Directory UMM :Data Elmu:jurnal:J-a:Journal of Econometrics:Vol98.Issue2.Oct2000:
Journal of Econometrics 98 (2000) 317}334
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 initial Jnconsistent 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
G
d"
1 if x@b !a !v'0,
0
0
0 otherwise,
* Tel.: 852-2358-7602; fax: 852-2358-2084.
E-mail address: [email protected] (S. Chen).
0304-4076/00/$ - see front matter ( 2000 Elsevier Science S.A. All rights reserved.
PII: S 0 3 0 4 - 4 0 7 6 ( 0 0 ) 0 0 0 2 1 - X
(1)
318
S. Chen / Journal of Econometrics 98 (2000) 317}334
where d is the indicator of the response (or participation) variable, x3Rq is
a vector of exogenous variables, a 3R and b 3Rq are unknown parameters,
0
0
and the error term v is 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 for v. 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 b "1, where b is the "rst component of b . Abusing notation
01
01
0
slightly, we still use (a , b@ )@ to denote the normalized parameters. Recently,
0 0
there have been several semi-parametric estimators proposed for (a , b@ )@ in the
0 0
literature. Ahn et al. (1996), Cavanagh and Sherman (1998), Cosslett (1983), Han
(1987), HaK rdle and Stoker (1989), Horowitz and HaK rdle (1996), Ichimura (1993),
Klein and Spady (1993), Powell et al. (1989) and Sherman (1993), among others,
considered estimating the slope parameter b under a single index or indepen0
dence restriction. The intercept term a is not identi"ed under the single index
0
or independence restriction because no centering assumption is made about the
distribution of v. 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
n is the sample size). Under mean restrictions Lewbel (1997, 1999) considered
Jn-consistent estimation of the intercept term a and/or the slope parameter
0
b by imposing a strong restriction on the support of the linear index x@b
0
0
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 of a and
0
b under symmetry with general heteroscedasticity.
0
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
a Jn-consistent estimator for the slope parameter. The new estimator is a rankbased 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
parameter. In particular, nonparametric smoothing could be avoided altogether
S. Chen / Journal of Econometrics 98 (2000) 317}334
319
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 population 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 a Jn consistent estimator for a , our results will complement
0
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 rule1 for the response variable d is (see, e.g., Greene, 1994)
as follows:
dK "1 if x@bK !a( '0,
where (a( , bK @)@ is some estimate for (a , b@ )@. As a result, estimation of a is needed
0
0 0
for this purpose.2 Such a prediction rule is also used in constructing some
goodness-of-"t tests for discrete-choice models (Amemiya, 1981; Maddala,
1 Note that xbK !a( '0 is equivalent to F(xbK !a( #a )'0.5 where F( ) ) the participation prob0
ability function to be de"ned later. An alternative predition rule without estimating a is to set dK "1
0
if FK (xbK )'0.5 where FK ( ) ) is a nonparametric estimator for F( ) ); however, with a root-n consistent
estimator a( , F(xbK !a( #a ) converges faster than FK (xbK ) for a given x. Furthermore, the predition
0
rule based on estimated linear indices is easier to implement.
2 Manski's (1985) estimator is devised by maximizing the number of correct predictions based on
this prediction rule.
320
S. Chen / Journal of Econometrics 98 (2000) 317}334
1983). Also, knowing the intercept is helpful for estimating choice probabilities
under the symmetry restriction; in particular, based on the notation and argument in the next section, we have
E(d D x"x6 )"E(d D x@b "x6 @b )"E(1!d D x@b "2a !x6 @b )
0
0
0
0
0
thus, with knowledge of a and b , nonparametric estimation of E(d D x"x6 ) can
0
0
be based on observations for which x@b are in the neighborhood of x6 @b and
0
0
that of 2a !x6 b , which could lead to possible e$ciency gains, especially when
0
0
the "rst neighborhood contains only few observations.3
Consistent estimation of the intercept term a is also useful for estimating the
0
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@b !a !v'0N,
(2)
0
0
where for simplicity, b is taken to be the normalized parameter vector (see,
0
Cosslett, 1983; Manski, 1985; Ichimura, 1993, for detailed discussions). Let
z"x@b . Let F(t, b)"E(d D x@b"t), for any possible value of b, and
0
F(t)"F(t, b ). When v is independent of x, then F(t) is the cumulative distribu0
tion function of a #v. Here we assume that the error term v satis"es a mono0
tonic index and symmetry condition as in Chen (1999b). Speci"cally, the
conditional density of v, f (v D x), depends on x only through D z!a D , the
0
3 In fact, Chen (1999b) considered e$cient estimation of a and b under symmetry by combining
0
0
this idea and the approach of Klein and Spady (1993).
4 Andrews and Schafgans provided a consistent estimator for the intercept term in the outcome
equation by relying on the &identi"cation at in"nity'; as a result, their estimator converges at a rate
slower than Jn.
S. Chen / Journal of Econometrics 98 (2000) 317}334
321
magnitude of the linear index x@b !a , and symmetric around 0, i.e.,
0
0
f (v D x)"f (v DD z!a D ) and f (!v D x)"f (v D x); in addition, we assume that F(t) is
0
strictly increasing in t3R. Therefore, we have
P
F(t)#F(2a !t)"
0
P
(a0 ~t)
(t~a0 )
f (s DD a !tD ds#
f (s DD a !t D ) ds"1
0
0
~=
~=
(3)
for any t.
Let Md , x@ N@, for i"1, 2,2, n, be a random sample, and z "x@ b . To motii 0
i
i i
vate our estimator, suppose that z #z '2a , then it is easy to deduce from (3)
i
j
0
that
E(d #d D z , z )"F(z )#F(z )'F(z )#F(2a !z )"1.
i
j i j
i
j
i
0
i
Similarly, we have
(4)
E(d #d D z , z )"F(z )#F(z )(1
(5)
i
j i j
i
j
if z #z (2a . In other words, it is more likely than not that d #d '1
i
j
0
i
j
whenever z #z '2a . Similar to Cavanagh and Sherman (1998), Chen (1998),
i
j
0
Han (1987), and Sherman (1993), we propose the following rank estimator for
a , a( , which maximizes
0
1
+ [(d #d !1)1Mx@ bK #x@ bK '2aN
j
i
i
j
n(n!1)
iEj
(6)
# (1!d !d )1Mx@ bK #x@ bK (2aN]
j
i
i
j
with respect to a over an appropriate compact interval A, where bK is a "rst-step
consistent estimator for b .
0
Note that direct implementation of the estimation procedure requires O(n2)
operations for each evaluation of the objective function. However, the estimator
can be calculated in a much more e$cient manner. For the original sample of
size n, we construct an arti"cial sample of size 2n by de"ning
dH"d
i
i
and dH "1!d
i`n
i
for i"1,2, n
and
and zH "2a!x@ bK for i"1,2, n.
zH"x@ bK
i
i`n
i
i
Then similar to Cavanagh and Sherman (1998), our estimator can be de"ned
equivalently by maximizing +2n dHR (zH), where R (a ) denotes the rank of
2n i
i/1 i 2n i
a for real numbers a , a ,2, a . Consequently, our procedure can be implei
1 2
2n
mented with only O(n log n) operations for each evaluation of the objective
function, resulting substantial savings in computation time.
322
S. Chen / Journal of Econometrics 98 (2000) 317}334
Now we consider large sample properties of the estimator. Let x"(1, x8 @)@ and
b "(1, bI @ )@. We make the following assumptions.
0
0
Assumption 1. The vectors (d , x@ )@ are independent and identically distributed
i i
across i.
Assumption 2. The conditional density of v, f (v D x), depends on x only through
D z!a D , and symmetric around 0, i.e., f (v D x)"f (v DD z!a D ) and f (!v D x)"
0
0
f (v D x).
Assumption 3. (i) The support of the distribution of x is not contained in any
proper linear subspace of Rq. (ii) For almost every x8 "(x ,2, x )@, the distribu2
q
tion of x conditional on x8 has everywhere positive density with respect to
1
Lebesgue measure.
Assumption 4. The true value of a is an interior point of a compact set A.
0
Assumption 5. The linear index z"x@b has positive density at a and the
0
0
conditional density of the error term satis"es f (0 D x)'f , for a positive constant
0
f .
0
De"ne
q(w, a, b)"E[h(w, w , a, b)],
i
where
h(w , w , a, b)"(d #d !1)1Mx@ b#x@ b'2aN
2
1
1 2
1
2
# (1!d !d )1Mx@ b#x@ b(2aN
2
1
1
2
for w "(d , x@ )@ and w "(d , x@ )@.
2
2 2
1
1 1
Assumption 6. For each w, all mixed third partial derivatives of q(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 ))(R.
i
Assumption 7. The preliminary estimator, bK "(1, bM @)@, for b "(1, bI @ )@ is Jn0
0
consistent, and has the asymptotic linear representation
1
bM "bI # +t #o (n~1@2)
i
1
0 n
for some t "t(d , x ) such that Et(d , x )"0 and EDDt(d , x ),D 2)R.
i
i i
i i
i i
Assumptions 1 and 2 describe the model and the data. Note that certain
unknown heteroscedasticity is allowed here. Assumption 3 is used for identifying
S. Chen / Journal of Econometrics 98 (2000) 317}334
323
the slope parameter by Manski (1985) under a conditional quantile restriction,
which is certainly su$cient under our symmetry condition. It implies that x has
at least one continuously distributed component, and that this component has
unbounded support. However, this assumption of unbounded support can be
relaxed following the arguments in Horowitz (1998). Assumption 4 is a standard
assumption in the literature. Assumption 5 essentially is an identi"cation condition for the intercept term; as in Chen (1999b, c), it requires a portion of the
population with participation probability below 0.5, and another portion with
participation probability above 0.5. In particular, some algebra will show that
Rank estimation of a location parameter in the
binary choice model
Songnian Chen*
Department of Economics, The Hong Kong University of Science and Technology, Clear Water Bay,
Kowloon, Hong Kong, People's Republic of China
Received 1 September 1998; received in revised form 1 January 2000; accepted 13 March 2000
Abstract
This paper proposes a rank-based estimator for a location parameter in the binary
choice model under a monotonic index and symmetry condition, given an initial Jnconsistent estimator for the slope parameter. The estimator converges at the usual
parametric rate. Compared with existing estimators, no nonparametric smoothing is
needed here. A small Monte Carlo study illustrates the usefulness of the estimator. We
also point out that the location and slope parameters can be jointly estimated. ( 2000
Elsevier Science S.A. All rights reserved.
JEL classixcation: C21; C25
Keywords: Binary choice; Location parameter; Semiparametric estimation
1. Introduction
In this paper we consider semi-parametric estimation of the binary choice
model de"ned by
G
d"
1 if x@b !a !v'0,
0
0
0 otherwise,
* Tel.: 852-2358-7602; fax: 852-2358-2084.
E-mail address: [email protected] (S. Chen).
0304-4076/00/$ - see front matter ( 2000 Elsevier Science S.A. All rights reserved.
PII: S 0 3 0 4 - 4 0 7 6 ( 0 0 ) 0 0 0 2 1 - X
(1)
318
S. Chen / Journal of Econometrics 98 (2000) 317}334
where d is the indicator of the response (or participation) variable, x3Rq is
a vector of exogenous variables, a 3R and b 3Rq are unknown parameters,
0
0
and the error term v is 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 for v. 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 b "1, where b is the "rst component of b . Abusing notation
01
01
0
slightly, we still use (a , b@ )@ to denote the normalized parameters. Recently,
0 0
there have been several semi-parametric estimators proposed for (a , b@ )@ in the
0 0
literature. Ahn et al. (1996), Cavanagh and Sherman (1998), Cosslett (1983), Han
(1987), HaK rdle and Stoker (1989), Horowitz and HaK rdle (1996), Ichimura (1993),
Klein and Spady (1993), Powell et al. (1989) and Sherman (1993), among others,
considered estimating the slope parameter b under a single index or indepen0
dence restriction. The intercept term a is not identi"ed under the single index
0
or independence restriction because no centering assumption is made about the
distribution of v. 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
n is the sample size). Under mean restrictions Lewbel (1997, 1999) considered
Jn-consistent estimation of the intercept term a and/or the slope parameter
0
b by imposing a strong restriction on the support of the linear index x@b
0
0
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 of a and
0
b under symmetry with general heteroscedasticity.
0
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
a Jn-consistent estimator for the slope parameter. The new estimator is a rankbased 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
parameter. In particular, nonparametric smoothing could be avoided altogether
S. Chen / Journal of Econometrics 98 (2000) 317}334
319
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 population 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 a Jn consistent estimator for a , our results will complement
0
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 rule1 for the response variable d is (see, e.g., Greene, 1994)
as follows:
dK "1 if x@bK !a( '0,
where (a( , bK @)@ is some estimate for (a , b@ )@. As a result, estimation of a is needed
0
0 0
for this purpose.2 Such a prediction rule is also used in constructing some
goodness-of-"t tests for discrete-choice models (Amemiya, 1981; Maddala,
1 Note that xbK !a( '0 is equivalent to F(xbK !a( #a )'0.5 where F( ) ) the participation prob0
ability function to be de"ned later. An alternative predition rule without estimating a is to set dK "1
0
if FK (xbK )'0.5 where FK ( ) ) is a nonparametric estimator for F( ) ); however, with a root-n consistent
estimator a( , F(xbK !a( #a ) converges faster than FK (xbK ) for a given x. Furthermore, the predition
0
rule based on estimated linear indices is easier to implement.
2 Manski's (1985) estimator is devised by maximizing the number of correct predictions based on
this prediction rule.
320
S. Chen / Journal of Econometrics 98 (2000) 317}334
1983). Also, knowing the intercept is helpful for estimating choice probabilities
under the symmetry restriction; in particular, based on the notation and argument in the next section, we have
E(d D x"x6 )"E(d D x@b "x6 @b )"E(1!d D x@b "2a !x6 @b )
0
0
0
0
0
thus, with knowledge of a and b , nonparametric estimation of E(d D x"x6 ) can
0
0
be based on observations for which x@b are in the neighborhood of x6 @b and
0
0
that of 2a !x6 b , which could lead to possible e$ciency gains, especially when
0
0
the "rst neighborhood contains only few observations.3
Consistent estimation of the intercept term a is also useful for estimating the
0
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@b !a !v'0N,
(2)
0
0
where for simplicity, b is taken to be the normalized parameter vector (see,
0
Cosslett, 1983; Manski, 1985; Ichimura, 1993, for detailed discussions). Let
z"x@b . Let F(t, b)"E(d D x@b"t), for any possible value of b, and
0
F(t)"F(t, b ). When v is independent of x, then F(t) is the cumulative distribu0
tion function of a #v. Here we assume that the error term v satis"es a mono0
tonic index and symmetry condition as in Chen (1999b). Speci"cally, the
conditional density of v, f (v D x), depends on x only through D z!a D , the
0
3 In fact, Chen (1999b) considered e$cient estimation of a and b under symmetry by combining
0
0
this idea and the approach of Klein and Spady (1993).
4 Andrews and Schafgans provided a consistent estimator for the intercept term in the outcome
equation by relying on the &identi"cation at in"nity'; as a result, their estimator converges at a rate
slower than Jn.
S. Chen / Journal of Econometrics 98 (2000) 317}334
321
magnitude of the linear index x@b !a , and symmetric around 0, i.e.,
0
0
f (v D x)"f (v DD z!a D ) and f (!v D x)"f (v D x); in addition, we assume that F(t) is
0
strictly increasing in t3R. Therefore, we have
P
F(t)#F(2a !t)"
0
P
(a0 ~t)
(t~a0 )
f (s DD a !tD ds#
f (s DD a !t D ) ds"1
0
0
~=
~=
(3)
for any t.
Let Md , x@ N@, for i"1, 2,2, n, be a random sample, and z "x@ b . To motii 0
i
i i
vate our estimator, suppose that z #z '2a , then it is easy to deduce from (3)
i
j
0
that
E(d #d D z , z )"F(z )#F(z )'F(z )#F(2a !z )"1.
i
j i j
i
j
i
0
i
Similarly, we have
(4)
E(d #d D z , z )"F(z )#F(z )(1
(5)
i
j i j
i
j
if z #z (2a . In other words, it is more likely than not that d #d '1
i
j
0
i
j
whenever z #z '2a . Similar to Cavanagh and Sherman (1998), Chen (1998),
i
j
0
Han (1987), and Sherman (1993), we propose the following rank estimator for
a , a( , which maximizes
0
1
+ [(d #d !1)1Mx@ bK #x@ bK '2aN
j
i
i
j
n(n!1)
iEj
(6)
# (1!d !d )1Mx@ bK #x@ bK (2aN]
j
i
i
j
with respect to a over an appropriate compact interval A, where bK is a "rst-step
consistent estimator for b .
0
Note that direct implementation of the estimation procedure requires O(n2)
operations for each evaluation of the objective function. However, the estimator
can be calculated in a much more e$cient manner. For the original sample of
size n, we construct an arti"cial sample of size 2n by de"ning
dH"d
i
i
and dH "1!d
i`n
i
for i"1,2, n
and
and zH "2a!x@ bK for i"1,2, n.
zH"x@ bK
i
i`n
i
i
Then similar to Cavanagh and Sherman (1998), our estimator can be de"ned
equivalently by maximizing +2n dHR (zH), where R (a ) denotes the rank of
2n i
i/1 i 2n i
a for real numbers a , a ,2, a . Consequently, our procedure can be implei
1 2
2n
mented with only O(n log n) operations for each evaluation of the objective
function, resulting substantial savings in computation time.
322
S. Chen / Journal of Econometrics 98 (2000) 317}334
Now we consider large sample properties of the estimator. Let x"(1, x8 @)@ and
b "(1, bI @ )@. We make the following assumptions.
0
0
Assumption 1. The vectors (d , x@ )@ are independent and identically distributed
i i
across i.
Assumption 2. The conditional density of v, f (v D x), depends on x only through
D z!a D , and symmetric around 0, i.e., f (v D x)"f (v DD z!a D ) and f (!v D x)"
0
0
f (v D x).
Assumption 3. (i) The support of the distribution of x is not contained in any
proper linear subspace of Rq. (ii) For almost every x8 "(x ,2, x )@, the distribu2
q
tion of x conditional on x8 has everywhere positive density with respect to
1
Lebesgue measure.
Assumption 4. The true value of a is an interior point of a compact set A.
0
Assumption 5. The linear index z"x@b has positive density at a and the
0
0
conditional density of the error term satis"es f (0 D x)'f , for a positive constant
0
f .
0
De"ne
q(w, a, b)"E[h(w, w , a, b)],
i
where
h(w , w , a, b)"(d #d !1)1Mx@ b#x@ b'2aN
2
1
1 2
1
2
# (1!d !d )1Mx@ b#x@ b(2aN
2
1
1
2
for w "(d , x@ )@ and w "(d , x@ )@.
2
2 2
1
1 1
Assumption 6. For each w, all mixed third partial derivatives of q(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 ))(R.
i
Assumption 7. The preliminary estimator, bK "(1, bM @)@, for b "(1, bI @ )@ is Jn0
0
consistent, and has the asymptotic linear representation
1
bM "bI # +t #o (n~1@2)
i
1
0 n
for some t "t(d , x ) such that Et(d , x )"0 and EDDt(d , x ),D 2)R.
i
i i
i i
i i
Assumptions 1 and 2 describe the model and the data. Note that certain
unknown heteroscedasticity is allowed here. Assumption 3 is used for identifying
S. Chen / Journal of Econometrics 98 (2000) 317}334
323
the slope parameter by Manski (1985) under a conditional quantile restriction,
which is certainly su$cient under our symmetry condition. It implies that x has
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 condition for the intercept term; as in Chen (1999b, c), it requires a portion 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