, C14, C34

##  1 Introduction

Since it was first introduced by the seminal papers of Heckman (1974), Gronau
(1974), and Lewis (1974), the sample selection model has been increasingly
widely applied in empirical studies to address potentially nonrandom samples
that may arise from a variety of causes, such as improper sampling design,
self-selectivity, nonresponse on survey questions, and attrition from social
programs. Several important applications of the sample selection model, such
as the estimation of average treatment effects and the decomposition of wage
differentials, rely on an estimate of the intercept.

The intercept of the sample selection model is conventionally estimated along
with the slope coefficients by means of the parametric maximum likelihood
approach or the likelihood-based two-step procedure (Heckman, 1979). When the
distribution of the model disturbance is misspecified, however, the parametric
likelihood-based estimators for limited dependent variable models may possess
evident bias and are likely inconsistent for the true value of interest (e.g.,
Arabmazar and Schmidt, 1982). This finding spawned considerable and
influential literature on semiparametric identification and estimation
approaches that do not rest on parametric specification of the disturbance
distribution, among which Chamberlain (1986) invented the notion of
“identification at infinity”, building identification upon the unbounded
support of the regressor distribution. According to the identification at
infinity, Heckman (1990) proposed a semiparametric estimator for the intercept
of the sample selection model. However, Heckman’s estimator contains
discontinuous indicator functions, which greatly complicate the analysis of
the statistical properties. Andrews and Schafgans (1998) suggested replacing
the indicator function with a smoothed version and established the consistency
and asymptotic normality of their modified estimator. Since then, the
identification-at-infinity method has been standard for semiparametrically
estimating the intercept of the sample selection model (see Schafgans, 1998,
2000; Hussinger, 2008; Mulligan and Rubinstein, 2008; Liu et al., 2009; Shen,
2013, among others).

Nevertheless, identification-at-infinity estimation must choose a smoothing
parameter or bandwidth controlling for the proportion of observations used,
and no bandwidth selection algorithm that is both theoretically valid and
practically tractable in the context of the intercept estimation has yet been
developed. This is mainly because the asymptotic biases and variances of the
estimators of Heckman (1990) and Andrews and Schafgans (1998) are all implicit
functions of the bandwidth. To choose a theoretically valid bandwidth, one
must impose high-level assumptions on the tail behaviors of the regressor and
disturbance distributions in the selection equation and then carefully
estimate some sort of tail indices, as in Klein et al. (2015). However, a
precise estimate of the tail index of the selection disturbance distribution
is difficult to obtain for two reasons. First, only the information in the
tails, which is limited, is useful for estimating tail indices. Second, there
is additional information loss when estimating the tail index of the selection
disturbance distribution because one can observe only the binary selection
outcome instead of the continuous selection disturbance. The imprecise
estimation of the tail index makes it difficult to choose an eligible
bandwidth in practice (Tan and Zhang, 2018). In empirical studies, researchers
typically report several estimates of the intercept according to different
values of bandwidth selected by simple rules, such as by sample quantiles of
the selection linear index. However, these simple rules lack theoretical
justification, and an inappropriately selected bandwidth may lead to sizable
estimation bias and misleading inference results (Schafgans, 2004).
Additionally, confusion will arise if different bandwidths lead to
contradictory conclusions.

In this paper, I extend the identification-at-infinity estimators to a kernel
regression estimator at the boundary and provide a simple bandwidth selection
algorithm that is theoretically optimal in the sense of minimizing the
asymptotic mean squared estimation error. By transforming the identification
at infinity into identification at the boundary, I propose a kernel regression
estimator for the intercept of the sample selection model that includes the
estimators of Heckman (1990) and Andrews and Schafgans (1998) as special cases
by taking particular forms of transformation. I suggest using the cumulative
distribution function (CDF) of the selection linear index as the
transformation, which makes the proposed kernel estimator substantially
different from the existing identification-at-infinity estimators. Under this
specific form of transformation, the asymptotic bias and variance of the
kernel estimator are explicit functions of the bandwidth, motivating a simple
plug-in procedure of optimal bandwidth selection. In practice, however, the
CDF of the linear index is unknown. I adopt the empirical CDF estimation and
show that the induced estimation error is asymptotically negligible under
regularity conditions, implying that the kernel estimator with the empirical
CDF follows the same asymptotic distribution as if the true CDF were known. As
a result, the proposed bandwidth selection algorithm is theoretically valid.

Careful consideration of the asymptotic distribution of the kernel regression
estimator for the intercept indicates a boundary effect in that the asymptotic
bias has a larger order than the nonparametric kernel regression estimator in
the interior because the estimator is, in nature, a Nadaraya-Watson or local
constant estimator at the boundary. Hence, I further consider the local linear
estimation of the intercept for bias reduction and show that it achieves a
univariate nonparametric rate. The consistency and asymptotic normality of the
estimator are established, and an optimal bandwidth selection algorithm
tailored to the local linear case is presented.

The main contributions of this paper are as follows. First, I provide a novel
interpretation of identification at infinity. Although infinity is intractable
or at least irregular in most econometric models, it is merely an ordinary
boundary point of the extended real line. Under the monotonic transformation
that serves to construct a metric, the identification at infinity is converted
into identification at the boundary, and kernel-type estimators are naturally
derived. This approach may be helpful in other irregular identification
problems involving infinity (see, e.g., Khan and Tamer, 2010). Second, I
provide a theoretically justified procedure for bandwidth selection that is
easy to implement. By implementing a special form of transformation, the
asymptotic biases and variances of the kernel-type estimators become explicit
functions of the bandwidth, based on which a fully data-driven algorithm is
developed to choose the optimal bandwidth. The regularization method in Imbens
and Kalyanaraman (2012) is employed to keep the random denominator of the
estimated bandwidth away from zero. Third, in the course of developing the
statistical properties of the estimators, I provide a rigorous treatment of
the estimation error induced by the empirical CDF. By means of a decomposition
into an empirical process component and a discontinuous error component, the
estimation error of the empirical CDF can be delicately controlled by the
uniform rate of convergence of the empirical process over shrinking intervals
(e.g., Stute, 1982) and by the convergence results developed by Schafgans and
Zinde-Walsh (2002).

The rest of this paper is organized as follows. Section 2 presents the sample
selection model and reviews the semiparametric estimators for the intercept in
the literature. Section 3 motivates the kernel regression estimator, provides
the regularity conditions under which the estimator is consistent and
asymptotically normal, discusses the choice of the transformation, and
presents the optimal bandwidth selection method. Section 4 proposes local
linear estimation to eliminate the boundary effect. Section 5 reports the
results from a small simulation and Section 6 concludes. All the technical
proofs are left to the Appendix.

##  2 The model and existing estimators

###  2.1 The model

The sample selection model has the following three-equation form:

$\displaystyle D_{i}$ $\displaystyle=$ $\displaystyle
1\left\\{X_{i}^{\prime}\beta_{0}>\varepsilon_{i}\right\\},$ $\displaystyle
Y_{i}^{\ast}$ $\displaystyle=$
$\displaystyle\mu_{0}+Z_{i}^{\prime}\theta_{0}+U_{i},$ (1) $\displaystyle
Y_{i}$ $\displaystyle=$ $\displaystyle Y_{i}^{\ast}D_{i},$

where $D_{i}$ is a binary selection variable, $Y_{i}^{\ast}$ is the latent
outcome, and $Y_{i}$ is the observed outcome. $X_{i}$ and $Z_{i}$ are random
vectors of regressors (excluding the constant term) of the selection and
outcome equations, respectively. $\varepsilon_{i}$ and $U_{i}$ are scalar
disturbances that are possibly correlated with each other. $\beta_{0}$ and
$\theta_{0}$ are vectors of the slope coefficients, and $\mu_{0}$ is a scalar
intercept coefficient, both of which are the estimands of the model. $U_{i}$
is assumed to have zero mean to ensure identifiability of the intercept.

This paper is primarily concerned with the estimation of $\mu_{0}$ in the
sample selection model (1) without imposing a normal (or any other parametric)
distribution restriction on the disturbances. The estimation of $\mu_{0}$,
however, relies on preliminary $\sqrt{n}$-consistent estimators for
$\beta_{0}$ and $\theta_{0}$, which are denoted by $\hat{\beta}$ and
$\hat{\theta}$. Several such estimators are available in the literature. For
instance, $\hat{\beta}$ can be one of the semiparametric estimators for the
binary response model (e.g., Klein and Spady, 1993; Lewbel, 2000) or for the
single-index model (e.g., Powell et al., 1989; Ichimura, 1993), and
$\hat{\theta}$ can be one of the semiparametric slope estimators for the
sample selection model (e.g., Gallant and Nychka, 1987; Chen and Lee, 1998;
Powell, 2001; Lewbel, 2007; Newey, 2009; Chen and Zhou, 2010, to name a few).
For notational convenience, denote $W_{i}=X_{i}^{\prime}\beta_{0}$ and
$\hat{W}_{i}=X_{i}^{\prime}\hat{\beta}$, hence
$D_{i}=1\left\\{W_{i}>\varepsilon_{i}\right\\}$.

###  2.2 The identification-at-infinity estimators

A natural strategy for identifying the intercept $\mu_{0}$ starts from the
expectation of $Y_{i}-Z_{i}^{\prime}\theta_{0}$ conditional on the observation
being selected:

$E\left[Y_{i}-Z_{i}^{\prime}\theta_{0}\left|D_{i}=1\right.\right]=E\left[Y_{i}^{\ast}-Z_{i}^{\prime}\theta_{0}\left|D_{i}=1\right.\right]=\mu_{0}+E\left[U_{i}\left|D_{i}=1\right.\right].$

In the context of sample selection, where $U_{i}$ is correlated with
$\varepsilon_{i}$, the selectivity bias term
$E\left[U_{i}\left|D_{i}=1\right.\right]$ is generally nonzero and
contaminates the identification of $\mu_{0}$. Inspired by the fact that
$E\left[U_{i}\left|D_{i}=1\right.\right]$ will become arbitrarily close to
zero for those $W_{i}$ such that $\Pr\left(D_{i}=1\left|W_{i}\right.\right)$
is arbitrarily close to unity, Heckman (1990) suggested identifying $\mu_{0}$
at infinity, i.e., when $W_{i}$ takes arbitrarily large values. This idea can
be heuristically formulated as

$E\left[Y_{i}-Z_{i}^{\prime}\theta_{0}\left|D_{i}=1,W_{i}=+\infty\right.\right]=\mu_{0}+E\left[U_{i}\left|D_{i}=1\right.,W_{i}=+\infty\right]=\mu_{0}+E\left[U_{i}\left|W_{i}=+\infty\right.\right]=\mu_{0},$

provided the conditional mean independence is assumed. An intuitive estimator
based on the above identification strategy is

$\hat{\mu}^{H}=\frac{\sum_{i=1}^{n}\left(Y_{i}-Z_{i}^{\prime}\hat{\theta}\right)D_{i}\cdot
1\left\\{\hat{W}_{i}>\gamma_{n}\right\\}}{\sum_{i=1}^{n}D_{i}\cdot
1\left\\{\hat{W}_{i}>\gamma_{n}\right\\}},$ (2)

where $\left\\{\gamma_{n}\right\\}$ is a sequence of positive smoothing
parameters such that $\gamma_{n}\rightarrow\infty$ as $n\rightarrow\infty$.
The estimator $\hat{\mu}^{H}$ is essentially a sample average of
$\mu_{0}+U_{i}$ over a decreasingly small fraction of all observations. The
effective sample size depends on the proportion of data censoring, the degree
of tail heaviness of $\hat{W}_{i}$’s distribution, and the choice of
$\gamma_{n}$.

To facilitate the development of distribution theory of the identification-at-
infinity estimation, Andrews and Schafgans (1998) replaced the indicator
function in Heckman’s estimator (2) with a smoothed function and proposed a
weighted sample average estimator:

$\hat{\mu}^{AS}=\frac{\sum_{i=1}^{n}\left(Y_{i}-Z_{i}^{\prime}\hat{\theta}\right)D_{i}\cdot
s\left(\hat{W}_{i}-\gamma_{n}\right)}{\sum_{i=1}^{n}D_{i}\cdot
s\left(\hat{W}_{i}-\gamma_{n}\right)},$ (3)

where $s\left(\cdot\right)$ is a nondecreasing $\left[0,1\right]$-valued
function that has a third derivative bounded over $R$ and satisfies
$s\left(w\right)=0$ for $w\leq 0$ and $s\left(w\right)=1$ for $w\geq b$, for
some $b>0$. By taking advantage of the smoothness of $s\left(\cdot\right)$,
Andrews and Schafgans (1998) showed the asymptotic negligibility of the
estimation error induced by the preliminary estimators $\hat{\beta}$ and
$\hat{\theta}$ and established the consistency and asymptotic normality of
$\hat{\mu}^{AS}$ under mild regularity conditions. The rate of convergence of
$\hat{\mu}^{AS}$ depends both upon the tail heaviness of $W_{i}$’s
distribution and upon the rate of divergence of $\gamma_{n}$. However,
theoretically valid procedures for choosing $\gamma_{n}$ have not yet been
developed.

###  2.3 Other existing estimators

The semiparametric literature has neglected, to a certain extent, the
intercept estimation of the sample selection model. This is mainly because the
intercept is absorbed into the selectivity bias correction term in the course
of estimating the slope coefficients. The only exceptions, besides
$\hat{\mu}^{H}$ and $\hat{\mu}^{AS}$, are the estimators of Gallant and Nychka
(1987), Lewbel (2007), and Chen and Zhou (2010).

By employing Hermite series to approximate the unknown bivariate density
function of the disturbances, Gallant and Nychka (1987) considered
semiparametric maximum likelihood estimation of the intercept along with the
slopes. The consistency of their estimator requires complicated continuity
conditions on the distributions of the disturbances and regressors that are
difficult to verify. Moreover, the asymptotic distribution of their estimator
has not been established. Lewbel (2007) achieved identification of the
intercept by the presence of a special regressor that has a large support.
When the support is infinite, his identification strategy would be essentially
equivalent to the identification at infinity. Chen and Zhou (2010) proposed a
kernel-weighted pairwise-difference-type estimator for the intercept and
showed its consistency and asymptotic normality. However, their estimation
method requires the disturbances to be jointly symmetrically distributed; this
joint symmetry condition may not hold in practice.

##  3 Kernel regression estimation

The kernel approach to estimating the intercept $\mu_{0}$ in the sample
selection model (1) is grounded in the identification at infinity:

$\mu_{0}=E\left[Y_{i}-Z_{i}^{\prime}\theta_{0}\left|D_{i}=1,W_{i}=+\infty\right.\right].$
(4)

Let $F\left(\cdot\right)$ be an absolutely continuous CDF that is strictly
increasing over $R$; then, the infinity condition $W_{i}=+\infty$ is
equivalent to a boundary condition
$F\left(W_{i}\right)=F\left(+\infty\right)=1$. Therefore, the identification
at infinity (4) can be written as identification at the boundary:

$\mu_{0}=E\left[Y_{i}-Z_{i}^{\prime}\theta_{0}\left|D_{i}=1,F\left(W_{i}\right)=1\right.\right].$
(5)

In other words, $\mu_{0}$ is identified by the conditional expectation of
$Y_{i}-Z_{i}^{\prime}\theta_{0}$ given that the observation is selected and
that the transformation of $W_{i}$, $F\left(W_{i}\right)$, takes a boundary
value. The identification-at-boundary equation (5) motivates the kernel
regression estimation of $\mu_{0}$:

$\tilde{\mu}=\frac{\displaystyle\sum_{i=1}^{n}\left(Y_{i}-Z_{i}^{\prime}\hat{\theta}\right)D_{i}k\left(\frac{1-F\left(\hat{W}_{i}\right)}{h_{n}}\right)}{\displaystyle\sum_{i=1}^{n}D_{i}k\left(\frac{1-F\left(\hat{W}_{i}\right)}{h_{n}}\right)},$
(6)

where $\hat{W}_{i}=X_{i}^{\prime}\hat{\beta}$, $\hat{\beta}$ and
$\hat{\theta}$ are preliminary estimators for $\beta_{0}$ and $\theta_{0}$,
$k\left(\cdot\right)$ is a kernel function defined on $\left[0,\infty\right)$,
and $\left\\{h_{n}\right\\}$ is a sequence of positive bandwidth parameters
such that $h_{n}\rightarrow 0$ as $n\rightarrow\infty$. The values of the
kernel over the negative reals are irrelevant since the argument of it is
always positive. Some examples of $k\left(\cdot\right)$ are given in Table 1.
$\tilde{\mu}$ is a kernel regression estimator at the boundary. Alternatively,
it can be viewed as a kernel regression estimator at infinity over the
extended real line
$\bar{R}=\left[-\infty,+\infty\right]=R\cup\left\\{-\infty,+\infty\right\\}$.
A distance function with which $\bar{R}$ would become a (compact) metric space
can be defined as
$d\left(w_{1},w_{2}\right)=\left|F\left(w_{1}\right)-F\left(w_{2}\right)\right|$
for any $w_{1},w_{2}\in\bar{R}$. Accordingly, the numerator within the kernel
is the distance between $\hat{W}_{i}$ and infinity since
$d\left(\hat{W}_{i},+\infty\right)=\left|F\left(\hat{W}_{i}\right)-1\right|=1-F\left(\hat{W}_{i}\right)$.
From this perspective, our estimation method is an extension of kernel
regression to the extended real line.

Similar to the identification-at-infinity estimators $\hat{\mu}^{H}$ and
$\hat{\mu}^{AS}$ defined in (2) and (3), the kernel estimator $\tilde{\mu}$ is
also a (weighted) sample average over a vanishingly small subset of the entire
data set. However, the local average in $\tilde{\mu}$ is over observations for
$F\left(\hat{W}_{i}\right)$ in a left neighborhood of the right boundary
point, rather than for $\hat{W}_{i}$ near positive infinity. Notably,
$\tilde{\mu}$ generalizes the identification-at-infinity estimators in the
sense that it nests $\hat{\mu}^{H}$ and $\hat{\mu}^{AS}$ as special cases by
taking specific forms of $F\left(\cdot\right)$, $k\left(\cdot\right)$ and
$h_{n}$.

######  Proposition.

(i) For any strictly increasing $F\left(\cdot\right)$, if
$k\left(u\right)=1\left\\{0\leq u\leq 1\right\\}$ and
$h_{n}=1-F\left(\gamma_{n}\right)$, then $\tilde{\mu}=\hat{\mu}^{H}$. (ii) If
$F\left(\cdot\right)$ is the standard Laplacian CDF such that
$F\left(w\right)=1-\left(1\left/2\right.\right)\exp\left(-w\right)$ for $w\geq
0$, if $k\left(u\right)=s\left(-\log u\right)$, and if
$h_{n}=1-F\left(\gamma_{n}\right)=\left(1\left/2\right.\right)\exp\left(-\gamma_{n}\right)$,
then $\tilde{\mu}=\hat{\mu}^{AS}$.

###  3.1 Asymptotic property

This subsection investigates the consistency and asymptotic normality of the
kernel estimator $\tilde{\mu}$ under general $F\left(\cdot\right)$. Several
regularity assumptions are made first. Denote
$k_{ni}=k\left(\left.\left(1-F\left(W_{i}\right)\right)\right/h_{n}\right)$,
where $W_{i}=X_{i}^{\prime}\beta_{0}$.

Assumption 1. (i)
$\left\\{\left(Y_{i},D_{i},X_{i},Z_{i}\right)\right\\}_{i=1}^{n}$ is a random
sample of $n$ observations from Model (1). (ii) The model disturbance
$\left(\varepsilon_{i},U_{i}\right)$ is independent of the regressors $X_{i}$
and $Z_{i}$. (iii) $EU_{i}=0$. (iv) There exists a positive constant $c_{1}$
such that $E\left|U_{i}\right|^{2+c_{1}}<\infty$,
$E\left\|X_{i}\right\|^{4+c_{1}}<\infty$, and
$E\left\|Z_{i}\right\|^{2+c_{1}}<\infty$.

Assumption 2. The kernel function $k\left(\cdot\right)$ is defined on
$\left[0,\infty\right)$ and satisfies that (i) it is nonnegative and supported
on $\left[0,1\right]$, (ii) it is bounded above by $\bar{k}>0$, and (iii) it
is twice continuously differentiable over $\left[0,\infty\right)$ and its
derivatives $k^{\prime}\left(\cdot\right)$ and
$k^{\prime\prime}\left(\cdot\right)$ are bounded above by $\bar{k^{\prime}}$
and $\bar{k^{\prime\prime}}$, respectively.

Assumption 3. (i) The transformation function $F\left(\cdot\right)$ is a CDF
for a continuously distributed random variable whose support is right-
unbounded. (ii) The probability density function (PDF) $f\left(\cdot\right)$
corresponding to $F\left(\cdot\right)$ is absolutely continuous with respect
to the Lebesgue measure, with its derivative $f^{\prime}\left(\cdot\right)$
bounded almost everywhere by $\bar{f^{\prime}}$. (iii) For a large positive
constant $C$, the tail hazard rate
$H_{C}\left(\cdot\right)=1\left\\{\cdot>C\right\\}f\left(\cdot\right)\left/\left[1-F\left(\cdot\right)\right]\right.$
and the tail score (of location parameter)
$S_{C}\left(\cdot\right)=1\left\\{\cdot>C\right\\}\left.f^{\prime}\left(\cdot\right)\right/f\left(\cdot\right)$
corresponding to $F\left(\cdot\right)$ satisfy
$E\left[\sup_{\beta\in\mathcal{N}\left(\beta_{0}\right)}H_{C}^{4}\left(X_{i}^{\prime}\beta\right)\right]<\infty$
and
$E\left[\sup_{\beta\in\mathcal{N}\left(\beta_{0}\right)}S_{C}^{4}\left(X_{i}^{\prime}\beta\right)\right]<\infty$
for a neighborhood $\mathcal{N}\left(\beta_{0}\right)$ of $\beta_{0}$.

Assumption 4. The preliminary estimators $\hat{\beta}$ and $\hat{\theta}$ are
$\sqrt{n}$-consistent, that is,
$\sqrt{n}\left\|\hat{\beta}-\beta_{0}\right\|=O_{p}\left(1\right)$ and
$\sqrt{n}\left\|\hat{\theta}-\theta_{0}\right\|=O_{p}\left(1\right)$.

Assumption 5. The bandwidth satisfies that $0<h_{n}\leq 1/2$ and that as
$n\rightarrow\infty$, (i) $h_{n}\rightarrow 0$, (ii)
$nEk_{ni}^{2}\rightarrow\infty$, and (iii)
$\left.\left[\Pr\left(F\left(W_{i}\right)>1-h_{n}\right)\right]^{1+c}\right/Ek_{ni}^{2}\rightarrow
0$ for any $c>0$.

Assumption 1 describes the model and data. Assumptions 2 and 3 impose
smoothness and boundedness conditions on the kernel function
$k\left(\cdot\right)$ and the transformation function $F\left(\cdot\right)$.
The compactness of the kernel’s support is assumed to simplify the technical
proofs, and the theoretical results in this paper are still supposed to hold
when using kernels that decay sufficiently fast in the tails. Assumption
3.(iii) is a joint condition on the tail behaviors of $F\left(\cdot\right)$
and $X_{i}$’s distribution. When $F\left(\cdot\right)$ has a power-type upper
tail, e.g., Pareto($\lambda$) tail such that111Here and below,
$g\left(t\right)\sim h\left(t\right)$ means that the ratios
$g\left(t\right)\left/h\left(t\right)\right.$ and
$h\left(t\right)\left/g\left(t\right)\right.$ are $O\left(1\right)$ as
$t\rightarrow+\infty$. $1-F\left(t\right)\sim t^{-\lambda}$ for some
$\lambda>0$, we have $H_{C}\left(t\right)\sim t^{-1}$ and
$S_{C}\left(t\right)\sim t^{-1}$, and Assumption 3.(iii) holds for any
distribution of $X_{i}$. When $F\left(\cdot\right)$ has an exponential-type
upper tail, e.g., Weibull($\lambda$) tail such that
$1-F\left(t\right)\sim\exp\left(-c_{0}t^{\lambda}\right)$ for some $\lambda>0$
and $c_{0}>0$, we have $H_{C}\left(t\right)\sim t^{\lambda-1}$ and
$S_{C}\left(t\right)\sim t^{\lambda-1}$. If $0<\lambda\leq 1$, Assumption
3.(iii) still holds for any distribution of $X_{i}$. For example, in the
special case of Andrews and Schafgans (1998)’s estimator that corresponds to a
Laplacian $F\left(\cdot\right)$ (i.e., $\lambda=1$), Assumption 3.(iii) is
automatically satisfied. If $\lambda>1$, then Assumption 3.(iii) will require
$E\left\|X_{i}\right\|^{4\left(\lambda-1\right)}<\infty$. For example, if
$F\left(\cdot\right)$ is the normal CDF, we need
$E\left\|X_{i}\right\|^{4}<\infty$, which is already guaranteed by Assumption
1.(iv). When the tail of $F\left(\cdot\right)$ decays more rapidly as
$1-F\left(t\right)\sim\exp\left(-\exp\left(c_{0}t^{\lambda}\right)\right)$, a
sufficient condition of Assumption 3.(iii) is that the distribution of
$\left\|X_{i}\right\|$ has a Weibull($\theta$) tail with $\theta>\lambda$.
Assumption 4 imposes the $\sqrt{n}$-consistency of the preliminary slope
estimators. Examples of such estimators are listed in Subsection 2.1.

According to Assumption 5, the bandwidth $h_{n}$ is required to go to zero as
the sample size goes to infinity, but the speed of the decline is not allowed
to be excessively fast. To see this, note that
$Ek_{ni}^{2}\leq\bar{k}^{2}\Pr\left(F\left(W_{i}\right)>1-h_{n}\right)$ by
Assumption 2. As $h_{n}$ goes to zero,
$\Pr\left(F\left(W_{i}\right)>1-h_{n}\right)$ will also go to zero. If the
speed at which $h_{n}$ declines is so fast that
$\Pr\left(F\left(W_{i}\right)>1-h_{n}\right)$ goes to zero at a rate faster
than $n^{-1}$, we will have $nEk_{ni}^{2}\rightarrow 0$, which violates
Assumption 5.(ii). Another implicit requirement of Assumption 5.(ii) is the
upper unboundedness of $W_{i}$’s support, for if $W_{i}$ has bounded support
from above, $\Pr\left(F\left(W_{i}\right)>1-h_{n}\right)$ will be exactly zero
for sufficiently small $h_{n}$. Assumption 5.(iii) requires the distribution
of $W_{i}$ to be not too thin upper tailed relative to $F\left(\cdot\right)$,
as illustrated by the following example.

######  Example.

Suppose $F\left(\cdot\right)$ is the standard Laplacian CDF, as is the case
with $\hat{\mu}^{AS}$. Then, Assumption 5.(iii) holds if

(i) the distribution of $W_{i}$ has a power-type upper tail;

(ii) the distribution of $W_{i}$ has an exponential-type upper tail;

(iii) the distribution of $W_{i}$ has a “double”-exponential-type but not
excessively thin upper tail, namely,
$\Pr\left(W_{i}>t\right)\sim\exp\left(-\exp\left(c_{0}t^{\lambda}\right)\right)$
with $\lambda<1$.

######  Theorem 1.

Under Assumptions 1-5, the kernel regression estimator $\tilde{\mu}$ defined
in (6) is consistent and asymptotically normal:

$\frac{\sqrt{n}Ek_{ni}}{\sqrt{Ek_{ni}^{2}}}\left(\tilde{\mu}-\mu_{0}-\frac{EU_{i}D_{i}k_{ni}}{ED_{i}k_{ni}}\right)\rightarrow
N\left(0,\sigma_{U}^{2}\right),$

where $\sigma_{U}^{2}=EU_{i}^{2}$.

Theorem 1 generalizes Theorems 1-2 of Andrews and Schafgans (1998) by allowing
general forms of the transformation function $F\left(\cdot\right)$. At first
sight, introducing a general $F\left(\cdot\right)$ appears to help little in
the choice of $h_{n}$ because the asymptotic bias and variance of
$\tilde{\mu}$ are still unknown functions of $h_{n}$. However, as will be seen
below, a particular data-dependent choice of $F\left(\cdot\right)$ can largely
facilitate the subsequent choice of $h_{n}$.

###  3.2 Choice of the transformation $F\left(\cdot\right)$

A natural choice of $F\left(\cdot\right)$ would be the CDF of
$W_{i}=X_{i}^{\prime}\beta_{0}$, under which $F\left(W_{i}\right)$ follows a
uniform distribution and several assumptions of Theorem 1 have a simpler form.
For example, Assumption 3.(iii) will be automatically fulfilled in this case
as long as $W_{i}$ is continuously distributed, and Assumption 3 is implied by

Assumption 3’. $W_{i}$ is continuously distributed with right-unbounded
support. In addition, $W_{i}$’s PDF $f_{W}\left(\cdot\right)$ is absolutely
continuous with a derivative that is bounded almost everywhere.

Assumption 3’ presumes that the underlying data is continuous, as in standard
kernel methods. When encountering discrete regressors, the kernel estimator
can be adapted using the frequency-based method or the smoothing method
(Racine and Li, 2004) at the cost of more complicated notations. Assumption
5.(iii) will also be automatically fulfilled for
$F\left(\cdot\right)=F_{W}\left(\cdot\right)$ because in this case
$\Pr\left(F\left(W_{i}\right)>1-h_{n}\right)=h_{n}$ and

$Ek_{ni}^{r}=\int_{0}^{1}k^{r}\left(\frac{1-t}{h_{n}}\right)dt=h_{n}\int_{0}^{1\left/h_{n}\right.}k^{r}\left(s\right)ds=h_{n}\int_{0}^{1}k^{r}\left(s\right)ds$
(7)

for any $r>0$. Therefore, Assumption 5 is simplified to $h_{n}\rightarrow 0$
and $nh_{n}\rightarrow\infty$ as $n\rightarrow\infty$.

Now, consider the asymptotic bias of $\tilde{\mu}$,
$EU_{i}D_{i}k_{ni}\left/ED_{i}k_{ni}\right.$, under
$F\left(\cdot\right)=F_{W}\left(\cdot\right)$. Define

$G\left(t\right)=E\left[U_{i}\left|D_{i}=1,F_{W}\left(W_{i}\right)=t\right.\right]=E\left[U_{i}\left|\varepsilon_{i}<F_{W}^{-1}\left(t\right)\right.\right]$
(8)

for $t\in\left(0,1\right)$, and define $G\left(1\right)=\lim_{t\rightarrow
1}G\left(t\right)=0$.

######  Lemma.

Let
$F\left(\cdot\right)=F_{W}\left(\cdot\right)=\Pr\left(W_{i}\leq\cdot\right)$.
Suppose $G\left(t\right)$ is continuously differentiable over
$t\in\left(1-\delta,1\right]$ for a small $\delta>0$, with its derivative
function $g\left(t\right)$ being bounded above over
$t\in\left(1-\delta,1\right]$. Then, we have

$\frac{EU_{i}D_{i}k_{ni}}{ED_{i}k_{ni}}=-\frac{\kappa_{1}g\left(1\right)}{\kappa_{0}}h_{n}+o\left(h_{n}\right),$

where $\kappa_{r}=\int_{0}^{1}t^{r}k\left(t\right)dt$.

Since by a calculation

$g\left(t\right)=\frac{dG\left(t\right)}{dt}=\frac{f_{\varepsilon}\left(F_{W}^{-1}\left(t\right)\right)}{f_{W}\left(F_{W}^{-1}\left(t\right)\right)}\cdot\frac{E\left[U_{i}\left|\varepsilon_{i}=F_{W}^{-1}\left(t\right)\right.\right]-G\left(t\right)}{F_{\varepsilon}\left(F_{W}^{-1}\left(t\right)\right)},$

it can be seen that the finiteness of $g\left(1\right)=\lim_{t\rightarrow
1}g\left(t\right)$ assumed by the Lemma essentially requires the selection
index $W_{i}$ to have a heavier upper tail than the selection disturbance
$\varepsilon_{i}$. The relatively heavy tail of $W_{i}$ is also required by
$\hat{\mu}^{AS}$, as illustrated by the examples of Andrews and Schafgans
(1998).

######  Corollary.

Let
$F\left(\cdot\right)=F_{W}\left(\cdot\right)=\Pr\left(W_{i}\leq\cdot\right)$.
Suppose (i) Assumptions 1-5 hold, (ii) the assumption of the Lemma holds, and
(iii) $nh_{n}^{3}=O\left(1\right)$. Then, the kernel regression estimator
$\tilde{\mu}$ defined in (6) is consistent and asymptotically normal:

$\sqrt{nh_{n}}\left(\tilde{\mu}-\mu_{0}+\frac{\kappa_{1}g\left(1\right)}{\kappa_{0}}h_{n}\right)\rightarrow
N\left(0,\frac{\chi_{0}\sigma_{U}^{2}}{\kappa_{0}^{2}}\right),$

where $\chi_{0}=\int_{0}^{1}k^{2}\left(t\right)dt$.

The corollary shows that, by setting
$F\left(\cdot\right)=F_{W}\left(\cdot\right)$, the asymptotic bias and
variance of the kernel estimator become explicit functions of $h_{n}$, based
on which we can select a theoretically optimal bandwidth. In practice,
however, $F_{W}\left(\cdot\right)$ is unknown and must be estimated
beforehand. A simple estimator for $F_{W}\left(\cdot\right)$ is the empirical
CDF of $\hat{W}_{i}=X_{i}^{\prime}\hat{\beta}$. With this specific choice, the
kernel estimator becomes

$\hat{\mu}=\frac{\displaystyle\sum_{i=1}^{n}\left(Y_{i}-Z_{i}^{\prime}\hat{\theta}\right)D_{i}k\left(\frac{1-\hat{F}_{n}\left(\hat{W}_{i}\right)}{h_{n}}\right)}{\displaystyle\sum_{i=1}^{n}D_{i}k\left(\frac{1-\hat{F}_{n}\left(\hat{W}_{i}\right)}{h_{n}}\right)},$
(9)

where $\hat{F}_{n}\left(\hat{W}_{i}\right)=\frac{1}{n-1}\sum_{j\neq
i}1\left\\{\hat{W}_{j}\leq\hat{W}_{i}\right\\}$. To control the estimation
error induced by the empirical CDF, the existing assumptions should be
strengthened, and one new assumption should be imposed. Partition
$X_{i}=\left(X_{i1},X_{i,(-1)}^{\prime}\right)^{\prime}$,
$\beta_{0}=\left(\beta_{01},\beta_{0,(-1)}^{\prime}\right)^{\prime}$, and
$\hat{\beta}=\left(\hat{\beta}_{1},\hat{\beta}_{(-1)}^{\prime}\right)^{\prime}$,
with $X_{i1}$, $\beta_{01}$, and $\hat{\beta}_{1}$ being the respective first
components.

Assumption 1’. Assumption 1 holds and $E\left\|X_{i}\right\|^{6}<\infty$;
$\beta_{01}\neq 0$. Without loss of generality, set $\beta_{01}=1$ to achieve
scale normalization of $\beta_{0}$.

Assumption 2’. Assumption 2 holds and $k\left(\cdot\right)$ is six times
continuously differentiable over $\left[0,\infty\right)$ with all of its
derivatives bounded above.

Assumption 4’. Assumption 4 holds and $\hat{\beta}_{1}=1$.

Assumption 5’. The bandwidth $h_{n}\in\left(0,1/2\right]$ satisfies (i)
$h_{n}\rightarrow 0$, (ii) $nh_{n}^{3}=O\left(1\right)$, and (iii)
$nh_{n}^{13/5}\rightarrow\infty$, as $n\rightarrow\infty$.

Assumption 6. The PDF of $W_{i}$, $f_{W}\left(\cdot\right)$, and the
conditional PDF of $W_{i}$ given $X_{i,(-1)}$,
$f_{W\left|X_{(-1)}\right.}\left(\cdot\left|X_{i,(-1)}\right.\right)$,
satisfies that (i)
$\left.f_{W}\left(F_{W}^{-1}\left(1-h_{n}\right)-n^{-1/10}h_{n}^{-1/18}\right)\right/h_{n}^{2/3}\rightarrow
0$, (ii) $f_{W\left|X_{(-1)}\right.}\left(\cdot\left|X_{i,(-1)}\right.\right)$
is uniformly bounded by $\overline{f_{W\left|X_{(-1)}\right.}}$, and (iii)
there exists a large constant $C$ such that
$f_{W\left|X_{(-1)}\right.}\left(w\left|X_{i,(-1)}\right.\right)$ declines
monotonically for $w>C$ almost surely.

Since $\beta_{0}$ is identified only up to scale, the normalization
$\beta_{01}=\hat{\beta}_{1}=1$ is postulated by most semiparametric estimators
for the binary response model. Assumption 6, which is comparable to Assumption
A of Schafgans and Zinde-Walsh (2002), is imposed to address the non-
differentiable indicator function contained in the empirical CDF. Because the
expectation of the generalized derivative of the indicator function equals the
value of the PDF, Assumption 6 involves restrictions on the PDF and
conditional PDF of $W_{i}$. Assumption 6.(i) relates to the upper tail
behavior of the distribution of $W_{i}$. Under the assumption of
$nh_{n}^{13/5}\rightarrow\infty$ that implies
$n^{-1/10}h_{n}^{-1/18}\rightarrow 0$, it can be shown that Assumption 6.(i)
is satisfied if $W_{i}$ has a power- or exponential-type upper tail.

######  Theorem 2.

Under Assumptions 1’-5’ and 6, the kernel regression estimator $\hat{\mu}$
defined in (9) is consistent and asymptotically normal:

$\sqrt{nh_{n}}\left(\hat{\mu}-\mu_{0}+\frac{\kappa_{1}g\left(1\right)}{\kappa_{0}}h_{n}\right)\rightarrow
N\left(0,\frac{\chi_{0}\sigma_{U}^{2}}{\kappa_{0}^{2}}\right),$

where $\kappa_{r}=\int_{0}^{1}t^{r}k\left(t\right)dt$ and
$\chi_{r}=\int_{0}^{1}t^{r}k^{2}\left(t\right)dt$.

Theorem 2 reveals that, under slightly stronger conditions, the estimation
error induced by the empirical CDF is asymptotically negligible; thus, the
asymptotic distribution of $\hat{\mu}$ is the same as if the true CDF
$F_{W}\left(\cdot\right)$ were known.

###  3.3 Bandwidth selection

An important implication of Theorem 2 is that, under the specific choice of
$F\left(\cdot\right)=F_{W}\left(\cdot\right)$, the kernel estimator for the
intercept follows a standard asymptotic distribution as an ordinary kernel
regression estimator at the boundary point. As a result, we can borrow
approaches of bandwidth selection from the nonparametric regression literature
(see, e.g., Li and Racine, 2007). However, the widely used cross-validation
method based on the integrated mean squared error (MSE) criteria takes into
account the global performance of the regression function estimation and is
thus not suitable for the problem at hand. A closely related problem is the
choice of bandwidth for the nonparametric regression discontinuity estimator,
which is the difference between two regression estimators evaluated at
boundary points. I follow the plug-in method proposed by Imbens and
Kalyanaraman (2012) and suggest a data-dependent bandwidth selection procedure
that is tailored to $\hat{\mu}$.

By Theorem 2, we know that

$\text{Abias}\left(\hat{\mu}\right)=\left(-\frac{\kappa_{1}g\left(1\right)}{\kappa_{0}}\right)h_{n},\text{
\
Avar}\left(\hat{\mu}\right)=\left(\frac{\chi_{0}\sigma_{U}^{2}}{\kappa_{0}^{2}}\right)\frac{1}{nh_{n}},$

where $g\left(1\right)=\left.\left.dG\left(t\right)\right/dt\right|_{t=1}$
with $G\left(t\right)$ defined in (8), and
$\sigma_{U}^{2}=Var\left(U_{i}\right)$. Provided $g\left(1\right)\neq 0$, the
optimal bandwidth for $\hat{\mu}$ is defined as a minimizer of its asymptotic
MSE:

$\displaystyle h_{opt}$ $\displaystyle=$
$\displaystyle\arg\min_{h}\left\\{\text{AMSE}_{h}\left(\hat{\mu}\right)=\left(\frac{\kappa_{1}g\left(1\right)}{\kappa_{0}}\right)^{2}h^{2}+\left(\frac{\chi_{0}\sigma_{U}^{2}}{\kappa_{0}^{2}}\right)\frac{1}{nh}\right\\}$
$\displaystyle=$ $\displaystyle
c_{k}\left(\sigma_{U}^{2}\left/g^{2}\left(1\right)\right.\right)^{1/3}n^{-1/3},$

where
$c_{k}=\left(\chi_{0}\left/\left(2\kappa_{1}^{2}\right)\right.\right)^{1/3}$
is a functional of the kernel $k\left(\cdot\right)$. If $g\left(1\right)=0$,
the bias converges to zero faster, allowing for estimation of the intercept at
a faster rate of convergence. However, it is difficult to exploit the improved
convergence rate resulting from this condition in practice; hence, I focus on
the optimal bandwidth given $g\left(1\right)\neq 0$.

A natural choice of the estimator for the optimal bandwidth $h_{opt}$ is to
replace $\sigma_{U}^{2}$ and $g\left(1\right)$ with their consistent
estimators $\hat{\sigma}_{U}^{2}$ and $\hat{g}\left(1\right)$, respectively.
One potential problem with this choice is that $\hat{g}\left(1\right)$ may
occasionally be very close to zero due to the stochastic estimation error,
even if $g\left(1\right)\neq 0$. In such cases, the estimated bandwidth will
be imprecisely and unstably large, which may in turn lead to large finite
sample bias of $\hat{\mu}$ because observations that are far from the boundary
will be included in the kernel estimation. To alleviate this problem, I employ
the regularization method (Imbens and Kalyanaraman, 2012, Subsection 4.1.1)
and propose the following bandwidth estimator:

$\hat{h}_{opt}=c_{k}\left(\frac{\hat{\sigma}_{U}^{2}}{\hat{g}^{2}\left(1\right)+3\widehat{Var}\left(\hat{g}\left(1\right)\right)}\right)^{1/3}n^{-1/3}.$
(10)

By means of regularization, $\hat{h}_{opt}$ will not become infinite even in
the case of $\hat{g}\left(1\right)=0$. Moreover, the leading term of
$E\left[1\left/\left(\hat{g}^{2}\left(1\right)+3Var\left(\hat{g}\left(1\right)\right)\right)\right.-1\left/\hat{g}^{2}\left(1\right)\right.\right]$
cancels out the leading term of
$E\left[1\left/\hat{g}^{2}\left(1\right)\right.-1\left/g^{2}\left(1\right)\right.\right]$.
Therefore, the bias of
$1\left/\left(\hat{g}^{2}\left(1\right)+3Var\left(\hat{g}\left(1\right)\right)\right)\right.$
for the reciprocal of $g^{2}\left(1\right)$ is of lower order than the bias of
the naive $1\left/\hat{g}^{2}\left(1\right)\right.$. It remains to construct
consistent estimators for the components of the plug-in bandwidth, namely,
$\hat{\sigma}_{U}^{2}$, $\hat{g}^{2}\left(1\right)$, and
$\widehat{Var}\left(\hat{g}\left(1\right)\right)$, which is deferred to the
next section for expositional convenience.

##  4 Local linear estimation

Theorem 2 finds that the kernel regression estimator $\hat{\mu}$ for the
intercept suffers from a boundary effect in the sense that its order of bias,
$O\left(h_{n}\right)$, is larger than that of the nonparametric regression
estimator in the interior, which is typically $O\left(h_{n}^{2}\right)$. This
is simply because $\hat{\mu}$ is in nature a Nadaraya-Watson or local constant
estimator at the boundary. In this section, I resort to the local polynomial
regression method (Fan and Gijbels, 1996) for bias reduction. In particular, I
focus on the local linear estimation because of its asymptotic minimax
efficiency properties (Cheng et al., 1997) and attractive practical
performance (Gelman and Imbens, 2019).

Denote $\hat{F}_{n}\left(\hat{W}_{i}\right)=\frac{1}{n-1}\sum_{j\neq
i}1\left\\{\hat{W}_{j}\leq\hat{W}_{i}\right\\}$ as the empirical estimate of
$F_{W}\left(W_{i}\right)$ as before. The local linear estimator
$\hat{\mu}^{L}$ for the intercept is defined via locally weighted least
squares regression:

$\left(\hat{\mu}^{L},\hat{b}^{L}\right)=\arg\min_{\mu,b}\sum_{i=1}^{n}\left[Y_{i}-Z_{i}^{\prime}\hat{\theta}-\mu-\left(\hat{F}_{n}\left(\hat{W}_{i}\right)-1\right)b\right]^{2}D_{i}k\left(\frac{1-\hat{F}_{n}\left(\hat{W}_{i}\right)}{h_{n}}\right).$
(11)

To establish the asymptotic properties of $\hat{\mu}^{L}$, Assumption 5’ must
be modified to accommodate the local linear case.

Assumption 5”. The bandwidth $h_{n}\in\left(0,1/2\right]$ satisfies (i)
$h_{n}\rightarrow 0$, (ii) $nh_{n}^{5}=O\left(1\right)$, and (iii)
$nh_{n}^{3}\rightarrow\infty$, as $n\rightarrow\infty$.

######  Theorem 3.

Suppose Assumptions 1’-4’, 5”, and 6 hold. In addition, suppose
$G\left(t\right)$ given in (8) is twice continuously differentiable over
$t\in\left(1-\delta,1\right]$ for a small $\delta>0$, with its first and
second derivative functions $g\left(t\right)$ and $g^{\prime}\left(t\right)$
being bounded above over $t\in\left(1-\delta,1\right]$. Then, the local linear
estimator defined in (11) is consistent for
$\left(\mu_{0},g\left(1\right)\right)$ and asymptotically normal:

$\left(\begin{array}[]{cc}\sqrt{nh_{n}}&\\\
&\sqrt{nh_{n}^{3}}\end{array}\right)\left[\left(\begin{array}[]{c}\hat{\mu}^{L}\\\
\hat{b}^{L}\end{array}\right)-\left(\begin{array}[]{c}\mu_{0}\\\
g\left(1\right)\end{array}\right)+\left(\begin{array}[]{c}\frac{\left(\kappa_{1}\kappa_{3}-\kappa_{2}^{2}\right)g^{\prime}\left(1\right)}{2\left(\kappa_{0}\kappa_{2}-\kappa_{1}^{2}\right)}h_{n}^{2}\\\
\frac{\left(\kappa_{0}\kappa_{3}-\kappa_{1}\kappa_{2}\right)g^{\prime}\left(1\right)}{2\left(\kappa_{0}\kappa_{2}-\kappa_{1}^{2}\right)}h_{n}\end{array}\right)\right]\rightarrow
N\left(0,\sigma_{U}^{2}\Omega^{L}\right),$

where $\sigma_{U}^{2}=EU_{i}^{2}$ and

$\Omega^{L}=\frac{1}{\left(\kappa_{0}\kappa_{2}-\kappa_{1}^{2}\right)^{2}}\left(\begin{array}[]{cc}\kappa_{2}^{2}\chi_{0}+\kappa_{1}^{2}\chi_{2}-2\kappa_{1}\kappa_{2}\chi_{1}&\kappa_{1}\kappa_{2}\chi_{0}+\kappa_{0}\kappa_{1}\chi_{2}-\left(\kappa_{0}\kappa_{2}+\kappa_{1}^{2}\right)\chi_{1}\\\
\kappa_{1}\kappa_{2}\chi_{0}+\kappa_{0}\kappa_{1}\chi_{2}-\left(\kappa_{0}\kappa_{2}+\kappa_{1}^{2}\right)\chi_{1}&\kappa_{1}^{2}\chi_{0}+\kappa_{0}^{2}\chi_{2}-2\kappa_{0}\kappa_{1}\chi_{1}\end{array}\right).$

Theorem 3 shows that the local linear estimation of the intercept
automatically eliminates the boundary effect, and its asymptotic bias is of
the same order as that in the interior. More interestingly, the local linear
procedure generates a consistent estimate of $g\left(1\right)$ as a byproduct,
which is a key ingredient of the optimal bandwidth for the kernel estimator
$\hat{\mu}$. There would be no technical difficulty in extending the results
of Theorem 3 to local polynomial estimation with higher order, except for more
complicated notation and more involved mathematical derivations.

###  4.1 Bandwidth selection

Provided $g^{\prime}\left(1\right)\neq 0$, the optimal bandwidth for
$\hat{\mu}^{L}$ is analogously defined by minimizing the asymptotic MSE of
$\hat{\mu}^{L}$:

$\displaystyle h_{opt}^{L}$ $\displaystyle=$
$\displaystyle\arg\min_{h}\left\\{\text{AMSE}_{h}\left(\hat{\mu}^{L}\right)=\left[\frac{\left(\kappa_{1}\kappa_{3}-\kappa_{2}^{2}\right)g^{\prime}\left(1\right)}{2\left(\kappa_{0}\kappa_{2}-\kappa_{1}^{2}\right)}\right]^{2}h^{4}+\frac{\left(\kappa_{2}^{2}\chi_{0}+\kappa_{1}^{2}\chi_{2}-2\kappa_{1}\kappa_{2}\chi_{1}\right)\sigma_{U}^{2}}{\left(\kappa_{0}\kappa_{2}-\kappa_{1}^{2}\right)^{2}}\frac{1}{nh}\right\\}$
$\displaystyle=$ $\displaystyle
c_{k}^{L}\left(\sigma_{U}^{2}\left/\left[g^{\prime}\left(1\right)\right]^{2}\right.\right)^{1/5}n^{-1/5},$

where

$c_{k}^{L}=\left[\frac{\kappa_{2}^{2}\chi_{0}+\kappa_{1}^{2}\chi_{2}-2\kappa_{1}\kappa_{2}\chi_{1}}{\left(\kappa_{1}\kappa_{3}-\kappa_{2}^{2}\right)^{2}}\right]^{1/5}.$
(12)

As in Subsection 3.3, I adopt the regularization method and propose the
following estimator for $h_{opt}^{L}$:

$\hat{h}_{opt}^{L}=c_{k}^{L}\left(\frac{\hat{\sigma}_{U}^{2}}{\left[\hat{g}^{\prime}\left(1\right)\right]^{2}+3\widehat{Var}\left(\hat{g}^{\prime}\left(1\right)\right)}\right)^{1/5}n^{-1/5}.$
(13)

To implement the plug-in bandwidth selection procedures (10) and (13), one
must estimate the limits of the derivative functions, $g\left(1\right)$ and
$g^{\prime}\left(1\right)$, the regularization terms,
$Var\left(\hat{g}\left(1\right)\right)$ and
$Var\left(\hat{g}^{\prime}\left(1\right)\right)$, and the variance of the
disturbance, $\sigma_{U}^{2}$. I first construct $\hat{g}\left(1\right)$ and
$\hat{g}^{\prime}\left(1\right)$ by fitting a quadratic function to the
observations near the boundary:

$\left(\hat{\mu}^{Q},\hat{g}\left(1\right),\hat{g}^{\prime}\left(1\right)\right)=\arg\min_{b_{0},b_{1},b_{2}}\sum_{i=1}^{n}\left[Y_{i}-Z_{i}^{\prime}\hat{\theta}-\sum_{r=0}^{2}\frac{\left(\hat{F}_{n}\left(\hat{W}_{i}\right)-1\right)^{r}}{r!}b_{r}\right]^{2}D_{i}k\left(\frac{1-\hat{F}_{n}\left(\hat{W}_{i}\right)}{h_{1n}}\right),$
(14)

where $h_{1n}$ is a pilot bandwidth. Similar to Theorem 3, one can show that
under $nh_{1n}^{7}=O\left(1\right)$, $nh_{1n}^{5}\rightarrow\infty$, and the
regularity conditions, the local quadratic estimator is consistent and
asymptotically normal:

$\left(\begin{array}[]{ccc}\sqrt{nh_{1n}}&&\\\ &\sqrt{nh_{1n}^{3}}&\\\
&&\sqrt{nh_{1n}^{5}}\end{array}\right)\left[\left(\begin{array}[]{c}\hat{\mu}^{Q}\\\
\hat{g}\left(1\right)\\\
\hat{g}^{\prime}\left(1\right)\end{array}\right)-\left(\begin{array}[]{c}\mu_{0}\\\
g\left(1\right)\\\
g^{\prime}\left(1\right)\end{array}\right)+\left(\begin{array}[]{c}B_{1}^{Q}g^{\prime\prime}\left(1\right)h_{1n}^{3}\\\
B_{2}^{Q}g^{\prime\prime}\left(1\right)h_{1n}^{2}\\\
B_{3}^{Q}g^{\prime\prime}\left(1\right)h_{1n}\end{array}\right)\right]\rightarrow
N\left(0,\sigma_{U}^{2}\Omega^{Q}\right).$

As a result, the regularization terms can be estimated by

$\widehat{Var}\left(\hat{g}\left(1\right)\right)=\frac{1}{nh_{1n}^{3}}\hat{\sigma}_{U}^{2}\Omega_{22}^{Q},\text{
\
}\widehat{Var}\left(\hat{g}^{\prime}\left(1\right)\right)=\frac{1}{nh_{1n}^{5}}\hat{\sigma}_{U}^{2}\Omega_{33}^{Q},$

where

$\displaystyle\Omega_{22}^{Q}$ $\displaystyle=$
$\displaystyle\frac{\left\\{\begin{array}[]{c}\left(\kappa_{1}\kappa_{4}-\kappa_{2}\kappa_{3}\right)^{2}\chi_{0}-2\left(\kappa_{1}\kappa_{4}-\kappa_{2}\kappa_{3}\right)\left(\kappa_{0}\kappa_{4}-\kappa_{2}^{2}\right)\chi_{1}\\\
+\left[\left(\kappa_{0}\kappa_{4}-\kappa_{2}^{2}\right)^{2}+2\left(\kappa_{1}\kappa_{4}-\kappa_{2}\kappa_{3}\right)\left(\kappa_{0}\kappa_{3}-\kappa_{1}\kappa_{2}\right)\right]\chi_{2}\\\
-\left(\kappa_{0}\kappa_{4}-\kappa_{2}^{2}\right)\left(\kappa_{0}\kappa_{3}-\kappa_{1}\kappa_{2}\right)\chi_{3}+\left(\kappa_{0}\kappa_{3}-\kappa_{1}\kappa_{2}\right)^{2}\chi_{4}\end{array}\right\\}}{\left(\kappa_{0}\kappa_{2}\kappa_{4}-\kappa_{0}\kappa_{3}^{2}-\kappa_{1}^{2}\kappa_{4}+2\kappa_{1}\kappa_{2}\kappa_{3}-\kappa_{2}^{3}\right)^{2}},$
(18) $\displaystyle\Omega_{33}^{Q}$ $\displaystyle=$
$\displaystyle\frac{\left\\{\begin{array}[]{c}\left(\kappa_{1}\kappa_{3}-\kappa_{2}^{2}\right)^{2}\chi_{0}-2\left(\kappa_{1}\kappa_{3}-\kappa_{2}^{2}\right)\left(\kappa_{0}\kappa_{3}-\kappa_{1}\kappa_{2}\right)\chi_{1}\\\
+\left[\left(\kappa_{0}\kappa_{3}-\kappa_{1}\kappa_{2}\right)^{2}+2\left(\kappa_{1}\kappa_{3}-\kappa_{2}^{2}\right)\left(\kappa_{0}\kappa_{2}-\kappa_{1}^{2}\right)\right]\chi_{2}\\\
-\left(\kappa_{0}\kappa_{3}-\kappa_{1}\kappa_{2}\right)\left(\kappa_{0}\kappa_{2}-\kappa_{1}^{2}\right)\chi_{3}+\left(\kappa_{0}\kappa_{2}-\kappa_{1}^{2}\right)^{2}\chi_{4}\end{array}\right\\}}{\left(\kappa_{0}\kappa_{2}\kappa_{4}-\kappa_{0}\kappa_{3}^{2}-\kappa_{1}^{2}\kappa_{4}+2\kappa_{1}\kappa_{2}\kappa_{3}-\kappa_{2}^{3}\right)^{2}}.$
(22)

Last, I estimate the variance of the disturbance by

$\hat{\sigma}_{U}^{2}=\frac{\displaystyle\sum_{i=1}^{n}\left(Y_{i}-\hat{\mu}^{Q}-Z_{i}^{\prime}\hat{\theta}\right)^{2}D_{i}k\left(\frac{1-\hat{F}_{n}\left(\hat{W}_{i}\right)}{h_{2n}}\right)}{\displaystyle\sum_{i=1}^{n}D_{i}k\left(\frac{1-\hat{F}_{n}\left(\hat{W}_{i}\right)}{h_{2n}}\right)},$
(23)

where $\hat{\mu}^{Q}$ is the initial estimate of $\mu_{0}$ given in (14) and
$h_{2n}$ is another pilot bandwidth. Following Theorem 2, it can be shown that

$\hat{\sigma}_{U}^{2}\overset{p}{\rightarrow}E\left[\left.\left(Y_{i}-\mu_{0}-Z_{i}^{\prime}\theta_{0}\right)^{2}\right|D_{i}=1,F_{W}\left(W_{i}\right)=1\right]=EU_{i}^{2}=\sigma_{U}^{2}.$

For the pilot bandwidths, simply setting

$h_{1n}=n^{-1/7},\text{ }h_{2n}=n^{-1/3}$

is sufficient to ensure the consistency of $\hat{h}_{opt}$ for $h_{opt}$ and
of $\hat{h}_{opt}^{L}$ for $h_{opt}^{L}$. In practice, the suggested bandwidth
selection algorithm is fairly robust to the choice of pilot bandwidth, which
is not surprising given the presence of the power $1/3$ or $1/5$ in the
expressions for the optimal bandwidths.

##  5 Simulation

This section examines the finite sample properties of the kernel regression
estimator $\hat{\mu}$ defined in (9) and the local linear estimator
$\hat{\mu}^{L}$ defined in (11), in comparison with the parametric two-step
estimator (Heckman, 1979) and the semiparametric identification-at-infinity
estimators (Heckman, 1990; Andrews and Schafgans, 1998). The parametric two-
step procedure implements probit estimation for the selection equation in the
first step and least squares estimation for the outcome equation with a
correction term using the uncensored observations in the second step. This
approach is commonly applied in empirical studies due to its computational
ease. However, it is likely to be inconsistent when the true distribution of
the model disturbance is nonnormal. In contrast, the consistency of Heckman
(1990)’s estimator (henceforth the Heckman estimator) and Andrews and
Schafgans (1998)’s estimator (henceforth the AS estimator) does not rely on
parametric specification of the disturbance distribution, but it is difficult
to choose an appropriate smoothing parameter for these estimators. To
investigate the robustness of their practical performance, a wide range of
smoothing parameters is considered. Following the literature, the choices
considered are based on the percentage of uncensored observations used in the
estimation. Specifically, the smoothing parameter takes the values of various
quantiles of the selection linear index in the uncensored subsample.

The simulation setting mainly follows Schafgans (2004), and the data
generating process is

$\displaystyle D_{i}$ $\displaystyle=$ $\displaystyle
1\left\\{c_{0}+X_{1i}+X_{2i}>\varepsilon_{i}\right\\},$ $\displaystyle
Y_{i}^{\ast}$ $\displaystyle=$ $\displaystyle\mu_{0}+U_{i},$ $\displaystyle
Y_{i}$ $\displaystyle=$ $\displaystyle Y_{i}^{\ast}D_{i},\text{
}i=1,2,\cdots,n,$

where $\mu_{0}=0$, $X_{1i}$ follows the standard normal distribution, $X_{2i}$
follows the standardized (zero-mean, unit-variance) Student’s $t$ distribution
with three degrees of freedom, $\varepsilon_{i}$ and $U_{i}$ are zero-mean
random variables described below, and only
$\left(Y_{i},D_{i},X_{1i},X_{2i}\right)$ is observed. In the simulation, the
outcome equation does not contain any nonconstant regressors, implying that
the intercept $\mu_{0}$ of primary concern represents the population mean of
the latent outcome $Y_{i}^{\ast}$. Different designs are constructed by
varying the disturbance distribution and the value of $c_{0}$.
$\varepsilon_{i}$ follows three different distributions, namely, the standard
normal distribution, the standardized Student’s $t$ distribution with three
degrees of freedom, and the standardized chi-square distribution with three
degrees of freedom. $U_{i}$ is generated by $U_{i}=\varepsilon_{i}+e_{i}$,
where $e_{i}$ is a standard normal random variable independent of
$\varepsilon_{i}$. The constant $c_{0}$ controls for the amount of censoring.
In the benchmark design, $c_{0}=0$, producing approximately 50% censoring.
Different values of $c_{0}$ are chosen so that
$\Pr\left(c_{0}+X_{1i}+X_{2i}>\varepsilon_{i}\right)$ is equal to 0.8 and 0.2,
corresponding to 20% and 80% proportions of zero observations, respectively.
The sample size $n$ is set to 250, 1000, 4000, and the simulation is
replicated 1000 times for each design.

Before calculating the semiparametric estimators for $\mu_{0}$, a
distribution-free estimate for the selection equation is necessary. Since the
simulation results of Schafgans (2004) show little sensitivity to the
particular choice of this estimate, I employ the average derivative estimation
(Powell et al., 1989) for computational convenience. When implementing the
Heckman and AS estimators, the smoothing parameter is equal to the 0.99, 0.95,
0.9, 0.8, 0.7, and 0.5 quantiles of
$\hat{W}_{i}=\hat{\beta}_{1}X_{1i}+\hat{\beta}_{2}X_{2i}$ in the uncensored
subsample, corresponding to 1%, 5%, 10%, 20%, 30%, and 50% uncensored
observations used in the estimation. The value of the smoothing parameter
declines with the proportion of uncensored observations. For consistency, the
smoothing parameter is required to approach infinity as $n$ goes to infinity
such that the estimation is based on only the observations for which
$\Pr\left(\left.D_{i}=1\right|X_{i}\right)$ is close to one and in the limit
is equal to one. Following the suggestion of Andrews and Schafgans (1998), the
smoothed function in the AS estimator is

$s\left(w\right)=\left\\{\begin{array}[]{c}0\\\
1-\exp\left(-\frac{w}{b-w}\right)\\\ 1\end{array}\begin{array}[]{l}\text{for
}w\leq 0,\\\ \text{for }0<w\leq b,\\\ \text{for }w>b,\end{array}\right.$

where $b$ is set equal to 1. Note that the AS estimator with $b=0$ is
equivalent to the Heckman estimator. For the proposed kernel-type estimators,
the bandwidth is chosen by the plug-in algorithm given in Subsection 4.1. The
commonly used Gaussian and Epanechnikov kernel functions are applied. However,
the Gaussian kernel is not compactly supported, and the Epanechnikov kernel is
not smooth. Therefore, I also consider the polynomial and polyweight kernels,
both of degree seven, which possess sufficient smoothness required by
Assumption 2’. The definition of these kernel functions and their relevant
functionals are presented in Table 1.

Table 1: Several kernel functions and their relevant functionals

Kernel | Definition |  $\kappa_{r}=\int_{0}^{\infty}t^{r}k\left(t\right)dt$, $r\in\mathbb{N}$ |  $\chi_{r}=\int_{0}^{\infty}t^{r}k^{2}\left(t\right)dt$, $r\in\mathbb{N}$
---|---|---|---
Gaussian | $\displaystyle k\left(t\right)=\frac{1}{\sqrt{2\pi}}\exp\left(-\frac{t^{2}}{2}\right)1\left\\{t\geq 0\right\\}$ | $\displaystyle\sqrt{\frac{2^{r-2}}{\pi}}\gamma\left(\frac{r+1}{2}\right)$ | $\displaystyle\frac{1}{4\pi}\gamma\left(\frac{r+1}{2}\right)$
Epanechnikov | $\displaystyle k\left(t\right)=\left\\{\QATOP{1-t^{2}\text{ if }0\leq t\leq 1}{0\text{ \ if }t>1}\right.$ | $\displaystyle\frac{2}{\left(r+1\right)\left(r+3\right)}$ | $\displaystyle\frac{8}{\left(r+1\right)\left(r+3\right)\left(r+5\right)}$
Polynomial of degree 7 | $\displaystyle k\left(t\right)=\left\\{\QATOP{\left(1-t\right)^{7}\text{ if }0\leq t\leq 1}{\text{ \ }0\text{\ \ \ if }t>1}\right.$ | $\displaystyle B\left(r+1,8\right)=\frac{7!r!}{\left(r+8\right)!}$ | $\displaystyle B\left(r+1,15\right)=\frac{14!r!}{\left(r+15\right)!}$
Polyweight of degree 7 | $\displaystyle k\left(t\right)=\left\\{\QATOP{\left(1-t^{2}\right)^{7}\text{ if }0\leq t\leq 1}{\text{ \ }0\text{\ \ \ \ if }t>1}\right.$ | $\displaystyle\frac{1}{2}B\left(\frac{r+1}{2},8\right)=\frac{2^{7}7!\left(r-1\right)!!}{\left(r+15\right)!!}$ | $\displaystyle\frac{1}{2}B\left(\frac{r+1}{2},15\right)=\frac{2^{14}14!\left(r-1\right)!!}{\left(r+29\right)!!}$
Kernel |  $c_{k}^{L}$ in (12) |  $\Omega_{22}^{Q}$ in (18) |  $\Omega_{33}^{Q}$ in (22)
Gaussian | $\left[\frac{\left(\pi+1-2\sqrt{2}\right)\sqrt{\pi}}{\left(4-\pi\right)^{2}}\right]^{1/5}\approx\allowbreak 1.\allowbreak 259$ | $\frac{\left(4\pi+11-12\sqrt{2}\right)\sqrt{\pi}}{8\left(\pi-3\right)^{2}}\approx\allowbreak 72.\allowbreak 89$ | $\frac{3\pi^{2}-\left(16-4\sqrt{2}\right)\pi+\left(44-24\sqrt{2}\right)}{16\sqrt{\pi}\left(\pi-3\right)^{2}}\approx\allowbreak 12.\allowbreak 62$
Epanechnikov | $3.200$ | $4913.0$ | $6327.8$
Polynomial of degree 7 | $8.175$ | $8477.5$ | $48857.0$
Polyweight of degree 7 | $5.396$ | $8645.0$ | $29139.5$

Note: $r!!$, $\gamma\left(r\right)$, and $B\left(r_{1},r_{2}\right)$ denote
the double factorial, gamma function, and beta function, respectively.

The summary statistics for the simulation are the estimators’ bias, standard
deviation (SD), root mean squared error (RMSE) ratio, and rejection rate of
the t test, over 1000 replications. The RMSE can be calculated from the bias
and SD and is thus omitted from the tables. Instead, the RMSE ratio defined by
the RMSE of the semiparametric estimators over that of the parametric two-step
estimator is reported. Although the RMSE (ratio) is the primary criterion used
to compare consistent estimators, it is useless if there are both consistent
and inconsistent estimators. An inconsistent estimator having a small RMSE
implies that the estimator is highly concentrated in a narrow interval
centered at a biased value, consequently leading to incorrect inference. As a
complement to the RMSE (ratio), I consider the simulated probability of
rejecting the null hypothesis $H_{0}:\mu_{0}=0$ against $H_{1}:\mu_{0}\neq 0$
at a 5% level of significance using t tests (or, more precisely, z tests),
based on the asymptotic variances given in Heckman (1979), Schafgans and
Zinde-Walsh (2002), Andrews and Schafgans (1998), and Theorems 2-3.

Tables 2-4 report the simulation results when the model disturbance follows a
normal distribution, a $t\left(3\right)$ distribution that is symmetric but
fat-tailed, and a $\chi^{2}\left(3\right)$ distribution that is skewed,
respectively, under approximately 50% censoring. Table 2 shows that, under
normal disturbance, the parametric two-step estimator is asymptotically
unbiased, as expected, and converges at a $\sqrt{n}$ rate (its SD halves when
the sample size quadruples). In contrast, all the considered semiparametric
estimators converge at slower than $\sqrt{n}$ rates, as their RMSE ratios
increase with the sample size. For the Heckman and AS estimators, the bias
increases and the SD decreases when the proportion of uncensored observations
used in the estimation increases or, equivalently, the smoothing parameter
decreases. The optimal smoothing parameter in terms of RMSE depends upon the
estimator, the sample size, and, by comparing across Tables 2-4, the
disturbance distribution. If the smoothing parameter is improperly chosen, the
RMSE may be several times larger than the smallest RMSE, and the rejection
rate may be far larger than the specified level of significance. Moreover, the
optimal smoothing parameter in terms of RMSE usually disagrees with that in
terms of the rejection rate. For instance, in the case of $n=1000$ for the
Heckman estimator and the case of $n=4000$ for the AS estimator, the optimal
smoothing parameter in terms of RMSE would use 30% uncensored observations in
the estimation, but the corresponding simulated rejection rates are more than
twice the real level. In these two cases, a more sensible choice would be to
use 20% uncensored observations, sacrificing a little RMSE but leading to a
rejection rate very close to 0.05. In practice, however, the RMSE and
rejection rate are not known; therefore, we are never aware of whether we have
made a good choice.

Table 2: Simulation results when $\varepsilon_{i}\sim N\left(0,1\right)$

$\Pr\left(Y_{i}=0\right)=0.5$ |  | $n=250$ |  | $n=1000$ |  | $n=4000$
---|---|---|---|---|---|---
Estimator |  | Bias | SD |  RMSE ratio |  Rejection rate |  | Bias | SD |  RMSE ratio |  Rejection rate |  | Bias | SD |  RMSE ratio |  Rejection rate
Heckman (1979)’s parametric two-step estimator
|  | -0.004 | 0.186 |  1 |  0.052 |  | -0.002 | 0.093 |  1 |  0.048 |  | 0.000 | 0.044 |  1 |  0.043
Heckman (1990)’s semiparametric estimator with various smoothing parameters
$1\%$ observations |  | -0.007 | 1.025 |  5.519 |  0.390 |  | 0.028 | 0.599 |  6.444 |  0.128 |  | 0.006 | 0.301 |  6.812 |  0.062
$5\%$ observations |  | -0.003 | 0.549 |  2.955 |  0.125 |  | 0.001 | 0.286 |  3.074 |  0.076 |  | -0.006 | 0.140 |  3.173 |  0.056
$10\%$ observations |  | -0.016 | 0.389 |  2.093 |  0.088 |  | -0.021 | 0.195 |  2.108 |  0.064 |  | -0.016 | 0.096 |  2.209 |  0.062
$20\%$ observations |  | -0.046 | 0.275 |  1.503 |  0.070 |  | -0.049 | 0.136 |  1.558 |  0.059 |  | -0.044 | 0.067 |  1.801 |  0.088
$30\%$ observations |  | -0.077 | 0.228 |  1.294 |  0.082 |  | -0.085 | 0.111 |  1.499 |  0.114 |  | -0.079 | 0.055 |  2.185 |  0.278
$50\%$ observations |  | -0.168 | 0.175 |  1.306 |  0.178 |  | -0.164 | 0.084 |  1.982 |  0.518 |  | -0.164 | 0.041 |  3.821 |  0.983
Andrews and Schafgans (1998)’s semiparametric estimator with various smoothing
parameters
$1\%$ observations |  | -0.055 | 1.302 |  7.013 |  0.616 |  | 0.037 | 0.697 |  7.498 |  0.144 |  | 0.010 | 0.341 |  7.710 |  0.065
$5\%$ observations |  | -0.007 | 0.689 |  3.711 |  0.165 |  | 0.007 | 0.338 |  3.632 |  0.077 |  | -0.005 | 0.165 |  3.737 |  0.048
$10\%$ observations |  | -0.012 | 0.480 |  2.585 |  0.106 |  | -0.007 | 0.241 |  2.589 |  0.074 |  | -0.008 | 0.116 |  2.628 |  0.050
$20\%$ observations |  | -0.027 | 0.331 |  1.787 |  0.076 |  | -0.029 | 0.163 |  1.775 |  0.064 |  | -0.024 | 0.079 |  1.857 |  0.051
$30\%$ observations |  | -0.045 | 0.263 |  1.438 |  0.076 |  | -0.052 | 0.130 |  1.508 |  0.070 |  | -0.046 | 0.063 |  1.769 |  0.101
$50\%$ observations |  | -0.106 | 0.198 |  1.211 |  0.111 |  | -0.110 | 0.096 |  1.569 |  0.214 |  | -0.106 | 0.047 |  2.629 |  0.596
Kernel regression (local constant) estimator with various kernel functions
Gaussian |  | -0.031 | 0.302 |  1.635 |  0.058 |  | -0.027 | 0.165 |  1.797 |  0.072 |  | -0.016 | 0.090 |  2.073 |  0.058
Epanechnikov |  | -0.009 | 0.550 |  2.959 |  0.038 |  | 0.004 | 0.312 |  3.357 |  0.052 |  | -0.005 | 0.171 |  3.874 |  0.042
7th polynomial |  | -0.017 | 0.653 |  3.514 |  0.064 |  | 0.012 | 0.357 |  3.843 |  0.052 |  | -0.001 | 0.192 |  4.333 |  0.044
7th polyweight |  | -0.012 | 0.605 |  3.259 |  0.046 |  | 0.009 | 0.344 |  3.696 |  0.049 |  | -0.004 | 0.187 |  4.222 |  0.039
Local linear estimator with various kernel functions
Gaussian |  | 0.032 | 0.284 |  1.539 |  0.092 |  | 0.029 | 0.148 |  1.616 |  0.091 |  | 0.032 | 0.077 |  1.894 |  0.102
Epanechnikov |  | 0.019 | 0.424 |  2.284 |  0.057 |  | 0.014 | 0.235 |  2.535 |  0.053 |  | 0.014 | 0.125 |  2.843 |  0.045
7th polynomial |  | -0.001 | 0.553 |  2.979 |  0.089 |  | 0.015 | 0.312 |  3.353 |  0.076 |  | 0.006 | 0.167 |  3.791 |  0.053
7th polyweight |  | 0.005 | 0.491 |  2.643 |  0.065 |  | 0.013 | 0.278 |  2.989 |  0.059 |  | 0.008 | 0.151 |  3.424 |  0.063

Table 3: Simulation results when $\varepsilon_{i}\sim t\left(3\right)$

$\Pr\left(Y_{i}=0\right)=0.5$ |  | $n=250$ |  | $n=1000$ |  | $n=4000$
---|---|---|---|---|---|---
Estimator |  | Bias | SD |  RMSE ratio |  Rejection rate |  | Bias | SD |  RMSE ratio |  Rejection rate |  | Bias | SD |  RMSE ratio |  Rejection rate
Heckman (1979)’s parametric two-step estimator
|  | 0.014 | 0.181 |  1 |  0.059 |  | 0.027 | 0.089 |  1 |  0.058 |  | 0.025 | 0.047 |  1 |  0.114
Heckman (1990)’s semiparametric estimator with various smoothing parameters
$1\%$ observations |  | -0.052 | 0.981 |  5.403 |  0.386 |  | 0.003 | 0.592 |  6.363 |  0.155 |  | -0.001 | 0.296 |  5.594 |  0.069
$5\%$ observations |  | -0.050 | 0.533 |  2.945 |  0.121 |  | -0.030 | 0.271 |  2.934 |  0.070 |  | -0.029 | 0.135 |  2.603 |  0.056
$10\%$ observations |  | -0.049 | 0.367 |  2.035 |  0.073 |  | -0.033 | 0.186 |  2.029 |  0.056 |  | -0.039 | 0.097 |  1.975 |  0.081
$20\%$ observations |  | -0.061 | 0.260 |  1.468 |  0.063 |  | -0.052 | 0.133 |  1.532 |  0.065 |  | -0.056 | 0.067 |  1.654 |  0.137
$30\%$ observations |  | -0.076 | 0.210 |  1.226 |  0.073 |  | -0.068 | 0.105 |  1.351 |  0.092 |  | -0.072 | 0.054 |  1.701 |  0.274
$50\%$ observations |  | -0.120 | 0.163 |  1.115 |  0.113 |  | -0.109 | 0.080 |  1.452 |  0.257 |  | -0.111 | 0.042 |  2.238 |  0.770
Andrews and Schafgans (1998)’s semiparametric estimator with various smoothing
parameters
$1\%$ observations |  | -0.089 | 1.285 |  7.083 |  0.627 |  | 0.006 | 0.711 |  7.646 |  0.197 |  | 0.005 | 0.346 |  6.541 |  0.072
$5\%$ observations |  | -0.059 | 0.659 |  3.641 |  0.137 |  | -0.018 | 0.325 |  3.503 |  0.069 |  | -0.022 | 0.157 |  2.999 |  0.060
$10\%$ observations |  | -0.054 | 0.476 |  2.633 |  0.089 |  | -0.028 | 0.227 |  2.460 |  0.060 |  | -0.031 | 0.113 |  2.209 |  0.068
$20\%$ observations |  | -0.051 | 0.311 |  1.733 |  0.069 |  | -0.039 | 0.156 |  1.727 |  0.058 |  | -0.045 | 0.080 |  1.727 |  0.087
$30\%$ observations |  | -0.061 | 0.246 |  1.395 |  0.063 |  | -0.052 | 0.123 |  1.440 |  0.062 |  | -0.057 | 0.064 |  1.616 |  0.161
$50\%$ observations |  | -0.089 | 0.184 |  1.123 |  0.075 |  | -0.080 | 0.093 |  1.319 |  0.142 |  | -0.083 | 0.048 |  1.813 |  0.438
Kernel regression (local constant) estimator with various kernel functions
Gaussian |  | -0.061 | 0.278 |  1.564 |  0.051 |  | -0.041 | 0.153 |  1.705 |  0.066 |  | -0.042 | 0.086 |  1.798 |  0.090
Epanechnikov |  | -0.053 | 0.547 |  3.021 |  0.041 |  | -0.021 | 0.303 |  3.264 |  0.048 |  | -0.020 | 0.163 |  3.099 |  0.058
7th polynomial |  | -0.058 | 0.642 |  3.545 |  0.061 |  | -0.013 | 0.355 |  3.824 |  0.059 |  | -0.014 | 0.190 |  3.596 |  0.055
7th polyweight |  | -0.055 | 0.605 |  3.341 |  0.037 |  | -0.016 | 0.335 |  3.610 |  0.053 |  | -0.017 | 0.181 |  3.427 |  0.054
Local linear estimator with various kernel functions
Gaussian |  | -0.020 | 0.270 |  1.488 |  0.072 |  | -0.009 | 0.147 |  1.583 |  0.069 |  | -0.018 | 0.082 |  1.587 |  0.067
Epanechnikov |  | -0.034 | 0.399 |  2.200 |  0.046 |  | -0.016 | 0.222 |  2.397 |  0.046 |  | -0.022 | 0.125 |  2.403 |  0.060
7th polynomial |  | -0.044 | 0.541 |  2.987 |  0.076 |  | -0.013 | 0.301 |  3.238 |  0.071 |  | -0.015 | 0.164 |  3.112 |  0.070
7th polyweight |  | -0.039 | 0.474 |  2.616 |  0.050 |  | -0.014 | 0.265 |  2.850 |  0.056 |  | -0.019 | 0.147 |  2.803 |  0.068

Table 4: Simulation results when $\varepsilon_{i}\sim\chi^{2}\left(3\right)$

$\Pr\left(Y_{i}=0\right)=0.5$ |  | $n=250$ |  | $n=1000$ |  | $n=4000$
---|---|---|---|---|---|---
Estimator |  | Bias | SD |  RMSE ratio |  Rejection rate |  | Bias | SD |  RMSE ratio |  Rejection rate |  | Bias | SD |  RMSE ratio |  Rejection rate
Heckman (1979)’s parametric two-step estimator
|  | -0.146 | 0.167 |  1 |  0.180 |  | -0.145 | 0.082 |  1 |  0.459 |  | -0.146 | 0.041 |  1 |  0.913
Heckman (1990)’s semiparametric estimator with various smoothing parameters
$1\%$ observations |  | -0.008 | 0.977 |  4.405 |  0.413 |  | 0.004 | 0.578 |  3.465 |  0.134 |  | -0.024 | 0.299 |  1.978 |  0.076
$5\%$ observations |  | -0.016 | 0.514 |  2.321 |  0.125 |  | -0.047 | 0.258 |  1.569 |  0.067 |  | -0.052 | 0.128 |  0.913 |  0.073
$10\%$ observations |  | -0.057 | 0.351 |  1.603 |  0.080 |  | -0.084 | 0.183 |  1.206 |  0.087 |  | -0.083 | 0.088 |  0.797 |  0.140
$20\%$ observations |  | -0.117 | 0.243 |  1.215 |  0.099 |  | -0.129 | 0.124 |  1.072 |  0.189 |  | -0.126 | 0.059 |  0.917 |  0.530
$30\%$ observations |  | -0.162 | 0.200 |  1.160 |  0.152 |  | -0.165 | 0.098 |  1.148 |  0.395 |  | -0.163 | 0.048 |  1.124 |  0.911
$50\%$ observations |  | -0.229 | 0.149 |  1.232 |  0.354 |  | -0.230 | 0.073 |  1.445 |  0.869 |  | -0.229 | 0.037 |  1.535 |  1.000
Andrews and Schafgans (1998)’s semiparametric estimator with various smoothing
parameters
$1\%$ observations |  | -0.009 | 1.219 |  5.497 |  0.605 |  | 0.021 | 0.694 |  4.162 |  0.180 |  | -0.009 | 0.350 |  2.315 |  0.078
$5\%$ observations |  | -0.009 | 0.624 |  2.816 |  0.153 |  | -0.026 | 0.315 |  1.893 |  0.067 |  | -0.040 | 0.156 |  1.061 |  0.068
$10\%$ observations |  | -0.037 | 0.440 |  1.990 |  0.102 |  | -0.057 | 0.217 |  1.343 |  0.066 |  | -0.063 | 0.108 |  0.822 |  0.097
$20\%$ observations |  | -0.084 | 0.291 |  1.365 |  0.081 |  | -0.099 | 0.148 |  1.068 |  0.113 |  | -0.097 | 0.072 |  0.799 |  0.251
$30\%$ observations |  | -0.119 | 0.231 |  1.173 |  0.106 |  | -0.131 | 0.116 |  1.048 |  0.210 |  | -0.127 | 0.056 |  0.919 |  0.609
$50\%$ observations |  | -0.182 | 0.171 |  1.124 |  0.206 |  | -0.185 | 0.084 |  1.217 |  0.582 |  | -0.185 | 0.042 |  1.250 |  0.991
Kernel regression (local constant) estimator with various kernel functions
Gaussian |  | -0.087 | 0.278 |  1.316 |  0.093 |  | -0.092 | 0.157 |  1.091 |  0.131 |  | -0.079 | 0.088 |  0.783 |  0.199
Epanechnikov |  | -0.019 | 0.526 |  2.372 |  0.051 |  | -0.028 | 0.302 |  1.817 |  0.067 |  | -0.038 | 0.167 |  1.128 |  0.068
7th polynomial |  | -0.015 | 0.624 |  2.816 |  0.071 |  | -0.017 | 0.355 |  2.131 |  0.072 |  | -0.030 | 0.196 |  1.308 |  0.062
7th polyweight |  | -0.016 | 0.580 |  2.615 |  0.045 |  | -0.020 | 0.337 |  2.024 |  0.069 |  | -0.033 | 0.186 |  1.245 |  0.066
Local linear estimator with various kernel functions
Gaussian |  | -0.066 | 0.258 |  1.202 |  0.095 |  | -0.070 | 0.134 |  0.907 |  0.122 |  | -0.065 | 0.069 |  0.627 |  0.183
Epanechnikov |  | -0.027 | 0.387 |  1.751 |  0.056 |  | -0.044 | 0.222 |  1.355 |  0.066 |  | -0.039 | 0.122 |  0.846 |  0.071
7th polynomial |  | -0.007 | 0.530 |  2.392 |  0.090 |  | -0.019 | 0.302 |  1.813 |  0.072 |  | -0.026 | 0.167 |  1.116 |  0.071
7th polyweight |  | -0.012 | 0.466 |  2.102 |  0.063 |  | -0.029 | 0.266 |  1.605 |  0.059 |  | -0.031 | 0.148 |  0.997 |  0.072

By contrast, the kernel regression and local linear estimators have no
difficulty in choosing the bandwidth parameter because fully data-driven
procedures of selecting the optimal bandwidths have been explicitly proposed.
Table 2 shows that the proposed bandwidths perform satisfactorily in that the
rejection rates for the kernel regression and local linear estimators are all
close to 0.05 for different sample sizes and different kernel functions. In
terms of RMSE, the local linear estimator is superior to the kernel regression
estimator, as expected. The RMSE of the local linear estimator with Gaussian
kernel is comparable to the optimal RMSE of the Heckman and AS estimators.
However, the Gaussian kernel may not be the best choice because over-rejection
for the t test appears to be a problem. As an alternative, the local linear
estimator with Epanechnikov kernel achieves a satisfactory balance between
RMSE and rejection rate. The polynomial and polyweight kernels also do not
lead to size distortion of the t test but are clearly outperformed by the
simple Epanechnikov kernel in terms of RMSE.

Table 3 investigates the finite sample behavior of the estimators under
$t\left(3\right)$ distributed disturbance. The parametric two-step estimator
performs well for small sample sizes but deteriorates as the sample size
increases because its nonvanishing bias due to nonnormality becomes large
relative to its declining SD. By contrast, the performance of the
semiparametric estimators is robust to nonnormality, and the main findings are
almost the same as in the normal case. First, if the smoothing parameter of
the Heckman and AS estimators is improperly chosen, their RMSEs may be
increased by several factors and the t tests may yield misleading inferences.
Second, the smoothing parameter giving rise to the smallest RMSE may lead to
evident over-rejection for the t test. These facts indicate the importance of
selecting a proper smoothing parameter for the identification-at-infinity
estimators, but this task is difficult since the RMSE and rejection rate are
unobserved in practice. Third, the bandwidth selection algorithms proposed for
the kernel-type estimators perform well in terms of rejection rate. Fourth,
the local linear estimator dominates the kernel regression estimator. Fifth,
the RMSE of the local linear estimator with Gaussian kernel is comparable to
(when $n=4000$, even smaller than) the smallest RMSE of the identification-at-
infinity estimators. Sixth, using the Epanechnikov kernel results in more
accurate rejection rate.

Table 4 considers the case of $\chi^{2}\left(3\right)$ distributed
disturbance. In this case, the parametric two-step estimator has notable bias,
and the probability of making a type I error rapidly approaches one.
Comparison with Table 3 indicates that the skewness of the nonnormal
disturbance exerts a worse influence on the parametric estimator than does the
fat tail. The Heckman and AS estimators become less robust to the smoothing
parameter under this design, in the sense that the rejection rate is below 0.1
for only a narrow range of smoothing parameters. On the other hand, the local
linear estimator with Epanechnikov kernel still has desirable finite sample
properties in terms of both RMSE and rejection rate.

Tables 5-6 investigate the effect of the proportion of censoring. Table 5
shows that, under mild censoring, all the considered estimators behave better.
The parametric two-step estimator becomes less biased in nonnormal designs.
The Heckman and AS estimators are more robust to the smoothing parameter, and
the rejection rates for the kernel regression and local linear estimators are
closer to the true level, even when the Gaussian kernel is used. Table 6
reveals the reverse side of the coin, with an unchanged conclusion being the
superiority of the local linear estimator with Epanechnikov kernel.

Table 5: Simulation results when $n=1000$ and the amount of zero observations
is small

$\Pr\left(Y_{i}=0\right)=0.2$ |  | $\varepsilon_{i}\sim N\left(0,1\right)$ |  | $\varepsilon_{i}\sim t\left(3\right)$ |  | $\varepsilon_{i}\sim\chi^{2}\left(3\right)$
---|---|---|---|---|---|---
Estimator |  | Bias | SD |  RMSE ratio |  Rejection rate |  | Bias | SD |  RMSE ratio |  Rejection rate |  | Bias | SD |  RMSE ratio |  Rejection rate
Heckman (1979)’s parametric two-step estimator
|  | -0.001 | 0.065 |  1 |  0.055 |  | -0.015 | 0.068 |  1 |  0.070 |  | -0.071 | 0.066 |  1 |  0.239
Heckman (1990)’s semiparametric estimator with various smoothing parameters
$1\%$ observations |  | -0.006 | 0.483 |  7.447 |  0.103 |  | -0.014 | 0.466 |  6.719 |  0.099 |  | -0.004 | 0.477 |  4.920 |  0.102
$5\%$ observations |  | 0.006 | 0.224 |  3.449 |  0.065 |  | -0.017 | 0.218 |  3.145 |  0.061 |  | -0.021 | 0.206 |  2.134 |  0.057
$10\%$ observations |  | 0.002 | 0.163 |  2.510 |  0.063 |  | -0.027 | 0.153 |  2.231 |  0.049 |  | -0.036 | 0.145 |  1.545 |  0.054
$20\%$ observations |  | -0.001 | 0.113 |  1.743 |  0.056 |  | -0.035 | 0.132 |  1.970 |  0.058 |  | -0.056 | 0.106 |  1.242 |  0.081
$30\%$ observations |  | -0.012 | 0.095 |  1.474 |  0.057 |  | -0.042 | 0.102 |  1.589 |  0.082 |  | -0.075 | 0.087 |  1.183 |  0.153
$50\%$ observations |  | -0.036 | 0.073 |  1.252 |  0.093 |  | -0.054 | 0.074 |  1.317 |  0.130 |  | -0.111 | 0.067 |  1.336 |  0.434
Andrews and Schafgans (1998)’s semiparametric estimator with various smoothing
parameters
$1\%$ observations |  | -0.006 | 0.573 |  8.841 |  0.126 |  | 0.002 | 0.536 |  7.723 |  0.109 |  | -0.016 | 0.573 |  5.917 |  0.136
$5\%$ observations |  | 0.002 | 0.264 |  4.077 |  0.061 |  | -0.012 | 0.258 |  3.714 |  0.056 |  | -0.010 | 0.258 |  2.661 |  0.070
$10\%$ observations |  | 0.006 | 0.190 |  2.924 |  0.061 |  | -0.021 | 0.182 |  2.642 |  0.061 |  | -0.026 | 0.172 |  1.797 |  0.051
$20\%$ observations |  | 0.000 | 0.132 |  2.037 |  0.060 |  | -0.030 | 0.130 |  1.918 |  0.060 |  | -0.042 | 0.121 |  1.325 |  0.064
$30\%$ observations |  | -0.005 | 0.107 |  1.651 |  0.054 |  | -0.036 | 0.116 |  1.755 |  0.074 |  | -0.057 | 0.100 |  1.185 |  0.095
$50\%$ observations |  | -0.020 | 0.081 |  1.285 |  0.066 |  | -0.046 | 0.088 |  1.423 |  0.091 |  | -0.088 | 0.074 |  1.189 |  0.243
Kernel regression (local constant) estimator with various kernel functions
Gaussian |  | 0.002 | 0.160 |  2.466 |  0.051 |  | -0.025 | 0.155 |  2.259 |  0.063 |  | -0.031 | 0.147 |  1.553 |  0.049
Epanechnikov |  | -0.002 | 0.302 |  4.663 |  0.043 |  | -0.011 | 0.300 |  4.322 |  0.054 |  | -0.005 | 0.305 |  3.150 |  0.059
7th polynomial |  | -0.004 | 0.345 |  5.321 |  0.056 |  | -0.007 | 0.342 |  4.920 |  0.057 |  | -0.010 | 0.359 |  3.711 |  0.074
7th polyweight |  | -0.004 | 0.330 |  5.094 |  0.043 |  | -0.012 | 0.329 |  4.748 |  0.056 |  | -0.005 | 0.337 |  3.482 |  0.063
Local linear estimator with various kernel functions
Gaussian |  | 0.017 | 0.143 |  2.222 |  0.070 |  | -0.024 | 0.147 |  2.151 |  0.067 |  | -0.020 | 0.128 |  1.334 |  0.057
Epanechnikov |  | 0.005 | 0.234 |  3.608 |  0.057 |  | -0.016 | 0.227 |  3.280 |  0.053 |  | -0.011 | 0.217 |  2.247 |  0.044
7th polynomial |  | 0.003 | 0.301 |  4.635 |  0.056 |  | -0.008 | 0.299 |  4.310 |  0.063 |  | -0.005 | 0.304 |  3.135 |  0.077
7th polyweight |  | 0.008 | 0.274 |  4.227 |  0.054 |  | -0.010 | 0.273 |  3.933 |  0.056 |  | -0.008 | 0.265 |  2.734 |  0.061

Table 6: Simulation results when $n=1000$ and the amount of zero observations
is large

$\Pr\left(Y_{i}=0\right)=0.8$ |  | $\varepsilon_{i}\sim N\left(0,1\right)$ |  | $\varepsilon_{i}\sim t\left(3\right)$ |  | $\varepsilon_{i}\sim\chi^{2}\left(3\right)$
---|---|---|---|---|---|---
Estimator |  | Bias | SD |  RMSE ratio |  Rejection rate |  | Bias | SD |  RMSE ratio |  Rejection rate |  | Bias | SD |  RMSE ratio |  Rejection rate
Heckman (1979)’s parametric two-step estimator
|  | 0.001 | 0.168 |  1 |  0.045 |  | 0.195 | 0.231 |  1 |  0.193 |  | -0.234 | 0.132 |  1 |  0.428
Heckman (1990)’s semiparametric estimator with various smoothing parameters
$1\%$ observations |  | -0.040 | 0.889 |  5.285 |  0.316 |  | 0.030 | 0.879 |  2.907 |  0.343 |  | -0.067 | 0.855 |  3.186 |  0.325
$5\%$ observations |  | -0.035 | 0.403 |  2.404 |  0.086 |  | -0.044 | 0.415 |  1.380 |  0.097 |  | -0.081 | 0.391 |  1.485 |  0.098
$10\%$ observations |  | -0.066 | 0.289 |  1.762 |  0.080 |  | -0.056 | 0.292 |  0.984 |  0.064 |  | -0.133 | 0.277 |  1.141 |  0.114
$20\%$ observations |  | -0.154 | 0.208 |  1.539 |  0.149 |  | -0.099 | 0.212 |  0.772 |  0.089 |  | -0.212 | 0.188 |  1.053 |  0.219
$30\%$ observations |  | -0.237 | 0.166 |  1.716 |  0.324 |  | -0.143 | 0.171 |  0.737 |  0.145 |  | -0.269 | 0.148 |  1.140 |  0.445
$50\%$ observations |  | -0.388 | 0.124 |  2.419 |  0.876 |  | -0.228 | 0.134 |  0.875 |  0.452 |  | -0.362 | 0.113 |  1.409 |  0.891
Andrews and Schafgans (1998)’s semiparametric estimator with various smoothing
parameters
$1\%$ observations |  | -0.021 | 1.085 |  6.449 |  0.428 |  | 0.024 | 1.011 |  3.345 |  0.442 |  | -0.062 | 1.039 |  3.869 |  0.436
$5\%$ observations |  | -0.022 | 0.497 |  2.955 |  0.105 |  | -0.032 | 0.487 |  1.614 |  0.104 |  | -0.056 | 0.485 |  1.814 |  0.108
$10\%$ observations |  | -0.039 | 0.347 |  2.072 |  0.075 |  | -0.048 | 0.353 |  1.178 |  0.077 |  | -0.094 | 0.335 |  1.294 |  0.094
$20\%$ observations |  | -0.093 | 0.243 |  1.545 |  0.080 |  | -0.070 | 0.251 |  0.860 |  0.071 |  | -0.159 | 0.230 |  1.040 |  0.136
$30\%$ observations |  | -0.155 | 0.195 |  1.480 |  0.150 |  | -0.098 | 0.203 |  0.744 |  0.088 |  | -0.209 | 0.181 |  1.027 |  0.233
$50\%$ observations |  | -0.282 | 0.144 |  1.878 |  0.526 |  | -0.162 | 0.152 |  0.736 |  0.199 |  | -0.291 | 0.131 |  1.187 |  0.610
Kernel regression (local constant) estimator with various kernel functions
Gaussian |  | -0.142 | 0.216 |  1.535 |  0.207 |  | -0.110 | 0.193 |  0.734 |  0.156 |  | -0.201 | 0.198 |  1.050 |  0.349
Epanechnikov |  | -0.052 | 0.306 |  1.843 |  0.062 |  | -0.055 | 0.296 |  0.994 |  0.066 |  | -0.115 | 0.294 |  1.174 |  0.098
7th polynomial |  | -0.041 | 0.355 |  2.122 |  0.069 |  | -0.043 | 0.343 |  1.142 |  0.062 |  | -0.094 | 0.359 |  1.380 |  0.094
7th polyweight |  | -0.042 | 0.335 |  2.005 |  0.062 |  | -0.050 | 0.325 |  1.087 |  0.060 |  | -0.098 | 0.329 |  1.276 |  0.089
Local linear estimator with various kernel functions
Gaussian |  | -0.101 | 0.188 |  1.264 |  0.224 |  | -0.012 | 0.182 |  0.602 |  0.139 |  | -0.169 | 0.181 |  0.920 |  0.328
Epanechnikov |  | -0.035 | 0.244 |  1.467 |  0.087 |  | -0.023 | 0.230 |  0.764 |  0.068 |  | -0.134 | 0.217 |  0.947 |  0.136
7th polynomial |  | -0.005 | 0.317 |  1.885 |  0.082 |  | -0.022 | 0.306 |  1.015 |  0.078 |  | -0.087 | 0.306 |  1.182 |  0.104
7th polyweight |  | -0.011 | 0.284 |  1.686 |  0.081 |  | -0.023 | 0.272 |  0.904 |  0.067 |  | -0.106 | 0.263 |  1.054 |  0.112

##  6 Conclusion

This paper rephrases the identification at infinity into an identification at
the boundary via a CDF transformation and accordingly proposes a kernel
approach to semiparametrically estimate the intercept of the sample selection
model. The proposed kernel regression estimator with generic transformation is
a generalization of the identification-at-infinity estimators and thus
inherits the disadvantage that the asymptotic bias and variance are implicit
functions of the bandwidth parameter. To select a bandwidth that minimizes the
asymptotic mean squared error, I use a specific transformation, namely, the
empirical CDF of the selection index, under which the asymptotic bias and
variance become explicit with respect to the bandwidth. A plug-in bandwidth
selection algorithm with regularization is therefore suggested. For the
purpose of bias reduction, I further propose a local linear estimator and an
associated analogous bandwidth selection algorithm. A simulation study
illustrates the effectiveness of the selected bandwidths. Comparison of the
finite sample performance of the estimators indicates that the local linear
estimator with Epanechnikov kernel is superior to the parametric two-step
estimator under nonnormal disturbance and to the identification-at-infinity
estimators in most cases.

##  7 Appendix

###  7.1 Proofs of the Example, Lemma, and Corollary

###### Proof of the Example.

By Assumption 2, there exists a $\delta\in\left(0,1\right)$ and a positive
$C_{\delta}$ such that $k\left(u\right)\geq C_{\delta}$ for any
$u\in\left[0,\delta\right]$. Therefore,

$\displaystyle Ek_{ni}^{2}$ $\displaystyle=$
$\displaystyle\int_{0}^{1}k^{2}\left(\frac{1-t}{h_{n}}\right)f_{F\left(W\right)}\left(t\right)dt=h_{n}\int_{0}^{1\left/h_{n}\right.}k^{2}\left(u\right)f_{F\left(W\right)}\left(1-uh_{n}\right)du$
$\displaystyle=$ $\displaystyle
h_{n}\int_{0}^{1}k^{2}\left(u\right)f_{F\left(W\right)}\left(1-uh_{n}\right)du\geq
C_{\delta}^{2}h_{n}\int_{0}^{\delta}f_{F\left(W\right)}\left(1-uh_{n}\right)du$
$\displaystyle=$ $\displaystyle C_{\delta}^{2}\int_{1-\delta
h_{n}}^{1}f_{F\left(W\right)}\left(s\right)ds=C_{\delta}^{2}\Pr\left(F\left(W_{i}\right)>1-\delta
h_{n}\right).$

Since $F\left(\cdot\right)$ is the standard Laplacian CDF, the ratio in
Assumption 5.(iii) is bounded by

$\frac{\left[\Pr\left(F\left(W_{i}\right)>1-h_{n}\right)\right]^{1+c}}{Ek_{ni}^{2}}\leq\frac{\left[\Pr\left(F\left(W_{i}\right)>1-h_{n}\right)\right]^{1+c}}{C_{\delta}^{2}\Pr\left(F\left(W_{i}\right)>1-\delta
h_{n}\right)}=\frac{\left[\Pr\left(W_{i}>-\log\left(2h_{n}\right)\right)\right]^{1+c}}{C_{\delta}^{2}\Pr\left(W_{i}>-\log\left(2\delta
h_{n}\right)\right)}.$

Denote $t_{n}=-\log\left(2h_{n}\right)\rightarrow+\infty$.

(i) If the distribution of $W_{i}$ has a power-type upper tail such that
$\Pr\left(W_{i}>t\right)\sim t^{-\lambda}$ for some $\lambda>0$, we have

$\frac{\left[\Pr\left(W_{i}>-\log\left(2h_{n}\right)\right)\right]^{1+c}}{C_{\delta}^{2}\Pr\left(W_{i}>-\log\left(2\delta
h_{n}\right)\right)}\sim\frac{\left(t_{n}-\log\delta\right)^{\lambda}}{t_{n}^{\lambda\left(1+c\right)}}\rightarrow
0$

for any $c>0$ and Assumption 5.(iii) holds.

(ii) If the distribution of $W_{i}$ has an exponential-type upper tail such
that $\Pr\left(W_{i}>t\right)\sim\exp\left(-c_{0}t^{\lambda}\right)$ for some
$\lambda>0$ and $c_{0}>0$, we have

$\displaystyle\frac{\left[\Pr\left(W_{i}>-\log\left(2h_{n}\right)\right)\right]^{1+c}}{C_{\delta}^{2}\Pr\left(W_{i}>-\log\left(2\delta
h_{n}\right)\right)}$ $\displaystyle\sim$
$\displaystyle\frac{\exp\left\\{-\left(1+c\right)c_{0}t_{n}^{\lambda}\right\\}}{\exp\left\\{-c_{0}\left(t_{n}-\log\delta\right)^{\lambda}\right\\}}$
$\displaystyle=$
$\displaystyle\exp\left\\{c_{0}t_{n}^{\lambda}\left[\left(1+\frac{-\log\delta}{t_{n}}\right)^{\lambda}-\left(1+c\right)\right]\right\\}$
$\displaystyle\rightarrow$ $\displaystyle 0$

for any $c>0$ and Assumption 5.(iii) holds.

(iii) If the upper tail of $W_{i}$’s distribution decays as rapidly as
$\Pr\left(W_{i}>t\right)\sim\exp\left(-\exp\left(c_{0}t^{\lambda}\right)\right)$,
however, Assumption 5.(iii) is guaranteed to hold only for $\lambda<1$,
because in this case

$\displaystyle\frac{\left[\Pr\left(W_{i}>-\log\left(2h_{n}\right)\right)\right]^{1+c}}{C_{\delta}^{2}\Pr\left(W_{i}>-\log\left(2\delta
h_{n}\right)\right)}$ $\displaystyle\sim$
$\displaystyle\frac{\exp\left\\{-\left(1+c\right)\exp\left(c_{0}t_{n}^{\lambda}\right)\right\\}}{\exp\left\\{-\exp\left[c_{0}\left(t_{n}-\log\delta\right)^{\lambda}\right]\right\\}}$
$\displaystyle=$
$\displaystyle\exp\left\\{\exp\left(c_{0}t_{n}^{\lambda}\right)\left[\exp\left[c_{0}t_{n}^{\lambda}\left(\left(1+\frac{-\log\delta}{t_{n}}\right)^{\lambda}-1\right)\right]-\left(1+c\right)\right]\right\\}$
$\displaystyle\rightarrow$ $\displaystyle\left\\{\begin{array}[]{ll}0&\text{if
}\left(\lambda<1\right)\text{ or }\left(\lambda=1\text{ and
}c>\delta^{-c_{0}}-1\right)\\\ 1&\text{if }\lambda=1\text{ and
}c=\delta^{-c_{0}}-1\\\ +\infty&\text{if }\left(\lambda>1\right)\text{ or
}\left(\lambda=1\text{ and }c<\delta^{-c_{0}}-1\right)\end{array}\right.$

###### Proof of the Lemma.

For the numerator of the bias term, we have

$\displaystyle EU_{i}D_{i}k_{ni}$ $\displaystyle=$ $\displaystyle
E\left[k_{ni}E\left[U_{i}D_{i}\left|W_{i}\right.\right]\right]=E\left[k_{ni}G\left(F_{W}\left(W_{i}\right)\right)G_{1}\left(F_{W}\left(W_{i}\right)\right)\right]$
$\displaystyle=$
$\displaystyle\int_{0}^{1}k\left(\frac{1-t}{h_{n}}\right)G\left(t\right)G_{1}\left(t\right)dt$
$\displaystyle=$ $\displaystyle
h_{n}\int_{0}^{1}k\left(s\right)G\left(1-h_{n}s\right)G_{1}\left(1-h_{n}s\right)ds$
$\displaystyle=$ $\displaystyle
h_{n}\int_{0}^{1}k\left(s\right)h_{n}s\left[-g\left(1\right)+o\left(1\right)\right]ds$
$\displaystyle=$
$\displaystyle\left[-\kappa_{1}g\left(1\right)+o\left(1\right)\right]h_{n}^{2},$

where
$G_{1}\left(t\right)=E\left[D_{i}\left|F_{W}\left(W_{i}\right)=t\right.\right]=\Pr\left(F_{W}\left(\varepsilon_{i}\right)<t\right)$
and $G_{1}\left(1\right)=1$. For the denominator, it follows from Lemma A.1
and Equation (7) that

$ED_{i}k_{ni}=Ek_{ni}\left(1+o\left(1\right)\right)=\kappa_{0}h_{n}\left(1+o\left(1\right)\right).$

Therefore,

$\frac{EU_{i}D_{i}k_{ni}}{ED_{i}k_{ni}}=\frac{\left[-\kappa_{1}g\left(1\right)+o\left(1\right)\right]h_{n}}{\kappa_{0}\left(1+o\left(1\right)\right)}=-\frac{\kappa_{1}g\left(1\right)}{\kappa_{0}}h_{n}+o\left(h_{n}\right).$

###### Proof of the Corollary.

It follows immediately from Theorem 1, Equation (7), and the Lemma.

###  7.2 Proof of Theorem 1

Denote
$\tilde{k}_{ni}=k\left(\left.\left(1-F\left(\hat{W}_{i}\right)\right)\right/h_{n}\right)$
and
$k_{ni}=k\left(\left.\left(1-F\left(W_{i}\right)\right)\right/h_{n}\right)$.
And define the infeasible kernel estimator as

$\mu_{n}=\frac{\sum_{i=1}^{n}\left(Y_{i}-Z_{i}^{\prime}\theta_{0}\right)D_{i}k_{ni}}{\sum_{i=1}^{n}D_{i}k_{ni}}.$

The asymptotic normality of $\tilde{\mu}$ follows by first establishing

$\frac{\sqrt{n}Ek_{ni}}{\sqrt{Ek_{ni}^{2}}}\left(\mu_{n}-\mu_{0}-\frac{EU_{i}D_{i}k_{ni}}{ED_{i}k_{ni}}\right)\rightarrow
N\left(0,\sigma_{U}^{2}\right)$ (25)

and then proving the asymptotic negligibility of $\tilde{\mu}-\mu_{n}$.

  * •

First step: proving

$\frac{\sqrt{n}Ek_{ni}}{\sqrt{Ek_{ni}^{2}}}\left(\mu_{n}-\mu_{0}-\frac{EU_{i}D_{i}k_{ni}}{ED_{i}k_{ni}}\right)\rightarrow
N\left(0,\sigma_{U}^{2}\right),$

where

$\mu_{n}=\frac{\sum_{i=1}^{n}\left(Y_{i}-Z_{i}^{\prime}\theta_{0}\right)D_{i}k_{ni}}{\sum_{i=1}^{n}D_{i}k_{ni}}=\mu_{0}+\frac{\sum_{i=1}^{n}U_{i}D_{i}k_{ni}}{\sum_{i=1}^{n}D_{i}k_{ni}}.$

For the triangular array $\left\\{U_{i}D_{i}k_{ni}:i\leq n,n\geq 1\right\\}$,
since

$Var\left(\sum_{i=1}^{n}U_{i}D_{i}k_{ni}\right)=nVar\left(U_{i}D_{i}k_{ni}\right)\leq
nE\left[U_{i}^{2}D_{i}k_{ni}^{2}\right]\leq
n\sigma_{U}^{2}\bar{k}^{2}<\infty,$

it follows from Lindeberg’s central limit theorem (e.g., Shao, 2003, Theorem
1.15) and Lemma A.5 that

$\frac{\sum_{i=1}^{n}\left(U_{i}D_{i}k_{ni}-EU_{i}D_{i}k_{ni}\right)}{\sqrt{nVar\left(U_{i}D_{i}k_{ni}\right)}}\rightarrow
N\left(0,1\right).$

Therefore, by Lemmas A.1, A.3 and A.4

$\displaystyle\frac{\sqrt{n}Ek_{ni}}{\sqrt{Ek_{ni}^{2}}}\left(\mu_{n}-\mu_{0}-\frac{EU_{i}D_{i}k_{ni}}{\left(\left.1\right/n\right)\sum_{i=1}^{n}D_{i}k_{ni}}\right)$
$\displaystyle=$
$\displaystyle\sigma_{U}\left(\frac{Ek_{ni}}{\left(\left.1\right/n\right)\sum_{i=1}^{n}D_{i}k_{ni}}\right)\sqrt{\frac{Var\left(U_{i}D_{i}k_{ni}\right)}{\sigma_{U}^{2}Ek_{ni}^{2}}}\frac{\sum_{i=1}^{n}\left(U_{i}D_{i}k_{ni}-EU_{i}D_{i}k_{ni}\right)}{\sqrt{nVar\left(U_{i}D_{i}k_{ni}\right)}}$
$\displaystyle=$
$\displaystyle\sigma_{U}\left(1+o_{p}\left(1\right)\right)\frac{\sum_{i=1}^{n}\left(U_{i}D_{i}k_{ni}-EU_{i}D_{i}k_{ni}\right)}{\sqrt{nVar\left(U_{i}D_{i}k_{ni}\right)}}\rightarrow
N\left(0,\sigma_{U}^{2}\right).$

It remains to show that

$\frac{\sqrt{n}Ek_{ni}}{\sqrt{Ek_{ni}^{2}}}\left(\frac{EU_{i}D_{i}k_{ni}}{\left(\left.1\right/n\right)\sum_{i=1}^{n}D_{i}k_{ni}}-\frac{EU_{i}D_{i}k_{ni}}{ED_{i}k_{ni}}\right)\overset{p}{\rightarrow}0$

for this step of proof. Note that
$\sum_{i=1}^{n}\left(D_{i}k_{ni}-ED_{i}k_{ni}\right)\left/\sqrt{nEk_{ni}^{2}}\right.$
is bounded in probability because it has mean zero and variance

$\frac{Var\left(\sum_{i=1}^{n}D_{i}k_{ni}\right)}{nEk_{ni}^{2}}=\frac{Var\left(D_{i}k_{ni}\right)}{Ek_{ni}^{2}}\leq\frac{ED_{i}k_{ni}^{2}}{Ek_{ni}^{2}}\leq
1.$

As a result, we have

$\displaystyle\frac{\sqrt{n}Ek_{ni}}{\sqrt{Ek_{ni}^{2}}}\left(\frac{EU_{i}D_{i}k_{ni}}{\left(\left.1\right/n\right)\sum_{i=1}^{n}D_{i}k_{ni}}-\frac{EU_{i}D_{i}k_{ni}}{ED_{i}k_{ni}}\right)$
$\displaystyle=$
$\displaystyle-\frac{Ek_{ni}}{\left(\left.1\right/n\right)\sum_{i=1}^{n}D_{i}k_{ni}}\cdot\frac{EU_{i}D_{i}k_{ni}}{ED_{i}k_{ni}}\cdot\frac{\sum_{i=1}^{n}\left(D_{i}k_{ni}-ED_{i}k_{ni}\right)}{\sqrt{nEk_{ni}^{2}}}$
$\displaystyle=$ $\displaystyle-\left(1+o_{p}\left(1\right)\right)\cdot
o\left(1\right)\cdot O_{p}\left(1\right)=o_{p}\left(1\right),$

where the second equality follows from Lemmas A.1, A.2 and A.4.

  * •

Second step: proving

$\frac{\sqrt{n}Ek_{ni}}{\sqrt{Ek_{ni}^{2}}}\left(\tilde{\mu}-\mu_{n}\right)\overset{p}{\rightarrow}0.$

To this end, write the left-hand side as

$C_{n}\left(\frac{\tilde{A}_{n}}{\tilde{B}_{n}}-\frac{A_{n}}{B_{n}}\right)=C_{n}\frac{\tilde{A}_{n}-A_{n}}{B_{n}}\cdot\frac{B_{n}}{\tilde{B}_{n}}-C_{n}\frac{\tilde{B}_{n}-B_{n}}{B_{n}}\cdot\frac{A_{n}}{B_{n}}\cdot\frac{B_{n}}{\tilde{B}_{n}},$

where

$\displaystyle C_{n}$ $\displaystyle=$
$\displaystyle\sqrt{n}Ek_{ni}\left/\sqrt{Ek_{ni}^{2}}\right.,$
$\displaystyle\tilde{A}_{n}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{n}\left(Y_{i}-Z_{i}^{\prime}\hat{\theta}\right)D_{i}\tilde{k}_{ni},\text{
}\tilde{B}_{n}=\sum_{i=1}^{n}D_{i}\tilde{k}_{ni},$ $\displaystyle A_{n}$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n}\left(Y_{i}-Z_{i}^{\prime}\theta_{0}\right)D_{i}k_{ni},\text{
}B_{n}=\sum_{i=1}^{n}D_{i}k_{ni}.$

It is sufficient to show that

$\text{(a) }\frac{\tilde{B}_{n}}{B_{n}}\overset{p}{\rightarrow}1\text{, (b)
}C_{n}\frac{\tilde{A}_{n}-A_{n}}{B_{n}}\overset{p}{\rightarrow}0\text{, (c)
}\frac{A_{n}}{B_{n}}=O_{p}\left(1\right)\text{, (d)
}C_{n}\frac{\tilde{B}_{n}-B_{n}}{B_{n}}\overset{p}{\rightarrow}0.$ (26)

Note that (d) implies (a) because $C_{n}\geq\sqrt{n}E\left[k_{ni}\cdot
k_{ni}\left/\bar{k}\right.\right]\left/\sqrt{Ek_{ni}^{2}}\right.=\left.\sqrt{nEk_{ni}^{2}}\right/\bar{k}\rightarrow\infty$
by Assumption 5.(ii). Next I will prove (b), (c) and (d), respectively.

For (b), the left-hand side can be further decomposed as

$C_{n}\frac{\tilde{A}_{n}-A_{n}}{B_{n}}=\left(\frac{Ek_{ni}}{\left(\left.1\right/n\right)\sum_{i}D_{i}k_{ni}}\right)\left[\underset{\text{(I)}}{\underbrace{\frac{\sum_{i}\left(\mu_{0}+U_{i}\right)D_{i}\left(\tilde{k}_{ni}-k_{ni}\right)}{\sqrt{nEk_{ni}^{2}}}}}-\underset{\text{(II)}}{\underbrace{\frac{\left(\hat{\theta}-\theta_{0}\right)^{\prime}\sum_{i}Z_{i}D_{i}\tilde{k}_{ni}}{\sqrt{nEk_{ni}^{2}}}}}\right].$

It follows from Lemmas A.1 and A.4 that
$Ek_{ni}\left/\left[\left(\left.1\right/n\right)\sum_{i}D_{i}k_{ni}\right]\right.\overset{p}{\rightarrow}1$.
For the first term in the bracket, a Taylor expansion about $\beta_{0}$ yields

$\displaystyle\left|\text{(I)}\right|$ $\displaystyle\leq$
$\displaystyle\left|\sum_{i}\frac{\left(\mu_{0}+U_{i}\right)D_{i}}{\sqrt{nEk_{ni}^{2}}}k^{\prime}\left(\frac{1-F\left(W_{i}\right)}{h_{n}}\right)\frac{f\left(W_{i}\right)}{h_{n}}\left(\hat{W}_{i}-W_{i}\right)\right|$
$\displaystyle+\left|\frac{1}{2}\sum_{i}\frac{\left(\mu_{0}+U_{i}\right)D_{i}}{\sqrt{nEk_{ni}^{2}}}k^{\prime\prime}\left(\frac{1-F\left(W_{i}^{\ast}\right)}{h_{n}}\right)\left[\frac{f\left(W_{i}^{\ast}\right)}{h_{n}}\right]^{2}\left(\hat{W}_{i}-W_{i}\right)^{2}\right|$
$\displaystyle+\left|\frac{1}{2}\sum_{i}\frac{\left(\mu_{0}+U_{i}\right)D_{i}}{\sqrt{nEk_{ni}^{2}}}k^{\prime}\left(\frac{1-F\left(W_{i}^{\ast}\right)}{h_{n}}\right)\frac{f^{\prime}\left(W_{i}^{\ast}\right)}{h_{n}}\left(\hat{W}_{i}-W_{i}\right)^{2}\right|$
$\displaystyle\leq$
$\displaystyle\frac{\sqrt{n}\left\|\hat{\beta}-\beta_{0}\right\|}{\sqrt{Ek_{ni}^{2}}}\frac{1}{n}\sum_{i}\left(\left|\mu_{0}\right|+\left|U_{i}\right|\right)\left|k^{\prime}\left(\frac{1-F\left(W_{i}\right)}{h_{n}}\right)\right|\frac{f\left(W_{i}\right)}{h_{n}}\left\|X_{i}\right\|$
$\displaystyle+\frac{\left(\sqrt{n}\left\|\hat{\beta}-\beta_{0}\right\|\right)^{2}}{2\sqrt{nEk_{ni}^{2}}}\frac{1}{n}\sum_{i}\left(\left|\mu_{0}\right|+\left|U_{i}\right|\right)\left|k^{\prime\prime}\left(\frac{1-F\left(W_{i}^{\ast}\right)}{h_{n}}\right)\right|\left[\frac{f\left(W_{i}^{\ast}\right)}{h_{n}}\right]^{2}\left\|X_{i}\right\|^{2}$
$\displaystyle+\frac{\left(\sqrt{n}\left\|\hat{\beta}-\beta_{0}\right\|\right)^{2}}{2\sqrt{nEk_{ni}^{2}}}\frac{1}{n}\sum_{i}\left(\left|\mu_{0}\right|+\left|U_{i}\right|\right)\left|k^{\prime}\left(\frac{1-F\left(W_{i}^{\ast}\right)}{h_{n}}\right)\right|\frac{\left|f^{\prime}\left(W_{i}^{\ast}\right)\right|}{h_{n}}\left\|X_{i}\right\|^{2},$

where $W_{i}^{\ast}=X_{i}^{\prime}\beta^{\ast}$ with $\beta^{\ast}$ lying on
the line segment joining $\hat{\beta}$ and $\beta_{0}$. Note that
$\left|k^{\prime}\left(u\right)\right|\leq\bar{k^{\prime}}1\left\\{0\leq u\leq
1\right\\}$ and
$\left|k^{\prime\prime}\left(u\right)\right|\leq\bar{k^{\prime\prime}}1\left\\{0\leq
u\leq 1\right\\}$ by Assumption 2. For large enough $n$ such that $h_{n}\leq
1-F\left(C\right)$, we have

$\displaystyle\left|\text{(I)}\right|$ $\displaystyle\leq$
$\displaystyle\frac{\bar{k^{\prime}}\sqrt{n}\left\|\hat{\beta}-\beta_{0}\right\|}{\sqrt{Ek_{ni}^{2}}}\frac{1}{n}\sum_{i}\left(\left|\mu_{0}\right|+\left|U_{i}\right|\right)1\left\\{F\left(W_{i}\right)>1-h_{n}\right\\}H_{C}\left(W_{i}\right)\left\|X_{i}\right\|$
$\displaystyle+\frac{\left(\sqrt{n}\left\|\hat{\beta}-\beta_{0}\right\|\right)^{2}}{2\sqrt{nEk_{ni}^{2}}}\frac{1}{n}\sum_{i}\left(\left|\mu_{0}\right|+\left|U_{i}\right|\right)\left[\bar{k^{\prime\prime}}H_{C}^{2}\left(W_{i}^{\ast}\right)+\bar{k^{\prime}}\left|S_{C}\left(W_{i}^{\ast}\right)\right|H_{C}\left(W_{i}^{\ast}\right)\right]\left\|X_{i}\right\|^{2},$

where $H_{C}\left(\cdot\right)$ and $S_{C}\left(\cdot\right)$ are defined in
Assumption 3.(iii). Note that
$\left.\left(\left.1\right/n\right)\sum_{i=1}^{n}\left|G_{ni}\right|\right/E\left|G_{ni}\right|=O_{p}\left(1\right)$
by Markov’s inequality for any i.i.d. random variables $\left\\{G_{ni}:i\leq
n\right\\}$ with $0<E\left|G_{ni}\right|<\infty$. Thus, it follows from
Assumption 4 that, with probability tending to one,

$\displaystyle\left|\text{(I)}\right|$ $\displaystyle\leq$
$\displaystyle\frac{O_{p}\left(1\right)}{\sqrt{Ek_{ni}^{2}}}\left(\left|\mu_{0}\right|+E\left|U_{i}\right|\right)E\left[1\left\\{F\left(W_{i}\right)>1-h_{n}\right\\}H_{C}\left(W_{i}\right)\left\|X_{i}\right\|\right]$
$\displaystyle+\frac{O_{p}\left(1\right)}{\sqrt{nEk_{ni}^{2}}}\left(\left|\mu_{0}\right|+E\left|U_{i}\right|\right)E\left[\sup_{\beta\in\mathcal{N}\left(\beta_{0}\right)}\left[H_{C}^{2}\left(X_{i}\beta\right)+\left|S_{C}\left(X_{i}\beta\right)\right|H_{C}\left(X_{i}\beta\right)\right]\left\|X_{i}\right\|^{2}\right]$
$\displaystyle\leq$ $\displaystyle
O_{p}\left(1\right)\sqrt{\frac{\left[\Pr\left(F\left(W_{i}\right)>1-h_{n}\right)\right]^{1+c_{1}\left/\left(8+2c_{1}\right)\right.}}{Ek_{ni}^{2}}}\left[EH_{C}^{4}\left(W_{i}\right)\right]^{1\left/4\right.}\left(E\left\|X_{i}\right\|^{4+c_{1}}\right)^{1\left/\left(4+c_{1}\right)\right.}$
$\displaystyle+\frac{O_{p}\left(1\right)}{\sqrt{nEk_{ni}^{2}}}\left(E\sup_{\beta\in\mathcal{N}\left(\beta_{0}\right)}H_{C}^{4}\left(X_{i}\beta\right)\right)^{1\left/2\right.}\left(E\left\|X_{i}\right\|^{4}\right)^{1\left/2\right.}$
$\displaystyle+\frac{O_{p}\left(1\right)}{\sqrt{nEk_{ni}^{2}}}\left(E\sup_{\beta\in\mathcal{N}\left(\beta_{0}\right)}S_{C}^{4}\left(X_{i}\beta\right)\right)^{1\left/4\right.}\left(E\sup_{\beta\in\mathcal{N}\left(\beta_{0}\right)}H_{C}^{4}\left(X_{i}\beta\right)\right)^{1\left/4\right.}\left(E\left\|X_{i}\right\|^{4}\right)^{1\left/2\right.}$
$\displaystyle=$ $\displaystyle o_{p}\left(1\right),$

where the second inequality follows from Hölder’s inequality and the last
equality follows from Assumptions 1.(iv), 3.(iii) and 5.

For the second term of (b), similarly, we have

$\displaystyle\left|\text{(II)}\right|$ $\displaystyle\leq$
$\displaystyle\frac{\sqrt{n}\left\|\hat{\theta}-\theta_{0}\right\|}{\sqrt{Ek_{ni}^{2}}}\frac{1}{n}\sum_{i=1}^{n}\left\|Z_{i}\right\|\left(k_{ni}+\left|\tilde{k}_{ni}-k_{ni}\right|\right)$
$\displaystyle\leq$
$\displaystyle\frac{O_{p}\left(1\right)}{\sqrt{Ek_{ni}^{2}}}\frac{1}{n}\sum_{i=1}^{n}\left\|Z_{i}\right\|\left[\bar{k}1\left\\{F\left(W_{i}\right)>1-h_{n}\right\\}+\bar{k^{\prime}}H_{C}\left(W_{i}^{\ast}\right)\left|\hat{W}_{i}-W_{i}\right|\right]$
$\displaystyle\leq$
$\displaystyle\frac{O_{p}\left(1\right)}{\sqrt{Ek_{ni}^{2}}}E\left[\left\|Z_{i}\right\|1\left\\{F\left(W_{i}\right)>1-h_{n}\right\\}\right]+\frac{O_{p}\left(1\right)}{\sqrt{nEk_{ni}^{2}}}E\left[\left\|Z_{i}\right\|\left\|X_{i}\right\|\sup_{\beta\in\mathcal{N}\left(\beta_{0}\right)}H_{C}\left(X_{i}\beta\right)\right]$
$\displaystyle\leq$ $\displaystyle
O_{p}\left(1\right)\sqrt{\frac{\left[\Pr\left(F\left(W_{i}\right)>1-h_{n}\right)\right]^{1+c_{1}\left/\left(2+c_{1}\right)\right.}}{Ek_{ni}^{2}}}\left(E\left\|Z_{i}\right\|^{2+c_{1}}\right)^{1\left/\left(2+c_{1}\right)\right.}$
$\displaystyle+\frac{O_{p}\left(1\right)}{\sqrt{nEk_{ni}^{2}}}\left(E\left\|Z_{i}\right\|^{2}\right)^{1\left/2\right.}\left(E\left\|X_{i}\right\|^{4}\right)^{1\left/4\right.}\left(E\sup_{\beta\in\mathcal{N}\left(\beta_{0}\right)}H_{C}^{4}\left(X_{i}\beta\right)\right)^{1\left/4\right.}$
$\displaystyle=$ $\displaystyle o_{p}\left(1\right),$

which completes the proof of (b).

For (c) in (26), we know by (25) that

$\frac{A_{n}}{B_{n}}=\mu_{n}=\mu_{0}+\frac{EU_{i}D_{i}k_{ni}}{ED_{i}k_{ni}}+O_{p}\left(\frac{1}{C_{n}}\right).$

It follows from Lemma A.2 and $C_{n}\rightarrow\infty$ that
$\left.A_{n}\right/B_{n}=\mu_{0}+o\left(1\right)+o_{p}\left(1\right)=O_{p}\left(1\right)$.
The proof of (d) in (26) follows exactly the same line as the analysis for the
first term (I) of (b), by replacing $\mu_{0}+U_{i}$ with $1$. Consequently, we
have $C_{n}\left(\tilde{\mu}-\mu_{n}\right)\overset{p}{\rightarrow}0$, which
in combination of (25) leads to the conclusion.

###  7.3 Proof of Theorem 2

Denote
$\hat{k}_{ni}=k\left(\left.\left(1-\hat{F}_{n}\left(\hat{W}_{i}\right)\right)\right/h_{n}\right)$
and
$k_{ni}=k\left(\left.\left(1-F_{W}\left(W_{i}\right)\right)\right/h_{n}\right)$.
By (25), Equation (7), the Lemma, and Assumption 5’.(ii), we have

$\sqrt{nh_{n}}\left(\mu_{n}-\mu_{0}+\frac{\kappa_{1}g\left(1\right)}{\kappa_{0}}h_{n}\right)\rightarrow
N\left(0,\frac{\chi_{0}\sigma_{U}^{2}}{\kappa_{0}^{2}}\right),$ (27)

where

$\mu_{n}=\frac{\sum_{i=1}^{n}\left(Y_{i}-Z_{i}^{\prime}\theta_{0}\right)D_{i}k_{ni}}{\sum_{i=1}^{n}D_{i}k_{ni}}.$

It remains to show that
$\sqrt{nh_{n}}\left(\hat{\mu}-\mu_{n}\right)\overset{p}{\rightarrow}0$.

Denote

$\displaystyle\hat{A}_{n}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{n}\left(Y_{i}-Z_{i}^{\prime}\hat{\theta}\right)D_{i}\hat{k}_{ni},\text{
}\hat{B}_{n}=\sum_{i=1}^{n}D_{i}\hat{k}_{ni},$ $\displaystyle A_{n}$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n}\left(Y_{i}-Z_{i}^{\prime}\theta_{0}\right)D_{i}k_{ni},\text{
}B_{n}=\sum_{i=1}^{n}D_{i}k_{ni},$

then

$\displaystyle\sqrt{nh_{n}}\left(\hat{\mu}-\mu_{n}\right)$ $\displaystyle=$
$\displaystyle\sqrt{nh_{n}}\left(\frac{\hat{A}_{n}}{\hat{B}_{n}}-\frac{A_{n}}{B_{n}}\right)$
$\displaystyle=$
$\displaystyle\sqrt{nh_{n}}\left(\frac{\hat{A}_{n}-A_{n}}{B_{n}}-\frac{\hat{B}_{n}-B_{n}}{B_{n}}\cdot\frac{A_{n}}{B_{n}}\right)\frac{B_{n}}{\hat{B}_{n}}$
$\displaystyle=$
$\displaystyle\sqrt{nh_{n}}\left(\frac{\hat{A}_{n}-A_{n}}{B_{n}}-\frac{\hat{B}_{n}-B_{n}}{B_{n}}\mu_{0}-o_{p}\left(\frac{\hat{B}_{n}-B_{n}}{B_{n}}\right)\right)\frac{B_{n}}{\hat{B}_{n}},$

where the last equality follows from the consistency of
$\left.A_{n}\right/B_{n}=\mu_{n}$ for $\mu_{0}$ implied by (27). It is
sufficient to show that

$\text{(a) }\frac{\hat{B}_{n}}{B_{n}}\overset{p}{\rightarrow}1\text{, (b)
}\sqrt{nh_{n}}\left(\frac{\hat{A}_{n}-A_{n}}{B_{n}}-\frac{\hat{B}_{n}-B_{n}}{B_{n}}\mu_{0}\right)\overset{p}{\rightarrow}0\text{,
(c)
}\sqrt{nh_{n}}\left(\frac{\hat{B}_{n}-B_{n}}{B_{n}}\right)=O_{p}\left(1\right)\text{.}$

Note that (c) implies (a). For (b) and (c), the left-hand sides can be written
as

$\displaystyle\sqrt{nh_{n}}\left(\frac{\hat{A}_{n}-A_{n}}{B_{n}}-\frac{\hat{B}_{n}-B_{n}}{B_{n}}\mu_{0}\right)$
(28) $\displaystyle=$
$\displaystyle\left(\frac{1}{\kappa_{0}}+o_{p}\left(1\right)\right)\left[\underset{\text{(b1)}}{\underbrace{\frac{1}{\sqrt{nh_{n}}}\sum_{i=1}^{n}U_{i}D_{i}\left(\hat{k}_{ni}-k_{ni}\right)}}-\underset{\text{(b2)}}{\underbrace{\frac{\left(\hat{\theta}-\theta_{0}\right)^{\prime}}{\sqrt{nh_{n}}}\sum_{i=1}^{n}Z_{i}D_{i}\hat{k}_{ni}}}\right]$

and

$\sqrt{nh_{n}}\left(\frac{\hat{B}_{n}-B_{n}}{B_{n}}\right)=\left(\frac{1}{\kappa_{0}}+o_{p}\left(1\right)\right)\left[\underset{\text{(c1)}}{\underbrace{\frac{1}{\sqrt{nh_{n}}}\sum_{i=1}^{n}D_{i}\left(\hat{k}_{ni}-k_{ni}\right)}}\right],$
(29)

because $B_{n}=nh_{n}\left(\kappa_{0}+o_{p}\left(1\right)\right)$ by Lemma
A.6.

First consider
$\left(1\left/\sqrt{nh_{n}}\right.\right)\sum_{i}\pi_{i}D_{i}\left(\hat{k}_{ni}-k_{ni}\right)$,
where $\pi_{i}$ is either $U_{i}$ or $1$, corresponding to (b1) and (c1),
respectively. A sixth-order Taylor expansion yields

$\displaystyle\hat{k}_{ni}-k_{ni}$ $\displaystyle=$
$\displaystyle\sum_{r=1}^{5}\frac{1}{r!h_{n}^{r}}k^{\left(r\right)}\left(\frac{1-F_{W}\left(W_{i}\right)}{h_{n}}\right)\left[\left(1-\hat{F}_{n}\left(\hat{W}_{i}\right)\right)-\left(1-F_{W}\left(W_{i}\right)\right)\right]^{r}$
$\displaystyle+\frac{1}{6!h_{n}^{6}}k^{\left(6\right)}\left(\frac{1-F_{i}^{\ast}}{h_{n}}\right)\left[\left(1-\hat{F}_{n}\left(\hat{W}_{i}\right)\right)-\left(1-F_{W}\left(W_{i}\right)\right)\right]^{6},$

where $F_{i}^{\ast}$ lies between $\hat{F}_{n}\left(\hat{W}_{i}\right)$ and
$F_{W}\left(W_{i}\right)$. Denote
$\alpha_{ij}\left(\beta\right)=1\left\\{X_{ij}^{\prime}\beta<0\right\\}-1\left\\{X_{ij}^{\prime}\beta_{0}<0\right\\}$,
where $X_{ij}=X_{i}-X_{j}$, then

$\left(1-\hat{F}_{n}\left(\hat{W}_{i}\right)\right)-\left(1-F_{W}\left(W_{i}\right)\right)=\frac{1}{n-1}\sum_{j\neq
i}\left\\{\left[1\left\\{W_{j}>W_{i}\right\\}-\left(1-F_{W}\left(W_{i}\right)\right)\right]+\alpha_{ij}\left(\hat{\beta}\right)\right\\}.$
(30)

It follows from the Taylor expansion and the above equality that

$\frac{1}{\sqrt{nh_{n}}}\sum_{i=1}^{n}\pi_{i}D_{i}\left(\hat{k}_{ni}-k_{ni}\right)=\text{(I)}+\text{(II)}+\text{(III)}+\text{(IV)},$
(31)

where

(I) $\displaystyle=$
$\displaystyle\sqrt{\frac{n}{h_{n}^{3}}}\frac{1}{n\left(n-1\right)}\sum_{i}\sum_{j\neq
i}\pi_{i}D_{i}k^{\prime}\left(\frac{1-F_{W}\left(W_{i}\right)}{h_{n}}\right)\left[1\left\\{W_{j}>W_{i}\right\\}-\left(1-F_{W}\left(W_{i}\right)\right)\right]$
(II) $\displaystyle=$
$\displaystyle\frac{1}{\left(n-1\right)\sqrt{nh_{n}^{3}}}\sum_{i}\sum_{j\neq
i}\pi_{i}D_{i}k^{\prime}\left(\frac{1-F_{W}\left(W_{i}\right)}{h_{n}}\right)\alpha_{ij}\left(\hat{\beta}\right)$
(III) $\displaystyle=$
$\displaystyle\sum_{r=2}^{5}\sum_{s=0}^{r}\frac{1}{s!\left(r-s\right)!}\frac{1}{\sqrt{nh_{n}^{2r+1}}}\sum_{i=1}^{n}\pi_{i}D_{i}k^{\left(r\right)}\left(\frac{1-F_{W}\left(W_{i}\right)}{h_{n}}\right)$
$\displaystyle\cdot\left\\{\frac{1}{n-1}\sum_{j\neq
i}\left[1\left\\{W_{j}>W_{i}\right\\}-\left(1-F_{W}\left(W_{i}\right)\right)\right]\right\\}^{r-s}\left\\{\frac{1}{n-1}\sum_{j\neq
i}\alpha_{ij}\left(\hat{\beta}\right)\right\\}^{s}$ (IV) $\displaystyle=$
$\displaystyle\frac{1}{6!\sqrt{nh_{n}^{13}}}\sum_{i=1}^{n}\pi_{i}D_{i}k^{\left(6\right)}\left(\frac{1-F_{i}^{\ast}}{h_{n}}\right)\left[\left(1-\hat{F}_{n}\left(\hat{W}_{i}\right)\right)-\left(1-F_{W}\left(W_{i}\right)\right)\right]^{6}.$

I will consider these four terms, respectively.

Denote $\xi_{i}$ as the $i$-th observation and

$m_{n}\left(\xi_{i},\xi_{j}\right)=h_{n}^{-3/2}\pi_{i}D_{i}k^{\prime}\left(\frac{1-F_{W}\left(W_{i}\right)}{h_{n}}\right)\left[1\left\\{W_{j}>W_{i}\right\\}-\left(1-F_{W}\left(W_{i}\right)\right)\right],$

then

$\text{(I)}=\frac{\sqrt{n}}{n\left(n-1\right)}\sum_{i}\sum_{j\neq
i}m_{n}\left(\xi_{i},\xi_{j}\right).$

Note that
$E\left[\left.m_{n}\left(\xi_{i},\xi_{j}\right)\right|\xi_{i}\right]=0$ and
$E\left[m_{n}\left(\xi_{i},\xi_{j}\right)\right]=0$. Since

$E\left[m_{n}^{2}\left(\xi_{i},\xi_{j}\right)\right]\leq
h_{n}^{-3}\left(\bar{k}^{\prime}\right)^{2}E\pi_{i}^{2}\Pr\left(F_{W}\left(W_{i}\right)>1-h_{n}\right)=O\left(h_{n}^{-2}\right)=o\left(n\right)$

by Assumption 5’, it follows from the projection method for U-statistics
(e.g., Powell et al., 1989, Lemma 3.1) that

$\text{(I)}=\frac{1}{\sqrt{n}}\sum_{j=1}^{n}E\left[\left.m_{n}\left(\xi_{i},\xi_{j}\right)\right|\xi_{j}\right]+o_{p}\left(1\right).$

Denote
$G_{\pi}\left(t\right)=E\left[\pi_{i}D_{i}\left|F_{W}\left(W_{i}\right)=t\right.\right]=E\left[\pi_{i}1\left\\{F_{W}\left(\varepsilon_{i}\right)<t\right\\}\right]$
with $\pi$ being either $U$ or $1$, then

$\displaystyle
E\left[\left.m_{n}\left(\xi_{i},\xi_{j}\right)\right|\xi_{j}\right]$
$\displaystyle=$ $\displaystyle
h_{n}^{-3/2}E\left[\left.G_{\pi}\left(F_{W}\left(W_{i}\right)\right)k^{\prime}\left(\frac{1-F_{W}\left(W_{i}\right)}{h_{n}}\right)\left[1\left\\{W_{j}>W_{i}\right\\}-\left(1-F_{W}\left(W_{i}\right)\right)\right]\right|\xi_{j}\right]$
$\displaystyle=$ $\displaystyle
h_{n}^{-3/2}\int_{0}^{1}G_{\pi}\left(t\right)k^{\prime}\left(\frac{1-t}{h_{n}}\right)\left[1\left\\{t<F_{W}\left(W_{j}\right)\right\\}-\left(1-t\right)\right]dt$
$\displaystyle=$ $\displaystyle
h_{n}^{-1/2}\int_{0}^{1/h_{n}}G_{\pi}\left(1-h_{n}s\right)k^{\prime}\left(s\right)\left[1\left\\{1-h_{n}s<F_{W}\left(W_{j}\right)\right\\}-h_{n}s\right]ds$
$\displaystyle=$ $\displaystyle
h_{n}^{-1/2}\int_{0}^{1}\left[G_{\pi}\left(1-h_{n}s\right)-G_{\pi}\left(1\right)\right]k^{\prime}\left(s\right)\left[1\left\\{1-F_{W}\left(W_{j}\right)<h_{n}s\right\\}-h_{n}s\right]ds$
$\displaystyle-
h_{n}^{-1/2}G_{\pi}\left(1\right)\left(k_{nj}-h_{n}\kappa_{0}\right),$

therefore

$\text{(I)}=-\frac{G_{\pi}\left(1\right)}{\sqrt{nh_{n}}}\sum_{i=1}^{n}\left(k_{ni}-h_{n}\kappa_{0}\right)+R_{1n}+o_{p}\left(1\right),$
(32)

where

$\displaystyle R_{1n}$ $\displaystyle=$
$\displaystyle\int_{0}^{1}\left[G_{\pi}\left(1-h_{n}s\right)-G_{\pi}\left(1\right)\right]k^{\prime}\left(s\right)\frac{1}{\sqrt{nh_{n}}}\sum_{i=1}^{n}\left[1\left\\{1-F_{W}\left(W_{i}\right)<h_{n}s\right\\}-h_{n}s\right]ds,$
$\displaystyle\left|R_{1n}\right|$ $\displaystyle\leq$
$\displaystyle\bar{k}^{\prime}\int_{0}^{1}\left|G_{\pi}\left(1-h_{n}s\right)-G_{\pi}\left(1\right)\right|ds\cdot\sup_{s\in\left[0,1\right]}\left|\frac{1}{\sqrt{nh_{n}}}\sum_{i=1}^{n}\left[1\left\\{1-F_{W}\left(W_{i}\right)<h_{n}s\right\\}-h_{n}s\right]\right|.$

Since $\left|G_{\pi}\left(1-h_{n}s\right)-G_{\pi}\left(1\right)\right|\leq
E\left|\pi_{i}\right|=O\left(1\right)$ and
$\lim_{n\rightarrow\infty}\left|G_{\pi}\left(1-h_{n}s\right)-G_{\pi}\left(1\right)\right|=0$
for any $s\in\left[0,1\right]$, it follows from the dominated convergence
theorem (e.g., Shao, 2003, Theorem 1.1.(ii)) that
$\lim_{n\rightarrow\infty}\int_{0}^{1}\left|G_{\pi}\left(1-h_{n}s\right)-G_{\pi}\left(1\right)\right|ds=0$.
For the supremum term, since $1-F_{W}\left(W_{i}\right)$ follows a uniform
distribution on the unit interval, it follows from Stute (1982, Lemma 2.3)
that

$\sup_{u\in\left[0,1\right]}\left|\frac{1}{\sqrt{nh_{n}}}\sum_{i=1}^{n}\left[1\left\\{1-F_{W}\left(W_{i}\right)<h_{n}s\right\\}-h_{n}s\right]\right|=O_{p}\left(1\right),$

thus the remainder term satisfies

$\left|R_{1n}\right|=o\left(1\right)\cdot
O_{p}\left(1\right)=o_{p}\left(1\right).$ (33)

For the first term of (32), since $Ek_{ni}=h_{n}\kappa_{0}$ and
$Var\left(k_{ni}\right)=Ek_{ni}^{2}-\left(Ek_{ni}\right)^{2}=h_{n}\chi_{0}-\left(h_{n}\kappa_{0}\right)^{2}$,
we have

$E\left[\frac{1}{\sqrt{nh_{n}}}\sum_{i=1}^{n}\left(k_{ni}-h_{n}\kappa_{0}\right)\right]^{2}=\frac{Var\left(k_{ni}\right)}{h_{n}}=\chi_{0}-h_{n}\kappa_{0}^{2}=O\left(1\right).$
(34)

Substituting (33) and (34) into (32) gives

$\text{(I)}=O_{p}\left(1\right)\cdot
G_{\pi}\left(1\right)+o_{p}\left(1\right)=\left\\{\begin{array}[]{cc}o_{p}\left(1\right)&\text{for
}\pi_{i}=U_{i}\\\ O_{p}\left(1\right)&\text{for }\pi_{i}=1,\end{array}\right.$
(35)

by noting that $G_{\pi}\left(1\right)=E\pi_{i}$.

Next consider (II) and (III). For (III), we have

$\displaystyle\left|\text{(III)}\right|$ $\displaystyle\leq$
$\displaystyle\sum_{r=2}^{5}\sum_{s=0}^{r}\frac{1}{s!\left(r-s\right)!}\frac{\overline{k^{\left(r\right)}}}{\sqrt{nh_{n}^{2r+1}}}\left(\frac{n}{n-1}\right)^{r-s}$
$\displaystyle\cdot\left(\sup_{0<w<h_{n}}\left|\frac{1}{n}\sum_{j=1}^{n}\left[1\left\\{1-F_{W}\left(W_{j}\right)<w\right\\}-w\right]\right|\right)^{r-s}$
$\displaystyle\cdot\sum_{i=1}^{n}\left|\pi_{i}\right|1\left\\{F_{W}\left(W_{i}\right)>1-h_{n}\right\\}\left(\frac{1}{n-1}\sum_{j\neq
i}\left|\alpha_{ij}\left(\hat{\beta}\right)\right|\right)^{s}.$

It follows from Stute (1982, Lemma 2.3) that

$\sup_{0<w<h_{n}}\left|\frac{1}{n}\sum_{j=1}^{n}\left[1\left\\{1-F_{W}\left(W_{j}\right)<w\right\\}-w\right]\right|=O_{p}\left(\sqrt{\frac{h_{n}}{n}}\right),$

so that

$\left|\text{(III)}\right|\leq\sum_{r=2}^{5}\sum_{s=0}^{r}O_{p}\left(\left(nh_{n}\right)^{-\left.\left(r+s+1\right)\right/2}\right)\sum_{i=1}^{n}\left|\pi_{i}\right|1\left\\{F_{W}\left(W_{i}\right)>1-h_{n}\right\\}\left(\sum_{j\neq
i}\left|\alpha_{ij}\left(\hat{\beta}\right)\right|\right)^{s}.$

For the terms of $s=0$, since
$\left.\left(\left.1\right/n\right)\sum_{i=1}^{n}\left|G_{ni}\right|\right/E\left|G_{ni}\right|=O_{p}\left(1\right)$
for any i.i.d. random variables $\left\\{G_{ni}:i\leq n\right\\}$ with
$0<E\left|G_{ni}\right|<\infty$,

$\displaystyle
O_{p}\left(\left(nh_{n}\right)^{-\left.\left(r+1\right)\right/2}\right)\sum_{i=1}^{n}\left|\pi_{i}\right|1\left\\{F_{W}\left(W_{i}\right)>1-h_{n}\right\\}$
$\displaystyle=$ $\displaystyle
O_{p}\left(\left(nh_{n}\right)^{-\left.\left(r+1\right)\right/2}\right)\cdot
O_{p}\left(n\right)E\left|\pi_{i}\right|\Pr\left(F_{W}\left(W_{i}\right)>1-h_{n}\right)$
$\displaystyle=$ $\displaystyle
O_{p}\left(\left(nh_{n}\right)^{-\left.\left(r-1\right)\right/2}\right)=o_{p}\left(1\right).$

For the terms of $s\geq 1$, since
$\left|\alpha_{ij}\left(\hat{\beta}\right)\right|^{s}=\left|\alpha_{ij}\left(\hat{\beta}\right)\right|$,

$\displaystyle\left(\sum_{j\neq
i}\left|\alpha_{ij}\left(\hat{\beta}\right)\right|\right)^{s}=\sum_{j_{1}\neq
i}\left|\alpha_{ij_{1}}\left(\hat{\beta}\right)\right|\left[\left|\alpha_{ij_{1}}\left(\hat{\beta}\right)\right|+\sum_{j\neq
i,j\neq j_{1}}\left|\alpha_{ij}\left(\hat{\beta}\right)\right|\right]^{s-1}$
$\displaystyle=$ $\displaystyle\sum_{j_{1}\neq
i}\left|\alpha_{ij_{1}}\left(\hat{\beta}\right)\right|\sum_{t_{1}=0}^{s-1}\binom{s-1}{t_{1}}\left|\alpha_{ij_{1}}\left(\hat{\beta}\right)\right|^{s-1-t_{1}}\left[\sum_{j\neq
i,j_{1}}\left|\alpha_{ij}\left(\hat{\beta}\right)\right|\right]^{t_{1}}$
$\displaystyle=$ $\displaystyle\sum_{j_{1}\neq
i}\left|\alpha_{ij_{1}}\left(\hat{\beta}\right)\right|+\sum_{t_{1}=1}^{s-1}\binom{s-1}{t_{1}}\sum_{j_{1}\neq
i}\left|\alpha_{ij_{1}}\left(\hat{\beta}\right)\right|\sum_{j_{2}\neq
i,j_{1}}\left|\alpha_{ij_{2}}\left(\hat{\beta}\right)\right|\left[\left|\alpha_{ij_{2}}\left(\hat{\beta}\right)\right|+\sum_{j\neq
i,j_{1},j_{2}}\left|\alpha_{ij}\left(\hat{\beta}\right)\right|\right]^{t_{1}-1}$
$\displaystyle=$ $\displaystyle\sum_{j_{1}\neq
i}\left|\alpha_{ij_{1}}\left(\hat{\beta}\right)\right|+\sum_{t_{1}=1}^{s-1}\binom{s-1}{t_{1}}\sum_{j_{1}\neq
i}\left|\alpha_{ij_{1}}\left(\hat{\beta}\right)\right|\sum_{j_{2}\neq
i,j_{1}}\left|\alpha_{ij_{2}}\left(\hat{\beta}\right)\right|$
$\displaystyle+\sum_{t_{1}=1}^{s-1}\sum_{t_{2}=1}^{t_{1}-1}\binom{s-1}{t_{1}}\binom{t_{1}-1}{t_{2}}\sum_{j_{1}\neq
i}\left|\alpha_{ij_{1}}\left(\hat{\beta}\right)\right|\sum_{j_{2}\neq
i,j_{1}}\left|\alpha_{ij_{2}}\left(\hat{\beta}\right)\right|\left[\sum_{j\neq
i,j_{1},j_{2}}\left|\alpha_{ij}\left(\hat{\beta}\right)\right|\right]^{t_{2}}$
$\displaystyle=$
$\displaystyle\cdots=\sum_{t=1}^{s}C_{t}\prod_{t^{\prime}=1}^{t}\sum_{j_{t^{\prime}}\neq
i,j_{1},\cdots,j_{t^{\prime}-1}}\left|\alpha_{ij_{t^{\prime}}}\left(\hat{\beta}\right)\right|,$

where $C_{t}$ is a constant depending only on $t$. Therefore,

$\left|\text{(III)}\right|\leq\sum_{r=2}^{5}\sum_{s=1}^{r}\sum_{t=1}^{s}O_{p}\left(\left(nh_{n}\right)^{-\frac{r+s+1}{2}}\right)\sum_{i=1}^{n}\left|\pi_{i}\right|1\left\\{F_{W}\left(W_{i}\right)>1-h_{n}\right\\}\prod_{t^{\prime}=1}^{t}\sum_{j_{t^{\prime}}\neq
i,j_{1},\cdots,j_{t^{\prime}-1}}\left|\alpha_{ij_{t^{\prime}}}\left(\hat{\beta}\right)\right|+o_{p}\left(1\right).$

Denote

$J_{rsti}\left(\beta\right)=\left(nh_{n}\right)^{-\left.\left(r+s+1\right)\right/2}\left|\pi_{i}\right|1\left\\{F_{W}\left(W_{i}\right)>1-h_{n}\right\\}\prod_{t^{\prime}=1}^{t}\sum_{j_{t^{\prime}}\neq
i,j_{1},\cdots,j_{t^{\prime}-1}}\left|\alpha_{ij_{t^{\prime}}}\left(\beta\right)\right|$
(36)

for any $1\leq t\leq s\leq r\leq 5$ and $i=1,\cdots,n$, then

$\left|\text{(II)}\right|+\left|\text{(III)}\right|\leq
O_{p}\left(1\right)\sum_{r=1}^{5}\sum_{s=1}^{r}\sum_{t=1}^{s}\sum_{i=1}^{n}J_{rsti}\left(\hat{\beta}\right)+o_{p}\left(1\right).$

Following the method of Schafgans and Zinde-Walsh (2002), I will prove that
for any $1\leq t\leq s\leq r\leq 5$, we have
$\sum_{i}J_{rsti}\left(\hat{\beta}\right)=o_{p}\left(1\right)$, namely
$\Pr\left(\sum_{i}J_{rsti}\left(\hat{\beta}\right)>\delta\right)\rightarrow 0$
for any $\delta>0$. Let $M_{n}$ be a slowly divergent sequence, then

$\displaystyle\Pr\left(\sum_{i=1}^{n}J_{rsti}\left(\hat{\beta}\right)>\delta\right)$
$\displaystyle\leq$
$\displaystyle\Pr\left(\sum_{i=1}^{n}J_{rsti}\left(\hat{\beta}\right)>\delta,\left\|\hat{\beta}-\beta_{0}\right\|\leq\frac{M_{n}}{\sqrt{n}}\right)+\Pr\left(\left\|\hat{\beta}-\beta_{0}\right\|>\frac{M_{n}}{\sqrt{n}}\right)$
$\displaystyle\leq$ $\displaystyle\sup_{\left\|\beta-\beta_{0}\right\|\leq
M_{n}\left/\sqrt{n}\right.}\Pr\left(\sum_{i=1}^{n}J_{rsti}\left(\beta\right)>\delta\right)+o\left(1\right)$
$\displaystyle\leq$
$\displaystyle\frac{n}{\delta}\left(nh_{n}\right)^{-\frac{r+s+1}{2}}E\left|\pi_{i}\right|\sup_{\left\|\beta-\beta_{0}\right\|\leq\frac{M_{n}}{\sqrt{n}}}E\left[1\left\\{F_{W}\left(W_{i}\right)>1-h_{n}\right\\}\prod_{t^{\prime}=1}^{t}\sum_{j_{t^{\prime}}\neq
i,j_{1},\cdots,j_{t^{\prime}-1}}\left|\alpha_{ij_{t^{\prime}}}\left(\beta\right)\right|\right]+o\left(1\right),$

where the second inequality follows from the $\sqrt{n}$-consistency of
$\hat{\beta}$ and the third inequality follows from Markov’s inequality. As in
the proof of Lemma A.7, we can show that under
$\left\|\beta-\beta_{0}\right\|\leq M_{n}\left/\sqrt{n}\right.$, for
$j_{t^{\prime}}\neq i,j_{1},\cdots,j_{t^{\prime}-1}$,

$\displaystyle
E\left[\left.\left|\alpha_{ij_{t^{\prime}}}\left(\beta\right)\right|\right|X_{i},X_{j_{1}},\cdots,X_{j_{t^{\prime}-1}}\right]=E\left[\left.E\left[\left.\left|\alpha_{ij}\left(\beta\right)\right|\right|X_{i},X_{j,(-1)}\right]\right|X_{i}\right]$
$\displaystyle=$ $\displaystyle
E\left[\left.\begin{array}[]{c}1\left\\{X_{ij,(-1)}^{\prime}\left(\beta_{(-1)}-\beta_{0,(-1)}\right)>0\right\\}\int_{W_{i}}^{W_{i}+X_{ij,(-1)}^{\prime}\left(\beta_{(-1)}-\beta_{0,(-1)}\right)}f_{W\left|X_{(-1)}\right.}\left(w\left|X_{j,(-1)}\right.\right)dw\\\
+1\left\\{X_{ij,(-1)}^{\prime}\left(\beta_{(-1)}-\beta_{0,(-1)}\right)<0\right\\}\int_{W_{i}+X_{ij,(-1)}^{\prime}\left(\beta_{(-1)}-\beta_{0,(-1)}\right)}^{W_{i}}f_{W\left|X_{(-1)}\right.}\left(w\left|X_{j,(-1)}\right.\right)dw\end{array}\right|X_{i}\right]$
$\displaystyle\leq$
$\displaystyle\overline{f_{W\left|X_{(-1)}\right.}}E\left[\left.\left|X_{ij,(-1)}^{\prime}\left(\beta_{(-1)}-\beta_{0,(-1)}\right)\right|\right|X_{i}\right]$
$\displaystyle\leq$
$\displaystyle\frac{\overline{f_{W\left|X_{(-1)}\right.}}M_{n}}{\sqrt{n}}\left(\left\|X_{i}\right\|+E\left\|X_{i}\right\|\right).$

Therefore, under $\left\|\beta-\beta_{0}\right\|\leq
M_{n}\left/\sqrt{n}\right.$,

$\displaystyle
E\left[1\left\\{F_{W}\left(W_{i}\right)>1-h_{n}\right\\}\prod_{t^{\prime}=1}^{t}\sum_{j_{t^{\prime}}\neq
i,j_{1},\cdots,j_{t^{\prime}-1}}\left|\alpha_{ij_{t^{\prime}}}\left(\beta\right)\right|\right]$
$\displaystyle\leq$
$\displaystyle\frac{\overline{f_{W\left|X_{(-1)}\right.}}M_{n}}{\sqrt{n}}\left(n-t\right)E\left[1\left\\{F_{W}\left(W_{i}\right)>1-h_{n}\right\\}\left(\left\|X_{i}\right\|+E\left\|X_{i}\right\|\right)\prod_{t^{\prime}=1}^{t-1}\sum_{j_{t^{\prime}}\neq
i,j_{1},\cdots,j_{t^{\prime}-1}}\left|\alpha_{ij_{t^{\prime}}}\left(\beta\right)\right|\right]$
$\displaystyle\leq$
$\displaystyle\cdots\leq\left(\frac{\overline{f_{W\left|X_{(-1)}\right.}}M_{n}}{\sqrt{n}}\right)^{t-1}\prod_{t^{\prime}=1}^{t}\left(n-t^{\prime}\right)E\left[1\left\\{F_{W}\left(W_{i}\right)>1-h_{n}\right\\}\left(\left\|X_{i}\right\|+E\left\|X_{i}\right\|\right)^{t-1}\left|\alpha_{ij_{1}}\left(\beta\right)\right|\right].$

It follows from Lemma A.7 that by properly restricting the divergence rate of
$M_{n}$,

$\displaystyle\Pr\left(\sum_{i=1}^{n}J_{rsti}\left(\hat{\beta}\right)>\delta\right)$
$\displaystyle\leq$ $\displaystyle
O\left(n\left(nh_{n}\right)^{-\left.\left(r+s+1\right)\right/2}\left(\frac{M_{n}}{\sqrt{n}}\right)^{t-1}n^{t}\right)\left(\frac{M_{n}}{\sqrt{nh_{n}^{2}}}\right)^{-\left(t-1\right)}\left(\sqrt{\frac{n}{h_{n}^{3}}}\right)^{-1}J_{n}^{\left(t-1\right)}\left(E\left\|X_{i}\right\|\right)$
$\displaystyle=$ $\displaystyle
O\left(\left(nh_{n}\right)^{-\left.\left(r+s-2t\right)\right/2}\right)\cdot
o\left(1\right)$ $\displaystyle=$ $\displaystyle o\left(1\right)$

for any $1\leq t\leq s\leq r\leq 5$. Consequently,

$\left|\text{(II)}\right|+\left|\text{(III)}\right|=o_{p}\left(1\right).$ (38)

Now consider (IV). Write

$\displaystyle\left(1-\hat{F}_{n}\left(\hat{W}_{i}\right)\right)-\left(1-F_{W}\left(W_{i}\right)\right)$
$\displaystyle=$ $\displaystyle\left\\{\frac{1}{n-1}\sum_{j\neq
i}1\left\\{X_{ij}^{\prime}\hat{\beta}<0\right\\}-\left.E\left[\left.1\left\\{X_{ij}^{\prime}\beta<0\right\\}\right|X_{i}\right]\right|_{\beta=\hat{\beta}}\right\\}$
$\displaystyle+\left\\{\left.E\left[\left.1\left\\{X_{ij}^{\prime}\beta<0\right\\}\right|X_{i}\right]\right|_{\beta=\hat{\beta}}-E\left[\left.1\left\\{X_{ij}^{\prime}\beta_{0}<0\right\\}\right|X_{i}\right]\right\\}.$

For the first term, it follows from the VC property of
$\left\\{1\left\\{X_{j}^{\prime}\beta>t\right\\}:\beta\in R^{d_{x}},t\in
R\right\\}$ and the Donsker theorem that

$\displaystyle\sup_{i}\left|\frac{1}{n-1}\sum_{j\neq
i}1\left\\{X_{ij}^{\prime}\hat{\beta}<0\right\\}-\left.E\left[\left.1\left\\{X_{ij}^{\prime}\beta<0\right\\}\right|X_{i}\right]\right|_{\beta=\hat{\beta}}\right|$
$\displaystyle\leq$ $\displaystyle\sup_{\beta\in R^{d_{x}},t\in
R}\left|\frac{1}{n}\sum_{j=1}^{n}1\left\\{X_{j}^{\prime}\beta>t\right\\}-E\left[1\left\\{X_{j}^{\prime}\beta>t\right\\}\right]\right|+\frac{1}{n}$
$\displaystyle=$ $\displaystyle O_{p}\left(\frac{1}{\sqrt{n}}\right).$

For the second term, since

$\displaystyle
E\left[\left.1\left\\{X_{ij}^{\prime}\beta<0\right\\}\right|X_{i}\right]$
$\displaystyle=$ $\displaystyle
E\left[\left.1\left\\{W_{j}>W_{i}+X_{ij,(-1)}^{\prime}\left(\beta_{(-1)}-\beta_{0,(-1)}\right)\right\\}\right|X_{i}\right]$
$\displaystyle=$ $\displaystyle
E\left[\left.1-F_{\left.W\right|X_{(-1)}}\left(\left.W_{i}+X_{ij,(-1)}^{\prime}\left(\beta_{(-1)}-\beta_{0,(-1)}\right)\right|X_{j,(-1)}\right)\right|X_{i}\right],$

we have

$\partial_{\beta_{(-1)}}E\left[\left.1\left\\{X_{ij}^{\prime}\beta<0\right\\}\right|X_{i}\right]=-E\left[\left.f_{\left.W\right|X_{(-1)}}\left(\left.W_{i}+X_{ij,(-1)}^{\prime}\left(\beta_{(-1)}-\beta_{0,(-1)}\right)\right|X_{j,(-1)}\right)X_{ij,(-1)}\right|X_{i}\right].$

It follows that

$\displaystyle\left|\left.E\left[\left.1\left\\{X_{ij}^{\prime}\beta<0\right\\}\right|X_{i}\right]\right|_{\beta=\hat{\beta}}-E\left[\left.1\left\\{X_{ij}^{\prime}\beta_{0}<0\right\\}\right|X_{i}\right]\right|$
$\displaystyle\leq$
$\displaystyle\overline{f_{\left.W\right|X_{(-1)}}}\left\|\hat{\beta}_{(-1)}-\beta_{0,(-1)}\right\|E\left[\left.\left\|X_{ij,(-1)}\right\|\right|X_{i}\right]$
$\displaystyle\leq$ $\displaystyle
O_{p}\left(\frac{1}{\sqrt{n}}\right)\left(\left\|X_{i}\right\|+E\left\|X_{i}\right\|\right).$

uniformly over $i=1,\cdots,n$. Therefore,

$\left|\left(1-\hat{F}_{n}\left(\hat{W}_{i}\right)\right)-\left(1-F_{W}\left(W_{i}\right)\right)\right|\leq
O_{p}\left(\frac{1}{\sqrt{n}}\right)\left(\left\|X_{i}\right\|+1\right)$ (39)

uniformly over $i=1,\cdots,n$, and by Assumption 5’,

$\displaystyle\left|\text{(IV)}\right|$ $\displaystyle\leq$
$\displaystyle\frac{\overline{k^{\left(6\right)}}}{6!\sqrt{nh_{n}^{13}}}\sum_{i=1}^{n}\left|\pi_{i}\right|\left|\left(1-\hat{F}_{n}\left(\hat{W}_{i}\right)\right)-\left(1-F_{W}\left(W_{i}\right)\right)\right|^{6}$
(40) $\displaystyle\leq$ $\displaystyle
O_{p}\left(\frac{1}{n^{3}\sqrt{nh_{n}^{13}}}\right)\sum_{i=1}^{n}\left|\pi_{i}\right|\left(\left\|X_{i}\right\|+1\right)^{6}$
$\displaystyle=$ $\displaystyle
O_{p}\left(\frac{n}{n^{3}\sqrt{nh_{n}^{13}}}\right)E\left|\pi_{i}\right|E\left[\left(\left\|X_{i}\right\|+1\right)^{6}\right]$
$\displaystyle=$ $\displaystyle o_{p}\left(1\right).$

Inserting (35), (38), and (40) into (31) obtains

$\text{(b1)}=o_{p}\left(1\right),\text{\ (c1)}=O_{p}\left(1\right),$

where (b1) and (c1) are defined in (28) and (29), respectively.

It remains to show that (b2) converges to zero in probability. It follows from
(39) and Assumption 5’ that

$\displaystyle\left|\text{(b-2)}\right|$ $\displaystyle\leq$ $\displaystyle
O_{p}\left(\frac{1}{\sqrt{h_{n}}}\right)\frac{1}{n}\sum_{i=1}^{n}\left\|Z_{i}\right\|\left(\left|k_{ni}\right|+\left|\hat{k}_{ni}-k_{ni}\right|\right)$
$\displaystyle\leq$ $\displaystyle
O_{p}\left(\frac{1}{\sqrt{h_{n}}}\right)\frac{1}{n}\sum_{i=1}^{n}\left\|Z_{i}\right\|1\left\\{F_{W}\left(W_{i}\right)>1-h_{n}\right\\}$
$\displaystyle+O_{p}\left(\frac{1}{\sqrt{h_{n}^{3}}}\right)\frac{1}{n}\sum_{i=1}^{n}\left\|Z_{i}\right\|1\left\\{F_{W}\left(W_{i}\right)>1-h_{n}\right\\}\left|\left(1-\hat{F}_{n}\left(\hat{W}_{i}\right)\right)-\left(1-F_{W}\left(W_{i}\right)\right)\right|$
$\displaystyle+O_{p}\left(\frac{1}{\sqrt{h_{n}^{5}}}\right)\frac{1}{n}\sum_{i=1}^{n}\left\|Z_{i}\right\|\left|\left(1-\hat{F}_{n}\left(\hat{W}_{i}\right)\right)-\left(1-F_{W}\left(W_{i}\right)\right)\right|^{2}$
$\displaystyle\leq$ $\displaystyle
O_{p}\left(\frac{1}{\sqrt{h_{n}}}\right)\cdot
O_{p}\left(E\left[\left\|Z_{i}\right\|1\left\\{F_{W}\left(W_{i}\right)>1-h_{n}\right\\}\right]\right)$
$\displaystyle+O_{p}\left(\frac{1}{\sqrt{nh_{n}^{3}}}\right)\frac{1}{n}\sum_{i=1}^{n}\left\|Z_{i}\right\|1\left\\{F_{W}\left(W_{i}\right)>1-h_{n}\right\\}\left(\left\|X_{i}\right\|+1\right)$
$\displaystyle+O_{p}\left(\frac{1}{\sqrt{n^{2}h_{n}^{5}}}\right)\frac{1}{n}\sum_{i=1}^{n}\left\|Z_{i}\right\|\left(\left\|X_{i}\right\|+1\right)^{2}$
$\displaystyle\leq$ $\displaystyle
O_{p}\left(\frac{1}{\sqrt{h_{n}}}\right)\cdot
O_{p}\left(h_{n}^{1-1\left/\left(2+c_{1}\right)\right.}\right)+O_{p}\left(\frac{1}{\sqrt{nh_{n}^{3}}}\right)\cdot
O_{p}\left(h_{n}^{1-1/2-1/6}\right)+O_{p}\left(\frac{1}{\sqrt{n^{2}h_{n}^{5}}}\right)$
$\displaystyle=$ $\displaystyle
O_{p}\left(h_{n}^{c_{1}\left/\left(4+2c_{1}\right)\right.}\right)+O_{p}\left(\left(nh_{n}^{7/3}\right)^{-1/2}\right)+O_{p}\left(\left(nh_{n}^{5/2}\right)^{-1}\right)$
$\displaystyle=$ $\displaystyle o_{p}\left(1\right),$

which completes the proof.

###  7.4 Proof of Theorem 3

Denote
$\hat{k}_{ni}=k\left(\left.\left(1-\hat{F}_{n}\left(\hat{W}_{i}\right)\right)\right/h_{n}\right)$
and
$k_{ni}=k\left(\left.\left(1-F_{W}\left(W_{i}\right)\right)\right/h_{n}\right)$.
And define the infeasible local linear estimator as

$\left(\mu_{n}^{L},b_{n}^{L}\right)=\arg\min_{\mu,b}\sum_{i=1}^{n}\left[Y_{i}-Z_{i}^{\prime}\theta_{0}-\mu-\left(F_{W}\left(W_{i}\right)-1\right)b\right]^{2}D_{i}k_{ni}.$
(41)

I will first establish the asymptotic normality of
$\left(\mu_{n}^{L},b_{n}^{L}\right)$ by the standard argument for the
nonparametric local linear estimation, and then prove the asymptotic
negligibility of the difference of $\left(\hat{\mu}^{L},\hat{b}^{L}\right)$
and $\left(\mu_{n}^{L},b_{n}^{L}\right)$ in a way similar to the proof of
Theorem 2.

  * •

First step: proving

$\left(\begin{array}[]{cc}\sqrt{nh_{n}}&\\\
&\sqrt{nh_{n}^{3}}\end{array}\right)\left[\left(\begin{array}[]{c}\mu_{n}^{L}\\\
b_{n}^{L}\end{array}\right)-\left(\begin{array}[]{c}\mu_{0}\\\
g\left(1\right)\end{array}\right)-\left(\begin{array}[]{c}\frac{\left(\kappa_{2}^{2}-\kappa_{1}\kappa_{3}\right)g^{\prime}\left(1\right)}{2\left(\kappa_{0}\kappa_{2}-\kappa_{1}^{2}\right)}h_{n}^{2}\\\
\frac{\left(\kappa_{1}\kappa_{2}-\kappa_{0}\kappa_{3}\right)g^{\prime}\left(1\right)}{2\left(\kappa_{0}\kappa_{2}-\kappa_{1}^{2}\right)}h_{n}\end{array}\right)\right]\rightarrow
N\left(0,\sigma_{U}^{2}\Omega^{L}\right),$

where the infeasible local linear estimator