*Tel.: 852-2358-7602; fax: 852-2358-2084. E-mail address:[email protected] (S. Chen).
Rank estimation of a location parameter in the
binary choice model
Songnian Chen*
Department of Economics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, People's Republic of China
Received 1 September 1998; received in revised form 1 January 2000; accepted 13 March 2000
Abstract
This paper proposes a rank-based estimator for a location parameter in the binary
choice model under a monotonic index and symmetry condition, given an initialJ
n-consistent estimator for the slope parameter. The estimator converges at the usual parametric rate. Compared with existing estimators, no nonparametric smoothing is needed here. A small Monte Carlo study illustrates the usefulness of the estimator. We
also point out that the location and slope parameters can be jointly estimated. ( 2000
Elsevier Science S.A. All rights reserved.
JEL classixcation: C21; C25
Keywords: Binary choice; Location parameter; Semiparametric estimation
1. Introduction
In this paper we consider semi-parametric estimation of the binary choice
model de"ned by
d"
G
1 if x@b0 !a0!v'0,
0 otherwise, (1)
where d is the indicator of the response (or participation) variable, x3Rq is
a vector of exogenous variables, a03R andb03Rq are unknown parameters,
and the error term vis assumed to satisfy a monotonic index and symmetry
condition.
For the binary choice model, traditionally, the logit and probit estimation methods are the most popular approaches by restricting the error distribution to
parametric families. For this model, however, misspeci"cation of the parametric
distributions, in general, will result in inconsistent estimates for likelihood-based
approaches. In addition, speci"c functional forms for the error distribution
cannot usually be justi"ed by economic theory. Here we consider estimating the
binary choice model without assuming a parametric distribution forv. It is well
known that some scale normalization is necessary in order to identify the intercept and slope parameters. We adopt a common normalization scheme by
setting b01"1, where b01 is the "rst component of b0. Abusing notation
slightly, we still use (a0,b@0)@ to denote the normalized parameters. Recently,
there have been several semi-parametric estimators proposed for (a0, b@0)@in the
literature. Ahn et al. (1996), Cavanagh and Sherman (1998), Cosslett (1983), Han
(1987), HaKrdle and Stoker (1989), Horowitz and HaKrdle (1996), Ichimura (1993),
Klein and Spady (1993), Powell et al. (1989) and Sherman (1993), among others,
considered estimating the slope parameterb0under a single index or
indepen-dence restriction. The intercept terma0is not identi"ed under the single index
or independence restriction because no centering assumption is made about the
distribution ofv. Manski (1985) and Horowitz (1992) considered estimating both
the slope and intercept terms under a conditional median restriction, but their
estimators converge at rates slower than the usual parametric rate-Jn(where
nis the sample size). Under mean restrictions Lewbel (1997, 1999) considered
Jn-consistent estimation of the intercept term a0 and/or the slope parameter
b0 by imposing a strong restriction on the support of the linear index x@b0
relative to that of v; in particular, Lewbel (1999) also allows for mismeasured
regressors and general heteroscedastic errors. Chen (1999b, 2000) considered
Jn-consistent estimation of the intercept term and e$cient estimation of both
the intercept and slope parameters under a symmetry and monotonic index restriction, by strengthening both the index and conditional mean and median
restrictions. More recently, Chen (1999c) considered the estimation ofa0 and
b0 under symmetry with general heteroscedasticity.
In this paper we propose a Jn-consistent estimator for the intercept term
under the symmetry and monotonic index assumption as in Chen (1999b), given
aJn-consistent estimator for the slope parameter. The new estimator is a
rank-based one. Therefore, compared with existing estimators for the intercept term, the main advantage of our approach here is that no nonparametric local
smoothing is needed, given an initial Jn-consistent estimator for the slope
1Note thatxbK!a('0 is equivalent toF(xbK!a(#a
0)'0.5 whereF()) the participation
prob-ability function to be de"ned later. An alternative predition rule without estimatinga0is to setdK"1 ifFK(xbK)'0.5 whereFK()) is a nonparametric estimator forF()); however, with a root-nconsistent
estimatora(,F(xbK!a(#a
0) converges faster thanFK(xbK) for a givenx. Furthermore, the predition
rule based on estimated linear indices is easier to implement.
2Manski's (1985) estimator is devised by maximizing the number of correct predictions based on this prediction rule.
for estimating both the slope and intercept terms if our approach is combined with the rank estimators for the slope parameter mentioned above (Cavanagh and Sherman, 1998; Han, 1987; Sherman, 1993). The symmetry restriction
imposed here relaxes the parametric speci"cations such as the probit and logit.
In semi-parametric estimation literature symmetry has be been widely used as a common shape restriction on the error distribution (see, e.g., Chen, 1999a, b, c,
2000; Cosslett, 1987; HonoreH et al., 1997; Lee, 1996; Linton, 1993; Manski, 1988;
Newey, 1988, 1991; Powell, 1986). As indicated below, the full symmetry can be relaxed to some extent. We will also point out that our approach can be extended to estimate the slope and intercept terms simultaneously.
For the binary choice model, estimation of the intercept term is useful in many empirical applications. As pointed out by Lewbel (1997), estimation of the both the intercept and slope parameters are of direct interest in determining the distribution of reservation prices in consumer demand analysis; in particular, the mean of reservation prices and the average consumer surplus in the popula-tion depends on both the intercept and slope parameters. In the context of referendum contingent valuation in resource economics, Lewbel and McFadden (1997) considered estimating features of the distribution of values placed by consumers on a public good, which includes both the intercept and slope terms, and other measures that depend on these two parameters.
By obtaining aJnconsistent estimator fora0, our results will complement
the existing semi-parametric estimation literature, thus making semi-parametric estimators for the binary choice model fully comparable to their parametric
counterparts. Model speci"cation tests like the Hausman-type test can then be
based on the slope parameter as well as the intercept term, giving rise to the possibility that the resulting tests could be more powerful than similar ones based on the slope parameter alone.
A useful prediction rule1for the response variabledis (see, e.g., Greene, 1994)
as follows:
dK"1 if x@bK!a('0,
where (a(,bK@)@is some estimate for (a0, b@0)@. As a result, estimation ofa0is needed
for this purpose.2 Such a prediction rule is also used in constructing some
3In fact, Chen (1999b) considered e$cient estimation ofa0andb0under symmetry by combining this idea and the approach of Klein and Spady (1993).
4Andrews and Schafgans provided a consistent estimator for the intercept term in the outcome equation by relying on the&identi"cation at in"nity'; as a result, their estimator converges at a rate slower thanJn.
1983). Also, knowing the intercept is helpful for estimating choice probabilities under the symmetry restriction; in particular, based on the notation and argu-ment in the next section, we have
E(dDx"x6)"E(dDx@b0"x6 @b0)"E(1!dDx@b0"2a0!x6 @b0)
thus, with knowledge ofa0andb0, nonparametric estimation of E(dDx"x6) can
be based on observations for whichx@b0 are in the neighborhood ofx6 @b0 and
that of 2a0!x6 b0, which could lead to possible e$ciency gains, especially when
the"rst neighborhood contains only few observations.3
Consistent estimation of the intercept terma0is also useful for estimating the
intercept term of the outcome equation in the binary choice sample selection model. For the sample selection model, the intercept term of the outcome equation has important economic implications in assessing the impact on earnings of job training and union status, among other applications (see, for
example, Andrews and Schafgans, 1998).4 Recently, Chen (1999a) considered
Jn-consistent estimation for both the intercept and slope parameters in the
sample selection model under a symmetry and index restriction. However,
Chen's (1999a) approach requires an initial Jn-consistent estimator for the
intercept term of the binary selection equation.
The paper is organized as follows. The next section introduces the estimator and provides the large sample properties. Section 3 contains a small Monte Carlo study to illustrate the usefulness of the estimator. Section 4 concludes.
2. The estimator
Recall the binary choice model
d"1Mx@b0!a
0!v'0N, (2)
where for simplicity,b0 is taken to be the normalized parameter vector (see,
Cosslett, 1983; Manski, 1985; Ichimura, 1993, for detailed discussions). Let
z"x@b
0. Let F(t,b)"E(dDx@b"t), for any possible value of b, and
F(t)"F(t,b0). Whenvis independent ofx, thenF(t) is the cumulative
distribu-tion funcdistribu-tion ofa0#v. Here we assume that the error termvsatis"es a
mono-tonic index and symmetry condition as in Chen (1999b). Speci"cally, the
conditional density of v, f(vDx), depends on x only through Dz!a
magnitude of the linear index x@b0!a
0, and symmetric around 0, i.e.,
f(vDx)"f(vDDz!a
0D) andf(!vDx)"f(vDx); in addition, we assume thatF(t) is
strictly increasing int3R. Therefore, we have
F(t)#F(2a0!t)"
P
(t~a0)vate our estimator, suppose thatz
i#zj'2a0, then it is easy to deduce from (3)
Han (1987), and Sherman (1993), we propose the following rank estimator for
a0,a(, which maximizes
with respect toaover an appropriate compact intervalA, wherebK is a"rst-step
consistent estimator forb0.
Note that direct implementation of the estimation procedure requires O(n2)
operations for each evaluation of the objective function. However, the estimator
can be calculated in a much more e$cient manner. For the original sample of
sizen, we construct an arti"cial sample of size 2nby de"ning
Then similar to Cavanagh and Sherman (1998), our estimator can be de"ned
equivalently by maximizing+2i/1n dHiR
2n(zHi), whereR2n(ai) denotes the rank of a
i for real numbersa1,a2,2,a2n. Consequently, our procedure can be
imple-mented with only O(nlogn) operations for each evaluation of the objective
Now we consider large sample properties of the estimator. Letx"(1,x8 @)@and
b0"(1,bI@0)@. We make the following assumptions.
Assumption 1. The vectors (d
i,x@i)@ are independent and identically distributed
acrossi.
Assumption 2. The conditional density ofv,f(vDx), depends onxonly through
Dz!a
0D, and symmetric around 0, i.e.,f(vDx)"f(vDDz!a0D) andf(!vDx)"
f(vDx).
Assumption 3. (i) The support of the distribution ofx is not contained in any
proper linear subspace ofRq. (ii) For almost everyx8"(x
2,2,xq)@, the
distribu-tion of x
1 conditional on x8 has everywhere positive density with respect to
Lebesgue measure.
Assumption 4. The true value ofa0 is an interior point of a compact setA.
Assumption 5. The linear index z"x@b
0 has positive density at a0 and the
conditional density of the error term satis"esf(0Dx)'f
0, for a positive constant
f
0. De"ne
q(w, a, b)"E[h(w, w
i, a, b)], where
h(w
1,w2, a, b)"(d1#d2!1)1Mx@1b#x@2b'2aN
#(1!d
1!d2)1Mx@1b#x@2b(2aN
forw
1"(d1, x@1)@andw2"(d2,x@2)@.
Assumption 6. For eachw, all mixed third partial derivatives ofq(w, a, b) with
respect to (a, b@)@exist, and the absolute value of the derivatives are bounded by
M(w) such that E(M(w
i))(R.
Assumption 7. The preliminary estimator,bK"(1,bM@)@, for b0"(1,bI@0)@ is Jn -consistent, and has the asymptotic linear representation
bM"bI
0#
1
n+ti#o1(n~1@2)
for someti"t(d
the slope parameter by Manski (1985) under a conditional quantile restriction,
which is certainly su$cient under our symmetry condition. It implies thatxhas
at least one continuously distributed component, and that this component has unbounded support. However, this assumption of unbounded support can be relaxed following the arguments in Horowitz (1998). Assumption 4 is a standard
assumption in the literature. Assumption 5 essentially is an identi"cation
condi-tion for the intercept term; as in Chen (1999b, c), it requires a porcondi-tion of the population with participation probability below 0.5, and another portion with participation probability above 0.5. In particular, some algebra will show that <"E[L2q(w
i,a0, b0)/L2a](0 under Assumption 5 and some mild
smooth-ness and boundedsmooth-ness conditions contained in Assumption 6. Assumption 6 can
be justi"ed by more primitive conditions on the distribution of the variables in
the model (see Lee, 1994; Sherman, 1994 for some discussions on similar
conditions). Several existing estimators for the slope parameter forb0 (Ahn et
al., 1996; Cavanagh and Sherman, 1998; Han, 1987, HaKrdle and Stoker, 1989;
Horowitz and HaKrdle, 1996; Ichimura, 1993; Klein and Spady, 1993; Powell
et al., 1989; Sherman, 1993, among others) satisfy Assumption 7.
Theorem 1. If Assumptions1}7hold,thena( is consistent and asymptotically normal
Jn(a(!a
Proof of Theorem 1. We"rst consider consistency. Let
G
a( follows from Amemiya (1985, pp. 106, 107) if we can show that (a):
sup
ADGn(a,bK)!G(a)D"o1(1), and (b):G(a) has a unique maximum ata0. De"ne a Euclidean class of functions as in Pakes and Pollard (1989, p. 1032).
Then it is easy to check that the class of functionsMh(w
1,w2, a, b),a3R, b3RN
is Euclidean with a constant envelop. By Theorem 3.1 of Arcones and GineH
To prove (b), note that
0. Therefore,a0is the unique maximizer forG(a). Consequently,
a( is consistent for a0.
We now establishJn-consistency and asymptotic normality. De"ne
hH(w
1, w2,a,b)"h(w1,w2, a, b)!h(w1, w2,a0, b).
Then,a( can be equivalently de"ned by maximizingGH
n(a, bK), where
Theorem 1 follows from the Lindeberg}Levy central limit theorem. h
While numerical derivatives can be used for estimating the asymptotic vari-ance based on the general method of Pakes and Pollard (1993, p. 1043), it is more sensible to adopt a kernel-based approach here, since our objective function is a step function of the unknown parameters, in the spirit of kernel
estimation of a density and its"rst derivatives (namely, in the estimation of the
"rst and second-order derivatives of the cumulative distribution function). De"ne
symmetric around zero, andMa
nNdenote a sequence of real numbers converging
to zero, withna3
Such sequences can be constructed corresponding to a variety of estimators
of b0 given above (see references for details). Let qn(w, a, b)"
Remark 1. Note that the full symmetry of error distribution has been assumed
for the validity of our estimation procedure, we now note that our procedure can
be modi"ed to allow for a partial symmetry restriction. As in Lee (1992), we
assume that the error density f(vDx) is symmetric up to $uaround 0 and
P(v(!u Dx)"P(v'u Dx) for a positive constantu. De"ne a weight function
=()) such that=(x@
ib0)"0 ifDx@ib0!a0D'u. Then we can de"ne a modi"ed
estimatora8fora0 by maximizing
where=K
i"=(x@ibK) and=Kj"=(x@jbK). Following the arguments in the proof of
Theorem 1, we can show thata8is consistent fora0and asymptotically normal
under this partial symmetry restriction.
Remark2. With the exception of Chen (1999c) and Lewbel (1999), most of the existing estimators, including the one proposed in this paper, requires two steps to estimate the slope and intercept terms. We now point out that joint
estima-tion is also possible. Speci"cally, we have, under the conditions given above,
E(d
i#djDxi,xj)'1 if and only if x@ib0#x@jb0!2a0'0
which, in turn, suggests that it is reasonable to estimate (a0, bI@0)@ byhK, which
maximizes
1
n(n!1) + iEj
h(w
i, wj, a, b)
with respect to h"(a, bI@)@ over a compact set H. Let
<
h"E[L2q(wi,a0, b0)/Lh Lh@] and uhi"2Lq(wi,a0,b0)/Lh. Suppose <h is
negative de"nite andh0"(a0, bI@0)@is an interior point ofH, then by following
the arguments of Theorem 1, Han (1987), Manski (1987) and Sherman (1993), we
can show that under Assumptions 1}3, 5 and 6, hK is consistent and
asymp-totically normal,
Jn(hK!h
0)P$ N(0,Rh)
whereR
h"<~1h Dhh<~1withDhh"Euhiu@hi.
3. A Monte Carlo study
In this section we present a small Monte Carlo study to illustrate the usefulness of the proposed estimator. The data is generated according to the following model
d"1Mx#a
0#v'0N
with a0"1. Note that for simplicity, we are treating the case in whichb0 is
assumed to be known. Ten di!erent designs are constructed by varying the
distributions of the error termvand regressorx, which are independent of each
other.
Here we consider the "nite-sample performance of our estimator, a(, the
estimatorsa(
1anda(2in Chen (1999b), along with Lewbel's estimator (1997),a(l.
(See Chen (1999b) and Lewbel (1997) for details in selecting kernel functions and
bandwidths h
Table 1 Design I
n"100 n"500
Estimators Mean SD RMSE Estimators Mean SD RMSE
a(
!MSE100"0.187, and MSE
500"0.085.
results from 500 replications for each design are presented with sample sizes of 100 and 500. For each estimator under consideration, we report the mean value (Mean), the standard deviation (SD), and the root mean square error (RMSE
when applicable). Since the identi"cation condition requires that !x have
positive density ata0, we search over (!x
0.975,!x0.025) in calculatinga(, where
x
0.025 andx0.975 are 0.025 and 0.975 sample quantiles forx.
Note that when the symmetry restriction on the error distribution is violated,
it is no longer clear what the exact interpretation of,plima(, the probability of
a( is, as it depends on both the error distribution and the regressor distribution.
In the absence of the symmetry restriction, two commonly used location
para-metersam0, the mean ofa0#v, andad0, the median ofa0#v, are, in general,
di!erent, andplima( is, typically, di!erent from am0 or ad0. In all the designs
considered here, Ev"0, thus am0"a
0, as required by Lewbel's procedure
(1997).
Tables 1}5 report the simulation results for the"rst"ve designs in whichxis
drawn from N(0,1). The error termvis drawn from N(0,1) (Design I) in the"rst
design, and from thet distribution with"ve degrees of freedom in the second
design (Design II). In these two cases, the symmetry restriction is satis"ed. In the
next three designs (Designs III}V),vis drawn from (e1#e2
2#e23#e24!3)/J7, (e1#e2
2!1)/J3, and (e21!1)/J2, respectively, where e1, e2, e3, ande4 are
independent standard normals. These three error distributions are all
asymmet-ric with corresponding skewness coe$cients equal to 1.296, 1.540 and 2.828,
respectively, with the corresponding medians of (a0#v) equal to 0.808, 0.849,
and 0.615, respectively. In Design I, all the estimators, especiallya(, perform well.
In Design II,a(,a(1anda(2have similar performances, whilea(
lhas relatively large
biases, even when the sample size is 500. In Designs III}V, under the type of
asymmetric error distributions considered here, as expected, a(
l clearly has
smaller biases fromam0, whereasa(, a(
Table 2 Design II
n"100 n"500
Estimators Mean SD RMSE Estimators Mean SD RMSE
a(
!MSE100"0.202, MSE
500"0.094.
Table 3
Design III (ad0"0.808)
n"100 n"500
Estimators Mean SD Estimators Mean SD
a(
100"0.169, MSE500"0.075.
5The small variances associated with Lewbel's estimator here are not unexpected since the estimated density function ofxenters as the denominator in his approach while in these designs
xhas larger densities in its support than in the"rst"ve designs. Note that whenxhas a"nite support smaller than that ofv, Lewbel's estimator essentially corresponds to a di!erent parameter other than
am0.
Tables 6}10 report the simulation results for the other"ve designs, which only
di!er from the"rst"ve designs in that xis drawn from N(0,1) conditional on
DxD(1.9. The purpose of this is to illustratethat, in comparison, Lewbel's
procedure5 can incur substantial biases when the regressors are bounded. In
Designs VI and VII where the symmetry restriction holds,a(
1, anda(2 perform
well, anda( clearly performs best, whilea(
Table 4
Design IV (ad0"0.849)
n"100 n"500
Estimators Mean SD Estimators Mean SD
a(
!MSE100"0.162, MSE
500"0.071.
Table 5
Design<(ad0"0.615)
n"100 n"500
Estimators Mean SD Estimators Mean SD
a(
100"0.133, MSE500"0.060.
6While these are di!erent estimation problems, the selection of the bandwidths based on the latter appear to perform reasonably well for the estimation of the asymptotic variance.
that in Designs VIII, IX, and X with asymmetric error distributions, all
estimators have similar biases froma
m0 andad0.
Similar to the Monte Carlo analyses of Lewbel (1997, 1999), we also report the Monte Carlo mean (MSE
n) for the estimated standard deviations based onRn.
As in any nonparametric kernel estimation procedure, choosing bandwidths is
always a di$cult task. Notice that, for a known b, the variance estimator
R
nreduces to<~2n Dnrr. AsDnrrand<ninvolve the kernel density functionk())
and its "rst derivative, similar to the kernel estimation of density and its
Table 6 Design VI
n"100 n"500
Estimators Mean SD RMSE Estimators Mean SD RMSE
a(
!MSE100"0.191, MSE
500"0.091.
Table 7 Design VII
n"100 n"500
Estimators Mean SD RMSE Estimators Mean SD RMSE
a(
100"0.199, MSE500"0.098.
7We tried bandwidthsc
1n~1@(s`3)andc2n~1@(s`5)fors"0, 1, 2, 3, forDnrrand<n, respectively,
together with other kernels such as Epanechnikov and Quartic kernels for which the bandwidths were adjusted using the kernel exchange rate in Table 2 of HaKrdle and Linton (1994). All produced similar results.
1986) in bandwidth selection used for the latter in this Monte Carlo experiment.
The resulting normal reference bandwidths7arec
1snn~1@5 andc2snn~1@7 with c
1"1.06 andc2"0.9686, forDnrrand<n, respectively, wheresnis the sample
standard deviation of x. Overall, the estimated standard deviations perform
satisfactorily, underestimating the Monte Carlo standard deviations by about
10}15%. It might be possible that some more sophisticated bandwidth selection
Table 8
Design VIII (ad0"0.808)
n"100 n"500
Estimators Mean SD Estimators Mean SD
a(
!MSE100"0.161, MSE
500"0.072.
Table 9
Design IX (ad0"0.849)
n"100 n"500
Estimators Mean SD Estimators Mean SD
a(
100"0.155, MSE500"0.070.
4. Concluding remarks
In this paper we have proposed a rank estimator for a location parameter in the binary choice model under a symmetry and monotonic index restric-tion.Compared with the estimators of Chen (1999b) and Lewbel (1997), the new
estimator does not require nonparametric smoothing, given a"rst-step
consis-tent estimator for the slope parameter. In particular, nonparametric smoothing could be avoided altogether if our approach is combined with the rank-based estimators for the slope parameter proposed by Cavanagh and Sherman (1998), Han (1987), and Sherman (1993). Note that our estimator relies crucially on the symmetry condition, and it is a testable restriction. The conditional moment
Table 10
Design X (ad0"0.615)
n"100 n"500
Estimators Mean SD Estimators Mean SD
a(
1,h2"0.100 0.768 0.133 a(1,h2"0.100 0.772 0.059
a(
1,h2"0.050 0.769 0.133 a(1,h2"0.050 0.775 0.059
a(
1,h2"0.025 0.770 0.133 a(1,h2"0.025 0.776 0.059
a(
2 0.768 0.133 a(2 0.774 0.059
a(
l,hl"0.315 0.816 0.128 a(l,hl"0.166 0.836 0.057 a(
l,hl"0.630 0.838 0.133 a(l,hl"0.331 0.846 0.058 a(
l,hl"1.260 0.881 0.140 a(l,hl"0.662 0.860 0.060
a(! 0.810 0.143 a(! 0.798 0.057
!MSE100"0.124, MSE
500"0.055.
by combining the conditional moment test (Newey, 1985) and nonparametric estimation of the choice probabilities, similar to Zheng (1998). Lewbel (1997) suggested other ways of testing for the symmetry restriction based on his estimator. We have also pointed out that the full symmetry restriction can be relaxed to a certain extent, and that joint estimation of the intercept and slope parameters is also possible.
Acknowledgements
I would like to thank Bo HonoreH, Roger Klein, Lung-Fei Lee, and two
anonymous referees for helpful comment.
References
Ahn, H., Ichimura, H., Powell, J.L., 1996. Simple estimators for monotone index models. Depart-ment of Economics, University of California, Berkeley.
Amemiya, T., 1981. Qualitative response models: a survey.. Journal of Economic Literature 18, 1483}1536.
Amemiya, T., 1985. Advanced Econometrics. Harvard University Press, Cambridge, MA. Andrews, D.W.K., Schafgans, M., 1998. Semiparametric estimation of the intercept of a sample
selection model. Reviews of Economic Studies 65, 497}517.
Arcones, M.A., GineH, E., 1993. Limit theorems for U-processes. Annals of Probability 21, 1494}1592. Cavanagh, C., Sherman, R., 1998. Rank estimators for monotonic index models. Journal of
Econometrics 84, 351}382.
Chen, S., 1998. Rank estimation of transformation models. The Hong Kong University of Science and Technology.
Chen, S., 1999b. Semiparametric estimation of a location parameter in the binary choice model. Econometric Theory 15, 79}98.
Chen, S., 1999c. Semiparametric estimation of heteroscedastic binary choice sample selection models under symmetry. The Hong Kong University of Science and Technology.
Chen, S., 2000. E$cient estimation of binary choice models under symmetry. Journal of Econo-metrics 96, 183}199.
Cosslett, S.R., 1983. Distribution-free maximum likelihood estimator of the binary choice model. Econometrica 51, 765}782.
Cosslett, S.R., 1987. E$ciency bounds for distribution-free estimators of the binary choice and censored regression models. Econometrica 55, 559}586.
Fan, J., Gijbels, I., 1995. Local Polynomial Modelling and Its Applications. Chapman and Hall, London.
Greene, W.H., 1994. Econometric Analysis. Prentice-Hall, Englewood Cli!s, NJ.
Han, A.K., 1987. Nonparametric analysis of a generalized regression model: the maximum rank correlation estimator. Journal of Econometrics 35, 303}316.
HaKrdle, W., Linton, O., 1994. Applied Nonparametric Methods. In: Engle, R., McFadden, D. (Eds.), Handbook of Econometrics, IV. Elsevier, Amsterdam.
HaKrdle, W., Stoker, T., 1989. Investigating smooth multiple regression by the method of average derivatives. Journal of the American Statistical Association 84, 986}995.
HonoreH, B., Kyriazidou, E., Udry, C., 1997. Estimation of type 3 Tobit models using symmetric and pairwise comparisons. Journal of Econometrics 64, 241}278.
Horowitz, J.L., 1992. A smooth maximum score estimator for the binary response model. Econo-metrica 60, 505}531.
Horowitz, J.L., 1998. Semiparametric Methods in Econometrics. Springer, New York.
Horowitz, J.L., HaKrdle, W., 1996. Direct semiparametric estimation of single-index models with discrete covariates. Journal of the American Statistical Association 91, 1632}1640.
Ichimura, H., 1993. Semiparametric least squares (SLS) and weighted SLS estimation of single-index models. Journal of Econometrics 58, 71}120.
Klein, R.W., Spady, R.S., 1993. An e$cient semiparametric estimator of the binary response model. Econometrica 61, 387}421.
Lee, L.F., 1994. Semiparametric instrumental variable estimation of simultaneous equation sample selection models. Journal of Econometrics 63, 341}388.
Lee, M.J., 1992. Winsorized mean estimator for censored regression. Journal of Econometrics 8, 368}382.
Lee, M.J., 1996. Nonparametric two stage estimation of simultaneous equations with limited endogenous regressors. Econometric Theory 12, 305}330.
Lewbel, A., 1997. Semiparametric estimation of location and other discrete choice moments. Econometric Theory 13, 32}51.
Lewbel, A., 1999. Semiparametric qualitative response model estimation with unknown hetero-scedasticity and instrumental variables. Journal of Econometrics, forthcoming.
Lewbel, A., McFadden, D., 1997. Estimating features of a distribution from binomial data. Depart-ment of Economics, University of California, Berkeley.
Linton, O.B., 1993. Adaptive estimation in ARCH models. Econometric Theory 9, 539}569. Maddala, G.S., 1983. Limited-Dependent and Qualitative Variables in Econometrics. Cambridge
University Press, Cambridge.
Manski, C.F., 1985. Semiparametric analysis of discrete response: asymptotic properties of the maximum score estimator. Journal of Econometrics 27, 313}333.
Manski, C.F., 1988. Identi"cation of binary response models. Journal of the American Statistical Association 83, 729}738.
Newey, W.K., 1991. E$cient estimation of Tobit models under symmetry. In: Barnett, W.A., Powell, J.L., Tauchen, G. (Eds.), Nonparametric and Semiparametric Methods in Econometrics and Statistics. Cambridge University Press, Cambridge.
Pakes, A., Pollard, D., 1993. Simulation and the asymptotics of optimization estimators. Econo-metrica 57, 1027}1057.
Pollard, D., 1985. New ways to prove central limit theorems. Econometric Theory 1, 295}314. Powell, J.L., 1986. Symmetrically trimmed least squares estimation for Tobit models. Econometrica
54, 1435}1460.
Powell, J.L., Stock, J.H., Stoker, T.M., 1989. Semiparametric estimation of weighted average derivatives. Econometrica 57, 1403}1430.
Sherman, R.P., 1993. The limiting distribution of the maximum rank correlation estimator. Econo-metrica 61, 123}138.
Sherman, R.P., 1994. U-processes in the analysis of a generalized semiparametric regression estimator. Econometric Theory 11, 372}395.
Silverman, B.W., 1986. Density Estimation for Statistics and Data Analysis. Chapman & Hall, London.
