Directory UMM :Data Elmu:jurnal:J-a:Journal of Econometrics:Vol95.Issue1.2000:
Journal of Econometrics 95 (2000) 25}56
Estimation of a censored regression panel data
model using conditional moment restrictions
e$ciently
Erwin Charlier!,",*, Bertrand Melenberg!, Arthur van Soest!
!Tilburg University, Department of Econometrics and CentER, P.O. Box 90153, 5000 LE,
Tilburg, The Netherlands
"ABP Investments, Fixed Income Europe, P.O. Box 2889, 6401 DJ, Heerlen, The Netherlands
Received 1 November 1995; received in revised form 1 January 1999; accepted 1 April 1999
Abstract
HonoreH has introduced a semiparametric estimator for the censored regression panel
data model with "xed individual e!ects. Newey has shown how to obtain e$cient
estimators under a given conditional moment restriction. We apply Newey's approach to
obtain a two-step GMM estimator which is more e$cient than the HonoreH estimator.
We compare this estimator to the HonoreH estimator and to parametric estimators
including Chamberlain's quasi-"xed e!ects estimator in a Monte Carlo experiment. We
also extend the HonoreH estimator and the two-step GMM estimator to the case of
a balanced or unbalanced panel of more than two waves. We apply the estimators to an
empirical example concerning earnings of married females, using data from the Dutch
Socio-Economic Panel. ( 2000 Elsevier Science S.A. All rights reserved.
JEL classixcation: C14; C23; C24
Keywords: Panel data; Censored regression; Semiparametric e$ciency; Unbalanced
panel
* Corresponding author. Tel.: #31-13-66-91-11; fax: #31-13-66-30-72.
E-mail address: [email protected] (E. Charlier)
0304-4076/00/$ - see front matter ( 2000 Elsevier Science S.A. All rights reserved.
PII: S 0 3 0 4 - 4 0 7 6 ( 9 9 ) 0 0 0 2 8 - 7
26
E. Charlier et al. / Journal of Econometrics 95 (2000) 25}56
1. Introduction
This paper considers estimation of the censored regression panel data model
with individual e!ects. This model has important applications in microeconometrics. The seminal example is the labor supply model of Heckman and
MaCurdy (1980), where nonparticipation leads to censoring at zero, and where
the individual e!ects have a clear economic interpretation in a life cycle context.
Other examples are Udry (1995), who analyzes grain sales and purchases of rural
households in Nigeria, and Alderman et al. (1995) and Udry (1996), who analyze
various types of labour inputs and manure input in agricultural production.
Two types of estimators for this model can be distinguished. Estimators which
require a parametric speci"cation of the model are discussed by Chamberlain
(1984). These models have the advantage that not only the parameters can be
estimated, but also quantities such as marginal e!ects of covariates on the
observed censored variable, which may be more interesting than the parameters
themselves from a policy point of view. On the other hand, they have the
drawback that estimates of the parameters of such e!ects will, in general, be
asymptotically biased if the parametric model is misspeci"ed. To overcome this
problem, HonoreH (1992) has introduced a semiparametric version of the model,
which avoids assumptions on the distributions of individual e!ects or error
terms. He derives consistent estimators for the parameters of this model, but
does not address estimation of the marginal e!ects.
In this paper, we introduce a new semiparametric estimator, which aims at
improving the e$ciency of one of the HonoreH (1992) estimators. HonoreH 's
estimator is based upon (unconditional) moment restrictions derived from
a conditional moment restriction. Following Newey (1993), we nonparametrically estimate the optimal moment restrictions for HonoreH 's conditional
moment restriction, and construct a two-step GMM estimator which will be
asymptotically more e$cient than the HonoreH estimator. Like HonoreH 's estimator, our estimator is easy to compute. It is e$cient in the class of estimators
based upon the given conditional moment restriction, but does not attain the
semiparametric e$ciency bound for HonoreH 's semiparametric model, since this
model leads to many more conditional moment restrictions, which our estimator does not exploit.
First, we consider the case of two panel waves. In a Monte Carlo experiment,
we compare the two-step GMM estimator with HonoreH 's estimator and with
some parametric estimators, including the Chamberlain (1984) estimator allowing for quasi-"xed e!ects. For the latter, we do not only look at parameter
estimates, but also analyze the estimates of the marginal e!ects of changes in the
covariates on the observed censored dependent variable. We then generalize the
estimator to the case of more than two waves, for balanced as well as unbalanced panels. Finally, we compare the semiparametric and parametric estimates
in an empirical example. We explain weekly earnings of Dutch married females,
E. Charlier et al. / Journal of Econometrics 95 (2000) 25}56
27
using data drawn from the Dutch Socio-Economic Panel (SEP). Stoker (1992)
uses earnings of married females as the prototype example of a censored
regression model in a cross-section framework. Since unsystematic earnings
variations mainly re#ect changes in labour supply, it is natural to add "xed
e!ects (Heckman and MaCurdy, 1980).
The remainder of the paper is structured as follows. In Section 2, we present
the model and discuss merits and drawbacks of existing parametric and
semiparametric estimators. Section 3 introduces our two step GMM estimator
and its properties for the case of two time periods. In Section 4, we compare
semiparametric and parametric estimators in a Monte Carlo experiment. Section 5 considers the empirical application for two time periods. In Section 6, the
GMM-estimator is extended to panel data with more than two waves, either
balanced or unbalanced. In Section 7, it is applied to the same empirical
example, using "ve waves of SEP data, from 1984 to 1988. This section also
presents the economic interpretation of the results. Section 8 concludes.
2. Parametric and semiparametric models and estimators
The censored regression model for panel data with individual e!ects is given
by
yH"a #b@ x #u , i"1,2, N, t"1,2, ¹,
0 it
it
it
i
y "maxM0, yHN.
it
it
Here i denotes the individual and t is the time period, y* is a latent variable, y is
it
it
the observed dependent variable, x is a vector of covariates, a is the individual
it
i
e!ect, u is an error term and b is a vector of unknown parameters to be
it
0
estimated. We observe (y , x ). We are interested in asymptotic results for
it it
NPR, but for "xed ¹. We assume independence across individuals, but not
necessarily over time. We discuss various models with di!erent assumptions and
corresponding estimators.
In models with random e!ects, a is assumed to be independent of
i
x "(x@ ,2, x@ )@. Several parametric random e!ects models and corresponding
iT
i1
i
estimators for b have been proposed in the literature. For example,
0
a &N(0, p2) and u "(u ,2, u )@&N(0, R), with R"p2I and I the ¹]¹
u T
T
a
i
i1
iT
i
identity matrix, yields the speci"cation of equicorrelation of Heckman and
Willis (1976). Here Maximum Likelihood (ML) can be applied. This requires
numerical integration in one dimension. If milder restrictions on R are imposed,
estimators can be based on ¹!1 dimensional numerical integration or simulation (such as simulated ML or simulated moments; see Gourieroux and
Monfort, 1993).
28
E. Charlier et al. / Journal of Econometrics 95 (2000) 25}56
If the assumptions on the distributions of error terms and individual e!ects
are satis"ed, the estimators are consistent and asymptotically normal. If the
assumptions of normality of (u , a ) or independence between (u , a ) and x are
i i
i i
i
not satis"ed, the estimators will generally be inconsistent (see, for example,
Arabmazar and Schmidt, 1981,1982).
Parametric models with "xed e!ects can be divided into two categories. In the
"rst, no restrictions on the distribution of a conditional on x are imposed. The
i
i
a are then usually considered as nuisance parameters. In the second category,
i
some restrictions on the distribution of a are imposed, which do not exclude
i
dependence between a and x .1
i
i
In models in the "rst category, it is usually assumed that u ,
it
i"1,2, N, t"1,2, ¹, are i.i.d. and independent of x . Since the a are parai
i
meters, models in this category su!er from the incidental parameter problem, see
Neyman and Scott (1948). ML estimates will generally be inconsistent. For
a speci"c distribution of the u , b could be estimated up to scale using
it 0
Chamberlain's conditional logit estimator, but this approach does not work in
general.
An example of a model in the second category is given by Chamberlain (1984).
It assumes that a "a@ x #w , with a an unknown vector of (nuisance) para0 i
i
0
i
meters, w &N(0, p2 ), u &N(0, R) without restrictions on R, and w , u and
w i
i i
i
x independent. Chamberlain proposes a two stage procedure to estimate b .
i
0
First the reduced-form model for each cross-section is estimated, ignoring
restrictions on the parameters across time. The second step is minimum distance, to take account of these restrictions. This model allows for a speci"c form
of correlation between a and x , but it retains the assumption of normality of
i
i
w and u . For a "0, it simpli"es to a random e!ects model.
i
i
0
The main goal of semiparametric estimation is to avoid the distributional and
independence assumptions discussed above, and to construct estimators of
b which are consistent under more general assumptions. HonoreH (1992) derives
0
various semiparametric estimators for the case ¹"2. He provides two representations of the identi"cation assumption. We use the representation stated in
HonoreH (1992, Assumption E.3, footnote 6). The starting point is the following
basic assumption (with the subscript i suppressed from now on):
(A1)
(Conditional exchangeability assumption).
The distribution of (u , u ), conditional on (a, x , x ), is absolutely con1 2
1 2
tinuous and u and u are conditionally interchangeable (i.e., have
1
2
conditional density f, with f (u , u Da, x , x )"f (u , u Da, x , x ) for all
1 2
1 2
2 1
1 2
(u , u ) and (a,x , x )).
1 2
1 2
1 Some studies only refer to the "rst category as "xed e!ects models, and refer to the second
category as random e!ects models (see Manski, 1987; Chamberlain, 1984).
E. Charlier et al. / Journal of Econometrics 95 (2000) 25}56
29
Assumption (A1) allows for nonnormality and dependence between the
errors and (a, x , x ), and imposes no restrictions on the distribution of a
1 2
conditional on (x , x ). In this sense it is more general than the assumptions
1 2
needed by Chamberlain (1984). On the other hand, the Chamberlain (1984)
model is not fully nested in that of HonoreH (1992), since Chamberlain (1984) does
not impose that the conditional distributions of u and u have the same
1
2
variance.
HonoreH derives two conditional moment restrictions (CMRs) from (A1). He
then constructs unconditional moment restrictions (UMRs) from these CMRs.
The value of the (generic) parameter vector b which satis"es the empirical
counterpart of these UMRs provides a consistent estimator of b . The UMRs
0
are chosen in such a way that these empirical counterparts of the UMRs are the
"rst-order conditions of minimizing a strictly convex objective function with
respect to b, implying that the estimates are easy to compute with a local search
algorithm.
Each CMR yields its own estimator for b ; HonoreH does not combine the two
0
CMRs. The two estimators share the property that, in the UMRs or the
objective function, (b,x , x ) appears only as b@(x !x ). This implies that
1 2
1
2
estimation hinges on variation in *x"x !x . The coe$cients of time-invari1
2
ant regressors are not identi"ed.
We will use HonoreH 's second estimator, for which the corresponding objective
function is everywhere continuously di!erentiable and twice di!erentiable in all
but a "nite number of points (the UMRs are given in (2.5) in HonoreH (1992)).
This makes it straightforward to derive the limit distribution of this estimator
and to estimate its covariance matrix. The corresponding CMR will be referred
to as the &smooth' CMR.
There are various strategies to construct more e$cient estimators than this
HonoreH estimator. The "rst would be to construct a semiparametrically e$cient
estimator based upon the e$cient scores corresponding to (A1). Estimation of
the e$cient scores appears to be hard in general, however. HonoreH (1993) needs
a speci"c distributional assumption concerning (u #a, u #a), conditional on
1
2
(x , x ). This approach can thus not be applied without making speci"c addi1 2
tional assumptions, and we will not use it.
A second strategy is suggested by Newey (1991). Following Chamberlain
(1987), this starts with noting that conditional exchangeability leads to in"nitely
many CMRs and in"nitely many UMRs. The idea is then to let the number of
CMRs used in estimation grow to in"nity at an appropriate rate as N tends to
in"nity. Newey shows that this approach leads to an estimator which attains the
semiparametric e$ciency bound for (A1). In "nite samples, however, this approach requires many choices: which of the in"nitely many CMRs should be
used, and which functions of the covariates should be used to form UMRs. We
compare some Monte Carlo results for one and two conditional moment
restrictions in Section 4. The Monte Carlo evidence suggests that increasing the
30
E. Charlier et al. / Journal of Econometrics 95 (2000) 25}56
number of CMRs used in estimation does not automatically lead to a large
increase in e$ciency, unless the data set is very large.
We will focus on the easier approach based on Newey (1993). Starting point is
the smooth CMR of HonoreH (1992). This CMR is used to construct optimal
UMRs, on which a GMM estimator is based. Thus, we do not aim at attaining
the semiparametric e$ciency bound for (A1). Our estimator will attain the
e$ciency bound for the class of models satisfying this single smooth CMR. This
class may be larger than the class of models satisfying the conditional exchangeability Assumption (A1), since (A1) implies more CMRs.
3. Identi5cation, consistency, e7ciency, and GMM estimation
Let ¹"2, y"(y , y ), x"(x , x ), and *x"x !x . In the remainder we
1 2
1 2
1
2
assume that HonoreH 's conditions (Assumption (A1) and regularity conditions
given in HonoreH (1992)) are satis"ed. The assumptions lead to in"nitely many
CMRs which can be presented compactly as in HonoreH and Powell (1994). To
do this, de"ne
e (b)"maxMa#u ,!b@x ,!b@x N"maxMy !b@ *x, 0N!b@x ,
12
1
2
1
1
2
e (b)"maxMa#u ,!b@x ,!b@x N"maxMy #b@ *x, 0N!b@x .
21
2
1
2
2
1
Then, under the exchangeability Assumption (A1),
e (b )!e (b )"maxMy !b@ *x, 0N!maxMy #b@ *x, 0N#b@ *x
0
0
0
2
12 0
21 0
1
"o(y, b@ *x)
0
(2)
is distributed symmetrically around zero conditional on x. This implies
EMm(e (b )!e (b ))DxN"0
12 0
21 0
(3)
for any odd function m. In this section we restrict attention to the smooth CMR
based on m(a)"a, used by HonoreH (1992):
EMo(y, b@ *x)DxN"0.
0
(4)
In Section 4 we will present some results for other choices for m(.). As shown
by HonoreH , CMR (4) identi"es b if and only if EM1(PMy '0,
0
1
y '0 D xN'0) *x *x@N has full rank. This excludes time constant regressors,
2
whose e!ects will be picked up by the "xed e!ects.
CMR (4) implies that, for any function A(x),
EMA(x)o(y, b@ *x)N"0.
0
(5)
E. Charlier et al. / Journal of Econometrics 95 (2000) 25}56
31
For a given choice for A(x), UMRs (5) can be used to apply GMM. A condition
for consistency of the GMM estimator is that b is the only value of b which
0
satis"es (5). This is di$cult to prove in general. HonoreH (1992) solves the
problem for this case: he chooses A(x)"*x, and constructs a strictly convex
objective function, whose "rst-order derivative is the sample analogue of the
UMRs. This guarantees identi"cation of the censored regression model with
UMRs (5), and, since (5) is implied by (4), also of the model de"ned by CMR (4).
It also guarantees consistency of the estimator obtained by minimizing the
strictly convex function. We denote the HonoreH (1992) estimator for b based on
0
A(x)"*x by bK . An estimator based on (5) for some arbitrary choice of A(x) is
H
denoted by bK . The limit distribution of bK is given by
JN(bK !b )P$N(0, G~1
Estimation of a censored regression panel data
model using conditional moment restrictions
e$ciently
Erwin Charlier!,",*, Bertrand Melenberg!, Arthur van Soest!
!Tilburg University, Department of Econometrics and CentER, P.O. Box 90153, 5000 LE,
Tilburg, The Netherlands
"ABP Investments, Fixed Income Europe, P.O. Box 2889, 6401 DJ, Heerlen, The Netherlands
Received 1 November 1995; received in revised form 1 January 1999; accepted 1 April 1999
Abstract
HonoreH has introduced a semiparametric estimator for the censored regression panel
data model with "xed individual e!ects. Newey has shown how to obtain e$cient
estimators under a given conditional moment restriction. We apply Newey's approach to
obtain a two-step GMM estimator which is more e$cient than the HonoreH estimator.
We compare this estimator to the HonoreH estimator and to parametric estimators
including Chamberlain's quasi-"xed e!ects estimator in a Monte Carlo experiment. We
also extend the HonoreH estimator and the two-step GMM estimator to the case of
a balanced or unbalanced panel of more than two waves. We apply the estimators to an
empirical example concerning earnings of married females, using data from the Dutch
Socio-Economic Panel. ( 2000 Elsevier Science S.A. All rights reserved.
JEL classixcation: C14; C23; C24
Keywords: Panel data; Censored regression; Semiparametric e$ciency; Unbalanced
panel
* Corresponding author. Tel.: #31-13-66-91-11; fax: #31-13-66-30-72.
E-mail address: [email protected] (E. Charlier)
0304-4076/00/$ - see front matter ( 2000 Elsevier Science S.A. All rights reserved.
PII: S 0 3 0 4 - 4 0 7 6 ( 9 9 ) 0 0 0 2 8 - 7
26
E. Charlier et al. / Journal of Econometrics 95 (2000) 25}56
1. Introduction
This paper considers estimation of the censored regression panel data model
with individual e!ects. This model has important applications in microeconometrics. The seminal example is the labor supply model of Heckman and
MaCurdy (1980), where nonparticipation leads to censoring at zero, and where
the individual e!ects have a clear economic interpretation in a life cycle context.
Other examples are Udry (1995), who analyzes grain sales and purchases of rural
households in Nigeria, and Alderman et al. (1995) and Udry (1996), who analyze
various types of labour inputs and manure input in agricultural production.
Two types of estimators for this model can be distinguished. Estimators which
require a parametric speci"cation of the model are discussed by Chamberlain
(1984). These models have the advantage that not only the parameters can be
estimated, but also quantities such as marginal e!ects of covariates on the
observed censored variable, which may be more interesting than the parameters
themselves from a policy point of view. On the other hand, they have the
drawback that estimates of the parameters of such e!ects will, in general, be
asymptotically biased if the parametric model is misspeci"ed. To overcome this
problem, HonoreH (1992) has introduced a semiparametric version of the model,
which avoids assumptions on the distributions of individual e!ects or error
terms. He derives consistent estimators for the parameters of this model, but
does not address estimation of the marginal e!ects.
In this paper, we introduce a new semiparametric estimator, which aims at
improving the e$ciency of one of the HonoreH (1992) estimators. HonoreH 's
estimator is based upon (unconditional) moment restrictions derived from
a conditional moment restriction. Following Newey (1993), we nonparametrically estimate the optimal moment restrictions for HonoreH 's conditional
moment restriction, and construct a two-step GMM estimator which will be
asymptotically more e$cient than the HonoreH estimator. Like HonoreH 's estimator, our estimator is easy to compute. It is e$cient in the class of estimators
based upon the given conditional moment restriction, but does not attain the
semiparametric e$ciency bound for HonoreH 's semiparametric model, since this
model leads to many more conditional moment restrictions, which our estimator does not exploit.
First, we consider the case of two panel waves. In a Monte Carlo experiment,
we compare the two-step GMM estimator with HonoreH 's estimator and with
some parametric estimators, including the Chamberlain (1984) estimator allowing for quasi-"xed e!ects. For the latter, we do not only look at parameter
estimates, but also analyze the estimates of the marginal e!ects of changes in the
covariates on the observed censored dependent variable. We then generalize the
estimator to the case of more than two waves, for balanced as well as unbalanced panels. Finally, we compare the semiparametric and parametric estimates
in an empirical example. We explain weekly earnings of Dutch married females,
E. Charlier et al. / Journal of Econometrics 95 (2000) 25}56
27
using data drawn from the Dutch Socio-Economic Panel (SEP). Stoker (1992)
uses earnings of married females as the prototype example of a censored
regression model in a cross-section framework. Since unsystematic earnings
variations mainly re#ect changes in labour supply, it is natural to add "xed
e!ects (Heckman and MaCurdy, 1980).
The remainder of the paper is structured as follows. In Section 2, we present
the model and discuss merits and drawbacks of existing parametric and
semiparametric estimators. Section 3 introduces our two step GMM estimator
and its properties for the case of two time periods. In Section 4, we compare
semiparametric and parametric estimators in a Monte Carlo experiment. Section 5 considers the empirical application for two time periods. In Section 6, the
GMM-estimator is extended to panel data with more than two waves, either
balanced or unbalanced. In Section 7, it is applied to the same empirical
example, using "ve waves of SEP data, from 1984 to 1988. This section also
presents the economic interpretation of the results. Section 8 concludes.
2. Parametric and semiparametric models and estimators
The censored regression model for panel data with individual e!ects is given
by
yH"a #b@ x #u , i"1,2, N, t"1,2, ¹,
0 it
it
it
i
y "maxM0, yHN.
it
it
Here i denotes the individual and t is the time period, y* is a latent variable, y is
it
it
the observed dependent variable, x is a vector of covariates, a is the individual
it
i
e!ect, u is an error term and b is a vector of unknown parameters to be
it
0
estimated. We observe (y , x ). We are interested in asymptotic results for
it it
NPR, but for "xed ¹. We assume independence across individuals, but not
necessarily over time. We discuss various models with di!erent assumptions and
corresponding estimators.
In models with random e!ects, a is assumed to be independent of
i
x "(x@ ,2, x@ )@. Several parametric random e!ects models and corresponding
iT
i1
i
estimators for b have been proposed in the literature. For example,
0
a &N(0, p2) and u "(u ,2, u )@&N(0, R), with R"p2I and I the ¹]¹
u T
T
a
i
i1
iT
i
identity matrix, yields the speci"cation of equicorrelation of Heckman and
Willis (1976). Here Maximum Likelihood (ML) can be applied. This requires
numerical integration in one dimension. If milder restrictions on R are imposed,
estimators can be based on ¹!1 dimensional numerical integration or simulation (such as simulated ML or simulated moments; see Gourieroux and
Monfort, 1993).
28
E. Charlier et al. / Journal of Econometrics 95 (2000) 25}56
If the assumptions on the distributions of error terms and individual e!ects
are satis"ed, the estimators are consistent and asymptotically normal. If the
assumptions of normality of (u , a ) or independence between (u , a ) and x are
i i
i i
i
not satis"ed, the estimators will generally be inconsistent (see, for example,
Arabmazar and Schmidt, 1981,1982).
Parametric models with "xed e!ects can be divided into two categories. In the
"rst, no restrictions on the distribution of a conditional on x are imposed. The
i
i
a are then usually considered as nuisance parameters. In the second category,
i
some restrictions on the distribution of a are imposed, which do not exclude
i
dependence between a and x .1
i
i
In models in the "rst category, it is usually assumed that u ,
it
i"1,2, N, t"1,2, ¹, are i.i.d. and independent of x . Since the a are parai
i
meters, models in this category su!er from the incidental parameter problem, see
Neyman and Scott (1948). ML estimates will generally be inconsistent. For
a speci"c distribution of the u , b could be estimated up to scale using
it 0
Chamberlain's conditional logit estimator, but this approach does not work in
general.
An example of a model in the second category is given by Chamberlain (1984).
It assumes that a "a@ x #w , with a an unknown vector of (nuisance) para0 i
i
0
i
meters, w &N(0, p2 ), u &N(0, R) without restrictions on R, and w , u and
w i
i i
i
x independent. Chamberlain proposes a two stage procedure to estimate b .
i
0
First the reduced-form model for each cross-section is estimated, ignoring
restrictions on the parameters across time. The second step is minimum distance, to take account of these restrictions. This model allows for a speci"c form
of correlation between a and x , but it retains the assumption of normality of
i
i
w and u . For a "0, it simpli"es to a random e!ects model.
i
i
0
The main goal of semiparametric estimation is to avoid the distributional and
independence assumptions discussed above, and to construct estimators of
b which are consistent under more general assumptions. HonoreH (1992) derives
0
various semiparametric estimators for the case ¹"2. He provides two representations of the identi"cation assumption. We use the representation stated in
HonoreH (1992, Assumption E.3, footnote 6). The starting point is the following
basic assumption (with the subscript i suppressed from now on):
(A1)
(Conditional exchangeability assumption).
The distribution of (u , u ), conditional on (a, x , x ), is absolutely con1 2
1 2
tinuous and u and u are conditionally interchangeable (i.e., have
1
2
conditional density f, with f (u , u Da, x , x )"f (u , u Da, x , x ) for all
1 2
1 2
2 1
1 2
(u , u ) and (a,x , x )).
1 2
1 2
1 Some studies only refer to the "rst category as "xed e!ects models, and refer to the second
category as random e!ects models (see Manski, 1987; Chamberlain, 1984).
E. Charlier et al. / Journal of Econometrics 95 (2000) 25}56
29
Assumption (A1) allows for nonnormality and dependence between the
errors and (a, x , x ), and imposes no restrictions on the distribution of a
1 2
conditional on (x , x ). In this sense it is more general than the assumptions
1 2
needed by Chamberlain (1984). On the other hand, the Chamberlain (1984)
model is not fully nested in that of HonoreH (1992), since Chamberlain (1984) does
not impose that the conditional distributions of u and u have the same
1
2
variance.
HonoreH derives two conditional moment restrictions (CMRs) from (A1). He
then constructs unconditional moment restrictions (UMRs) from these CMRs.
The value of the (generic) parameter vector b which satis"es the empirical
counterpart of these UMRs provides a consistent estimator of b . The UMRs
0
are chosen in such a way that these empirical counterparts of the UMRs are the
"rst-order conditions of minimizing a strictly convex objective function with
respect to b, implying that the estimates are easy to compute with a local search
algorithm.
Each CMR yields its own estimator for b ; HonoreH does not combine the two
0
CMRs. The two estimators share the property that, in the UMRs or the
objective function, (b,x , x ) appears only as b@(x !x ). This implies that
1 2
1
2
estimation hinges on variation in *x"x !x . The coe$cients of time-invari1
2
ant regressors are not identi"ed.
We will use HonoreH 's second estimator, for which the corresponding objective
function is everywhere continuously di!erentiable and twice di!erentiable in all
but a "nite number of points (the UMRs are given in (2.5) in HonoreH (1992)).
This makes it straightforward to derive the limit distribution of this estimator
and to estimate its covariance matrix. The corresponding CMR will be referred
to as the &smooth' CMR.
There are various strategies to construct more e$cient estimators than this
HonoreH estimator. The "rst would be to construct a semiparametrically e$cient
estimator based upon the e$cient scores corresponding to (A1). Estimation of
the e$cient scores appears to be hard in general, however. HonoreH (1993) needs
a speci"c distributional assumption concerning (u #a, u #a), conditional on
1
2
(x , x ). This approach can thus not be applied without making speci"c addi1 2
tional assumptions, and we will not use it.
A second strategy is suggested by Newey (1991). Following Chamberlain
(1987), this starts with noting that conditional exchangeability leads to in"nitely
many CMRs and in"nitely many UMRs. The idea is then to let the number of
CMRs used in estimation grow to in"nity at an appropriate rate as N tends to
in"nity. Newey shows that this approach leads to an estimator which attains the
semiparametric e$ciency bound for (A1). In "nite samples, however, this approach requires many choices: which of the in"nitely many CMRs should be
used, and which functions of the covariates should be used to form UMRs. We
compare some Monte Carlo results for one and two conditional moment
restrictions in Section 4. The Monte Carlo evidence suggests that increasing the
30
E. Charlier et al. / Journal of Econometrics 95 (2000) 25}56
number of CMRs used in estimation does not automatically lead to a large
increase in e$ciency, unless the data set is very large.
We will focus on the easier approach based on Newey (1993). Starting point is
the smooth CMR of HonoreH (1992). This CMR is used to construct optimal
UMRs, on which a GMM estimator is based. Thus, we do not aim at attaining
the semiparametric e$ciency bound for (A1). Our estimator will attain the
e$ciency bound for the class of models satisfying this single smooth CMR. This
class may be larger than the class of models satisfying the conditional exchangeability Assumption (A1), since (A1) implies more CMRs.
3. Identi5cation, consistency, e7ciency, and GMM estimation
Let ¹"2, y"(y , y ), x"(x , x ), and *x"x !x . In the remainder we
1 2
1 2
1
2
assume that HonoreH 's conditions (Assumption (A1) and regularity conditions
given in HonoreH (1992)) are satis"ed. The assumptions lead to in"nitely many
CMRs which can be presented compactly as in HonoreH and Powell (1994). To
do this, de"ne
e (b)"maxMa#u ,!b@x ,!b@x N"maxMy !b@ *x, 0N!b@x ,
12
1
2
1
1
2
e (b)"maxMa#u ,!b@x ,!b@x N"maxMy #b@ *x, 0N!b@x .
21
2
1
2
2
1
Then, under the exchangeability Assumption (A1),
e (b )!e (b )"maxMy !b@ *x, 0N!maxMy #b@ *x, 0N#b@ *x
0
0
0
2
12 0
21 0
1
"o(y, b@ *x)
0
(2)
is distributed symmetrically around zero conditional on x. This implies
EMm(e (b )!e (b ))DxN"0
12 0
21 0
(3)
for any odd function m. In this section we restrict attention to the smooth CMR
based on m(a)"a, used by HonoreH (1992):
EMo(y, b@ *x)DxN"0.
0
(4)
In Section 4 we will present some results for other choices for m(.). As shown
by HonoreH , CMR (4) identi"es b if and only if EM1(PMy '0,
0
1
y '0 D xN'0) *x *x@N has full rank. This excludes time constant regressors,
2
whose e!ects will be picked up by the "xed e!ects.
CMR (4) implies that, for any function A(x),
EMA(x)o(y, b@ *x)N"0.
0
(5)
E. Charlier et al. / Journal of Econometrics 95 (2000) 25}56
31
For a given choice for A(x), UMRs (5) can be used to apply GMM. A condition
for consistency of the GMM estimator is that b is the only value of b which
0
satis"es (5). This is di$cult to prove in general. HonoreH (1992) solves the
problem for this case: he chooses A(x)"*x, and constructs a strictly convex
objective function, whose "rst-order derivative is the sample analogue of the
UMRs. This guarantees identi"cation of the censored regression model with
UMRs (5), and, since (5) is implied by (4), also of the model de"ned by CMR (4).
It also guarantees consistency of the estimator obtained by minimizing the
strictly convex function. We denote the HonoreH (1992) estimator for b based on
0
A(x)"*x by bK . An estimator based on (5) for some arbitrary choice of A(x) is
H
denoted by bK . The limit distribution of bK is given by
JN(bK !b )P$N(0, G~1