Full text of DOI:10.1111/1468-0262.00430

Econometrica, Vol. 71, No. 3 (May, 2003), 905–932
NOTES AND COMMENTS
INFERENCE IN CENSORED MODELS
WITH ENDOGENOUS REGRESSORS
By Han Hong and Elie Tamer1
This paper analyzes the linear regression model y = xꢀ + ꢁ with a conditional median
assumption medꢂꢁ ꢀ zꢃ = 0, where z is a vector of exogenous instrument random variables.
We study inference on the parameter ꢀ when y is censored and x is endogenous. We treat
the censored model as a model with interval observation on an outcome, thus obtaining an
incomplete model with inequality restrictions on conditional median regressions. We ana-
lyze the identified features of the model and provide sufficient conditions for point identi-
fication of the parameter ꢀ. We use a minimum distance estimator to consistently estimate
the identified features of the model. We show that under point identification conditions
√
and additional regularity conditions, the estimator based on inequality restrictions is N-
normal and we derive its asymptotic variance. One can use our setup to treat the identifi-
cation and estimation of endogenous linear median regression models with no censoring.
A Monte Carlo analysis illustrates our estimator in the censored and the uncensored case.
Keywords: Endogeneity, interval data, modified minimum distance estimator, cen-
sored regression model, endogenous median regression.
1
ꢄ introduction
ꢁ
Consider the linear model y = x ꢀ + ꢁ where realizations of ꢂꢅy ꢆy ꢇꢆxꢆzꢃ are
0
1
ꢁ
observed such that medꢂꢁꢀzꢃ = 0 and Pꢂy ∈ ꢅy ꢆ y ꢇꢃ = 1. Moreover y = y = y if x ꢀ+ꢁ ≥ 0
0
1
0
1
ꢁ
and y = −ꢂꢆy = 0 if and only if x ꢀ+ꢁ < 0. This implies that for all z,
0
1
ꢁ
ꢁ
ꢁ
(1.1)
medꢂy −x ꢀꢀzꢃ ≤ medꢂy −x ꢀꢀzꢃ = 0 ≤ medꢂy −x ꢀꢀzꢃ
0
1
k
where ꢀ ∈ B, a compact subset of ꢀ , is the true parameter of interest. For exogenous
censored models where x = z, the pioneering work of Powell (1984) provided an estimator
for ꢀ under the conditional median assumption that is based on minimizing the least
ꢁ
absolute error Eꢀy1ꢂy ≥ 0ꢃ − maxꢂx ꢀꢆ0ꢃꢀ. In this paper, we use a different inferential
approach that accommodates cases where x is endogenous and also allows for general
interval measurement on the outcomes. Throughout this paper we maintain the following
assumptions:
Assumption 1: Pꢂy ∈ ꢅy ꢆy ꢇꢃ = 1.
0
1
1
We thank Bo Honoré, Shakeeb Khan, and Chuck Manski for discussions and Aprajit Mahajan
for excellent research assistance. We have benefited from comments by seminar participants at many
institutions. We are also grateful to the co-editor and three anonymous referees for helpful sugges-
tions on an earlier version of the paper. Tamer gratefully acknowledges financial support from the
NSF through Grant SES-0112311. All errors are ours.
905

9
06
Assumption 2: We have
h. hong and e. tamer
ꢁ
(1.2)
y = x ꢀ+ꢁ
where ꢁ is an unobserved random variable such that medꢅꢁꢀzꢇ = 0.
Assumption 1 is an enclosure assumption similar to the one used in Manski and Tamer
(
2002). In Assumption 2, we impose a semiparametric structure on the conditional distri-
ꢁ
ꢁ
bution of ꢂy −x ꢀꢃ given z, requiring that the conditional median of ꢂy −x ꢀꢃ given z be
zero. We allow for the vector x to be endogenous, where one can think of z as the vector
of instruments.
Due to the nonlinearity of the conditional median function, global identification con-
ditions of the model in (1.2) above are hard to derive. The attractiveness of the methods
provided in this paper stems from the fact that the estimator does not fail in the case of
nonidentification. We view the censored model as an incomplete model due to the inabil-
ity of the researcher to obtain complete information about the outcome variable. Hence
the estimator used here will converge to the identified feature of the model, the latter
being either a set or a point. We analyze the large sample properties of the estimator in
the point identified case, and to that effect the paper makes several contributions. For
the case of censoring and endogeneity, we analyze identification and provide sufficient
√
conditions for both point identification and n consistent estimation of the parameter
of interest. The model above maintains minimal plausible assumptions on the distribu-
tion of the data at the cost of more restrictive sufficient conditions for point identification.
We discuss cases where identification is reached by exploiting infinite support conditions
on the regressors. We use a modified minimum distance estimator first introduced in
Manski and Tamer (2002), which is based on inference in models with inequality restric-
tions (see also Tamer (2002)). This estimator consistently estimates the identified features
of the model. When there is positive probability that the censoring does not affect the
√
conditional median restriction, we show that our estimator is n consistent and asymp-
totically normal and derive its asymptotic variance. In addition, in the special case with
no censoring, our method reduces to an estimator in a linear model with endogenous
regressors and a conditional median restriction on the structural error term. This model
was first considered by Amemiya (1981) and Powell (1983) where they also imposed con-
ditional median assumptions on the error term in the reduced form equations for the
endogenous regressors in the structural equation. Hence from a theoretical perspective,
our estimator contrasts with the previous literature by using minimal plausible assump-
tions (no assumptions on the reduced form) at the cost of stronger sufficient conditions
for point identification.
There is a large literature in econometrics that deals with endogeneity in nonlinear
models. In median regressions with endogenous regressors and no censoring, Amemiya
(1981) and Powell (1983) introduced a two stage least absolute deviation estimator. Their
estimator requires additional assumptions on the reduced form equations that are not
present in our model. Existing methods for inference in censored models in the pres-
ence of endogeneity are based on the explicit parameterization of the endogeneity using
distributional assumptions. See Heckman (1978), Amemiya (1979), Blundell and Smith
(
1989), and Vella and Verbeek (1999) in panel data models. See also Chernozhukov and
Hansen (2002) and Honoré and Hu (1999). The recent paper by Blundell and Powell
2000) contains the latest developments in the literature on nonlinear semiparametric and
nonparametric regressions with endogeneity.
(

censored models with endogenous regressors
907
In Section 2, we study identification in the censored model with endogenous regressors.
We provide sufficient conditions for point identification. In Section 3, we use a modified
minimum distance estimator and show that it consistently estimates ꢀ. Under additional
regularity conditions, we show that the estimator is root-n normal and derive its asymp-
totic variance. Finally, a Monte Carlo simulation in Section 4 illustrates the small sample
performance of our estimator. A technical Appendix collects the proofs.
2
ꢄ censored regression model: identification
We study the identified features of the following model:
ꢁ
y = x ꢀ+ꢁ where medꢂꢁ ꢀ zꢃ = 0
and the latter quantity is defined by
ꢀ
ꢁ
1
medꢂꢁ ꢀ zꢃ = inf x ꢈ Pꢂꢁ ≤ xꢀzꢃ ≥ 2
k
d
where x ∈ ꢀ and z ∈ ꢀ and realizations of the variables ꢂꢅy ꢆy ꢇꢆxꢆzꢃ are observed
0
1
such that Assumptions 1 and 2 above are satisfied. To ensure unambiguous definition of
the conditional median we assume that it is uniquely defined for each z. One sufficient
condition is the following:
Assumption 3: The conditional density of ꢁ given x and z is continuous and bounded
away from 0 uniformly in a small neighborhood of 0 and uniformly in x and z. That is,
∃
f > 0ꢆc > 0ꢆ such that inf f ꢂꢉꢀzꢆxꢃ ≥ f ꢄ
ꢁ
ꢀ
ꢉꢀ≤c
It is a standard result that under Assumption 3, medꢂꢁꢀzꢃ is uniquely defined for each
z. It also allows us to translate a statement about the conditional median medꢂꢁꢀzꢃ into
an equivalent statement about the conditional probability distribution of ꢁ given z. When
1
ꢉ = medꢂꢁꢀzꢃꢆPꢂꢁ ≤ ꢉꢀzꢃ = 1/2, and for all ꢉ
ꢃ
ꢃ
. In the special
2
ꢁ
case where the interval is degenerate: Prꢅy = y = y ꢇ = 1, we have medꢅy−x ꢀ = ꢁ ꢀ zꢇ = 0.
0
1
This is a case of the linear LAD model with endogenous regressors. Assumption (1)
implies the following inequality restrictions on conditional median regressions:
ꢁ
ꢁ
(2.3)
medꢅy −x ꢀꢀzꢇ ≤ 0 ≤ medꢅy −x ꢀꢀzꢇ
0
1
for all instruments z almost everywhere. These inequalities provide all the identifying
restrictions available for the parameter ꢀ. Heuristically, the true parameter ꢀ is identified
if for all b
ꢃ= ꢀ the inequality restrictions above are violated on a set of z with positive
probability. This is formalized in the following definition.
Definition 1: Let b ∈ B. Let
b
d
ꢁ
ꢁ
Z = ꢊz ∈ ꢀ ꢈ medꢅy −x bꢀzꢇ > 0 or medꢅy −x bꢀzꢇ < 0ꢋꢄ
0
1
b
We say that ꢀ is identified relative to b if and only if PꢅZ ꢇ > 0. Let
b
V = ꢊb ∈ B ꢀ PꢅZ ꢇ = 0ꢋꢄ
V is the set of observationally equivalent parameter values. We say that the parameter ꢀ
is (point) identified if and only if V ≡ ꢊꢀꢋ.

9
08
h. hong and e. tamer
b
Note that under Assumption 3, Z can alternatively be defined as
ꢀ
ꢁ
1
2
1
2
b
d
ꢁ
ꢁ
(2.4)
Z = z ∈ ꢀ ꢈ Pꢂy −x b > 0ꢀzꢃ > or Pꢂy −x b > 0ꢀzꢃ <
ꢆ
0
1
b
which implies that V = ꢊꢀꢋ if and only if PꢂZ ꢃ > 0 for all b ꢃ= ꢀ. From the definition
above, the identified feature of the general censored model is the set V . The estimator
we propose in the next section will converge to the identified features of the model,
the latter being in general a set. In this paper, we use 0 as the censoring point. This is
an innocuous normalization that can be relaxed to allow arbitrary but known censoring
ꢁ
points. For example, when the censoring point is c, define y = c if x ꢀ + ꢁ < c. We also
1
briefly discuss the special case of no censoring where Prꢂy = y = y ꢃ = 1.
0
1
Sufficient Conditions for Point Identification.
We provide a set of sufficient conditions for point identification. Due to the nonlin-
earity of the conditional quantile function, necessary and sufficient conditions for point
identification of ꢀ are hard to obtain for the general case. A sufficient condition for point
√
identification that also allows for n consistent estimation under additional regularity
conditions is the existence of a set of the instrument vector z with positive probability such
that the regression function is in the uncensored region with probability one on this set.
This is similar to the identification condition obtained in Powell (1984) for the standard
censored model. In fact it reduces to the Powell (1984) condition when z = x. These con-
ditions are motivated by a set T that also plays an important role in the root-n consistent
estimation of the parameter ꢀ in the point identified case. Define
ꢁ
T = ꢊzꢀPꢂx ꢀ ≥ 0ꢀzꢃ = 1ꢋꢄ
An important property of the set above is shown in the following lemma.
Lemma 1: Under Assumptions 1, 2, and 3, we have
ꢁ
ꢁ
ꢁ
ꢁ
ꢁ
ꢁ
medꢂy −x ꢀꢀz ꢃ = medꢂy −x ꢀꢀz ꢃ = medꢂy −x ꢀꢀz ꢃ = 0
0
1
ꢁ
if and only if z ∈ T .
In Lemma 2 we show point identification of ꢀ (namely V = ꢊꢀꢋ) under the “regular”
condition that the set T is nonempty. This allows us to produce an estimator for ꢀ that is
√
n consistent and asymptotically normal under additional regularity conditions. We also
show that point identification of ꢀ is possible without the condition that T is nonempty.
This is a case of “identification at infinity” where convergence rates are usually slower
√
than n (see Andrews and Schafgans (1998) for an example).
Lemma 2: Let Assumptions 1, 2, and 3 hold. Let PꢂT ꢃ > 0. For any ꢌ
ꢃ= 0,
ꢂ
ꢌ
T
ꢁ
ꢁ
Z = zꢀz ∈ T ꢆ eitherꢅPꢂx ꢌ ≥ 0ꢀzꢃ = 1 and Pꢂx ꢌ > 0ꢀzꢃ > 0ꢇ
ꢃ
ꢁ
ꢁ
or ꢅPꢂx ꢌ ≤ 0ꢀzꢃ = 1 and Pꢂx ꢌ < 0ꢀzꢃ > 0ꢇ ꢄ
ꢌ
T
The parameter ꢀ is point identified (the set V ≡ ꢀ) if for all ꢌ
ꢃ

censored models with endogenous regressors
909
It is easy to see that the above sufficient conditions are satisfied in the case where we
have a single positive endogenous regressor x with a positive coefficient. In the case where
k
the endogenous variables x have continuous distributions with full support over R , and
ꢁ
ꢁ
ꢁ
ꢁ
for ꢀ
ꢃ= ꢌ, we require that either one of the sets ꢊx ꢀ > 0ꢆx ꢌ > 0ꢋ or ꢊx ꢀ > 0ꢆx ꢌ < 0ꢋ be
nonempty. A global sufficient condition by Lemma 2 is that it is possible to have a z such
that either one of these two sets of x has probability 1 conditional on z.
Identification at Infinity. Although the sufficient global identification conditions pre-
sented in Lemma 2 cover plausible cases, they also leave out some important cases in
which the supports of x and z are both unbounded. For example, in the case where x and
bz ua tr eT s ci sa lea mr sp ht ya .v ing a nondegenerate bivariate normal joint distributions we have V = ꢊꢀꢋ
Lemma 3: Let ꢂxꢆzꢃ be a bivariate nondegenerate normal random vector. Let Assump-
tions 1, 2, and 3 hold. Then the set T is empty and the parameter ꢀ is identified.
Besides requiring the median uniqueness Assumption 3, the crucial condition for the
ꢁ
arguments above is the possibility of moving z so that the conditional probability of x ꢀ > 0
ꢁ
given z becomes arbitrary close to 1, and the conditional probability of either x ꢌ > 0 or
ꢁ
x ꢌ < 0 given z becomes arbitrary close to 1. For our root N consistent estimator to be
well behaved, we require in Section 3 below that the set T be nonempty.
The case with no censoring (endogenous LAD model): As noted previously, in the
d
absence of censoring we have Pꢂy = y = y ꢃ = 1 and T = ꢀ . The sufficient condition
0
1
ꢌ
stated in Lemma 2 can thus be simplified to PꢂZ ꢃ > 0 where
ꢌ
ꢁ
ꢁ
Z = ꢊzꢀꢅPꢂx ꢌ ≥ 0ꢀzꢃ = 1 and Pꢂx ꢌ > 0ꢀzꢃ > 0ꢇ or
ꢁ
ꢁ
ꢅPꢂx ꢌ ≤ 0ꢀzꢃ = 1 and Pꢂx ꢌ < 0ꢀzꢃ > 0ꢇꢋꢄ
Notice that in the case where x = ꢂx ꢆ z ꢃ such that z has infinite support while x has
1
1
1
1
finite support, the above condition is satisfied.
3
ꢄ estimation
We present an estimator below that exploits the information contained in the inequality
restrictions (2.3) above. The estimator is based on the following proposition.
Proposition 1: Let V ≡ ꢊꢀꢋ (point identification condition). For every b ∈ B, define
the function
ꢄ
1
ꢂ
ꢃ
2
2
(
3.5)
Qꢂbꢃ = 2 ꢅꢉ ꢂzꢍbꢃꢇ 1ꢂꢉ ꢂzꢍbꢃ > 0ꢃ+ꢅꢉ ꢂzꢍbꢃꢇ 1ꢂꢉ ꢂzꢍbꢃ < 0ꢃ dP
0
0
1
1
z
ꢀ
ꢅꢆꢅ
ꢇ ꢈ
+
2
1
1
2
ꢁ
= 2
Ez
Pꢂy −x b > 0ꢀzꢃ−
0
ꢆꢅ
ꢇ ꢈ ꢇ
ꢁ
−
2
1
2
ꢁ
f ꢂzꢃ2
+
Pꢂy −x b > 0ꢀzꢃ−
1
ꢁ
1
ꢁ
1
where ꢉ ꢂzꢍbꢃ = ꢂPꢂy −x b > 0 ꢀ zꢃ− ꢃf ꢂzꢃ and ꢉ ꢂzꢆbꢃ = ꢂPꢂy −x b > 0 ꢀ zꢃ− ꢃf ꢂzꢃ.
0
0
2
1
1
2
+
−
We have used the notation that y = y 1ꢂy > 0ꢃ and y = y 1ꢂy < 0ꢃ. Then Qꢂbꢃ ≥ 0 for all
b ∈ B, and Qꢂbꢃ = 0 if and only if b = ꢀ.

9
10
h. hong and e. tamer
The point identification condition in the previous lemma can be removed and the con-
clusion of the lemma would hold on the identified set V , i.e., Qꢂbꢃ = 0 iff b ∈ V . The
estimator in this case would be a set based estimator. We also use a density weighting
scheme in the definition of the function Qꢂbꢃ. This eliminates the need for tedious trim-
ming arguments. We start with formally stating an assumption about the sampling scheme.
Assumption 4: In the censored model we observe an i.i.d. sample ꢂꢅy ꢆy ꢇꢆx ꢆz ꢃ of
0
i
1i
i
i
data such that y0i = y = y if y ≥ 0 and y = 0ꢆy0i = −ꢂ for y < 0.
i
1i
i
1i
i
By the analog estimation principle, the estimator of ꢀ is based on minimizing Q ꢂbꢃ,
n
defined in (3.6), which is a feasible variant of the sample analog of the function Qꢂbꢃ. First
we replace the functions ꢉ ꢂzꢍbꢃ and ꢉ ꢂzꢍbꢃ by consistent kernel based nonparametric
0
1
estimators ꢉˆ ꢂzꢍbꢃ and ꢉˆ ꢂzꢍbꢃ,
0
1
ꢉ
ꢉ
ꢋ
ꢋ
ꢌ
ꢌ
ꢍ
zj −zi
n
ꢂyj0 −xꢁ
d
d
1
ꢊ 1
s
j
b ≥ 0ꢃK
1
h
ˆ
ꢉˆ ꢂz ꢍbꢃ =
− 2 f ꢂz ꢃꢆ
0
i
i
nhd
ˆ
f ꢂz ꢃ
j=1
i
ꢍ
zj −zi
n
ꢂyj1 −xꢁ
1
nhd
ꢊ 1
s
j
b ≥ 0ꢃK
1
h
ˆ
ꢉˆ ꢂz ꢍbꢃ =
− 2 f ꢂz ꢃꢆ
1
i
i
ˆ
f ꢂz ꢃ
j=1
i
ꢅ
ꢇ
n
1
nhd
ꢊ
z
j
−z
h
i
d
ˆ
f ꢂz ꢃ =
K
ꢆ
i
j=1
ꢄ
ꢂ
1 Kꢎ
ꢏ
ꢄ
ꢂ
ꢎ
ꢏ
y
x
ꢄ
1sꢂx ≥ 0ꢃ =
dy =
kꢂyꢃdy ≡ 1−K
ꢆ
x
h
h
x/h
h
ꢄ
y
ꢄ
Kꢂyꢃ =
Kꢂuꢃduꢆ
−
ꢂ
d
1
2
d
where K ꢂxꢃ = Kꢂx ꢃKꢂx ꢃꢎ ꢎ ꢎ Kꢂx ꢃꢆKꢂ·ꢃ is a univariate “high order” kernel function
1
d
with bounded support, h is a bandwidth parameter, and x = ꢂx ꢆꢎ ꢎ ꢎ ꢆx ꢃ. Precise condi-
tions on the kernel function are given in the next section. For simplicity of illustration,
we use here a product kernel construction instead of a general d-dimensional multivari-
ate kernel. 1 ꢂ· ≥ 0ꢃ is a smoothed version of the indicator function 1ꢂ· ≥ 0ꢃ, using the
s
ˆ
integral of a kernel function. We then define ꢀ as
ˆ
(3.6)
ꢀ = argminQ ꢂbꢃ and
n
b∈B
ꢀ
ꢆ
ꢈ
ꢆ
ꢈꢁ
n
1
ꢊ
ꢉˆ ꢂz ꢍbꢃ
ꢉˆ ꢂz ꢍbꢃ
2
0
i
2
1
i
ꢄ
Q ꢂbꢃ =
ꢅꢉˆ ꢂz ꢍbꢃꢇ G
+ꢅꢉˆ ꢂz ꢍbꢃꢇ G
ꢄ
n
0
i
1
i
∗
∗
2
n
h
h
i=1
ꢄ
Gꢂxꢃ is a continuous function that smoothly approximates 1ꢂx > 0ꢃ, and Gꢂxꢃ is a con-
tinuous function that smoothly approximates 1ꢂx < 0ꢃ.
3
ꢄ1ꢄ Assumptions
In addition to Assumptions 1–4, we also introduce the following set of assumptions.
These are important regularity conditions to ensure that the estimator is well behaved in
large samples. In the absence of censoring (y = y = y ), one can also use the methods of
0
1
Ai and Chen (2000) to study the asymptotic properties of (3.6) (without the G’s).

censored models with endogenous regressors
911
k
Assumption 5: The parameter space B ⊂ ꢀ is compact. The true parameter value ꢀ is
an interior point of B and point identification holds: V = ꢊꢀꢋ.
Assumption 6: Prꢂꢁ ≤ t ꢀ zꢃ is continuous in t and z uniformly in z and t.
ꢁ
ꢁ
Assumption 7: ∃ꢉ > 0, such that Pꢂ0 < Pꢂx ꢀ < 0ꢀzꢃꢆPꢂx ꢀ < −ꢉꢀzꢃ ≤ hꢃf ꢂzꢃ = Oꢂhꢃ.
ꢁ
Moreover, ∃ꢁ > 0 such that ∀t ∈ ꢅ0ꢆꢁꢇꢆPꢂꢀx ꢀꢀ = tꢀzꢃ = 0 for almost surely all z.
Assumption 8: f ꢂꢁꢀxꢆzꢃ, f(x|z), and f ꢂzꢃ have at least m bounded continuous deriva-
ꢁ
tives with respect to ꢁ and z, uniformly in z and ꢁ in a small neighborhood of 0, such that
1
2
m > 4ꢂ d +2ꢃ+2.
Assumption 9: The kernel function Kꢂ·ꢃ and the smoothing parameter h satisfy the
following conditions:
(
i) Kꢂzꢃ vanishes outside a bounded support, and is twice continuously differentiable with
ꢐ
ꢐ
ꢏ
bounded derivatives. Kꢂzꢃdz = 1ꢆ z Kꢂzꢃdz = 0 for all 1 < ꢏ < m.
d+4
(ii) h → 0ꢆnh /logn → ꢂ; in addition for some ꢁ > 0,
ꢅ
ꢇ
√
3
8
logn
m−2
+ꢁ
nh
+n
→ 0ꢄ
1
n1/2
d+2
h
2
Assumption 10: The smoothed step function in (3.6) satisfies the following conditions:
i) Gꢂ·ꢃ is a four times continuously differentiable distribution function with bounded
(
ꢄ
derivatives such that Gꢂyꢃ = 0ꢆ∀y ≤ −1ꢆGꢂyꢃ = 1ꢆ∀y > −1/2. Gꢂ·ꢃ is a four times con-
ꢄ
tinuously differentiable function with bounded derivatives such that Gꢂyꢃ = 1ꢆ∀y ≤ 1/2ꢆ
ꢄ
Gꢂyꢃ = 0ꢆ∀y > 1.
(ii) Let ꢐ > 3/8 be such that for some small neighborhood ꢁ around ꢀ, both
−
ꢐ
supsupꢀꢉˆ ꢂzꢆbꢃ−ꢉ ꢂzꢆbꢃꢀ = o ꢂn
ꢃ
and
0
0
p
z
b∈ꢁ
−
ꢐ
supsupꢀꢉˆ ꢂzꢆbꢃ−ꢉ ꢂzꢆbꢃꢀ = o ꢂn ꢃꢄ
1
1
p
z
b∈ꢁ
∗
−ꢑ
Moreover, let h = n such that ꢑ > 1/4ꢆꢑ < ꢐ, and 1/2 −2ꢐ+ꢑ < 0.
Comments:
•
Sufficient conditions for point identification are given in the previous section. The
estimator is consistent without the additional regularity conditions to ensure n consis-
√
tency and asymptotic normality.
•
In Assumption 6, we require some continuity conditions that are needed for uni-
form convergence of empirical conditional probabilities.
Assumptions 7 and 8 specify the regularity conditions needed to derive the asymp-
•
totic distribution. They are similar to the regularity conditions for asymptotic normality
specified in Powell (1984). The first assumption ensures that given our objective func-
tion, the estimator is asymptotically normal. For example, in the exogenous case where
ꢁ
x = z, this condition amounts to excluding models where x ꢀ = 0 with positive probabil-
ity. Assumption 8 imposes the required smoothness conditions that are needed for a well
behaved asymptotic variance.

9
12
h. hong and e. tamer
•
The conditions in Assumption 9 are standard ones on the smoothness of the kernel
function and the bandwidth parameter. They combine with Assumption 8 to give the
desired uniform rate of nonparametric estimation. In particular, the second condition
d+2r
implies that for each r = 0ꢆ1ꢆ2ꢆnh
/logn → ꢂ, and
ꢅ
ꢇ
√
3
8
logn
m−r
+ꢁ
nh
+n
→ 0ꢄ
√nh
1
2
d+r
For consistency, only part 1 and the conditions on the smoothing parameter h in the first
line of part 2 are needed.
•
The same kernel function and bandwidth parameter in Assumption 9 are also used
to smooth the indicator function 1 ꢂx ≥ 0ꢃ.
s
ꢄ
•
Assumption 10 describes the functions Gꢂ·ꢃ and Gꢂ·ꢃ that are used to smooth the
step functions. To make sure that these functions “become” step functions as the sample
∗
size increases, we use a smoothing parameter h that goes to 0 at a certain rate that is a
function of n. As discussed in Lemma 6 in the Appendix, the existence of ꢐ in assumption
1
0(ii) is a direct consequence of Assumptions 8 and 9.
4
ꢄ consistency and asymptotic normality
We state the results for consistency and asymptotic normality of the estimator defined
in (3.6) above.
Theorem 1 (Consistency): Let Assumptions 1 through 6 hold. Moreover, let Assumption
d
∗
−ꢑ
8
, 9(i), and 10 hold. Assume also that h → 0, nh /logn → ꢂꢆh = n , and ꢑ < ꢐ where
p
ˆ
ꢐ is defined in Assumption 10. Then ꢀ −→ ꢀ.
The theorem above requires that the parameter ꢀ be point identified (Assumption 5).
Given point identification the estimator we propose is consistent, even in the case of
identification at infinity. Hence it does not require that the set T be nonempty. Next we
derive the asymptotic distribution of the estimator defined in (3.6) above. We first state
the theorem.
Theorem 2 (Asymptotic Normality): Let Assumptions 1–10 hold. Moreover, let
PꢂT ꢃ > 0. Then we have
√
d
nꢂꢀ−ꢀꢃ −→ Nꢂ0ꢆA 1ꢒA
−
−1
ꢃ
where
ˆ
ꢁ
2
A = EEꢂxf ꢂ0ꢀxꢆzꢃꢀzꢃEꢂxf ꢂ0ꢀxꢆzꢃꢀzꢃ f ꢂzꢃ1ꢂz ∈ T ꢃ
ꢁ
ꢁ
ꢁ
=
Eꢓꢉꢂzꢍꢀꢃꢓꢉꢂzꢍꢀꢃ 1ꢂz ∈ T ꢃꢆ
1
ꢁ
4
ꢒ = EEꢂxf ꢂ0ꢀxꢆzꢃꢀzꢃEꢂxf ꢂ0ꢀxꢆzꢃꢀzꢃ f ꢂzꢃ1ꢂz ∈ T ꢃ
ꢁ
ꢁ
4
1
4
2
ꢁ
=
Ef ꢂzꢃꢓꢉꢂzꢍꢀꢃꢓꢉꢂzꢍꢀꢃ 1ꢂz ∈ T ꢃꢄ
Moreover for
ꢅ
ꢇ
n
ꢊ
ˆ
ꢉˆ ꢂz ꢍꢀꢃ
1
n
ꢑ
ˆ
ˆ ˆ
ˆ
ꢁ
0
i
A =
ꢓꢉ ꢂz ꢍꢀꢃꢓꢉ ꢂz ꢍꢀꢃ G
0
i
0
i
∗
h
i=1

censored models with endogenous regressors
913
and
ꢅ
ꢇ
n
1 ꢊ
ꢉˆ ꢂz ꢍꢀꢃ
ˆ
1
4
2
ꢁ
0
i
ꢑ
ˆ
ˆ
ˆ ˆ
ˆ
ꢒ =
f ꢂz ꢃ ꢓꢉ ꢂz ꢍꢀꢃꢓꢉ ꢂz ꢍꢀꢃ G
i
0
i
0
i
∗
h
n
i=1
p
p
ꢑ
ꢑ
we have that A −→ A and ꢒ −→ ꢒ. The same conclusion holds if we replace ꢉ with ꢉ in
0
1
ꢑ
ꢑ
the definition of A and ꢒ.
ꢑ
ꢑ
In the definition of A and ꢒ we have
ꢔ
ꢔ
ˆ
ˆ
ꢓꢉ ꢂz ꢍbꢃ =
ꢉˆ ꢂz ꢍbꢃ and ꢓꢉ ꢂz ꢍbꢃ =
ꢉˆ ꢂz ꢍbꢃꢄ
0
i
0
i
1
i
1
i
ꢔb
ꢔb
The set T plays a crucial role for
√n convergence. This is similar to the case of censored
models with no endogeneity where only observations in T contribute to the asymptotic
variance (see, for example, page 311 of Powell (1984)). The first order condition of (3.6)
is given by
ꢀ
ꢅ
ꢇ
n
1
n
ꢊ
ꢉˆ ꢂz ꢍbꢃ
0
i
ˆ
(
4.7)
FOCꢂbꢃ =
ꢉˆ ꢂz ꢍbꢃꢓꢉ ꢂz ꢍbꢃG
0
i
0
i
∗
h
i=1
ꢒ
ꢓꢔ
ꢕ
ꢂaꢃ
ꢅ
ꢇ
ꢕ
ꢉˆ ꢂz ꢍbꢃ
1
i
ˆ
ꢄ
+
+
+
ꢉˆ ꢂz ꢍbꢃꢓꢉ ꢂz ꢍbꢃG
1
i
1
i
∗
h
ꢒ
ꢓꢔ
ꢂbꢃ
ꢅ
ꢅ
ꢇ
1
h
ꢉˆ ꢂz ꢍbꢃ
2
0
i
ˆ
ꢉˆ ꢂz ꢍbꢃ ꢓꢉ ꢂz ꢍbꢃ
g
0
i
0
i
∗
∗
2
h
ꢒ
ꢓꢔ
ꢕ
ꢂcꢃ
ꢇꢁ
1
ꢉˆ ꢂz ꢍbꢃ
2
1
i
ˆ
ꢉˆ ꢂz ꢍꢀꢃ ꢓꢉ ꢂz ꢍbꢃ
g¯
ꢄ
1
i
1
i
∗
∗
2
h
h
ꢒ
ꢓꢔ
ꢕ
ꢂdꢃ
ꢁ
ꢄ
ꢁ
In the above we have used the definitions gꢂxꢃ = G ꢂxꢃ and g¯ꢂxꢃ = G ꢂxꢃ. Hence, taking
a mean value expansion of the above equation around the true parameter ꢀ, we get
√
√
∗
ˆ
nꢂꢀ−ꢀꢃHꢂꢀ ꢃ = nFOCꢂꢀꢃ
−
∗
ˆ
2
where Hꢂbꢃ = ꢔFOCꢂbꢃ/ꢔb and ꢀ is between ꢀ and ꢀ. The objective is to first find the
√
asymptotic distribution of nFOCꢂꢀꢃ, which depends on the asymptotic behavior of terms
(
a), (b), (c), and (d) above evaluated at b = ꢀ. We then study the probability limit of the
Hessian matrix H. The proof of Theorem 2 below relies on a series of lemmas. The details
of these lemmas are given in the Appendix and only a sketch of the essential arguments
in the proof is given here. Lemma 6 in the Appendix, which also justifies Assumption
1
0, presents standard convergence results of the functions ꢉˆ and ꢉˆ . Lemma 9 shows
0
1
that only the terms (a) and (b) in (4.7) above contribute to the asymptotic distribution
2
∗
As pointed out by a referee, the intermediate value ꢀ is strictly speaking a set of intermediate
∗
values, one for each row Hꢂꢀ ꢃ.

9
14
h. hong and e. tamer
√
√
by showing that when evaluated at b = ꢀ, nꢂcꢃ = o ꢂ1ꢃ and nꢂdꢃ = o ꢂ1ꢃ. Lemmas 7
p
p
and 8 show that this variance depends only on observations on the set T . These together
allow us to write the first order condition as
n
1
n
ꢊ
FOCꢂꢀꢃ = √
ꢉˆꢂz ꢍꢀꢃꢓꢉꢂz ꢍꢀꢃ1ꢂz ∈ T ꢃ+o ꢂ1ꢃ
i
i
i
p
i=1
where ꢉˆꢂzꢍbꢃ and ꢓꢉꢂzꢍbꢃ are defined similarly to ꢉˆ ꢂzꢍbꢃ and ꢓꢉ ꢂzꢍbꢃ except that we
0
0
replace y with the unobserved latent y. Lemmas 10 through 16 show the local uniform
0
convergence of the Hessian to its counterpart. This is tedious because of the presence of
a nonparametric function that contains the parameter of interest inside a (smoothed) step
p
ˆ
function. Together these lemmas imply that for any ꢀ → ꢀ:
p
ꢁ
ˆ
Hꢂꢀꢃ −→ Eꢓꢉꢂzꢍꢀꢃꢓꢉꢂzꢍꢀꢃ 1ꢂz ∈ T ꢃꢄ
Lemma 16 provides the asymptotic linear representation of the first order condition.
Finally, we conjecture that the existence of sets like T with positive probability, on which
the left-hand side of the inequality in (2.3) is equal to its right-hand side (the inequalities
√
defining the imcomplete model), is a necessary condition for n consistent estimation of
the parameter of interest. Moreover, in the cases of identification at infinity where T is
empty, the rate of convergence of the estimator will depend on the tails of the joint distri-
bution of ꢂxꢆzꢃ (see, for example, Andrews and Schafgans (1998) for an example of such
a case in selection models).
5
ꢄ monte carlo simulations
In this section, we use a set of Monte Carlo simulations to illustrate our estimator and
study its small sample properties. In particular we use a simultaneous equation model such
as the one studied in Blundell and Smith (1994). The model consists of the reduced form
∗
ꢁ
y = x ꢕ +v ꢆ
1
1
1
∗
ꢁ
y = x ꢕ +v ꢆ
2
2
2
2
where ꢂv ꢆv ꢀxꢃ is bivariate normal. This gives rise to the simultaneous equation model
1
2
∗
ꢁ
y = ꢀ +y ꢌ+x ꢀ +u ꢄ
1
0
2
1
1
1
∗
∗
∗
1
Observations on ꢂy ꢆy ꢆx ꢆx ꢃ are observed such that y = y and y = gꢂy ꢃ. If gꢂy ꢃ =
2
1
1
2
2
2
1
1
∗
∗
∗
∗
y , we obtain the two stage lad model with endogeneity and when gꢂy ꢃ = 1ꢂy > 0ꢃy
1
1
1
1
we obtain the endogenous Tobit model. ꢂx ꢆx ꢃ are independent exogenous random vari-
1
2
ables such that x1 is standard normal and x2 is chi square distributed. We first draw
the disturbances ꢂu ꢆv ꢃ from a standard bivariate normal density with covariance ꢐ. We
1
2
ꢁ
then introduce heteroscedasticity by setting the random variable u = u ∗ F ꢂx ꢃ where
1
1
1
F ꢂ·ꢃ is the standard normal cumulative distribution function. For both models, we trun-
cate the distribution of v to lie between −3 and +3. This ensures that the regularity
2
√
3
conditions for n consistency and asymptotic normality hold. In the case where y is cen-
sored, we compare two estimators. The first estimator is the endogenous censored LAD
3
We also ran the above designs without the truncation and found that the results are only slightly
different.

censored models with endogenous regressors
915
(
eCLAD) estimator introduced in this paper. We compare it to the maximum likelihood
estimator. We report results both when the latter is correctly specified (MLE) and when
it is misspecified (mMLE) by ignoring heteroskedasticity. In the case where y is not cen-
sored, we also use two estimators. The endogenous LAD (eLAD) is the same as the
above eCLAD estimator with y = y = y . We compare the behavior of the eLAD to the
0
1
correctly specified two-stage LAD estimator (2SLAD) studied in Amemiya (1981) and
Powell (1983) and the misspecified two-stage LAD estimator (m2SLAD) where in the lat-
ter the second stage is misspecified (we let the second stage depend on the square of x2
as opposed to x ).
2
Throughout, we set the parameter vector ꢂꢀ ꢆꢌꢆꢀꢃ = ꢂ1ꢆꢄ5ꢆ1ꢃ. The experiments are
0
repeated 100 times. When y is censored, we report results for both the cases where
ꢐ = ꢄ5 and ꢐ = ꢄ9. For the designs above, it turns out that smoothing the step function
in the eCLAD estimator did not change the results much and hence was not used in the
simulations. For the eCLAD and the eLAD a plug-in smoothing parameter was chosen
similar to the one used for conditional density estimators and the standard gaussian kernel
is used where we numerically implement the definition of y = −ꢂ by setting it to a large
0
negative number. The optimization is done using simulated annealing. Table I summarizes
the experiments for the endogenous censored model. The eCLAD estimator performs
well both at 200 and at 800 observations. As compared to the MLE, the eCLAD estimator
is less efficient, especially when estimating ꢌ. However, a very small misspecification, like
ignoring heteroskedasticity, renders the MLE inconsistent, especially for estimating ꢌ in
TABLE I
Monte Carlo Simulations I
ꢀ
0
ꢀ1
ꢌ
eCLAD
MLE
mMLE
eCLAD
MLE
mMLE
eCLAD
MLE
mMLE
N = 200
Mean
Median
Std Dev
IQR
ꢐ = ꢄ9
1ꢄ001
1ꢄ000
1ꢄ13
1ꢄ03
1ꢄ026
ꢄ071
ꢄ996
ꢄ997
ꢄ015
ꢄ021
ꢄ9238
ꢄ927
ꢄ040
ꢄ056
ꢄ987
ꢄ977
ꢄ086
ꢄ088
1ꢄ131
1ꢄ131
ꢄ0218
ꢄ044
ꢄ455
ꢄ476
ꢄ102
ꢄ100
ꢄ501
ꢄ501
ꢄ008
ꢄ010
ꢄ401
ꢄ400
ꢄ027
ꢄ035
ꢄ0832
ꢄ017
N = 800
Mean
Median
Std Dev
IQR
ꢐ = ꢄ9
ꢄ999
1ꢄ000
ꢄ006
1ꢄ025
1ꢄ025
ꢄ035
ꢄ999
ꢄ998
ꢄ008
ꢄ112
ꢄ923
ꢄ9228
ꢄ023
ꢄ032
ꢄ988
ꢄ994
ꢄ044
ꢄ051
1ꢄ127
1ꢄ127
ꢄ021
ꢄ482
ꢄ485
ꢄ042
ꢄ047
ꢄ500
ꢄ500
3ꢄ3e-3
4ꢄ1e-3
ꢄ403
ꢄ403
ꢄ015
ꢄ0178
ꢄ051
8ꢄ8e-3
ꢄ024
N = 200
Mean
Median
Std Dev
IQR
ꢐ = ꢄ5
ꢄ9989
1ꢄ00
ꢄ0266
ꢄ0372
1ꢄ024
1ꢄ024
ꢄ0671
ꢄ0838
ꢄ9966
ꢄ999
ꢄ0345
ꢄ0441
ꢄ8799
ꢄ8824
ꢄ0553
ꢄ0754
ꢄ997
ꢄ993
ꢄ073
ꢄ097
1ꢄ135
1ꢄ135
ꢄ0581
ꢄ0755
ꢄ477
ꢄ488
ꢄ092
ꢄ0825
ꢄ5014
ꢄ5012
ꢄ0150
ꢄ0176
ꢄ4807
ꢄ4765
ꢄ0315
ꢄ0413
N = 800
Mean
Median
Std Dev
IQR
ꢐ = ꢄ5
1ꢄ00
1ꢄ00
ꢄ0166
ꢄ0139
1ꢄ017
1ꢄ019
ꢄ0351
ꢄ0485
1ꢄ00
1ꢄ00
ꢄ0154
ꢄ0201
ꢄ8849
ꢄ8824
ꢄ0233
ꢄ0312
ꢄ988
ꢄ986
ꢄ0481
ꢄ0614
1ꢄ135
1ꢄ135
ꢄ0287
ꢄ0367
ꢄ485
ꢄ485
ꢄ0392
ꢄ0496
ꢄ499
ꢄ499
ꢄ0067
ꢄ0077
ꢄ476
ꢄ475
ꢄ0150
ꢄ0189

9
16
h. hong and e. tamer
TABLE II
Monte Carlo Simulations II
ꢀ
0
ꢀ1
ꢌ
eLAD
2SLAD
m2SLAD
eLAD
2SLAD
m2SLAD
eLAD
2SLAD
m2SLAD
N = 200
Mean
Median
Std Dev
IQR
1ꢄ007
1ꢄ001
ꢄ0386
ꢄ0311
1ꢄ008
1ꢄ011
ꢄ094
ꢄ749
ꢄ745
ꢄ108
ꢄ16
1ꢄ002
1ꢄ001
ꢄ0244
ꢄ0161
1ꢄ013
1ꢄ013
ꢄ0657
ꢄ084
ꢄ996
ꢄ996
ꢄ064
ꢄ092
ꢄ482
ꢄ495
ꢄ0422
ꢄ0258
ꢄ495
ꢄ497
ꢄ0353
ꢄ0352
ꢄ482
ꢄ480
ꢄ084
ꢄ123
ꢄ151
N = 800
Mean
Median
Std Dev
IQR
1ꢄ00
1ꢄ00
ꢄ0135
ꢄ0133
ꢄ997
1ꢄ005
ꢄ0437
ꢄ052
ꢄ760
ꢄ763
ꢄ054
ꢄ072
1ꢄ00
1ꢄ00
ꢄ0073
ꢄ004
ꢄ998
1ꢄ002
ꢄ030
ꢄ999
ꢄ997
ꢄ033
ꢄ061
ꢄ491
ꢄ499
ꢄ016
ꢄ0111
ꢄ499
ꢄ500
ꢄ0155
ꢄ0208
ꢄ474
ꢄ474
ꢄ046
ꢄ038
ꢄ0396
the case where ꢐ = ꢄ9 and the structural equation error and the reduced form equation
error are highly correlated. Notice also that the standard error for the eCLAD estimator
falls roughly by a magnitude of 2 when we move from 200 to 800 observations reflecting
the root N convergence rate of the estimator. In Table II, we summarize the results for the
endogenous LAD model without censoring and compare our estimator to the two stage
LAD estimator. In this case, we see that the eLAD estimator performs exceptionally well
as compared to the 2SLAD. Notice also that the 2SLAD is inconsistent in this design if the
second stage is not well specified. Hence, from this perspective the eLAD is theoretically
more attractive. We also note that the standard deviation of the eLAD is considerably
lower than that for the 2SLAD except for ꢌ.
6
ꢄ conclusion
This paper provides an inferential approach to linear models with censored outcomes
and endogenous regressors that only imposes a conditional quantile restriction on the
errors. We study the identified feature of the model and provide sufficient identification
√
conditions that lead to n consistent and asymptotically normal estimation of the param-
eter of interest. Our proposed estimator consistently estimates the parameter of interest
under more general identification conditions that include “identification at infinity.” We
also derive the asymptotic distribution of the estimator in the regular case. As a by prod-
uct, our approach can also be used in linear models with no censoring and endogeneity
under a conditional quantile restriction on the error.
Economics Dept., Fisher Hall 110, Princeton University, Princeton, NJ 08544, U.S.A.;
doubleh@princeton.edu; www.princeton.edu/∼doubleh
and
Economics Dept., Fisher Hall 112, Princeton University, Princeton, NJ 08544, U.S.A.;
tamer@princeton.edu; www.princeton.edu/∼tamer
Manuscript received November, 2000; final revision received June, 2002.

censored models with endogenous regressors
917
APPENDIX A
Aꢄ1ꢄ Proofs of Some Lemmas
ꢁ
ꢁ
ꢁ
ꢁ
ꢁ
Proof of Lemma 1: Consider medꢂy − x ꢀꢀz ꢃ = medꢂmaxꢂꢁꢆ−x ꢀꢃꢀz ꢃ. For
z
such that
1
ꢁ
ꢁ
Pꢂx ꢀ ≥ 0ꢀz ꢃ = 1,
1 ꢄ
ꢁ
ꢁ
Pꢂy −x ꢀ ≥ 0ꢀzꢃ = Pꢂmaxꢂꢁꢆ−x ꢀꢃ ≥ 0ꢀzꢃ = Pꢂꢁ ≥ 0ꢀzꢃ =
1
2
ꢁ
Similarly for such z we have
1 ꢄ
ꢁ
ꢁ
ꢁ
ꢁ
ꢁ
ꢁ
ꢁ
Pꢂy −x ꢀ ≥ 0ꢀz ꢃ = Pꢂꢁ +x ꢀ ≥ 0ꢆꢁ ≥ 0ꢀz ꢃ = Pꢂꢁ ≥ −x ꢀꢆꢁ ≥ 0ꢀz ꢃ = Pꢂꢁ ≥ 0ꢀz ꢃ =
0
2
ꢁ
ꢁ
ꢁ
Hence medꢂy −x ꢀꢀz ꢃ = 0. For z
ꢃ
0
ꢁ
ꢁ
ꢁ
ꢁ
ꢁ
Pꢂy −x ꢀ ≥ 0ꢀzꢃ = Pꢂmaxꢂꢁꢆ−x ꢀꢃ ≥ 0ꢆx ꢀ > 0ꢀzꢃ+Pꢂmaxꢂꢁꢆ−x ꢀꢃ ≥ 0ꢆx ꢀ < 0ꢀzꢃ
1
ꢁ
ꢁ
=
Pꢂꢁ > 0ꢆx ꢀ > 0ꢀzꢃ+Pꢂx ꢀ < 0ꢀzꢃ
1
2
ꢁ
ꢁ
>
Pꢂꢁ > 0ꢆx ꢀ > 0ꢀzꢃ+Pꢂꢁ ≥ 0ꢆx ꢀ < 0ꢀzꢃ =
where the inequality holds strictly due to Assumption 3. Similarly,
ꢁ
ꢁ
Pꢂy −x ꢀ > 0ꢀzꢃ = Pꢂꢁ +x ꢀ ≥ 0ꢆꢁ ≥ 0 ꢀ zꢃ
0
ꢁ
ꢁ
ꢁ
=
Pꢂꢁ ≥ 0ꢆx ꢀ > 0ꢀzꢃ+Pꢂꢁ +x ꢀ ≥ 0ꢆꢁ ≥ 0ꢆx ꢀ < 0 ꢀ zꢃ
1
2
ꢁ
ꢁ
<
Pꢂꢁ ≥ 0ꢆx ꢀ > 0ꢀzꢃ+Pꢂꢁ ≥ 0ꢆx ꢀ < 0ꢀzꢃ =
where again the strict inequality is due to Assumption 3, which implies that
ꢁ
ꢁ
Pꢂꢁ +x ꢀ < 0ꢆꢁ ≥ 0ꢆx ꢀ < 0 ꢀ zꢃ > 0ꢄ
Note also that in case the censoring point is a instead of 0, the conditions on T should be modified
to ꢊzꢀPꢂxꢀ ≥ aꢀzꢃ = 1ꢋ. Therefore in the special case where a = −ꢂ and Prꢂy = y = yꢃ = 1, the set
0
1
d
T is ꢀ .
Q.E.D.
Proof of Lemma 2: Let b
ꢃ= ꢀ; define ꢌ = b −ꢀ. Suppose we can find z such that z ∈ T and
ꢁ
ꢁ
Pꢂx ꢌ ≥ 0ꢀzꢃ = 1ꢆ
Pꢂx ꢌ > 0ꢀzꢃ > 0ꢄ
ꢁ
1
2
Then we can show b ꢅ V by showing that Pꢂy −x b > 0ꢀzꢃ <
:
1
ꢁ
ꢁ
ꢁ
Pꢂy −x b > 0ꢀzꢃ = Pꢂmaxꢂꢁꢆ−x ꢀꢃ−x ꢌ > 0ꢀzꢃ
1
ꢁ
ꢁ
=
Pꢂmaxꢂꢁꢆ−x ꢀꢃ > x ꢌꢀzꢃ
1
2
ꢁ
=
Pꢂꢁ > x ꢌꢀzꢃ <
where the third equality holds by z ∈ T , and the strict inequality holds by Assumption 3. On the other
hand, if we can find a z ∈ T such that
ꢁ
ꢁ
(
A.8)
Pꢂx ꢌ ≤ 0ꢀzꢃ = 1
and
Pꢂx ꢌ < 0ꢀzꢃ > 0ꢆ
ꢁ
1
2
then we show b ꢅ V by showing that for this z, Pꢂy −x b > 0ꢀzꢃ >
:
0
1
2
ꢁ
ꢁ
ꢁ
ꢁ
ꢁ
Pꢂy −x b > 0ꢀzꢃ = Pꢂꢁ > −x ꢀꢆꢁ > x ꢌꢀzꢃ = Pꢂꢁ > maxꢂ−x ꢀꢆx ꢌꢃꢀzꢃ >
0

9
18
h. hong and e. tamer
where the strict inequality is due to Assumption 3 and the fact that
ꢁ
ꢁ
ꢁ
ꢁ
Pꢂmaxꢂ−x ꢀꢆx ꢌꢃ ≤ 0ꢀzꢃ = 1
and
Pꢂmaxꢂ−x ꢀꢆx ꢌꢃ < 0ꢀzꢃ > 0ꢄ
Both of these follow from z ∈ T and (A.8).
Q.E.D.
Proof of Lemma 3: Suppose ꢀ > 0 (the case where ꢀ < 0 can be similarly analyzed). Let ꢌ =
b −ꢀ > 0. Consider for M > 0:
ꢁ
ꢁ
ꢁ
Pꢂy −x b > 0ꢀzꢃ = Pꢂmaxꢂꢁꢆ−x ꢀꢃ > x ꢌꢀzꢃ
1
ꢁ
ꢁ
ꢁ
=
≤
Pꢂx > Mꢆꢁ > x ꢌꢀzꢃ+Pꢂx < Mꢆmaxꢂꢁꢆ−x ꢀꢃ > x ꢌꢀzꢃ
Pꢂꢁ > Mꢌꢀzꢃ+Pꢂx < Mꢀzꢃꢄ
Given M > 0, we first find ꢉ small enough (given ꢌ) such that
1
Pꢂꢁ > Mꢌꢀzꢃ < −ꢉ
2
for all z. Then given M and ꢌ, find z large enough such that Pꢂx < Mꢀzꢃ < ꢉ. This implies that
ꢁ
1
2
Pꢂy −x b > 0ꢀzꢃ < . On the other hand if ꢌ < 0, we can write
1
ꢁ
ꢁ
ꢁ
Pꢂy −x b > 0ꢀzꢃ = Pꢂꢁ > −x ꢀꢆꢁ > x ꢌꢀzꢃ
0
ꢁ
ꢁ
ꢁ
=
=
≥
=
Pꢂx > Mꢆꢁ > −x ꢀꢆꢁ > x ꢌꢀzꢃ+Pꢂx < Mꢆꢁ > −xꢀꢆꢁ > x ꢌꢀzꢃ
Pꢂx > Mꢆꢁ > maxꢂ−ꢀꢆꢌꢃxꢀzꢃ+Pꢂx < Mꢆꢁ > −xꢀꢆꢁ > x ꢌꢀzꢃ
Pꢂx > Mꢆꢁ > maxꢂ−ꢀꢆꢌꢃMꢀzꢃ+Pꢂx < Mꢆꢁ > −xꢀꢆꢁ > x ꢌꢀzꢃ
Pꢂꢁ > maxꢂ−ꢀꢆꢌꢃMꢀzꢃ+Pꢂx < Mꢆꢁ > −xꢀꢆꢁ > x ꢌꢀzꢃ
ꢁ
ꢁ
ꢁ
−
Pꢂx < Mꢆꢁ > maxꢂ−ꢀꢆꢌꢃMꢀzꢃ
≥
Pꢂꢁ > maxꢂ−ꢀꢆꢌꢃMꢀzꢃ−Pꢂx < Mꢀzꢃ
and given M > 0, we first find ꢉ small enough (depending on ꢌ) so that
1
2
Pꢂꢁ > maxꢂ−ꢀꢆꢌꢃMꢀzꢃ > +ꢉ
1
2
for all z, and then z large enough so that Pꢂx < Mꢀzꢃ < ꢉ. This leads again to Pꢂy −xb > 0ꢀzꢃ >
0
which contradicts the main model restriction in (2.3) above.
Q.E.D.
We make two brief notes before proceeding with the proofs of the theorems. First for all a ∈ R,
ꢖ
ꢖ
ꢖ
ꢖ
∗
∗
∗
∗
∗
Gꢂa/h ꢃ − 1ꢂa ≥ 0ꢃ ≤ Gꢂa/h ꢃ1ꢂ−h < a < 0ꢃ → 0 as h → 0. Second, if h is chosen such that it
converges to zero slower than ꢂꢉˆꢂzꢍꢀꢃ−ꢉꢂzꢍꢀꢃꢃ, for all z:
ꢖ
ꢖ
ꢖ
ꢖ
ꢖ
ꢖ
ꢅ
ꢇ
ꢅ
ꢇ
ꢖ
ꢖ
ꢖ
ꢖ
ꢖ
ꢖ
ꢖ
ꢉˆꢂzꢆꢀꢃ
ꢖ
ꢖꢉˆꢂzꢆꢀꢃ ꢉꢂzꢆꢀꢃ ꢖ
ꢖ
ꢉꢂzꢆꢀꢃ
ꢖ
ꢖG
−1ꢂꢉꢂzꢍꢀꢃ≥0ꢃꢖ≤Mꢖ
−
ꢖ+ꢖG
−1ꢂꢉꢂzꢍꢀꢃ≥0ꢃꢖ→0ꢆ
∗
∗
∗
∗
ꢖ
h
ꢖ
ꢖ
h
h
ꢖ
ꢖ
h
ꢖ
ꢄ
where the inequality follows from the assumption of a bounded derivative of Gꢂ·ꢃ. Similarly for Gꢂ·ꢃ,
ꢖ
ꢖ
ꢖ ꢄ
∗
ꢖ
∗
∗
we have Gꢂa/h ꢃ−1ꢂa ≤ 0ꢃ ≤ Gꢂa/h ꢃ1ꢂ0 < a < h ꢃ, and therefore
ꢖ
ꢅ
ꢇ
ꢖ
ꢖ
ꢖ
ꢖ
ꢅ
ꢇ
ꢖ
ꢖ
ꢉˆꢂzꢆꢀꢃ
ꢖ
ꢖꢉˆꢂzꢆꢀꢃ ꢉꢂzꢆꢀꢃ ꢖ
ꢖ
ꢉꢂzꢆꢀꢃ
ꢖ
ꢖ ꢄ
ꢖ
ꢖ
ꢖ
ꢖ
ꢖ
ꢖ ꢄ
ꢖ
G
−1ꢂꢉꢂzꢍꢀꢃ≤0ꢃ ≤M
−
+ G
−1ꢂꢉꢂzꢍꢀꢃ≤0ꢃ →0ꢄ
ꢖ
∗
ꢖ
∗
∗
ꢖ
∗
ꢖ
h
h
h
h

censored models with endogenous regressors
919
Aꢄ2ꢄ Proof of Theorem 1
In light of Assumption 5, it suffices to show uniform convergence of Q ꢂbꢃ to Qꢂbꢃ in probability.
n
For this purpose first note that the summands in ꢉˆ ꢂzꢍbꢃ and ꢉˆ ꢂzꢍbꢃ all belong to a Euclidean class
1
0
of function; hence using Lemma 22 of Nolan and Pollard (1987) and Lemma 2.37 of Pollard (1984),
uniformly in b and z:
ꢅ
ꢅ
ꢇ
ꢇ
ꢖ
ꢖ
ꢖ
ꢖ
logn
ꢖꢉˆ
ꢂzꢍbꢃ−Eꢉˆ
ꢂzꢍbꢃ−Eꢉˆ
ꢂzꢍbꢃꢖ = O
= o
= o
ꢂ1ꢃꢆ
ꢂ1ꢃꢄ
0
0
p
p
p
p
nhd
ꢖ
ꢖ
ꢖ
ꢖ
logn
nhd
ꢖꢉˆ
ꢂzꢍbꢃꢖ = O
1
1
Since ꢉ ꢂzꢍbꢃ is piecewise uniformly continuous in b and z, the following holds uniformly over b
0
and z:
ꢖ
ꢖ
ꢖ
ꢖ
ꢖEꢉˆ
ꢂzꢍbꢃ−ꢉ
ꢂzꢍbꢃꢖ
0
0
ꢄꢄ
ꢁ
ꢁ
≤
KꢂuꢃKꢂvꢃꢀPꢂy ≥ uh+x bꢍz = z+vhꢃ−Pꢂy ≥ x bꢍz = zꢃꢀdudv
0
0
=
Oꢂhꢃ → 0
as
h → 0ꢄ
Similarly, it is also true for ꢉ ꢂz ꢍbꢃ that uniformly over b and z,
1
i
ꢖ
ꢖ
ꢖ
ꢖ
ꢖEꢉˆ
ꢂzꢍbꢃ−ꢉ
ꢂzꢍbꢃꢖ = Oꢂhꢃ → 0
as
h → 0ꢄ
1
1
As a consequence, consider bounding ꢀQ ꢂbꢃ−Qꢂbꢃꢀ:
n
ꢖ
ꢖ
ꢖ
ꢖ
ꢖ
ꢖ
ꢖ
ꢖ
ꢖ
ꢖ
ꢖ
ꢄ
ꢖ ꢄ
Q ꢂbꢃ−Qꢂbꢃ ≤ Q ꢂbꢃ−Q ꢂbꢃ + Q ꢂbꢃ−Qꢂbꢃ
n
n
n
n
ꢗ
n
i=1
2
0
2
ꢄ
where Q ꢂbꢃ ≡ ꢂ1/nꢃ
ꢉ ꢂz ꢍbꢃ1ꢂꢉ ꢂz ꢍbꢃ > 0ꢃ+ꢉ ꢂz ꢍbꢃ1ꢂꢉ ꢂz ꢍbꢃ < 0ꢃ. We deal with each term
n
i
0
i
1
i
1
i
separately:
ꢖ
ꢖ
ꢖ
ꢖ
ꢖ
ꢖ
ꢅ
ꢇ
n
ꢖ
ꢖ
1 ꢊ
ꢉˆ ꢂz ꢍbꢃ
2
0
i
2
ꢖ
ꢄ
ꢖ
Q ꢂbꢃ−Q ꢂbꢃ ≤
ꢖꢉˆ ꢂz ꢍbꢃG
−ꢉ ꢂz ꢍbꢃ1ꢂꢉ ꢂz ꢍbꢃ > 0ꢃꢖ
n
n
0
i
0
i
0
i
∗
n
ꢖ
h
ꢖ
i=1
ꢖ
ꢖ
ꢅ
ꢇ
n
ꢖ
ꢖ
ꢖ
1
n
ꢊꢖ
ꢉˆ ꢂz ꢍbꢃ
2
ꢄ
0
i
2
1
+
ꢖꢉˆ ꢂz ꢍbꢃG
−ꢉ ꢂz ꢍbꢃ1ꢂꢉ ꢂz ꢍbꢃ > 0ꢃꢖꢄ
1
i
∗
i
1
i
ꢖ
h
ꢖ
i=1
It is sufficient to bound the first term. The arguments for the second term are identical.
ꢖ
ꢅ
ꢇ
ꢖ
ꢖ
ꢖ
ꢖ
n
1
n
ꢊꢖ
ꢉˆ ꢂzꢍbꢃ
0
2
ꢖ
2
ꢉˆ ꢂzꢍbꢃG
−ꢉ ꢂzꢍbꢃ1ꢂꢉ ꢂzꢍbꢃ > 0ꢃ
ꢖ
0
∗
0
0
h
i=1
ꢖ
ꢖ
ꢖ
ꢅ
ꢇ
ꢖ
ꢖ
ꢖ
ꢖ
ꢉˆ ꢂzꢍbꢃ
2
0
2
0
≤
sup ꢉˆ ꢂzꢍbꢃG
−ꢉˆ ꢂzꢍbꢃ1ꢂꢉˆ ꢂzꢍbꢃ > 0ꢃ
ꢖ
0
∗
0
zꢆb
h
ꢖ
ꢖ
ꢖ
ꢖ
2
0
2
0
+
+
sup ꢉˆ ꢂzꢍbꢃ1ꢂꢉˆ ꢂzꢍbꢃ > 0ꢃ−ꢉ ꢂzꢍbꢃ1ꢂꢉˆ ꢂzꢍbꢃ > 0ꢃ
0
0
zꢆb
ꢖ
ꢖ
ꢖ
2
2
0
ꢖ
sup ꢉ ꢂzꢍbꢃ 1ꢂꢉˆ ꢂzꢍbꢃ > 0ꢃ−ꢉ 1ꢂꢉ ꢂzꢍbꢃ > 0ꢃ ꢄ
0
0
0
zꢆb
The first term is simple:
ꢖ
ꢖ
ꢖ
ꢅ
ꢇ
ꢖ
ꢖ
ꢖ
ꢉˆ ꢂzꢍbꢃ
2
0
0
2
0
∗2
sup ꢉˆ ꢂzꢍbꢃG
−ꢉˆ ꢂzꢍbꢃ1ꢂꢉˆ ꢂz ꢍbꢃ > 0ꢃ ≤ h −→ 0ꢄ
ꢖ
∗
0
i
ꢖ
zꢆb
h

9
20
h. hong and e. tamer
p
2
0
2
0
The second term can be bounded by supzꢆb ꢀꢉˆ ꢂzꢍbꢃ−ꢉˆ ꢂzꢍbꢃꢀ −→ 0. The last term is also bounded
by
2
2
0
supꢀꢉˆ ꢂzꢍbꢃ 1ꢂꢉˆ ꢂzꢍbꢃ > 0ꢃ−ꢉ ꢂzꢍbꢃ1ꢂꢉ ꢂzꢍbꢃ > 0ꢃꢀ
0
0
0
zꢆb
p
2
2
≤
supꢀꢉˆ ꢂzꢍbꢃ−ꢉ ꢂzꢍbꢃꢀ+supꢂꢉˆ ꢂzꢍbꢃ−ꢉ ꢂzꢍbꢃꢃ −→ 0ꢄ
0
0
0
0
zꢆb zꢆb
ꢖ
ꢖ
ꢖ
p
ꢖ ꢄ
To finally conclude we need to show sup Q ꢂbꢃ−Qꢂbꢃ −→ 0. For each b, a standard law of large
b
n
p
ꢄ
numbers gives Q ꢂbꢃ −→ Qꢂbꢃ. We can conclude the uniformity by showing stochastic equicontinuity
n
ꢄ
of Q ꢂbꢃ, namely for each continuous piece of Q ꢂbꢃ,
n
n
ꢖ
ꢖ
ꢖ
p
ꢖ ꢄ
ꢄ
sup
Q ꢂb ꢃ−Q ꢂb ꢃ −→ 0ꢄ
n
1
n
2
ꢂb1ꢆb
ꢃꢈꢀb −b2ꢀ→0
2
1
Again it is sufficient to consider only the ꢉ ꢂzꢍbꢃ part of the summand:
0
n
1
n
ꢊꢖ
ꢖ
ꢖ
ꢖ
2
2
0
ꢉ ꢂz ꢍb ꢃ1ꢂꢉ ꢂz ꢍb > 0ꢃꢃ−ꢉ ꢂz ꢍb ꢃ1ꢂꢉ ꢂz ꢍb > 0ꢃꢃ
0
i
1
0
i
1
i
2
0
i
2
i=1
n
1
n
ꢊꢖ
ꢖ
ꢖ
ꢖ
2
0
2
0
2
≤
ꢉ ꢂz ꢍb ꢃ−ꢉ ꢂz ꢍb ꢃ +ꢂꢉ ꢂz ꢍb ꢃ−ꢉ ꢂz ꢍb ꢃꢃ
i
1
i
2
0
i
1
0
i
2
i=1
ꢖ
ꢖ
p
ꢖ
2
0
2
0
ꢖ
2
≤
sup ꢉ ꢂz ꢍb ꢃ−ꢉ ꢂz ꢍb ꢃ +supꢂꢉ ꢂz ꢍb ꢃ−ꢉ ꢂz ꢍb ꢃꢃ −→ 0ꢄ
i
1
i
2
0
i
1
0
i
2
z
z
The convergence is due to continuity.
Q.E.D.
Aꢄ3ꢄ Proof of Theorem 2
The proof in this section relies on a sequence of lemmas, which are used to deal with the presence
of the infinite dimensional parameter inside the step function. Recall that we replace the step function
by a smooth function that converges to it as n increases. The first step in proving asymptotic normality
is to find the asymptotic distribution of
ꢆ
ꢅ
ꢇ
ꢅ
ꢇꢈꢖ
n
1
ꢊ ꢔ
ꢉˆ ꢂz ꢍbꢃ
ꢉˆ ꢂz ꢍbꢃ
ꢖ
ꢖ
ꢖ
2
0
i
2
ꢄ
1
i
FOCꢂꢀꢃ = 2
√
ꢉˆ ꢂz ꢍbꢃ G
+ꢉˆ ꢂz ꢍbꢃ G
ꢄ
0
i
1
i
∗
∗
h
n
ꢔb
h
i=1
b=ꢀ
Due to the smoothing in the first order condition, several terms are present in the first order condi-
tion:
ꢀ
ꢅ
ꢇ
n
1
ꢊ
ꢉˆ ꢂz ꢍꢀꢃ
0
i
(
A.9)
FOCꢂꢀꢃ = √n
ꢉˆ ꢂz ꢍꢀꢃꢓꢉˆ ꢂz ꢍꢀꢃG
0
i
0
i
∗
h
i=1
ꢒ
ꢓꢔ
ꢕ
ꢂaꢃ
ꢅ
ꢇ
ꢕ
ꢉˆ ꢂz ꢍꢀꢃ
1
i
ꢄ
+
ꢉˆ ꢂz ꢍꢀꢃꢓꢉˆ ꢂz ꢍꢀꢃG
1
i
1
i
∗
h
ꢒ
ꢓꢔ
ꢂbꢃ
ꢅ
ꢅ
ꢇ
1
2
1
h
ꢉˆ ꢂz ꢍꢀꢃ
2
0
i
+
+
ꢉˆ ꢂz ꢍꢀꢃ ꢓꢉˆ ꢂz ꢍꢀꢃ
g
0
i
0
i
∗
∗
h
ꢒ
ꢓꢔ
ꢕ
ꢂcꢃ
ꢇꢁ
1
2
1
ꢉˆ ꢂz ꢍꢀꢃ
2
1
i
ꢉˆ ꢂz ꢍꢀꢃ ꢓꢉˆ ꢂz ꢍꢀꢃ g¯
ꢄ
1
i
1
i
∗
∗
h
h
ꢒ
ꢓꢔ
ꢕ
ꢂdꢃ

censored models with endogenous regressors
921
Our goal is to reduce this to the managable condition up to a term that is asymptotically negligible:
n
2
ꢊ
FOCꢂꢀꢃ = √n
ꢉˆꢂz ꢍꢀꢃꢓꢉꢂz ꢍꢀꢃ1ꢂz ∈ T ꢃ+o ꢂ1ꢃ
i
i
i
p
i=1
where ꢉꢂz ꢍꢀꢃ is defined in the same way as ꢉ ꢂz ꢍꢀꢃ with y replaced by the unobserved y . Define
i
0
i
i0
i
ꢉˆꢂz ꢍꢀꢃ in a similar way. We need to show that the first two terms above, (a) and (b), are well
i
behaved. This will be shown in the next two lemmas. Next we show that the contribution of the
smoothing terms (c) and (d) to FOCꢂꢀꢃ is asymptotically negligible. The main insight to the asymp-
totic representation is that at the true ꢀ, for all z, ꢉ ꢂzꢆꢀꢃ ≤ 0 ≤ ꢉ ꢂzꢆꢀꢃ, and that on the set T ,
0
1
according to its definition, ꢉ ꢂzꢆꢀꢃ = ꢉ ꢂzꢆꢀꢃ = ꢉꢂzꢆꢀꢃ = 0, which on the support of z is equivalent
0
1
to ꢉ ꢂz ꢍꢀꢃ = ꢉꢂz ꢍꢀꢃ = ꢉ ꢂz ꢍꢀꢃ. We first display the list of lemmas used in the proof. Their proofs
0
i
i
1
i
are collected at the end of the Appendix.
Lemma 4: Under Assumptions 3 and 7, both
Pꢂ0 < ꢉ ꢂzꢍꢀꢃ < hꢃ = Oꢂhꢃ
and
Pꢂ0 > ꢉ ꢂzꢍꢀꢃ > −hꢃ = Oꢂhꢃꢄ
1
0
Let
ꢔ
ꢔ
ꢓꢉ ꢂz ꢍbꢃ =
ꢉ ꢂz ꢍbꢃ
and
ꢓꢉ ꢂz ꢍbꢃ =
ꢉ ꢂz ꢍbꢃꢄ
1 i
0
i
0
i
1
i
ꢔb
ꢔb
ˆ
ˆ
Also let ꢓꢉ ꢂz ꢍbꢃ and ꢓꢉ ꢂz ꢍbꢃ denote the corresponding estimates, namely
0
i
1
i
ꢔ
ꢔ
ˆ
ˆ
ꢓꢉ ꢂz ꢍbꢃ =
ꢉˆ ꢂz ꢍbꢃ
and
ꢓꢉ ꢂz ꢍbꢃ =
ꢉˆ ꢂz ꢍbꢃꢄ
0
i
0
i
1
i
1
i
ꢔb
ꢔb
That these quantities are well defined is justified by Assumptions 7 and 8:
ꢁ
ꢁ
ꢁ
Lemma 5: Under Assumption 8, Pꢂy − x b > 0ꢀzꢃ, Pꢂy − x b > 0ꢀzꢃ, Pꢂ0 < ꢁ < −x ꢀꢀzꢃ, and
0
1
ꢁ
Pꢂꢁ < 0ꢆx ꢀ < 0ꢀzꢃ all have more than m bounded continuous derivatives with respect to b and z,
ꢋ
ꢌ
1
2
uniformly in z and in b in a small neighborhood of ꢀ , such that m > 4 d +2 +2.
0
The following lemma is a standard result regarding the nonparametric rates of convergence; see
for example Newey (1994).
Lemma 6: Under Assumptions 1–9, uniformly in z and b in a small neighborhood of ꢀ, for each
r = 0ꢆ1ꢆ2 and for some ꢐ > 3/8:
ꢘ
ꢘ
ꢙ
ꢙ
logn
r
r
+Oꢂhm−r ꢃ = o ꢂn ꢃꢆ
−ꢐ
ꢀ
ꢓ ꢉˆ ꢂzꢆbꢃ−ꢓ ꢉ ꢂzꢆbꢃꢀ = O
0
0
p
1
1
2
p
d+r
n
2
h
logn
r
r
+Oꢂhm−r ꢃ = o ꢂn ꢃꢄ
−ꢐ
ꢀ
ꢓ ꢉˆ ꢂzꢆbꢃ−ꢓ ꢉ ꢂzꢆbꢃꢀ = O
1
1
p
1
1
2
p
d+r
n
2
h
The convergence rates stated in the lemma allow for the choice of ꢑ in Assumption 10 such that
1
4
1 ꢄ
2
<
ꢑ < 2ꢐ−
Lemma 7: Evaluating FOC at the true ꢀ, only the terms (a) and (b) in (A.9) above contribute to
the asymptotic representation on the set T :
ꢅ
ꢇ
n
1
n
ꢊ
ꢉˆ ꢂz ꢍꢀꢃ
0
i
ˆ
√
ꢉˆ ꢂz ꢍꢀꢃꢓꢉ ꢂz ꢍꢀꢃG
0
i
0
i
∗
h
i=1
ꢅ
ꢇ
n
1
n
ꢊ
ꢉˆ ꢂz ꢍꢀꢃ
0
i
ˆ
=
√
ꢉˆ ꢂz ꢍꢀꢃꢓꢉ ꢂz ꢍꢀꢃG
1ꢂꢉ ꢂz ꢍꢀꢃ = 0ꢃ+o ꢂ1ꢃꢆ
0
i
0
i
0
i
p
∗
h
i=1

9
22
h. hong and e. tamer
ꢅ
ꢇ
n
1
n
ꢊ
ꢉˆ ꢂz ꢍꢀꢃ
1
i
ˆ
ꢄ
√
ꢉˆ ꢂz ꢍꢀꢃꢓꢉ ꢂz ꢍꢀꢃG
1
i
1
i
∗
h
i=1
ꢅ
ꢇ
n
1
n
ꢊ
ꢉˆ ꢂz ꢍꢀꢃ
1
i
ˆ
ꢄ
=
√
ꢉˆ ꢂz ꢍꢀꢃꢓꢉ ꢂz ꢍꢀꢃG
1ꢂꢉ ꢂz ꢍꢀꢃ = 0ꢃ+o ꢂ1ꢃꢄ
1
i
1
i
1
i
p
∗
h
i=1
The next lemma deals with the presence of estimated infinite dimensional parameters in ꢂaꢃ above.
Heuristically, as n increases, we should be able to replace ꢉˆ with ꢉ. This is made precise in the
statement of the lemma.
Lemma 8: The variation in the limiting distribution is driven by the observations z ∈ T :
i
ꢅ
ꢇ
n
1
n
ꢊ
ꢉˆ ꢂz ꢍꢀꢃ
0
i
ˆ
√
ꢉˆ ꢂz ꢍꢀꢃꢓꢉ ꢂz ꢍꢀꢃG
1ꢂꢉ ꢂz ꢍꢀꢃ = 0ꢃ
0
i
0
i
0
i
∗
h
i=1
n
1
n
ꢊ
=
√
ꢉˆꢂz ꢍꢀꢃꢓꢉꢂz ꢍꢀꢃ1ꢂz ∈ T ꢃ+o ꢂ1ꢃꢆ
i
i
i
p
i=1
ꢅ
ꢇ
n
1
ꢊ
ꢉˆ ꢂz ꢍꢀꢃ
1
i
ˆ
ꢄ
√
ꢉˆ ꢂz ꢍꢀꢃꢓꢉ ꢂz ꢍꢀꢃG
1ꢂꢉ ꢂz ꢍꢀꢃ = 0ꢃ
1
i
1
i
1
i
∗
h
n
i=1
n
1
n
ꢊ
=
√
ꢉˆꢂz ꢍꢀꢃꢓꢉꢂz ꢍꢀꢃ1ꢂz ∈ T ꢃ+o ꢂ1ꢃꢄ
i
i
i
p
i=1
Next we take care of the two negligible terms ꢂcꢃ and ꢂdꢃ in the first order condition (A.9).
Lemma 9:
ꢅ
ꢇ
ꢇ
n
1
n
ꢊ
1
h
ꢉˆ ꢂz ꢍꢀꢃ
2
0
i
√
ꢉˆ ꢂz ꢍꢀꢃ ꢓꢉˆ ꢂz ꢍꢀꢃ
g
0
i
0
i
∗
∗
h
i=1
ꢅ
1
ꢉˆ ꢂz ꢍꢀꢃ
2
1
i
+
ꢉˆ ꢂz ꢍꢀꢃ ꢓꢉˆ ꢂz ꢍꢀꢃ g¯
= o ꢂ1ꢃꢄ
1
i
1
i
p
∗
∗
h
h
Now we turn attention to the Jacobian term Hꢂbꢃ, which can be expanded as
ꢀ
ꢅ
ꢇ
ꢅ
ꢇ
n
1
n
ꢊ
ꢉˆ ꢂz ꢍbꢃ
ꢉˆ ꢂz ꢍbꢃ
ꢁ
0
i
0
i
ˆ
ˆ
ꢓꢉ ꢂz ꢍbꢃꢓꢉ ꢂz ꢍbꢃ G
+ꢉˆ ꢂz ꢍbꢃꢓꢓꢉˆ ꢂz ꢍbꢃG
0
i
0
i
0
i
0
i
∗
∗
h
h
i=1
ꢅ
ꢇ
1
h
ꢉˆ ꢂz ꢍbꢃ
ꢁ
0
i
+
2ꢉˆ ꢂz ꢍbꢃꢓꢉˆ ꢂz ꢍbꢃꢓꢉˆ ꢂz ꢍbꢃ
g
0
i
0
i
0
i
∗
∗
h
ꢅ
ꢇ
1
1
h
ꢉˆ ꢂz ꢍbꢃ
2
0
0
i
+ 2
ꢉˆ ꢂz ꢍbꢃꢓꢓꢉˆ ꢂz ꢍbꢃ
g
h
i
0
i
∗
∗
ꢅ
ꢇ
1
1
ꢉˆ ꢂz ꢍbꢃ
2
ꢁ
ꢁ
g
2
0
i
+ 2
ꢉˆ ꢂz ꢍbꢃ ꢓꢉˆ ꢂz ꢍbꢃꢓꢉˆ ꢂz ꢍbꢃ
h
0
i
0
i
0
i
∗
∗
h
ꢅ
ꢇ
ꢅ
ꢇ
ꢉˆ ꢂz ꢍbꢃ
ꢉˆ ꢂz ꢍbꢃ
ꢁ
ꢄ
1
i
ꢄ
1
i
+
+
ꢓꢉˆ ꢂz ꢍbꢃꢓꢉˆ ꢂz ꢍbꢃ G
+ꢉˆ ꢂz ꢍbꢃꢓꢓꢉˆ ꢂz ꢍbꢃG
ꢅ
1
i
1
i
1
i
1
i
∗
∗
h
h
ꢇ
1
h
ꢉˆ ꢂz ꢍbꢃ
ꢁ
1
i
2ꢉˆ ꢂz ꢍbꢃꢓꢉˆ ꢂz ꢍbꢃꢓꢉˆ ꢂz ꢍbꢃ
g
1
i
1
i
1
i
∗
∗
h
ꢅ
ꢇ
1
1
ꢉˆ ꢂz ꢍbꢃ
2
1
i
+ 2
ꢉˆ ꢂz ꢍbꢃ ꢓꢓꢉˆ ꢂz ꢍbꢃ g¯
∗
h
1
i
1
i
∗
h
ꢅ
ꢇꢁ
1
1
h
ꢉˆ ꢂz ꢍbꢃ
2
ꢁ
g¯ꢁ
1
i
+ 2
ꢉˆ ꢂz ꢍbꢃ ꢓꢉˆ ꢂz ꢍbꢃꢓꢉˆ ꢂz ꢍbꢃ
ꢄ
1
i
1
i
1
i
∗
2
∗
h
Each of these terms needs to be cleared. Although we need not worry about the rate of conver-
gence, we have to show local uniform convergence of each of the pieces in the Jacobian above to

censored models with endogenous regressors
923
its counterpart. This is tedious in this case due to the presence of the nonparametric nuisance param-
eters inside a smoothed step function.
p
Lemma 10: For b −→ ꢀ,
ꢅ
ꢇ
n
1
ꢊ
ꢉˆ ꢂz ꢍbꢃ
ꢁ
0
i
ꢁ
ˆ
ˆ
ꢓꢉ ꢂz ꢍbꢃꢓꢉ ꢂz ꢍbꢃ G
= Eꢓꢉꢂzꢍꢀꢃꢓꢉꢂzꢍꢀꢃ 1ꢂz ∈ T ꢃ+o ꢂ1ꢃꢄ
0
i
0
i
p
∗
h
n
i=1
p
Lemma 11: For b −→ ꢀ,
ꢅ
ꢇ
n
1
ꢊ
ꢉˆ ꢂz ꢍbꢃ
0
i
ꢉˆ ꢂz ꢍbꢃꢓꢓꢉˆ ꢂz ꢍbꢃG
= o ꢂ1ꢃꢄ
0
i
0
i
p
∗
n
h
i=1
p
Lemma 12: For b −→ ꢀ,
ꢅ
ꢇ
n
1
ꢊ
1
h
ꢉˆ ꢂz ꢍbꢃ
ꢁ
0
i
ꢉˆ ꢂz ꢍbꢃꢓꢉˆ ꢂz ꢍbꢃꢓꢉˆ ꢂz ꢍbꢃ
g
= o ꢂ1ꢃꢄ
0
i
0
i
0
i
p
∗
∗
h
n
i=1
p
Lemma 13: For b −→ ꢀ,
ꢅ
ꢇ
n
1
ꢊ
1
h
ꢉˆ ꢂz ꢍbꢃ
2
0
0
i
ꢉˆ ꢂz ꢍbꢃꢓꢓꢉˆ ꢂz ꢍbꢃ
g
= o ꢂ1ꢃꢄ
i
0
i
p
∗
∗
n
h
i=1
p
Lemma 14: For b −→ ꢀ,
ꢅ
ꢇ
n
1
ꢊ
1
h
ꢉˆ ꢂz ꢍbꢃ
2
ꢁ
ꢁ
0
i
ꢉˆ ꢂz ꢍbꢃ ꢓꢉˆ ꢂz ꢍbꢃꢓꢉˆ ꢂz ꢍbꢃ
g
= o ꢂ1ꢃꢄ
0
i
0
i
0
i
p
∗
2
∗
h
n
i=1
Lemma 15: The equalities in Lemmas 10 to 14 hold with ꢉ ꢂz ꢍbꢃ in the left-hand side replaced by
0
i
p
ꢄ
ꢉ ꢂz ꢍbꢃ, and with Gꢂ·ꢃ and gꢂ·ꢃ replaced by Gꢂ·ꢃ and g¯ꢂ·ꢃ, correspondingly, namely, as b −→ ꢀ:
1
i
ꢅ
ꢇ
n
1
n
ꢊ
ꢉˆ ꢂz ꢍbꢃ
ˆ
ˆ
ꢁ
ꢄ
1
i
ꢁ
ꢓꢉ ꢂz ꢍbꢃꢓꢉ ꢂz ꢍbꢃ G
= Eꢓꢉꢂzꢍꢀꢃꢓꢉꢂzꢍꢀꢃ 1ꢂz ∈ T ꢃ+o ꢂ1ꢃꢆ
1
i
1
i
p
∗
h
i=1
ꢅ
ꢇ
n
1
n
ꢊ
ꢉˆ ꢂz ꢍbꢃ
1
i
ꢄ
ꢉˆ ꢂz ꢍbꢃꢓꢓꢉˆ ꢂz ꢍbꢃG
= o ꢂ1ꢃꢆ
1
i
1
i
p
∗
h
i=1
ꢅ
ꢇ
n
1
n
ꢊ
1
ꢉˆ ꢂz ꢍbꢃ
ꢁ
1
i
ꢉˆ ꢂz ꢍbꢃꢓꢉˆ ꢂz ꢍbꢃꢓꢉˆ ꢂz ꢍbꢃ
g¯
= o ꢂ1ꢃꢆ
1
i
1
i
1
i
p
∗
∗
h
h
i=1
ꢅ
ꢇ
n
1
n
ꢊ
1
h
ꢉˆ ꢂz ꢍbꢃ
2
1
1
i
ꢉˆ ꢂz ꢍbꢃꢓꢓꢉˆ ꢂz ꢍbꢃ g¯
= o ꢂ1ꢃꢆ
i
1
i
p
∗
∗
h
i=1
ꢅ
ꢇ
n
1
n
ꢊ
1
ꢉˆ ꢂz ꢍbꢃ
2
ꢁ
ꢁ
1
i
ꢉˆ ꢂz ꢍbꢃ ꢓꢉˆ ꢂz ꢍbꢃꢓꢉˆ ꢂz ꢍbꢃ
g
= o ꢂ1ꢃꢄ
1
i
1
i
1
i
p
∗
2
∗
h
h
i=1
To complete the proof we only need a standard application of U-statistics projection results,
following the methods used in Lemma 3.1 of Powell, Stock, and Stoker (1989).
Lemma 16: Asymptotic linear representation of the first order condition:
n
1
n
ꢊ
√
ꢉˆꢂz ꢍꢀꢃꢓꢉꢂz ꢍꢀꢃ1ꢂz ∈ T ꢃ
i
i
i
i=1
ꢅ
ꢇ
n
1
n
ꢊ
1
2
2
ꢁ
=
=
√
f ꢂz ꢃ 1ꢂy −x ꢀ ≥ 0ꢃ−
ꢓꢉꢂz ꢍꢀꢃ1ꢂz ∈ T ꢃ+o ꢂ1ꢃ
i
i
i
i
i
p
i=1
n
1
ꢊꢚ
2
ꢁ
√
f ꢂz ꢃ1ꢂy −x ꢀ ≥ 0ꢃꢓꢉꢂz ꢍꢀꢃ1ꢂz ∈ T ꢃ
i
i
i
i
i
n
i=1
ꢛ
2
ꢁ
−
Ef ꢂz ꢃ1ꢂy −x ꢀ ≥ 0ꢃꢓꢉꢂz ꢍꢀꢃ1ꢂz ∈ T ꢃ +o ꢂ1ꢃꢄ
i
i
i
i
i
p

9
24
h. hong and e. tamer
As an immediate consequence the asymptotic approximation of the sampling distribution for point
identified parameters is a natural consequence of the combination of the preceeding lemmas. Using
√
∗
ˆ
a standard first order Taylor expansion, for a set of intermediate values b between b and ꢀ, nꢂꢀ−
∗
−1
ꢀ
ꢃ = ꢂHꢂb ꢃꢃ FOCꢂꢀꢃ. Combining Lemmas 10 to 16 shows that
p
∗
ꢁ
Hꢂb ꢃ −→ 2Eꢓꢉꢂzꢍꢀꢃꢓꢉꢂzꢍꢀꢃ 1ꢂz ∈ T ꢃ ≡ 2A
ꢁ
2
=
2EEꢂxf ꢂ0ꢀxꢆzꢃꢀzꢃEꢂxf ꢂ0ꢀxꢆzꢃꢀzꢃ f ꢂzꢃ1ꢂz ∈ T ꢃꢄ
ꢁ
ꢁ
d
On the other hand, combining Lemmas 7, 8, 9, and 16 shows that FOCꢂꢀꢃ −→ Nꢂ0ꢆ4ꢒꢃ for
1
ꢁ
4
ꢒ = 4 EEꢂxf ꢂ0ꢀxꢆzꢃꢀzꢃEꢂxf ꢂ0ꢀxꢆzꢃꢀzꢃ f ꢂzꢃ1ꢂz ∈ T ꢃ
ꢁ
ꢁ
1
4
2
ꢁ
=
Ef ꢂzꢃꢓꢉꢂzꢍꢀꢃꢓꢉꢂzꢍꢀꢃ 1ꢂz ∈ T ꢃꢄ
√
ꢋ
ꢌ
d
ˆ
−1
Hence we obtain the statement of the theorem n ꢀ − ꢀ −→ Nꢂ0ꢆA ꢒAꢃ. The components in
the limiting variance-covariance matrix can be consistently estimated by their sample analogs:
ꢅ ꢉˆ
ꢋ
z
ꢍꢀ
∗
ˆꢌꢇ
n
1
n
ꢊ
0
i
ꢑ
ˆ
ˆ ˆ
ˆ
ꢁ
A =
ꢓꢉ ꢂz ꢍꢀꢃꢓꢉ ꢂz ꢍꢀꢃ G
ꢆ
0
i
0
i
h
i=1
ꢅ
ꢇ
n
ˆ
1
4
1 ꢊ
ꢉˆ ꢂz ꢍꢀꢃ
ꢑ
ˆ
2
ˆ
ˆ ˆ
ˆ
ꢁ
0
i
ꢒ =
f ꢂz ꢃ ꢓꢉ ꢂz ꢍꢀꢃꢓꢉ ꢂz ꢍꢀꢃ G
ꢄ
i
0
i
0
i
∗
h
n
i=1
Alternatively, we can also estimate A and ꢒ using
ꢅ
ꢇ
n
ˆ
1
n
ꢊ
ꢉˆ ꢂz ꢍꢀꢃ
ꢁ
1
i
ꢑ
ˆ
ˆ ˆ
ˆ ꢄ
A =
ꢓꢉ ꢂz ꢍꢀꢃꢓꢉ ꢂz ꢍꢀꢃ G
ꢆ
1
i
1
i
∗
h
i=1
ꢅ
ꢇ
n
ˆ
1
4
1 ꢊ
ꢉˆ ꢂz ꢍꢀꢃ
2
ˆ
ˆ ˆ
ˆ
ꢁ
1
i
ꢑ
ˆ
ꢒ =
f ꢂz ꢃ ꢓꢉ ꢂz ꢍꢀꢃꢓꢉ ꢂz ꢍꢀꢃ G
ꢄ
i
1
i
1
i
∗
h
n
i=1
The last two lemmas establish the validity of sample estimates for the limiting variance matrix:
p
ꢑ ꢑ
Lemma 17: For either definition of A, A −→ A.
p
ꢑ
ꢑ
Lemma 18: For either definition of ꢒ, ꢒ −→ ꢒ.
Finally, we collect the proofs for all lemmas used in this section.
Proof of Lemma 4: First we note that Assumption 7 implies that
ꢁ
ꢁ
Pꢂ0 < Pꢂx ꢀ < 0ꢀzꢃf ꢂzꢃ < hꢃ = Oꢂhꢃ and Pꢂ0 < Pꢂx ꢀ < −ꢉꢀzꢃf ꢂzꢃ < hꢃ = Oꢂhꢃꢄ
Consider ꢉ ꢂzꢍꢀꢃ first, which can be written as
1
ꢅ
ꢇ
ꢅ
ꢇ
1
1
2
ꢁ
ꢁ
ꢁ
Pꢂmaxꢂꢁꢆx ꢀꢃ > 0ꢀzꢃ− 2 f ꢂzꢃ = Pꢂx ꢀ > 0ꢆꢁ > 0ꢀzꢃ+Pꢂx ꢀ < 0ꢀzꢃ−
f ꢂzꢃ
ꢁ
ꢁ
=
=
ꢂPꢂx ꢀ < 0ꢀzꢃ−Pꢂx ꢀ < 0ꢆꢁ > 0ꢀzꢃꢃf ꢂzꢃ
Pꢂꢁ < 0ꢆx ꢀ < 0ꢀzꢃf ꢂzꢃꢄ
ꢁ
We want to show that for some M > 0 but finite,
ꢁ
ꢁ
Pꢂ0 < Pꢂꢁ < 0ꢆx ꢀ < 0ꢀzꢃf ꢂzꢃ < hꢃ ≤ Pꢂ0 < Pꢂx ꢀ < 0ꢀzꢃf ꢂzꢃ < Mhꢃꢄ

censored models with endogenous regressors
925
ꢁ
ꢁ
Since Pꢂꢁ < 0ꢆx ꢀ < 0ꢀzꢃ > 0 ꢆ⇒ Pꢂx ꢀ < 0ꢀzꢃ > 0, it suffices to show that for some M > 0,
ꢁ
ꢁ
Pꢂꢁ < 0ꢆx ꢀ < 0ꢀzꢃf ꢂzꢃ < h ꢆ⇒ Pꢂx ꢀ < 0ꢀzꢃf ꢂzꢃ < Mhꢆ
ꢁ
ꢁ
which is implied by Pꢂꢁ < 0ꢆx ꢀ < 0ꢀzꢃ > ꢌPꢂx ꢀ < 0ꢀzꢃ for some ꢌ > 0. Using Assumption 3,
ꢄ
ꢁ
ꢁ
Pꢂꢁ < 0ꢆx ꢀ < 0ꢀzꢃ ≥
Pꢂ0 > ꢁ > −cꢀxꢆzꢃf ꢂxꢀzꢃdx ≥ cfP ꢂx ꢀ < 0ꢀzꢃꢄ
xꢁꢀ<0
Next consider ꢉ ꢂzꢍꢀꢃ, which is written as
0
ꢅ
ꢇ
ꢅ
ꢇ
1
1
2
ꢁ
ꢁ
Pꢂꢁ +x ꢀ > 0ꢆꢁ > 0ꢀzꢃ− 2 f ꢂzꢃ = Pꢂꢁ > maxꢂ−x ꢀꢆ0ꢃꢀzꢃ−
f ꢂzꢃ
ꢁ
ꢁ
ꢁ
=
=
ꢂPꢂꢁ > −x ꢀꢆx ꢀ < 0ꢀzꢃ−Pꢂꢁ > 0ꢆx ꢀ < 0ꢀzꢃꢃf ꢂzꢃ
−Pꢂ0 < ꢁ < −x ꢀꢀzꢃf ꢂzꢃꢄ
ꢁ
We want to show that for some M > 0,
ꢁ
ꢁ
ꢁ
ꢁ
Pꢂ0<Pꢂ0<ꢁ<−x ꢀꢆx ꢀ<0ꢀzꢃf ꢂzꢃ<hꢃ≤Pꢂ0<Pꢂx ꢀ<0ꢀzꢃꢆPꢂx ꢀ<−ꢉꢀzꢃf ꢂzꢃ<Mhꢃꢄ
ꢁ
ꢁ
ꢁ
Since Pꢂ0 < ꢁ < −x ꢀꢆx ꢀ < 0ꢀzꢃ > 0 ꢆ⇒ Pꢂx ꢀ < 0ꢀzꢃ > 0, we only need
ꢁ
ꢁ
ꢁ
Pꢂ0 < ꢁ < −x ꢀꢆx ꢀ < 0ꢀzꢃf ꢂzꢃ < h ꢆ⇒ Pꢂx ꢀ < −ꢉꢀzꢃf ꢂzꢃ < Mhꢍ
ꢁ
ꢁ
ꢁ
for this it suffices to show that for some ꢌ > 0, Pꢂ0 < ꢁ < −x ꢀꢆx ꢀ < 0ꢀzꢃ ≥ ꢌP ꢂx ꢀ < −ꢉꢀzꢃ. In fact
by Assumption 3,
ꢄ
ꢄ
ꢄ
−xꢁꢀ
ꢁ
ꢁ
Pꢂ0 < ꢁ < −x ꢀꢆx ꢀ < 0ꢀzꢃ =
f ꢂꢁꢀxꢆzꢃdꢁf ꢂxꢀzꢃdx
xꢁꢀ<0
0
ꢄ
minꢂꢉꢆcꢃ
ꢁ
≥
fdꢁf ꢂxꢀzꢃdx ≥ f minꢂꢉꢆcꢃPꢂx ꢀ < −ꢁꢀzꢃꢄ
Q.E.D.
−
xꢁꢀ<−ꢁ
0
This concludes the proof.
Proof of Lemma 5: Take for example ꢉꢂzꢍbꢃ and consider b in a neighborhood of ꢀ, which
ꢁ
can be written as, for t denoting xb −x ꢀ,
ꢄ ꢄ
ꢁ
ꢁ
Pꢂ0 < ꢁ < −x ꢀꢆx ꢀ < 0ꢀzꢃ =
f ꢂꢁꢀxꢆzꢃdꢁf ꢂxꢀzꢃdxꢄ
x
maxꢂtꢆ−xꢁꢀꢃ
ꢁ
ꢁ
The inner integral is differentiable at t if x ꢀ = t. Under Assumption 8, Pꢂx ꢀ = tꢀzꢃ = 0 a.s. in z;
hence by the dominated convergence theorem the outer integral is differentiable with respect to t in
a small neighborhood of 0 almost surely in z. The other terms are similar.
Q.E.D.
Proof of Lemma 7: By the boundedness Assumption 8 and uniform convergence of ꢉˆ ꢂzꢍꢀꢃ
0
to ꢉ ꢂzꢍꢀꢃ in Lemma 6, we can find M < ꢂ large enough such that w.p. −→ 1,
0
ꢀ
ꢓꢉˆ ꢂz ꢍꢀꢃꢀ < Mꢆ
ꢀꢓꢉˆ ꢂz ꢍꢀꢃꢀ < Mꢆ
∀i = 1ꢆꢎ ꢎ ꢎ ꢆnꢄ
0
i
1
i
Hence, the difference between the two sides of the first relation in Lemma 7 is, w.p. −→ 1,
ꢖ
ꢅ
ꢇ
ꢖ
ꢖ
ꢖ
ꢖ
n
ꢖ 1
ꢊ
ꢉˆ ꢂz ꢍꢀꢃ
0
i
ꢖ
√
ꢉˆ ꢂz ꢍꢀꢃꢓꢉˆ ꢂz ꢍꢀꢃG
1ꢂꢉ ꢂz ꢍꢀꢃ < 0ꢃ
ꢖ
0
i
0
i
∗
h
0
i
n
i=1
n
1
n
ꢊ
∗
≤
√
Mꢀꢉˆ ꢂz ꢍꢀꢃꢀ1ꢂꢉˆ ꢂz ꢍꢀꢃ > −h ꢃ1ꢂꢉ ꢂz ꢍꢀꢃ < 0ꢃ
0
i
0
i
0
i
i=1

9
26
h. hong and e. tamer
n
1
n
ꢊ
∗
−ꢐ
∗
−ꢐ
≤
√
M maxꢂh ꢆn ꢃ1ꢂ−h −n < ꢉ ꢂz ꢍꢀꢃ < 0ꢃ
0
i
i=1
n
1
ꢊ
∗
−ꢐ
∗
−ꢐ
≤
=
√
Mꢂh +n ꢃ1ꢂ−h −n < ꢉ ꢂz ꢍꢀꢃ < 0ꢃ
0
i
n
i=1
√
∗
−ꢐ
2
1/2−2ꢑ
+n1/2−2ꢑ ꢃ = o ꢂ1ꢃꢄ
Opꢂ nꢂh +n ꢃ ꢃ = O ꢂn
p
p
The first inequality holds by the definition of Gꢂ·ꢃ; the second inequality follows by Lemma 6. In
the first equality we use a Markov inequality and the fact that
∗
−ꢐ
∗
−ꢐ
Pꢂ−h −n < ꢉ ꢂz ꢍꢀꢃ < 0ꢃ = Oꢂh +n ꢃꢆ
0
i
which is proven in Lemma 4. Finally, the rate conditions in Assumption 10 (ii) deliver the result, since
ꢐ > ꢑ > 1/4. A completely analogous argument holds for the second relation, where the difference
is bounded by
ꢖ
ꢅ
ꢇ
ꢖ
ꢖ
ꢖ
ꢖ
n
ꢖ 1
ꢊ
ꢉˆ ꢂz ꢍꢀꢃ
1
i
ꢖ
ˆ
ꢄ
√
ꢉˆ ꢂz ꢍꢀꢃꢓꢉ ꢂz ꢍꢀꢃG
1ꢂꢉ ꢂz ꢍꢀꢃ > 0ꢃ
ꢖ
1
i
1
i
∗
h
1
i
n
i=1
n
1
n
ꢊ
√
p
∗
−ꢐ
∗
−ꢐ
∗
−ꢐ
2
≤
=
√
Mꢂh +n ꢃ1ꢂh +n > ꢉ ꢂz ꢍꢀꢃ > 0ꢃ = O ꢂ nꢂh +n ꢃ ꢃ
1
i
i=1
1/2−2ꢑ
+n1/2−2ꢐꢃ = o ꢂ1ꢃꢄ
O ꢂn
p
p
As for the first result we have used the condition that ꢉ ꢂz ꢍꢀꢃ is continuously distributed near 0.
1
i
Q.E.D.
Proof of Lemma 8: First of all 1ꢂz ∈ T ꢃ = 1ꢂꢉ ꢂz ꢍꢀꢃ = 0ꢃ, on which ꢓꢉ ꢂz ꢍꢀꢃ = ꢓꢉꢂz ꢍꢀꢃ.
i
0
i
0
i
i
1
2
∗
Since by Assumption 10 ꢐ > ꢑ, therefore w.p. −→ 1, ꢉˆ ꢂz ꢍꢀꢃ > − h , uniformly for z ∈ T ; hence
0
i
i
ꢅ
ꢇ
n
1
n
ꢊ
ꢉˆ ꢂz ꢍꢀꢃ
0
i
√
ꢉˆ ꢂz ꢍꢀꢃꢓꢉˆ ꢂz ꢍꢀꢃG
1ꢂz ∈ T ꢃ
0
i
0
i
i
∗
h
i=1
n
1
n
ꢊ
=
√
ꢉˆ ꢂz ꢍꢀꢃꢓꢉˆ ꢂz ꢍꢀꢃ1ꢂz ∈ T ꢃ+o ꢂ1ꢃ
0
i
0
i
i
p
i=1
n
1
ꢊ
=
√
ꢉˆ ꢂz ꢍꢀꢃꢓꢉ ꢂz ꢍꢀꢃ1ꢂz ∈ T ꢃ+o ꢂ1ꢃꢆ
0
i
0
i
i
p
n
i=1
where the last equation is evident by Lemma 6, since the difference is at most
ꢅ
ꢇ
√
√
ˆ
−2ꢐ
Op supꢀꢉˆ ꢂz ꢍꢀꢃꢀ O ꢂ nꢀꢓꢉ ꢂzꢍꢀꢃ−ꢓꢉ ꢂzꢍꢀꢃꢀꢃ = M ×O ꢂ nn ꢃ = o ꢂ1ꢃꢄ
0
i
P
0
0
p
p
z
Since ꢓꢉ ꢂz ꢍꢀꢃ = ꢓꢉꢂz ꢍꢀꢃ on T almost surely, what is left to be shown is that the following terms
0
i
i
are o ꢂ1ꢃ:
p
n
1
n
ꢊ
√
ꢓꢉꢂz ꢍꢀꢃꢂꢉˆ ꢂz ꢍꢀꢃ−ꢉˆꢂz ꢍꢀꢃꢃ1ꢂz ∈ T ꢃ
i
0
i
i
i
i=1
ꢅ
ꢇ
n
n
d
1
n
ꢊ
1
ꢊ
z
j
−z
i
ꢁ
ꢁ
=
√
ꢓꢉꢂz ꢍꢀꢃ
1ꢂ0 < ꢁ < −x ꢀꢆx ꢀ < 0ꢃK
1ꢂz ∈ T ꢃ
i
j
j
j
i
nhd
h
i=1
j=1
n
n
1
ꢊꢊ
≡
√
mꢂw ꢆw ꢃꢆ
i j
nn
i=1 j=1

censored models with endogenous regressors
927
where mꢂw ꢆw ꢃ is defined as
i
j
ꢅ
ꢇ
d
1
hd
z
j
−z
i
ꢁ
ꢁ
mꢂw ꢆw ꢃ = ꢓꢉꢂz ꢍꢀꢃ 1ꢂ0 < ꢁ < −x ꢀꢆx ꢀ < 0ꢃK
1ꢂz ∈ T ꢃꢄ
i
j
i
j
j
j
i
h
√
From standard mean variance calculation it suffices to show that each of
nꢂEmꢂw ꢆw ꢃꢃ,
i j
2
2
EꢂEꢂmꢂw ꢆw ꢃꢀw ꢃ ꢃ, and EꢂEꢂmꢂw ꢆw ꢃꢀw ꢃ ꢃ is oꢂ1ꢃ, since
i
j
i
i
j
j
ꢅ
ꢇ
n
n
1
nn
ꢊꢊ
o
1
1
n
2
2
2
var
√
mꢂw ꢆw ꢃ = n2 Emꢂw ꢆw ꢃ + Emꢂw ꢆw ꢃ +EꢂEꢂmꢂw ꢆw ꢃꢀw ꢃ ꢃ
i
j
i
j
i
j
i
j
i
i=1 j=1
2
+
EꢂEꢂmꢂw ꢆw ꢃꢀw ꢃ ꢃ+EꢂEꢂmꢂw ꢆw ꢃꢀw ꢃEꢂmꢂw ꢆw ꢃꢀw ꢃꢃ
i
j
j
i
j
i
j
i
i
and
ꢜ
ꢜ
2
2
EꢂEꢂmꢂw ꢆw ꢃꢀw ꢃEꢂmꢂw ꢆw ꢃꢀw ꢃꢃ ≤ EꢂEꢂmꢂw ꢆw ꢃꢀw ꢃ ꢃ EꢂEꢂmꢂw ꢆw ꢃꢀw ꢃ ꢃꢄ
i
j
i
j
i
i
i
j
i
j
i
i
ꢁ
They can be verified noting Pꢂx ꢀ < 0ꢀz ꢃ = 0 on T and using Assumptions 8 and 9:
j
j
ꢅ
ꢇ
ꢅ
ꢄ
ꢄ
ꢄ
1
z
j
−z
i
ꢁ
ꢁ
d
Emꢂw ꢆw ꢃ=
ꢓꢉꢂz ꢍꢀꢃdF ꢂz ꢃ 1ꢂ0<ꢁ <−x ꢀꢆx ꢀ<0ꢃK
dF ꢂꢁ ꢆx ꢆz ꢃ
i
j
i
i
j
j
j
j
j
j
hd
h
T
ꢇ
ꢄ
1
z
j
−z
i
ꢁ
ꢁ
d
= h
ꢓꢉꢂz ꢍꢀꢃdF ꢂz ꢃ Pꢂ0<ꢁ <−x ꢀꢆx ꢀ<0ꢀz ꢃf ꢂz ꢃK
dzj
i
i
j
j
j
j
j
d
T
h
ꢄ
ꢄ
ꢁ
ꢁ
d
=
=
ꢓꢉꢂz ꢍꢀꢃdF ꢂz ꢃ Pꢂ0<ꢁ <−x ꢀꢆx ꢀ<0ꢀz +uhꢃf ꢂz +uhꢃK ꢂuꢃdu
i
i
j
j
j
i
i
T
ꢅ 1
ꢇ
m
Oꢂh ꢃ=o
√
ꢄ
n
2
Exactly the same calculation can be used to verify that EꢂEꢂmꢂw ꢆw ꢃꢀw ꢃ ꢃ = oꢂ1ꢃ. Finally,
i
j
i
Eꢂmꢂw ꢆw ꢃꢀw ꢃ
i
j
j
ꢅ
ꢇ
ꢄ
z −z
1
d
i
j
=
f ꢂz ꢃꢓꢉꢂz ꢍꢀꢃ1ꢂ0 < ꢁ < −x ꢀꢆx ꢀ < 0ꢃK
1ꢂz ∈ T ꢃdzi
i
i
j
j
j
i
h
hd
ꢄ
d
=
1ꢂ0 < ꢁ < −x ꢀꢆx ꢀ < 0ꢃ f ꢂz +uhꢃꢓꢉꢂz +uhꢍꢀꢃK ꢂuꢃ1ꢂz +uh ∈ T ꢃduꢆ
j
j
j
j
j
j
ꢚ
ꢛ
2
E Eꢂmꢂw ꢆw ꢃꢀw ꢃ
i
j
j
ꢄ
=
Pꢂ0 < ꢁ < −x ꢀꢆx ꢀ < 0ꢀz ꢃ
j
j
j
j
ꢆ
ꢈ
ꢄ
2
d
×
f ꢂz +uhꢃꢓꢉꢂz +uhꢍꢀꢃK ꢂuꢃ1ꢂz +uh ∈ T ꢃdu dF ꢂz ꢃꢆ
j
j
j
j
which converges to 0 by iterated use of the dominated convergence theorem and the definition
of T . Q.E.D.
1
2
Proof of Lemma 9: Note that gꢂxꢃ
ꢃ
ꢖ
ꢅ
ꢇꢖ
ꢖ
ꢖ
ꢉˆ ꢂz ꢍꢀꢃ
0
i
ꢖ ˆ
ꢖ
ꢓꢉ ꢂz ꢍꢀꢃg
< Mꢆ
∀i = 1ꢆꢎ ꢎ ꢎ ꢆnꢄ
ꢖ
0
i
∗
ꢖ
h

9
28
h. hong and e. tamer
Then the term is handled by splitting it into two, with probability converging to 1:
ꢖ
ꢅ
ꢇꢖ
ꢖ
n
ꢖ 1
ꢊ
1
h
ꢉˆ ꢂz ꢍꢀꢃ
0
i
ꢖ
2
ˆ
ꢖ
√
ꢉˆ ꢂz ꢍꢀꢃ ꢓꢉ ꢂz ꢍꢀꢃ
g
ꢖ
0
i
0
i
∗
∗
h
ꢖ
n
i=1
ꢖ
ꢅ
ꢇ
n
ꢖ 1
ꢊ
1
h
ꢉˆ ꢂz ꢍꢀꢃ
ꢖ
2
ˆ
0
i
≤
√
ꢉˆ ꢂz ꢍꢀꢃ ꢓꢉ ꢂz ꢍꢀꢃ
g
ꢖ
0
i
0
i
∗
∗
n
h
i=1
ꢖ
ꢖ
ꢖ
ꢖ
∗
∗
×
1ꢂꢉ ꢂz ꢍꢀꢃ < 0ꢆ−h /2 > ꢉˆ ꢂz ꢍꢀꢃ > −h ꢃ
0
i
0
i
ꢖ
ꢅ
ꢇ
ꢖ
ꢖ
ꢖ
ꢖ
n
ꢖ 1
ꢊ
1
h
ꢉˆ ꢂz ꢍꢀꢃ
2
ˆ
0
i
ꢖ
+
√
ꢉˆ ꢂz ꢍꢀꢃ ꢓꢉ ꢂz ꢍꢀꢃ
g
1ꢂꢉ ꢂz ꢍꢀꢃ = 0ꢃ
ꢖ
0
i
0
i
∗
∗
h
0
i
n
i=1
n
2
1
ꢊ ꢉˆ ꢂz ꢍꢀꢃ
0
i
∗
−ꢐ
∗
≤
≤
M √
1ꢂ−h −n < ꢉ ꢂz ꢍꢀꢃ < 0ꢆꢀꢉˆ ꢂz ꢍꢀꢃꢀ < h ꢃ
0
i
0
i
∗
n
h
i=1
n
2
1
n
ꢊ ꢉˆ ꢂz ꢍꢀꢃ
0
i
+
M √
1ꢂz ∈ T ꢃ
i
∗
h
i=1
n
1
n
ꢊ
√
ꢋ
ꢌ
∗
∗
−ꢐ
−2ꢐ+ꢑ
n
M √
h 1ꢂ−h −n < ꢉ ꢂz ꢍꢀꢃ < 0ꢃ+M nOp
0
i
i=1
ꢋ√
ꢌ
ꢋ
ꢌ
∗
−ꢐ
∗
1/2−2ꢐ+ꢑ
n
=
=
Op nꢂh +n ꢃh +Op
ꢋ
ꢌ
1
/2−2ꢑ
+n1/2−ꢐ−ꢑ +n
1/2−2ꢐ+ꢑ
Op
n
= o ꢂ1ꢃ
p
where the second inequality and the last equality follow by Assumption 10 on ꢑ and ꢐ, and also
∗
−ꢐ
∗
−ꢐ
from, as previously, Pꢂ−h −n < ꢉ ꢂz ꢍꢀꢃ < 0ꢃ = Oꢂh +n ꢃ.
Q.E.D.
0
i
Proof of Lemma 10: It is straightforward to show that uniformly in b ꢈ ꢀb −ꢀꢀ → 0,
ꢅ
ꢇ
n
1
n
ꢊ
ꢉˆ ꢂz ꢍbꢃ
ꢁ
0
i
ˆ
ˆ
ꢓꢉ ꢂz ꢍbꢃꢓꢉ ꢂz ꢍbꢃ G
0
i
0
i
∗
h
i=1
ꢅ
ꢅ
ꢇ
ꢇ
n
1
n
ꢊ
ꢉˆ ꢂz ꢍbꢃ
ꢁ
0
i
=
ꢓꢉ ꢂz ꢍbꢃꢓꢉ ꢂz ꢍbꢃ G
+o ꢂ1ꢃ
0
i
0
i
p
∗
h
i=1
n
1
n
ꢊ
ꢉ ꢂz ꢍbꢃ
0 i
ꢁ
=
=
ꢓꢉ ꢂz ꢍbꢃꢓꢉ ꢂz ꢍbꢃ G
+o ꢂ1ꢃ
0
i
0
i
p
∗
h
i=1
ꢅ
ꢇ
n
1
n
ꢊ
ꢉ ꢂz ꢍbꢃ
0 i
ꢁ
ꢓꢉ ꢂz ꢍꢀꢃꢓꢉ ꢂz ꢍꢀꢃ G
+o ꢂ1ꢃ
0
i
0
i
p
∗
h
i=1
where the first equality follows from the uniform convergence rate and the third equality from uni-
form continuity. The second equality follows from that with probability converging to 1:
ꢖ
ꢅ
ꢇ
ꢅ
ꢇꢖ
ꢖ
n
n
ꢖ 1 ꢊ
ꢉˆ ꢂz ꢍbꢃ
1 ꢊ
ꢉ ꢂz ꢍbꢃ
0
i
ꢁ
ꢓꢉ ꢂz ꢍbꢃꢓꢉ ꢂz ꢍbꢃ G
0 i 0 i
0
i
ꢖ
ꢖ
ꢁ
ꢖ
sup
ꢓꢉ ꢂz ꢍbꢃꢓꢉ ꢂz ꢍbꢃ G
−
0
i
0
i
ꢖ
∗
∗
b
n
i=1
h
n
i=1
h
ꢖ
ꢅ
ꢇ
ꢖ
ꢖ
ꢖ
n
∗
ꢖ 1 ꢊ
1
h
ꢉˆ ꢂz ꢍbꢃ
ꢁ
0
i
ꢖ
=
sup
ꢓꢉ ꢂz ꢍbꢃꢓꢉ ꢂz ꢍbꢃ
g
ꢂꢉˆ ꢂz ꢍbꢃ−ꢉ ꢂz ꢍbꢃꢃ
ꢖ
0
i
0
i
∗
∗
0
i
0
i
ꢖ
b
n
i=1
h
1
≤
M2
supꢀꢉˆ ꢂz ꢍbꢃ−ꢉ ꢂz ꢍbꢃꢀ = o ꢂ1ꢃ
0
i
0
i
p
∗
h
b

censored models with endogenous regressors
929
where the supremum is taken over a small neighborhood of ꢀ. Following the same argument as in
Horowitz (1988), we know from Theorem 2.37 of Pollard (1984),
ꢖ
ꢅ
ꢅ
ꢇ
n
ꢖ 1 ꢊ
ꢉ ꢂz ꢍbꢃ
0
i
ꢖ
ꢖ
ꢁ
ꢓꢉ ꢂz ꢍꢀꢃꢓꢉ ꢂz ꢍꢀꢃ G
0 i 0 i
sup
∗
b
n
i=1
h
ꢇꢖ
ꢉ ꢂz ꢍbꢃ
ꢖ
ꢖ
ꢖ
ꢁ
0
i
−
Eꢓꢉ ꢂz ꢍꢀꢃꢓꢉ ꢂz ꢍꢀꢃ G
= o ꢂ1ꢃꢄ
0
i
0
i
p
∗
h
Hence we only need to show that
ꢖ
ꢖ
ꢖ
ꢅ
ꢇ
ꢉ ꢂz ꢍbꢃ
ꢁ
0
i
sup Eꢓꢉ ꢂz ꢍꢀꢃꢓꢉ ꢂz ꢍꢀꢃ G
ꢖ
0
i
0
i
∗
bꢈꢀb−ꢀꢀ→0
h
ꢖ
ꢖ
ꢖ
ꢁ
−
Eꢓꢉ ꢂz ꢍꢀꢃꢓꢉ ꢂz ꢍꢀꢃ 1ꢂꢉ ꢂz ꢍꢀꢃ ≥ 0ꢃ = oꢂ1ꢃꢄ
0
i
0
i
0
i
ꢖ
Breaking this into two parts, first the continuity result of Manski (1988, Lemma 7.2) gives
ꢖ
ꢖ
ꢖ
ꢁ
sup Eꢓꢉ ꢂz ꢍꢀꢃꢓꢉ ꢂz ꢍꢀꢃ 1ꢂꢉ ꢂz ꢍꢀꢃ ≥ 0ꢃ
ꢖ
0
i
0
i
0
i
bꢈꢀb−ꢀꢀ→0
ꢖ
ꢖ
ꢖ
ꢁ
−
Eꢓꢉ ꢂz ꢍꢀꢃꢓꢉ ꢂz ꢍꢀꢃ 1ꢂꢉ ꢂz ꢍbꢃ ≥ 0ꢃ = oꢂ1ꢃꢄ
0
i
0
i
0
i
ꢖ
Also we have
ꢖ
ꢖ
ꢖ
ꢖ
ꢅ
ꢅ
ꢇꢇꢖ
ꢉ ꢂz ꢍbꢃ
ꢖ
ꢖ
ꢖ
ꢁ
0
i
Eꢓꢉ ꢂz ꢍꢀꢃꢓꢉ ꢂz ꢍꢀꢃ 1ꢂꢉ ꢂz ꢍbꢃ ≥ 0ꢃ−G
0
i
0
i
0
i
∗
h
2
∗
≤
M Pꢂ−h ≤ ꢉ ꢂz ꢍbꢃ ≤ 0ꢃ → 0
0
i
∗
as h −→ 0 uniformly in b ꢈ ꢀb −ꢀꢀ −→ 0.
Q.E.D.
Proof of Lemma 11: Using arguments similar to those in the previous Lemma, the left-hand
side converges locally uniformly in b close to ꢀ to
Eꢉ ꢂz ꢍꢀꢃꢓꢓꢉ ꢂz ꢍꢀꢃ1ꢂꢉ ꢂz ꢍꢀꢃ ≥ 0ꢃ = 0ꢆ
0
i
0
i
0
i
since on 1ꢂꢉ ꢂz ꢍꢀꢃ ≥ 0ꢃ it is necessary that ꢉ ꢂz ꢍꢀꢃ = 0.
Q.E.D.
0
i
0
i
∗
∗
Proof of Lemma 12: The summand is nonzero only if ꢉˆ ꢂz ꢍbꢃ ∈ ꢂ−h ꢆ−h /2ꢃ; hence uni-
0
i
formly in b
ꢖ
ꢅ
ꢇꢖ
ꢖ
n
∗
ꢖ 1 ꢊ
1
h
ꢉˆ ꢂz ꢍbꢃ
ꢖ
ꢁ
0
i
ꢖ
ꢉˆ ꢂz ꢍbꢃꢓꢉˆ ꢂz ꢍbꢃꢓꢉˆ ꢂz ꢍbꢃ
g
ꢖ
0
i
0
i
0
i
∗
∗
ꢖ
n
i=1
h
n
1
n
ꢊ
M2
1ꢂ−h ≤ ꢉˆ ꢂz ꢍbꢃ ≤ −h /2ꢃꢄ
∗
∗
≤
0
i
i=1
Next we note that this can be in turn bounded by, with probability converging to 1,
ꢅ
ꢇ
n
1
n
ꢊ
1
2
1 −h −n ≤ ꢉ ꢂz ꢍbꢃ ≤ − h∗ +n−ꢐ
∗
−ꢐ
sup
0
i
b→ꢀ
i=1
ꢅ
ꢇ
n
1
ꢊ
1
4
∗
∗
≤
sup
1 −2 ×h ≤ ꢉ ꢂz ꢍbꢃ ≤ −
h
ꢆ
0
i
b→ꢀ
n
i=1
∗
which converges to 0 by the uniform law of large numbers and using h −→ 0.
Q.E.D.

9
30
h. hong and e. tamer
Proof of Lemma 13: The argument is simpler than the previous one. Uniformly in b
ꢖ
ꢅ
ꢇꢖ
n
ꢖ 1 ꢊ
1
h
ꢉˆ ꢂz ꢍbꢃ
ꢖ
0
i
∗
ꢖ
2
ꢖ
ꢉˆ ꢂz ꢍbꢃ ꢓꢓꢉˆ ꢂz ꢍbꢃ
g
≤ h −→ 0
ꢖ
0
i
0
i
∗
∗
ꢖ
n
i=1
h
∗
simply using the fact ꢀꢉˆ ꢂz ꢍbꢃꢀ < h whenever the summand is nonzero.
0
i
Proof of Lemma 14: This is very similar to the proof of Lemma 12; in fact
ꢖ
ꢅ
ꢇꢖ
n
ꢖ 1 ꢊ
1
h
ꢉˆ ꢂz ꢍbꢃ
ꢖ
0
i
ꢖ
2
ꢁ
ꢁ
g
ꢖ
ꢉˆ ꢂz ꢍbꢃ ꢓꢉˆ ꢂz ꢍbꢃꢓꢉˆ ꢂz ꢍbꢃ
ꢖ
0
i
0
i
0
i
∗2
∗
ꢖ
n
i=1
h
n
1
ꢊ
M2
1ꢂ−h ≤ ꢉˆ ꢂz ꢍbꢃ ≤ −h /2ꢃꢄ
∗
∗
≤
0
i
n
i=1
The same arguments as in (12) complete the proof.
Proof of Lemma 16: Note the left-hand side is represented as a U-statistic
Q.E.D.
n
n
1
nn
ꢊꢊ
√
mꢂw ꢆw ꢃꢆ
i j
i=1 j=1
where
ꢅ
ꢇ
ꢅ
ꢇ
1
2
1
hd
z
j
−z
i
ꢁ
d
mꢂw ꢆw ꢃ = 1ꢂy −x ꢀ ≥ 0ꢃ−
K
ꢓꢉꢂz ꢍꢀꢃ1ꢂz ∈ T ꢃꢄ
i
j
j
j
i
i
h
√
ꢁ
j
First the bias term can be shown to be negligible up to order o ꢂ1/ nꢃ; let ꢌ ≡ maxꢂꢁ ꢆ−x ꢀꢃ,
p
j
j
ꢅ
ꢇ
ꢄꢄꢄ
1
2
ꢇ
ꢁ
Emꢂw ꢆw ꢃ =
1ꢂy −x ꢀ ≥ 0ꢃ−
dF ꢂy ꢆx ꢀz ꢃ
i
j
j
j
j
j
j
ꢅ
1
z
j
−z
i
d
× h
K
dF ꢂz ꢃꢓꢉꢂz ꢍꢀꢃ1ꢂz ∈ T ꢃdF ꢂz ꢃ
j
i
i
i
d
h
ꢆ
ꢈ
ꢅ
ꢇ
ꢄꢄ
1
2
1
hd
z
j
−z
i
d
=
Pꢂꢌ > 0ꢀz ꢃ−
K
dF ꢂz ꢃꢓꢉꢂz ꢍꢀꢃ1ꢂz ∈ T ꢃdF ꢂz ꢃꢄ
j
j
j
i
i
i
h
m
The last term in the summand is O ꢂh ꢃ = o ꢂ1/
√nꢃ, by Assumptions 8 and 9. In fact
p
p
ꢆ
ꢈ
ꢅ
ꢇ
ꢄ
1
2
1
z
j
−z
i
d
Pꢂꢌ > 0ꢀz ꢃ−
K
dF ꢂzj ꢃ
j
j
hd
h
ꢅ
ꢇ
ꢄ
1
2
d
m
=
Pꢂꢌ > 0ꢀz +uhꢃ−
K ꢂuꢃdF ꢂz +uhꢃ = Oꢂh ꢃ
j
i
i
1
2
on T , since on T ꢈ Pꢂꢌ > 0ꢀz = z ꢃ = . The same calculation also shows that Eꢂmꢂw ꢆw ꢃꢀw ꢃ =
j
j
i
i
j
i
√
o ꢂ1/ nꢃ uniformly in w . Also the condition for Lemma 3.1 in Powell, Stock, and Stoker (1989) is
p
i
2
d
d
easily verified by noting that Em ꢂw ꢆw ꢃ = Oꢂ1/h ꢃ and nh → ꢂ. Hence we apply the projection
i
j
result to obtain that up to a term of o ꢂ1ꢃ the above U-statistic is approximated by
p
1
n
ꢊ
√
ꢂEꢂmꢂw ꢆw ꢃꢀw ꢃ−Emꢂw ꢆw ꢃꢃꢄ
i
j
j
i
j
j=1
Standard mean and variance calculation shows that
1
n
ꢊ
d
√
ꢂEꢂmꢂw ꢆw ꢃꢀw ꢃ−Emꢂw ꢆw ꢃꢃ −→ Nꢂ0ꢆV ꢃ
i
j
j
i
j
j=1

censored models with endogenous regressors
931
where
ꢋ
ꢌ
2
ꢒ = limE Eꢂmꢂw ꢆw ꢃꢀw ꢃ
i
j
j
ꢅ
ꢇ
2
1
2
ꢁ
2
=
E 1ꢂꢁ ≥ 0ꢃ−
ꢓꢉꢂzꢍꢀꢃꢓꢉꢂzꢍꢀꢃ f ꢂzꢃ 1ꢂz ∈ T ꢃꢄ
j
This gives the form of the limiting normal distribution given in the statement of the theorem. Q.E.D.
Proof of Lemma 17: This is an immediate consequence of Lemma 10.
Q.E.D.
Proof of Lemma 18: Up to terms of o ꢂ1ꢃ, one can replace ꢒ sequentially
p
ꢅ
ꢅ
ꢇ
ꢇ
n
ˆ
1
4
1 ꢊ
ꢉ ꢂz ꢍꢀꢃ
0 i
2
ˆ
ˆ
ꢁ
f ꢂz ꢃ ꢓꢉ ꢂz ꢍꢀꢃꢓꢉ ꢂz ꢍꢀꢃ G
ꢆ
ꢆ
i
0
i
0
i
∗
n
h
i=1
n
1
4
1 ꢊ
ꢉ ꢂz ꢍꢀꢃ
0 i
2
ꢁ
f ꢂz ꢃ ꢓꢉ ꢂz ꢍꢀꢃꢓꢉ ꢂz ꢍꢀꢃ G
i
0
i
0
i
∗
n
h
i=1
n
1
4
1 ꢊ
2
ꢁ
f ꢂz ꢃ ꢓꢉ ꢂz ꢍꢀꢃꢓꢉ ꢂz ꢍꢀꢃ 1ꢂꢉ ꢂz ꢍꢀꢃ ≥ 0ꢃꢆ
i
0
i
0
i
0
i
n
i=1
2
ꢁ
Ef ꢂzꢃ ꢓꢉ ꢂzꢍꢀꢃꢓꢉ ꢂzꢍꢀꢃ 1ꢂꢉ ꢂzꢍꢀꢃ ≥ 0ꢃꢄ
0
0
0
The proof is very similar to those of Lemma 17 and Lemma 10.
Q.E.D.
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