Directory UMM :Data Elmu:jurnal:J-a:Journal of Econometrics:Vol96.Issue1.May2000:

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*Tel.:#852-2358-7602; fax:#852-2358-2084. E-mail address:[email protected] (S. Chen)

E$cient estimation of binary choice models

under symmetry

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 July 1997; received in revised form 1 June 1999

Abstract

This paper proposes a semiparametric maximum likelihood estimator for both the intercept and slope parameters in a binary choice model under symmetry and index restrictions. The estimator attains the semiparametric e$ciency bound in Cosslett (1987) under the symmetry and independence restrictions. Compared with the estimator of Klein and Spady (1993), which attains the semiparametric e$ciency bound in Chamberlain (1986), and Cosslett (1987) under the independence restriction, we show that there are possible e$ciency gains in estimating the slope parameters by imposing the additional symmetry restriction. A small Monte Carlo study is carried out to illustrate the usefulness of our estimator. ( 2000 Elsevier Science S.A. All rights reserved.

JEL classixcation: C31; C34

Keywords: Binary choice; Symmetry; Semiparametric estimation; E$ciency bound

1. Introduction

In this paper we consider the estimation of the binary choice model de"ned by

d"1Mv(x,h₀)'uN,

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where 1MAN is the usual indicator function of the event A, x is a vector of exogenous variables, v(x,h₀) is a given parametric function of x with an unknown parameter vectorh₀, anduis an unobservable error term. In most applications v(x,h₀) is assumed to be a linear function, then the model (the linear binary choice model hereafter) becomes

d"1Ma

0#xb0'uN

with h₀"_(a

0,b@0)@, where a0 and b0 are the intercept and slope parameters,

respectively.

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. 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. This consideration has motivated recent interest in estimating the binary choice model by semiparametric methods which do not assume a parametric distribution foru. Most attention has been focused on the linear binary choice model (see Horowitz, 1993, for a survey) in semiparametric estimation literature. Under various weak distributional assumptions on the error termu, a number of semiparametric estimators have been proposed forh₀ (Ahn et al., 1996; Cavanagh and Sherman, 1998; Chen and Lee, 1998; Cosslett, 1983; Han, 1987; HaKrdle and Stoker, 1989; Horowitz and HaKrdle, 1996; Ichimura, 1993; Klein and Spady, 1993; Powell et al., 1989; Sherman, 1993), among others, for estimating the slope parameters with no location restriction onu, and Chen, 1998, 1999a; Horowitz, 1992; Lewbel, 1997; Manski, 1985 for estimating the intercept and/or slope parameter with location restrictions im-posed on u). Cosslett (1987) considered semiparametric e$ciency bounds for the parameters in the binary choice model under the assumption that the error term is independent of the regressors and various location restrictions. The estimators by Chen and Lee (1998) and Klein and Spady (1993), attain the e$ciency bound of Cosslett (1987) (and Chamberlain, 1986) for the slope parameter under the independence restriction. Cosslett (1987) further pointed out that under an additional symmetry restriction, not only is the location parameter identi"ed, but there is possible e$ciency gain for estimating the slope parameter. Therefore, exploiting the symmetry restriction could potentially lead to more e$cient estimator for the slope parameter than the estimator of Klein and Spady (1993).

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Chen's (1999b) approach requires an initial Jn-consistent estimator for the intercept term of the binary choice selection equation.

In this paper, we consider joint estimation of the intercept and slope parameters under index and symmetry restrictions. Following Klein and Spady (1993), we propose a semiparametric maximum likelihood estimator with both the index and symmetry restrictions taken into account in formulat-ing the semiparametric likelihood function. The resultformulat-ing estimator attains the e$ciency bound of Cosslett (1987) under the independence and symmetry restrictions.

The article is organized as follows. Section 2 introduces our estimator and considers its large sample properties. In Section 2, we also compare Cosslett's e$ciency bounds under the independence restriction with and without the symmetry restriction, and discuss possible e$ciency gain by imposing the symmetry restriction. In Section 3 a small Monte Carlo study is carried out to assess the"nite sample performance of the estimator. Section 4 concludes.

2. Motivation and the estimator

For the above binary choice model, it is well known that some scale and location normalization is necessary in order to identify h₀. In parametric models, such as the probit and logit models, scale normalization is accomplished by setting the variance ofuto one. In the semiparametric literature for the linear binary choice model, two common scale normalization schemes are to set either D b₀D"1 or the coe$cient of one component ofb₀to one. Location restrictions are imposed on various locational measures, for example, the conditional mean or median, of udirectly. In this paper, the location and scale restrictions are imposed by setting v(x,h₀)"_a

0#v1(x,b0) with v1(x,b0)"v1a(x1)# v

1b(x2,b0), and the error termubeing symmetrically distributed conditional onx.

Traditionally, the binary choice model is estimated by the maximum likeli-hood method, most notably for the logit and probit models. For a random sample of sizen, maximum likelihood estimator is de"ned by maximizing the log-likelihood function

ln¸ n(h)"

n

+

i/1

[d

ilnFx(v(xi,h))#(1!di) ln(1!Fx(v(xi,h)))],

where the error termuis assumed to have distributionF

x()) givenx. However, misspeci"cation of the parametric distribution function, in general, will lead to inconsistent estimates. Consequently, various semiparametric estimators have been proposed without a parametric speci"cation for the error distribution. In particular, under the index restriction that the error distribution depends on

x only through v

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1I thank one referee for pointing this out.

maximum likelihood estimator forb₀ by maximizing

ln¸ ,4(b)"

n

+

i/1

q_ni[d

ilnFn(v1(xi,b))#(1!di) ln(1!Fn(v1(xi,b)))],

whereF

nis a nonparametric estimator for the unknown distribution ofu!a0,

andq_niare some trimming functions adopted for technical convenience. Since no location restriction is imposed in their case, they only provide an estimator for b₀. Under the condition that the error term is independent of the regressors, they further show that their estimator attains Cosslett's (1987) (and Chamber-lain's, 1986) semiparametric e$ciency bound.

In this paper, we consider joint estimation of (a₀,b₀) under a location restriction in the form of conditional symmetry and an index restriction stronger than the one in Klein and Spady (1993). More speci"cally, we assume that the conditional density ofugivenx,f(uDx) is an even function ofu, and depends on

x only throughv2(x,h₀); that is,f(!uDx)"f(uDx) andf(uDx)"f(uDv2(x,h₀). Following Klein and Spady (1993), we propose a semiparametric maximum likelihood estimator; moreover, we take into account the symmetry as well as the index restriction in formulating the semiparametric likelihood function. Under our assumptions, the location parameter a₀ can be consistently esti-mated; furthermore, there are possible e$ciency gains in estimating b₀ by imposing the symmetry restriction.

To formulate the semiparametric likelihood function, let g(t,h) denote the density of v(x,h) evaluated at t, P(t,h)"E(dDv(x,h)"t), P

0(t,h)"

E(1!dDv(x,h)"t), g

1(t,h)"P(t,h)g(t,h) and g0(t,h)"P0(t,h)g(t,h)" g(t,h)!g

1(t,h) with P(t)"P(t,h0) and g(t)"g(t,h0). Notice that under the

symmetry and index restrictions, it can be easily shown that

P(t)"P

0(!t)

namely,

E(dDv(x,h₀)"t)"E((1!d)Dv(x,h₀)"!t)

which, in turn, suggests that under our conditionsG

n(t,h0) is a natural

non-parametric kernel estimator for P(t), whereG

n(t,h) is de"ned as the solution to the following problem1:

min

k

n

+

i/1

C

(d

i!k)2H

A

v(x

i,h)!t

a

n

B

#(1!d

i!k)2H

A

v(x

i,h)#t

a

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whereHis a kernel function anda

n is a sequence of bandwidth converging to zero as the sample size increases to in"nity, that is,

G

These nonparametric estimators will provide the basis for constructing the semiparametric likelihood function.

Following Klein and Spady (1993) we introduce probability and likelihood trimmings to deal with technical di$culties arising from small estimated densit-ies. For the probability trimming, let

d₁_n(t,h),ae{ then we de"ne the estimated probability function as

GK_n(t,h)"gn1(t,h)#gn0(!t,h)#d1n(t,h)#d0n(!t,h) g

n(t,h)#gn(!t,h)#dn(t,h)#dn(!t,h) .

We now consider the likelihood trimming. LethK₁ be a preliminary consistent estimate for h₀ for which D hK₁!_h

0D"O1(n~1@3). The estimators mentioned

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witheA(_e@(1. As in Klein and Spady (1993), the likelihood trimming is more severe than the probability trimming. Finally, we de"ne our semiparametric maximum likelihood estimator,hK, by maximizing

QK_n(h,q()"₊(q(

i(hK1)/2)[diln[GK 2n(v(xi,h),h] #(1!d

i) ln[(1!GK n(v(xi,h),h))2]. (1) Let D_rzm(z) denote the rth-order partial of m(z) with respect to z and

D0

zm(z)"m(z). Letcdenote a generic positive constant. We make the following assumptions.

Assumption 1. The observations (d

i,xi), i"1, 2,2,nare i.i.d. The conditional density ofu,f(uDx), depends onxonly throughv2(x,h₀), and symmetric around 0, namely, f(uDx)"f(uDv(x,h₀))"f(uDv2(x,h₀)) and f(!uDx)"f(uDx); and

f(uDv(x,h₀)"t) is four times continuously di!erentiable with respect to (u,t) and its derivatives are uniformly bounded.

Assumption 2. The parameter vectorhlies in a compact parameter space,H. The true value ofh,h₀, is in the interior ofH.

Assumption 3. There exist P, P that do not depend onxsuch that

0(P)E(dDx)4P(1.

Assumption 4. The index functionv(x,h) is smooth in that forhin a neighbor-hood ofh₀ and alls:

M DD*_hr+v(s;h)D,D LD*_hr+v(s;h)/LxD N(c (r"0, 1, 2, 3, 4).

Also, the conditional density ofx

1givenx2,fx1@x2(x1) is supported on [a,b] and smooth in that for allx

DD_rsf

x1@x2(s)D(c (r"0, 1, 2, 3, 4;s3[a,b]).

Assumption 5. The kernel functionH(v) is a symmetric function that integrates to one, four times continuously di!erentiable with bounded derivatives. Also,

H(v) is a higher-order kernel in that:v2H(v) dv"0, and:v4H(v) dv(R.

Assumption 6. The bandwidth sequence Ma

nN is chosen such that n~1@(6`2e{) (a

n(n~1@(8~eA).

Assumption 7. The model is identi"ed in that for almost allx,

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where

G(v(x,h),h)"g1(v(x,h),h)#g0(!v(x,h),h) g(v(x,h),h)#g(!v(x,h),h) .

Assumption 1 describes the model and the data. The symmetry restriction is a location normalization. Compared with the usual index restriction in the literature thatf(uDx)"f(uDv(x,h₀)) (see, e.g. Ichimura, 1993; Klein and Spady, 1993), our index restriction is slightly stronger; however, in both cases, the unknown form of heteroscedasticity allowed is very limited. Assumptions 2}6 are similar to those in Klein and Spady (1993) (see Klein and Spady, 1993 for detailed discussions); in particular, a higher-order bias reducing kernel is used here. However, following Klein and Spady (1993), we can also adopt an adaptive kernel function with local smoothing that has similar bias reduction properties. Speci"cally, we set the bandwidth to be variable and data dependent. For sample sizen, letlK

j"(1/nanp)+ni/1H((v(xi,h)!t)/anp) be a preliminary density

estimator with bandwidtha

np. For 0(e@(1, set the bandwidth for observation

jtoa

nj"anp((h)¸Kj, wherep((h) is the sample standard deviation ofv(x,h), ¸_K

j"M[lKj#(1!q(pj)/m]N with

q(

pj"q(lKj,ae{np)

andmbeing the geometric mean of thelK_j. As pointed out by Klein and Spady (1993), the estimated probabilities using local smoothing are always positive, whereas those based on higher-order kernels can be negative. Assumption 7 is an identi"cation condition. Note that G(v(x,h₀),h₀)"P(v(x,h₀)) under our index and symmetry restrictions. Klein and Spady (1993) provided two sets of models for which the slope parameters are identi"ed, allowing both monotonic and nonmonotonic models. We "rst consider the identi"cation of the slope parameter. For a"xedh"(a,b@)@, if

G(v(x,h),h)"G(v(x,h₀),h₀)"P(v(x,h₀))

for almost all x, then de"ne G_a(x,b)"G(a#v

1(x,b), (a,b@)@), which can be

viewed as a function ofv

1(x,b), then by following the arguments of Klein and

Spady (1993) we can show thatb"b

0if the assumptions in Theorems 1 or 2 of

Klein and Spady (1993) are satis"ed. Once the slope parameter is identi"ed, we only need to show that

G(v(x, (a,b@₀)@), (a,b@₀)@)"G(v(x,h₀),h₀)

for almost all x implies a"_a

0. It is straightforward to show that under our

index and symmetry restrictions

G(v(x, (a,b@₀)@), (a,b@₀)@)"g

w1(x,a)P(v(x,h0))

#g

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where

g

w1(x,a)"

g(v(x, (a,b@₀)@), (a,b@₀)@)

g(v(x, (a,b@₀)@), (a,b@₀)@), (a,b@₀)@)#g(!v(x, (a,b@₀)@), (a,b@₀)@)

" g(v(x,h0)) g(v(x,h₀)#g(!v(x,h₀)#2a

0!2a)

and

g

w2(x,a)"

g(!v(x, (a,b@₀)@), (a,b@₀)@)

g(v(x, (a,b@₀)@), (a,b@₀)@)#g(!v(x, (a,b@₀)@), (a,b@₀)@)

" g(!v(x,h0)#2a0!2a) g(v(x,h₀)#g(!v(x,h₀)#2a₀!2a)

withg

w1(x,a)#gw2(x,a)"1. Thus (2) is equivalent to g

w2(x,a)[P(v(x,h0)#2(a!a0))!P(v(x,h0))]"0. (3)

When the support of v(x,h₀) is large enough so that there exists an interval [a

v,bv], we haveg(t)g(!t#2a0!2a)'0 fort3[av,bv] and (a,b0)3H, then

identi"cation ofa₀follows if

P(t)"P(t#2(a!a

0)) (4)

fort 3[a

v,bv], impliesa"a0, which, in turn, obviously holds for monotonic

models. Even many nonmonotonic models, (4) typically ensures the identi"ca-tion of a₀; for example, identi"cation of a₀ follows easily from (4) when P(t)'0.5 if and only ift'0, and dP(t)/dtchanges sign only once fort'0.

Theorem 1. Under Assumptions 1}7, the estimator hK is consistent and asymp-totically normal,

Jn(hK!_h 0)

D

PN(0,R),

where

R"E

GC

LG Lh

LG

Lh@

DC

1

P(v)(1!P(v))

DH

~1

withv"v(x,h₀)andLG/Lh"[LG(v(x,h₀),h₀)]/Lh.

Proof. See the appendix. _h

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independence restriction. We will show that by also taking the symmetry restriction into account, our estimator attains the asymptotic e$ciency bound of Cosslett (1987) associated with both the independence and symmetry restrictions.

Lemma 2. For the binary choice model under consideration here,the asymptotic covariance ofhK achieves the ezciency bound as specixed in Cosslett(1987)under the condition that the error is independent of the regressors, and symmetrically distributed around the origin.

Following the arguments of Lemma 2 of Chen and Lee (1998) and Klein and Spady (1993), we can show that under the independence and symmetry restric-tions

L

Lh[E(dDv(xi,h)"v(x,h)]Dh/h0"f(v)

C

Lv

Lh!E

A

Lv

Lh

K

v

BD

and

L

Lh[E(dDv(xi,h)"!v(x,h)]Dh/h0"!f(v)

C

Lv

Lh#E

A

Lv

Lh

K

!v

BD

,

whereLv/Lh"[Lv(x,h₀)]/Lh. Therefore,

LG

Lh"f(v)

A

Lv

Lh!K(v)

B

,

where

K(v)" 1

g(v)#g(!v)

C

g(v)E

C

Lv

Lh

K

v

D

!g(!v)E

A

Lv

Lh

K

!v

BD

,

as speci"ed in Cosslett (1987). Therefore,

R_~1"E

G

f2(v) P(v)(1!P(v))

A

Lv

Lh!K(v)

BA

Lv

Lh!K(v)

B

@

H

1"

T

E

C

Lv

Lb

K

v

D

!K(v), E

C

Lv

Lb

K

v

D

!K(v)

U

!

T

E

C

Lv

Lb

K

v

D

!K(v),gH

U

SgH,gHT~1

]

T

gH, E

C

Lv

Lb

K

v

D

!K(v)

U

,

where gH"[2g(!v)]/[g(v)#g(!v)], and S),)T denotes an inner product such that

Sg

1,g2T"E

C

f2(v)

P(v)(1!P(v))g1g2

D

.

Consequently ((I~1₂ )_bb)~1!I

1, can be viewed as the corresponding norm of the

projection residual of E[Lv/Lb Dv]!K(v) onto gH, it is thus semi-de"nite, as expected. Whenv(x,h) is linear, andxhas a multivariate normal distribution, we have E(xDv)"a#bvand E(xD!v)"a!bv, thus

E

C

Lv

Lb

K

v

D

!K(v)"

g(!v)

g(v)#g(!v)

C

E

C

Homoscedastic design I

n Estimator Mean SD RMSE

100 a(

1" 0.000 0.149 0.149 a(

1" 1.026 0.238 0.239 a(

# !0.003 0.165 0.165

a(

,4 1.037 0.283 0.285

a(

1 !0.005 0.160 0.160

a(

2 1.030 0.248 0.250

300 a(

1" 0.005 0.091 0.091 a(

1" 1.000 0.131 0.131 a(

# 0.003 0.095 0.095

a(

,4 1.007 0.160 0.160

a(

1 0.003 0.095 0.095

a(

2 1.002 0.138 0.138

3. A small Monte Carlo study

In this section we perform some Monte Carlo simulations to investigate the "nite sample performance of our estimator. The data is generated according to the following model:

d"1Ma

0#x1#b0x2#u'0N

with (a₀,b₀)"(0, 1). Four di!erent designs are constructed by varying the distributions of the regressors and error term.

Notice that given the estimator forb₀, bK_,4, of Klein and Spady (1993),a₀can also be estimated bya(₁

c, de"ned by maximizingQK n(a,bK,4,q() in (1) with respect to

a. Here we consider the "nite sample performance of our estimator, (a(,bK), the probit estimator (a(

1",bK1"),a(1candbK,4de"ned above. We examine the e$ciency

loss relative to a correct parametric speci"cation and e$ciency gain in imposing symmetry and resulting joint estimation.

The results from 300 replications for each design are presented with sample sizes of 100 and 300. For each estimator under consideration, we report the mean value (Mean), the standard deviation (SD), and the root mean square error (RMSE). As in Klein and Spady (1993), we report results for the semiparametric estimators obtained without probability or likelihood trimming since estimates obtained are not sensitive to trimming. Also, as in Klein and Spady (1993), adaptive local smoothing with standard normal kernel is used for the semiparametric estimators, with local smoothing parameters de"ned without any trimming. The nonstochastic window widtha

nand the corresponding pilot

window componenta

npare set ton~1@7to satisfy the restriction speci"ed above. Tables 1 and 2 report the results for two homoscedastic designs in whichuis drawn from N(0, 1) independent of (x

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Table 3

Heteroscedastic design I

100 a(

1" !0.004 0.251 0.251 a(

100 a(

1" !0.004 0.148 0.148 a(

1 andx2are drawn from N(0, 1) independent of each other, whilex1 is

still drawn from N(0, 1) andx

2is drawn independently from a standardized

chi-squared distribution with 3 degrees of freedom with zero mean and unit variance in Homoscedastic Design II. The probit estimator, as expected, performs best in both designs, while our estimator is competitive and compared favorably with bK_,4, even though earlier discussions on e$ciency suggests that both the es-timator of Klein and Spady (1993) and our eses-timator attain the parametric e$ciency bound when (x

1,x2) is jointly normal. In both cases,a(c1is also quite

competitive compared with our estimator for the intercept.

Tables 3 and 4 report the results for two heteroscedastic designs in which

u"0.7uH(1#0.5v#0.25v2) for v"_a

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Table 4

Heteroscedastic design II

100 a(

1" !0.056 0.235 0.241 a(

1" 0.910 0.366 0.377 a(

# !0.009 0.223 0.223

a(

,4 0.987 0.336 0.336

a(

1 !0.014 0.216 0.217

a(

2 0.978 0.311 0.312

300 a(

1" 0.083 0.164 0.184 a(

1" 0.904 0.220 0.240 a(

# 0.000 0.136 0.136

a(

,4 1.014 0.193 0.194

a(

1 0.003 0.137 0.137

a(

2 1.010 0.190 0.190

N(0, 1) independent of (x

1,x2). In Heteroscedastic Design I, (x1,x2) is drawn

as in Homoscedastic Design I, and in Heteroscedastic Design II, (x 1,x2) is

drawn as in Homoscedastic Design II. In Heteroscedastic Design I, the probit estimator still has small bias despite misspeci"cation, but its bias increases signi"cantly in Heteroscedastic Design II. In both cases, the probit estimator has the largest RMSE, whereas all the semiparametric estimators perform well.

4. Conclusions

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Recently, Chamberlain (1986) derived the semiparametric e$ciency bound for the slope parameters for the binary choice sample selection model under the independence restriction. Various semiparametric estimators have been pro-posed for the slope parameters. In particular, the estimators by Ai (1997) and Chen and Lee (1998) attain the e$ciency bound. In view of the results obtained in the paper, there might be possible e$ciency gains in imposing an additional symmetry restriction. This is a topic for future research.

Acknowledgements

I would like to thank Bo HonoreH, Roger Klein, Lung-Fei Lee and Jim Powell and two anonymous referees for their helpful comments. I am also grateful to Roger Klein for providing Gauss codes.

Appendix. Proof of Theorem 1

Theorem 1 is proved by following Chen and Lee (1998), Klein and Spady (1993) and Lee (1994). Consistency follows from arguments similar to those in Klein and Spady (1993) by showing that the semiparametric likelihood function converges uniformly in hto a function uniquely maximized at h₀. To prove asymptotic normality, we"rst consider a Taylor expansion to obtain

Jn(hK!_h

0)"

C

!

L2QK(h`₁,q() Lh Lh@

D

~1

JnLQK(h0,q()

Lh , (A.1)

whereh`₁ lies betweenhK andh₀. Similar to Lemma 5 of Klein and Spady (1993), we can show that

!L2QK(h`1,q() Lh Lh@ "E

C

LG

Lh LG

Lh@ 1

P(v)(1!P(v))

D

#o1(1).

For the gradient term in (A.1), similar to Lemma 3 of Klein and Spady (1993), we have the following Taylor expansion for the likelihood trimming term

q(

i(hK1,eA)"q(i(h0,eA)#¸ni(h0,eA)(hK1!h0) #₁

2(hK1!h0)@Qni(h`2,eA)(hK1!h0),

where¸

ni(h,eA) andQni(h,eA) represent the"rst- and second-order partial deriva-tive functions andh`₂ lies betweenhK₁ andh₀. Thus, we obtain

JnLQK(h0,q()

Lh " 1

Jn

n

+

i/1

q( i(hK1,eA)

d

i!GK i

GK_i(1!GK_i) LGK_i

Lh

"S

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where

Following the arguments in Lee (1994, pp. 381}384), which involves Taylor expansion and repeated applications of Propositions 4 and 6 of Lee (1994),

we can show that S

n11"o1(1). For Sn12, by applying Taylor expansion

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show that

Lemma 5 in Klein and Spady (1993), we can show that

1

Then Theorem 1 follows by applying a central limit theorem (Ser#ing, 1980,

p. 32). K

References

Ai, C., 1997. A semiparametric maximum likelihood estimator. Econometrica 65, 933}963. Ahn, H., Ichimura, H., Powell, J.L., 1996. Simple estimators for monotone index models.

Depart-ment of Economics, University of California at Berkeley.

Cavanagh, C., Sherman, B., 1998. Rank estimators for monotonic index models. Journal of Econo-metrics 84, 351}382.

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Chen, S., 1998. Rank estimation of a location parameter in the binary choice model. Hong Kong University of Science and Technology.

Chen, S., 1999a. Semiparametric estimation of a location parameter in the binary choice model. Econometric Theory, 15, 77}98.

Chen, S., 1999b. Distribution-free estimation of the random coe$cient dummy endogenous variable model. Journal of Econometrics 91, 171}199.

Chen, S., Lee, L.F., 1998. E$cient semiparametric scoring estimation of sample selection models. Econometric Theory 14, 423}462.

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 the censored regression models. Econometrica 55, 559}587.

Han, A.K., 1987. Nonparametric analysis of a generalized regression model: the maximum rank correlation estimator. Journal of Econometrics 35, 303}316.

HaKrdle, W., Stoker, T., 1989. Investigating smooth multiple regression by the method of average derivative. Journal of the American Statistical Association 84, 986}995.

Horowitz, J.L., 1992. A smooth maximum score estimator for the binary response model. Econo-metrica 60, 505}531.

Horowitz, J.L., 1993. Semiparametric and nonparametric estimation of quantal response models. In: Maddala. G.S., Rao., C.R., Vinod, H.D. (Eds.), Handbook of Statistics, Vol. 11.

Horowitz, J.L., HaKrdle, W., 1996. Direct semiparametric estimation of single-index models with discrete covariates. Journal of the American Statistical Association 91, 1623}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.

Lewbel, A., 1997. Semiparametric estimation of location and other discrete choice moments. Econometric Theory 13, 32}51.

Manski, C.F., 1985. Semiparametric analysis of discrete response: asymptotic properties of the maximum score estimator. Journal of Econometrics 27, 313}333.

Newey, W.K., 1985. Maximum likelihood speci"cation testing and conditional moment tests. Econometrica 53, 1047}1070.

Powell, J.L., 1989. Semiparametric estimation of bivariate latent variable models. Social Research Institute, University of Wisconsin, Madison.

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.