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31
working paper
department
of
economics
Conditional
Moment
Restrictions in Censored
and Truncated Regression Models
Whitney K. Newey
No. 99-15
July 1999
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
WORKING PAPER
DEPARTMENT
OF ECONOMICS
Conditional
Moment
Restrictions in Censored
and Truncated Regression Models
Whitney K. Newey
No. 99-15
July 1999
MASSACHUSEHS
INSTITUTE OF
TECHNOLOGY
50
MEMORIAL DRIVE
CAMBRIDGE, MASS. 02142
MASSACHUbEHS
sWSTiTUTE
OFTECHf'OLOGY
CONDITIONAL MOMENT RESTRICTIONS IN
CENSORED AND TRUNCATED REGRESSION MODELS
Whitney K. Newey
Department of Economics
MIT
July,
The NSF provided financial support.
comments.
T.
1999
Rothenberg and
-1-
J.
Powell provided helpful
WP99-15
ABSTRACT
Censored and truncated regression models with unknown distribution are important in
Ttiis paper cinaracterizes the class of all conditional moment restrictions
econometrics.
that lead to
V«
-consistent estimators for these models. The semiparametric efficiency
each conditional moment restriction is derived. In the case of a nonzero
shown how an estimator can be constructed, and that an appropriately
weighted version can attain the efficiency bound. These estimators also work when the
bound
bound
for
it is
disturbance
is
independent of the
several estimators
efficiency.
in this
regressors.
case, as well as
The paper discusses selecting among
of combining them to improve
methods
Introduction
1.
Censored and truncated regression models are important for econometric data with a
Unlike regression models without censoring or truncation,
limited dependent variable.
consistency of
maximum
likelihood estimators depends on the distributional specification.
This property has motivated a search for estimators that are robust to distributional
This work includes Powell (1984, 1986a, 1986b),
assumptions.
Honore and Powell
1993),
(1992,
In this
(1994),
Newey
(1987,
and others.
paper we characterize the class of
conditional
all
moment
restrictions that
lead to v'n-consistent estimators for censored and truncated regression.
the semiparametric efficiency bound for each conditional
it
is
nonzero.
1989a), Lee
moment
We
derive
show when
restriction and
For the nonzero cases we describe how an estimator can be constructed,
and show that an appropriately weighted version can attain the semiparametric bound.
Because independence of disturbance and regressors will imply any conditional moment
restriction, all the estimators will
select
work
among several such estimators
in the
independence case.
in this case,
We
discuss
how
to
as well as methods of combining
them
Whether this approach can be used to attain the semiparametric
to improve efficiency.
efficiency bound in the independence case remains an open question.
In relation to previous
case generalize results of
work, the semiparametric efficiency bounds for the censored
Newey and Powell
(1990) and for the truncated case are new.
Also, the censored regression estimators given here are based on a conditional
restriction described by
(1986a).
Newey
(1989a), that generalizes the
moment
moment
restriction of Powell
Lee (1992) considered construction of -/n-consistent estimators from a special
case of these
to that of
moment
Newey
conditions.
(1987),
The truncated regression moment restriction
and generalizes Lee (1993).
assumption that was imposed by Lee (1992, 1993).
is
Here we dispense with the symmetry
Also,
we generalize previous
characterizing the entire class of useful moment conditions.
results by
This leads to estimators
that have improved properties over those previously proposed, including asymptotic
efficiency and ease of asymptotic variance estimation.
-2-
similar
The Model
2.
It
is
a conditional
y
moment
=
be some scalar function.
m(e)
Let
regression.
(2.1)
we consider
convenient to describe the class of models
X'pQ
if
terms of a latent
Then a latent regression equation with
restriction can be described as
E[m(c)|X] =
+ e,
0.
Each such condition corresponds to the location restriction
E[m(E-;_i(X)) |X] = 0.
in
For example,
m(G) = l(e>0)-l(e<0)
then
m{c) = c
if
then
for
/i(X)
solving
has conditional mean zero, while
c
has conditional median zero.
c
=
jiiX)
Other specifications of
correspond other location restrictions, some of which are less familiar than the
m(e)
median and mean.
Censored and truncated regression models are ones where
observed.
,X)
(y
is
only partially
For censored regression,
*
y = max{0,y
(2.2)
(Censored regression).
>;
For truncated regression we have
*
(y ,X)
(2.3)
*
only observed
These models are familiar
relatively simple.
in
if
y
>
(Truncated regression).
0;
econometrics, and
we focus on them
to keep the exposition
Our results can be extended to other models, including censored
regression where the censoring point varies with
X
or censoring occurs above as well as
below.
In the latent model,
use a conditional
moment
any vector of functions
will be satisfied at
/3
where
(y ,X)
is
always observed,
restriction to estimate
A(X)
= ^
the unconditional
/3
.
it
Equation
moment
is
well understood
(2.1)
restriction
(assuming expectations exist).
This
how
implies that for
E[A(X)m(y -X'/3)] =
moment condition could
be used to form a generalized method of moments (GMM) estimator in the usual way.
-3-
to
GMM
However, consistency of a
moment
equation, which
obtaining
from"
m(e).
It
To
w(X)
let
£;
is
this end,
q(e) =
let
m(u)du +
J~
C
E[w(X)Xm(e)] =
can be any constant.
Also,
will be the first order condition
extremum (min or max)
to have an
(3)]
d(X) = aE[m(e+a) |X]/aa|
chosen so that
is
where
C,
be a weight function that will be important for the efficiency discussion
E[w(X)q(y -X'
m(e)
This identification problem motivates
often easier to show that such an objective function has a unique
Then the moment restriction
below.
for
difficult to show.
is
be the unique solution to the
(3
from minimizing a corresponding objective function that "integrates back
^
minimum.
estimator requires that
^ ^
at
0.
Assume that the sign of
/3
Then
E[w(X)d(X)XX'
]
will be
*
positive semi-definite, the necessary second-order condition for
E[w(X)q(y -X'/3)]
to
*
have a minimum at
(3
Then
.
E[w(X)q(y -X'/3)]
corresponds to the moment restriction
becomes a function whose minimization
E[w(X)Xm(y -X' 13]] =
The sample analog to the
0.
*
minimizer of
E[w(X)q(y -X'/3)]
is
11
^ = argminX",w(X.)q(y. p^i=l
(2.4)
X'./3).
1
The identification condition for consistency of
has a unique minimum at
/3
which
is
this estimator is that
easier to
show than that
E[w(X)q(y-X'p)]
E[w(X)Xm(y-X'p)] =
has
a unique solution.
It
turns out that in censored and truncated models an analogous approach works for
some conditional moment
The nonexistence result
exists.
will follow
efficiency bound for this model.
(3
This bound
for regular parametric submodels
regular).
It
and that for the rest no Vn'-consistent estimator
restrictions,
and
in
let
is
the infimum of the information bounds for
see Newey, 1990, for the definition of
(e.g.
can often be computed by a projection.
the mean-square closure of the set of
(e,X)
from the form of the semiparametric
Define the tangent set
J
scores for parameters of the distribution of
all
parametric submodels passing through the truth and satisfying equation
S
denote the score for
from the mean-square projection of
Define the efficient score
(3.
S
to be
on
-4-
3",
assuming
J
is
S
(2.1)
to be the residual
linear.
If
E[SS'
]
is
singular then no \/n-consistent, regular estimator exists, while
nonsingular then
moment condition
We
(2.4).
Here we find that
S
is
zero except for certain cases where
leads to a Vn-consistent estimator analogous to that in equation
also find that the asymptotic variance of this estimator
semiparametric bound when
3.
is
]
inverse provides a bound on the asymptotic variance of regular
its
v^-consistent estimators.
the
E[SS'
if
w(X)
is
is
equal to the
chosen to have a certain form.
Censored Regression Models.
For censored regression any moment condition where
The fact that
small enough leads to Vn-consistent estimation.
when
certain value means that
X'/3
the same value at the censored
y
is
below that point,
set is empty.
let
as the latent
i.e.
equal the
I
I
= sup{e
= l(-X'/3
<
£){l(y
leading to the conditional
E[1(X'(3q
(3.2)
where
supremum
m(G)
Figure
1
will have
m(y-X'/3)
of all points
:£
e},
how
illustrates
where
v = X'/3
>
= 1(-X'P
<
m(c)
where we take
£){l(y = 0)m(-X'/3) + l(y >
< 0)m(£) + l(y
moment
>
0)m(y -X'/3)} =
l(X'/3
moment
this occurs.
is
£
constant
=
-oo
0)m(y -X'/3)}
>
-£)m(y*-X'|3),
restriction
-l)m{y-X'!3^)\X] = l(v>-£)E[m(£) |X] = 0,
.
As discussed
in Section 2,
better identification conditions.
integrating back to an objective function can lead to
To integrate back, note that for a scalar
-5-
e
all
constant below a
is
leading to the conditional
y,
mlc) = m{e) V e
constant for
Then
l(X'/3 > -£)m(y-X'/3)
(3.1)
:
is
large enough, the function
restriction being satisfied in the censored data.
To be precise,
m(c)
a,
if
the
l(a > -^)m(y - a) = -dq(y - max{a,-£})/dcx,
(3.3)
a =
except where
equation (3.2),
This means that
-I.
E[w(X)X*l(v>-£)m(y-v)] =
the first-order condition corresponding to minimization of
is
An estimator based on the sample analog of
E[w(X)q(y-max{X'/3, -•£})].
this minimization is
^ = argminp^^j:.^^w(X.)q(y.-max{X'./3 ,-m.
(3.4)
This estimator
the extension of that of equation (2.4) to censored regression.
is
The moment condition of equation
constant for
(3.2)
is
critically dependent on
X
distribution of
Assumption
3.1:
respect to
U x
f (u|X)du
S^
-00
E
p.
|X]
is
with distribution that
i.i.d.
p -almost
for
,
E[(l+IIXll^){l+J'[f
e
bounded
in a
Prob(v =
*
(X) =
is
)
(E[m(E)
-£)
2-1
|X])
Let
p.
denote the probability
X
denote Lebesgue measure.
XX
and
2
U
and
(e.,X'.
probability one,
w
being
This result follows from the form of the semiparametric information bound.
To derive that bound we impose the following condition.
E[m(c+a)
m(e)
Without this property, no yn-consistent estimator
small enough.
c
all
will exist.
Let
as implied by
0,
X
all
there
0,
d(X).
We
such that
G
(u |X)^/f (u |X)]du}]
and
absolutely continuous with
(c|X)
f
is
neighborhood of every
=
is
<
a
co,
as a function of
and
E[m(c)
E[IIXII^d(X)^/E[m(e)^|X]]
will also
f(e|X) =
<
2
|X] >
a,
with
oo.
impose Assumption A.l of the Appendix on
the parametric submodels.
Theorem
3.1:
If Assumptions
3.1
and
A.l
S = w''(X)X'l(X' I3^>-Vm(y-X'
If
E[SS'
]
is
nonsingular then
Since the efficient score
below some value
is
is
are satisfied then the efficient score is
(B^).
(E[SS'
])
is the
identically zero unless
£
semiparametric variance bound.
is finite,
m(e)
being constant
a necessary condition for existence of a (regular) Vn-consistent
-6-
That
estimator.
there will be no Vn'-consistent estimator unless
is,
from equation
p
*
(3.4)
is
variance of
will equal the bound.
/3
w(X) =
Furthermore, as shown below, for
available.
sense there
In this
available to be used in estimation of
could be that
it
approximate satisfaction of
(1986)
shows that
may apply
no additional information
is
is
/3
(3
the asymptotic
(X),
other than that used by
13
This result sidesteps the identification question for
/n-consistent estimator,
w
/3.
Despite the lack of a
.
identified "at infinity," by
E[m{y-X'/3 )|X] =
for large values of
X'/3
Chamberlain
.
this is possible for the sample selection model and similar reasoning
here.
To show /n-consistency
^
of
it
make additional assumptions.
useful to
is
Let
Q = E[w(X)d(X)l(v>-£)XX'].
Assumption
Also,
3.2:
E[m(e+£x)|X] ^ {^)
B
€ interior(£),
/3
w(X) ^
and
everywhere,
compact,
is
a
for
2:
and
(:£)
m(E)
is
that
m(e)
attention to bounded
constant anyway.
conditions.
is
m(E)
a =
3.2:
A simple
(i.e.
E[m(c+a)|X],
that its sign
sufficient condition for single
q(E)
is
Restricting
convex).
does not seem too stringent, because
lower
its
tail
must be
This restriction could be relaxed at the expense of complicating the
With this condition
If Assumptions
Furthermore, if
0.
monotonic increasing
in
normality result for the estimator.
Theorem
nonsingular.
is
bounded.
is
doesn't change on either side of
is
exists and
bounded and continuous almost
This condition imposes a "single crossing" property for
crossing
Q
3.1
place
Let
and
we can obtain
a consistency and asymptotic
2
2
Z = E[w(X) l(v>-£)m(e) XX'
3.2 are satisfied then
w(X) = w'(X) = d(X)/E[m(c)^ \X]
then
/n(/3-/3
V = (E[SS'
V = Q
and
]
)
—
>
-1-1
EQ
N(0, V).
]f\
This result also shows that the weighted m-estimator would attain the semiparametric
*
efficiency bound
if
the weight was equal to
-7-
w
(X).
In this sense there is
no
.
information lost from using an estimator
An asymptotic variance estimator
and
An estimator of
Z.
S
is
(3.4).
needed for large sample inference procedures
is
This can be formed
based on Theorem 3.2.
equation
like that of
way
the usual
in
Q
as
EQ
for estimators
Q
straightforward to construct, as
1111
S = y.",l(v.>-£)w(X.)^m(y.-v.)^X.X'./n,
^1=1
where
=
v.
1
1
An estimator
X'.B.
Q
of
more
is
because
difficult,
it
involves the
1
1
d(X) = dE[m{c+a]\X]/da.
derivative
m(e) =
If
C
continuous almost everywhere and a constant
Tm e (u)du
+
for some
C
m e (e)
that
is
then
,l(v.>-£)w(X.)m (y.-v.)X.X'./n
Q = y."
^1=1
1
1
e 1 1 1 1
Otherwise,
will do.
d(X)
numerical derivative, as
will need to be approximated.
This
may be done by
a
in
1111
= y;.'^,w(X.)X.X'. [q(y.-max{v.+5,-£})+q(y.-max{v.-5,-£})-2q(y.-max{v.,-£})]/(5^n).
Qs
^1=1
5
1
1
1
1
1
The following result shows consistency of the corresponding estimators.
Theorem
C
m
and
(l,x' )'
3.3:
,
(e)
Suppose
is
that
Assumptions
3.1
and
3.2 are satisfied.
continuous almost everywhere then
E[w(X)\\Xl\^] <
00,
d
-^
0,
and
n^^^5 -^
oo
Q
—^
ZQ
then
If
V.
Q'Jzq'J
o
o
m(c) = S^rn (u)du
U
of
6
in practice,
Also,
it
=
V.
Imposing the sixth moment condition simplifies the proof of this result, although
could probably be weakened.
X
Also, if
-^
e
it
would be useful to have guidelines for the choice
but these are beyond the scope of this paper.
Construction of an efficient semiparametric estimator, one that attains the bound,
would require nonparametric estimation of the optimal weight
generalize
Newey and Powell's
zero median.
w
*
(X).
Such a result would
(1990) efficient estimator for censored regression with
Derivation of such an estimator
-8-
is
beyond the scope of this paper.
+
As examples consider the conditional median and mean zero cases.
m(e) =
conditional median case, the function
score
4.
is
0) - 1(e
^
0)
<
S = 2f(0|X)X-l(X'/3 >0)m(y-X'/3
leading to the efficient score
m{e) = e
case, the function
l(e
not constant below any
is
is
zero
In the
constant below
In the conditional
).
0,
mean
and hence the efficient
I,
Consequently, no Vn-consistent regular estimator will exist in this case.
zero.
Truncated Regression Models.
For truncated regression the special characteristic that leads to a useful moment
condition
X'/3
is
m(e)
that
large enough,
distribution of
enough.
zero for
all
c
as
y
-X'/3) =
same
will be the
restriction
small enough.
That feature means that for
goes from
in
to
so the conditional
-co,
the truncated and latent data.
E[m(y-X'/3 )|X] =
will be satisfied for
Hence the
large
X'/3
Figure 2 illustrates this condition.
To be precise,
that point,
empty.
m(y
m(y-X'/3)
moment
conditional
is
E
k
k = sup{e
i.e.
Let
let
[
•
]
equal the
:
supremum
m{c) =
V e
:£
of all points
c},
where
where we take
m(e)
k =
denote the expectation for the latent model and
-co
E[
•
is
zero below
if
the set
is
the
]
*
expectation for the observed data, and
latent model.
Then, since
(4.1)
For
n(X) = E [l(y>0) X]
|
l(y:£0)l(v>-fc)m(y-v) = 0,
EinX' (S^>-k)m(.Y-X'
P (A)
the probability of an event
note that
E[
•
|
X] =
E
[l(y>0)(
• )
I
A
in the
X]/n(X).
we have
I3^)\X] = n(X)"^E [l(v>-«:)l(y>0)m{y-v) |X]
= n(X)~^E [l(v>-fc)m{c)|X] = n(X)"-^l(v>-^)E [m(c)|X] =
0.
Integrating back to an objective function as was done in Section 3 leads to the estimator
(4.2)
p = argmin^^^^^^w(X.)q(y.-max{X'.|3,-fc}).
this estimator is exactly analogous to those of
The analysis of the properties of
the censored regression case
Section
in
m(e)
If
3.
is
not zero below some value then
no \/n-consistent estimator will exist, and the semiparametric efficiency bound will
*
correspond to the asymptotic variance of
semiparametric efficiency bound
Theorem
If Assumption
4.1:
w
= d(X)/E[m(c)
(X)
2
The
|X].
given in the following result:
is
is
3.1
w(X) =
for
/3
satisfied and
P (y>0)
then the efficient score
>
is
S = w^(X)X-l(X' l3^>-k)m(y-X' ^^).
E[SS'
If
]
nonsingular then
is
(E[SS' ])
semiparametric efficiency bound.
is the
Asymptotic normality will also hold under similar conditions to those
Here
in Section 3.
let
V = Q~-^SQ~^
S = E[w(X)^l(v>-^)m(E)^XX'],
Q = E[w(X)l(v>-fc)d(X)XX'],
The following result gives asymptotic normality.
*
Theorem
-%
N(0, V].
Here
it
If Assumptions
4.2:
3.1
w(X)
Furthermore, if
should be noted that
and
m(c)
3.2 are satisfied
= w'^(X) =
and
P (y>0)
d(X)/E[m(c)^ \X]
_, „
then
>
then
vn(/3-p
V = (E[SS'
being zero below some value rules out
m(c)
])
I
being
monotonic, so that other primitive conditions for the "single crossing" identification
condition of Assumption 3.2 must be found.
distribution of
density and that
some choice of
X
given
c
m{e)
C,
is
is
symmetric
One such condition
(in
an odd function that
-q(c) = -J m(t)dt - C
density function, so that as in Bierens (1981)
e)
is
around zero with unimodal conditional
nonnegative for
Lemma
3.2.1,
E[m(E+a)|X] = -a{-E[q(E+a) X]}/aa
is
|
Then, for
unimodal symmetric
-E[q(e+a)|X]
a.
-10-
e ^ 0.
will be proportional to a
proportional to a unimodal density function as a function of
property will imply that
that the conditional
is
is
This unimodality
nonnegative for
a >
0,
and give the single crossing property of Assumption
3.2.
A consistent estimator of the asymptotic variance can be formed
to that for censored regression.
Let
1111
S = 7." ,l(v.>-fc)w(X.)^m(y.-v.)^X.X'./n,
^1=1
when
m{e)
Q
5
is
1
1
differentiable.
way analogous
a
in
Otherwise
1111
Q = y;."l(v.>-fc)w(X.)m
^^1=1
1
1
(y.-v.)X.X'./n,
e
1
1
1
1
let
= V." w(X.)X.X'. [q(y.-max{v.+6,-/tn+q(y.-max{v.-5,-ft})-2q(y.-max{v.,-^»]/{6^n).
^1=1
1
1
1
1
1
The following result shows consistency of the corresponding estimators.
Theorem
C
m
and
(hx'
)'
,
Suppose
4.3:
that
Assumptions
is
3.2 are satisfied.
continuous almost everywhere then
E[w(X)l\X\\^] <
CO,
5^0,
and
n^'^^S
—
>
co
Q
ZQ
then
a useful one, because no bandwidth
is
required for these estimators.
seem
and
"-2
(c)
The differentiable case
m(e)
3.1
to be the first
If
—
p
1
V.
>
Q~JtQ~J
o
(i.e.
o
5)
m(c) = S^rn (u)du +
Also, if
-^
X
=
V.
choice
is
For the truncated case the estimators for differentiable
moment condition estimators for the truncated model that avoid
the use of a bandwidth in estimating the asymptotic variance.
5.
Independence of
e
and
X.
The sensitivity of the efficiency bound results to the form of conditional location
restriction
is
troubling.
For instance, the parameters can be Vn'-consistently estimated
when the disturbance has conditional median
Some economic models do imply
zero, but not under conditional
mean
specific location restrictions, such as conditional
restrictions in rational expectations models.
Often though,
mean
we have no have strong
priori reasons for choosing one location restriction over another.
-11-
zero.
a
Things are different when the disturbance
case, corresponding to each
satisfies
moment
Consequently,
it.
moment
if
independent of the regressors.
is
restriction should be a location shift of
a constant
is
c
In that
that
included in the regressors, any conditional
restriction will be satisfied at the true slopes and some constant, including the
ones that lead to v/n-consistent estimation of censored or truncated regression.
any of the estimators will be Vn'-consistent for the slope coefficients
Thus,
in the
independence case.
To be
E[m(e-|u
(/3^,p^)'
Suppose that
0.
conformably with
X.
X =
(I,x'
such that there
p^^ = P^q+M^^
Then for
and
moment
restriction
we have considered
of the estimators
with
fi
/3
=
= (p^^.p^^)',
Consequently, any
for the censored and truncated regression models
that are v^-consistent under some conditional
under independence, for the slope parameters
moment
(3
.
also be V^-consistent for the original constant plus
It
J3^
satisfied in the latent data.
is
some
0.
|
a conditional
is
includes a constant, and partition
)'
E[m(y*-X'pQ)|X] = E[m(e-fi^) X] = E[m(e-)Li^)] =
(5.1)
So,
=
)]
m(c)
specific, consider any function
should be emphasized that the conditional
restriction should be \/n-consistent
The estimator of the constant will
u
m
.
moment
restriction in equation (5.1)
depends on independence, and not on any other restriction on the distribution of the
In particular,
disturbance.
with independence, the symmetry assumption of Lee (1993)
not needed for the estimation of a truncated regression.
Although symmetry
is
is
part of
the primitive conditions for the single crossing property for truncated regression given
in Section 4,
(5.1)
it
is
not the fundamental identification condition that leads to equation
being satisfied.
When
E
and
X
are independent, the asymptotic variance of the estimators
simplifies and the optimal weight
or
E[m(G)
IX]
depend on
X.
w(X)
Let
d
m
is
equal to
1.
By independence, neither
= 5E[m(e-ii +a]]/aoc\
Then for censored regression. Theorem 3.2
-12-
m
will hold
with
I
m
^
a=0
and
cr^
m
d(X)
= E[m(e-u
m
as defined analogously to
)^]
and
in Section 3
£
Q = d
(5.2)
m
E[w(X)l(v>-£
m )XX'
],
E = cr^E[w{X)^l(v>-£
m
m )XX'
].
V = d"^cT-^(E[w(X)l(v>-£ )XX'])~^E[w(X)^l(v>-£ )XX' ](E[w(X)l(v>-£ )XX'
m
m
m
m
m
Furthermore, by a standard result (see the proof of Theorem
w(X) =
the positive semi-definite semi-order) at
mm
V = d"^cr^(E[l(v>-£
(5.3)
m )XX'
1,
a
(5.4)
d
Section
in
m
m
])"^
way
to that
1
1
m(E)
is
i
i
V =
i
assumed to be differentiable
^1=1
1
1
mm
d~^(?-^(y.",l(v.>-£)X.X'./n)~\
^i=l
i
i
i
in
the definition of
d
m
.
The estimators
are the analog for the independence case of those given in Section 3 for
censored regression.
regression.
1
11
md m
V
1
1
1
[y.''l(v.>-£)m(y.-v.)^]/^." !(;.>-£),
^1=1
^1=1
1
1
6
and
in a similar
Let
3.
l(v.>-£)X.X'./n)"\
V, = d"^J-^(r."
^"1=1
o
(in
= [I
Uv.y-l)m {y.-v.)]/l Aiv.y-l),
^1=1
^1=1
1
1
c
1
^1=1
a^ =
V
minimized
= 5"^y." rq(y.-max{v.+5,-n)+q(y.-max{v.-5,-£})-2q(y.-max{v.,-£})]/y." ,l(v.>-£).
,
m6
where
is
where
The components of this asymptotic variance can be estimated
described
V
5.1),
])"^
Replacing
£
by
A;
leads to analogous estimators for truncated
The following result shows consistency of these estimators under
independence of
e
and
X:
-13-
Theorem
X
5.1:
For censored regression, if Assumptions
X
are independent, and
5.2.
V
Also, this
is
)'
- (l,x'
minimized
at
Vn(^-p
then
w(X) =
then
^
K^ —
5
replacing
V —
^
m(c)
while if
V,
for
N(0, V)
>
6
V
is
in
and
e
equation
continuously
—
and
>
n
P^(y>0) >
if
—
5
Furthermore, for truncated regression the same results hold with
V.
£
3.2 are satisfied,
In addition, if
1.
differentiate with bounded derivative then
—
)
and
3.1
>
k
0.
Because many estimators will be consistent for the slope coefficients under
independence,
is
it
possible to choose
from among a group of estimators the most
efficient one, or to combine several different estimators for improved efficiency.
Analogous methods are also available for the truncated regression model, and can be
obtained by replacing
k
with
£
in the
following discussion.
The efficient estimator from some class can be chosen by minimizing an estimator of
To describe
the asymptotic variance.
of functions
For each
m(e).
m(*),
regression estimator described
in
= KX'.S
r m
X'.
>
-I
m
),
and
v
.
mi
=
this type of estimator,
let
13
m
(plim/3
m
V^
2m
= (d"^^^/a
Var(x.|v
1
A member
.>-£
i
.>-£
mi
of the class
measure of the size of
m
=
)
ex
{p
V„
2m
m
Y.
m
:
e
If
.
A
tM}
m
.x.x'.
^i=l mi
Also, let
1.
1
.
mi
Then the block of the asymptotic variance
).
m m m )Var(x.|v mi
denote some family
w(X) =
Section 3 (or Section 4) for
i
M
denote the censored (or truncated)
estimator corresponding to the slope coefficients
(5.5)
let
is
)"\
a
/n - x
i
i
m
= Y.^ S
./n,
^i=l mi
x'
mm
x
,
m
= a
m
can be selected by choosing
Var(x.|v
1
.>-£
mi
m
choice could be obtained by minimizing the scalar
is
)
invariant to
--2-2
d
,1
^^1=1
.x./n.
mi
i
m
to minimize
m
then an equivalent
some
~
m m /a m
cr
y.
.
Powell (1986a) suggested such a procedure for choosing among different censored
regression quantile estimators.
conditions for the choice of
m
Also,
McDonald and Newey (1988) gave regularity
to have no effect on the asymptotic variance of
-14-
p
in
i
moment
the case of an uncensored regression model and a parametric class of
m
estimation of the best
Intuitively,
but for brevity
this result,
has no effect on the asymptotic variance because
does not depend on
p
the limit of
we do
that
is
minimum distance
We
m.
could specify regularity conditions for
not.
An alternative way of proceeding
efficient
is
to combine several estimators
asymptotically no less efficient than any of the individual ones as well as a
weighting matrix, as
well
is
For this approach we
m
known
to be a
M
let
problem
he a finite
with an element of the index set
=
{j
th
estimator,
j
moment
the constant point,
I.
A. A, .m.(E..)m,
^1=1
ji ki
between
S
ki
and
.
S,
jk
where
[0,1]
coefficients.
A ji-A,ki
the optimal
and we will drop the
J>,
...,
1,
/3
.>-£.),
J
will denote the slope
.
e
d
=
..
denote the derivative
.
y.-X'.
Ji
J
and
fi.,
1
1
c
.,
=
Jk
J
is
= dT^d"^£., [0,I](y;.",i..X.X'./n)"V.",l..l, .X.X'./n(y."
J
is
2
Then an estimator of the asymptotic covariance
^1=1
k
jk
^1=1
ji
^^1=1 ji ki
1
1
T,
.X.X'./n)"-^[0,I]',
^1=1 ki
1
1
1
1
a selection matrix that picks out the elements corresponding to slope
Q =
Then the partitioned matrix
asymptotic covariance matrix of
J(K-l)x(K-l)
8
1
of the optimal
estimation.
Similarly, let
1(X'.
that
k
J
Q.,
A/Y.
(e,
k
ji
J
=
1..
Ji
J
is
For notational simplicity we identify
Thus,
condition.
GMM
in
set.
subscript on the slope coefficient estimators.
coefficients for the
disadvantage
Its
may be adversely affected by estimation
the small sample distribution
y.
from a class using
This has the advantage of yielding an estimator
estimation.
simple test of independence between regressors and disturbance.
each
restrictions.
partitioned matrix
minimum distance
tt
=
(p'
made up
[^.,1
...,p')'.
of
estimator, from
is
Let
an estimator of the joint
H =
[1,1
1]'
be a
dimensional identity matrices.
K-1
minimizing
(tt
- UfS)'
Q
(tt
- H/3),
Then
and the
associated asymptotic variance estimator and overidentification test statistic are
(5.6)
p^ =
(H'Q"^H)"^H'Q"^rt,
V^
= (H'Q"^H)~\
Under the regularity conditions previously stated
-15-
it
T^
= nin - Hp^)'Q"^rt - Hp^).
will be the case that
VEC^^
(5.7)
T^
Here
- Pq)
^
V^
N(0,V^),
^
^
T^
V^,
a:2((J-l)(K-l)).
provides a statistic for testing independence that depends on
different estimators
^
.
close the
are to each other, analogous to the tests of Koenker and
Also, the estimator
Bassett (1982).
how
linear combination of the estimators
an optimal (asymptotic variance minimizing)
is
/3
^
,
...,
|3
For brevity we do not state these
.
results as a theorem.
The estimator selection and efficient minimum distance approaches require multiple
These may be constructed as described
estimators of the slope coefficients.
3 and
but this requires optimization of multiple objective functions.
4,
convenient computational approach
a single initial /n-consistent one.
is
This
is
= argmin
m
For
v.
=
1
replacing
To describe
let
Y. ,q(y.-max{x'.
^-^1=1
1
1
fi
tt
+ ^,
method
this
m
-£
let
be an initial
/3
be obtained as
m
}).
x'.
8 + j
1
m
let
d
m
on
be the same as given in equation (5.4) with
v.
^
and
v.
]3
= 3 + d"^[0,I](y." ,l(V.>-£
^1=1
m
m
1
By the usual one-step arguments
is
from
only minimization over a scalar, and can easily be carried out, e.g. by grid
search.
that
A more
to use a linearization to construct estimators
coefficients and
estimator of the slope
^
y
in Sections
obtained from the
full,
)X.X'.fW.^Uv.>-l )X.m(y.-v.).
mil
^i=l
m
i
i
i
i
this estimator will be asymptotically equivalent to
|3
global minimization of the objective function.
These estimators lead to computationally simpler versions of selecting an efficient
estimator
If
in
in
some class and optimal minimum distance estimation.
One could use
(3
and
computing each of the asymptotic variance estimators for selecting an efficient
estimator, and then use the linearized estimator
Also, one could use the linearized estimators
minimizers
in
constructing the optimal
(/3
S
at the efficient choice of
m
p
)
in
place of the full global
minimum distance estimator.
-16-
m.
This replacement
will
produce asymptotically equivalent estimators and test statistics.
The choice of the class
M
will affect the efficiency of the estimators.
choose them so that the location parameter
quantiles of Powell (1986a).
tail
m
varies,
as in the censored regression
^
This choice leads to a tradeoff between the location in the
of the disturbance distribution and the proportion of observations used by the
l(x'.
u
As
estimator.
but
u.
One might
S^ +
1
11
m
m
increases, the variation in
>-•£) =
!
m(e)
will be located
more
in the tail,
more often so that more observations are included.
An interesting open question concerns the efficiency of the optimal minimum distance
estimator relative to the efficiency bound derived by Cosslett (1987).
semiparametric models
it
is
possible to combine
minimum distance estimator approximately
discussed
in
Newey
{1989b).
Here,
it
is
moment
In
some
restrictions so that an optimal
attains the semiparametric efficiency bound, as
difficult to verify this result because of the
complicated nature of the efficiency bound and the moment conditions.
-17-
APPENDIX
C
Throughout the Appendix
will denote a generic constant that
The following Assumption
different uses.
may
be different in
useful in the proofs of the semiparametric
is
information bounds.
Assumption
such that
f(E|X,T))f(X|-r7)
f(E|X,7))
X,
is
bounded
Prob^(y
is
in a
0)
>
f(E X.t)
a density for
for all
A.l:
By
Proof:
latent model is
Lemma
A. 2 of
5"
A. 5 of
L =
I >
I.
{e £ -v}.
E[l(L)(l,m(£))'(l,m(e))|X]
Also,
A.l
in
in
t
1(IID(X)II
all
2
and satisfies
7}
is
the
(1990)
E[5] = 0,
(1990), for
it
JmlE) f(E|X,'n)dE
e = (/3',t}')',
i?(6)
m{e)
is
nonsingular.
For
>
M
0,
>
£ M)-l(v<-J),
= -X-1j^-{1(e>-v)s(e,X) - D(X)l(e<-v)(I,m(E))'}.
Note that
-18-
l(vs-i)S
G J.
E[II5II
0,
= E[5|y,x],
it
of the set
nonconstant on
Var(m(E) |L,X)
Lemma:
follows that the tangent set of the
E[m(E)5|X] =
mean square closure
v > -£,
is
:
of the following
D(X) = E[l(E>-v)(l,m(E))s(E,X)|X]A(X)"\
M
and for almost
(t)',t)')'
the truncated case, for
and otherwise
L, =
in
are satisfied then
Newey and Powell
v ^ -I,
Then for
smooth
smooth
make use
and
3.1
Newey and Powell
Then for
.
3.1 will
= {8 = dlcX)
set of the observed data
any
t)
is
is
)
9.
If Assumptions
Lemma
that
c
neighborhood of
>
)f (X |t)
|
The proof of Theorem
Lemma
flE.Xlrj) =
Parametric submodels correspond to latent densities
A.l:
(-oo,-v],
oo}.
Also, by
follows that the tangent
{6'(5)
:
5 6
D(X) =
Consider
3"}.
because -v ^
and hence
let
<
]
£
>£..
A(X) =
if
P(L|X) =
Let
E[t(l,m(E))|X] = -X-lj^-E[{l(c>-v)s(c,X) - D(X]l(E<-v)(l,m(e))' }(l,m(c)) X] = 0,
|
where the
last equality follows by the definition of
by Assumption
so that
3.1,
=
t
= -X-lj^{l(y>0)s(G,X)
t
€
tJlt)
D(X).
Also,
has second moments
t
Furthermore,
3".
- l(y=0)D(x)E[(l,m(e))' |y,X]}
= -X-L ,0(y>0)s(E,X) - l(y=0)E[l(E>-v)s(c,X)|X]/P(L|X)>
M
= -X-lj^{l(y>0)s(E,X)
It
-^
E[lll(v<-£)S -tll^] = E[IIS
follows that
CO.
+ l(y=0)E[s(E,X)|L,X]}
Therefore,
e J.
l(v<-£)S
IS.
=
ll^l(IID(X)ll>M)l(v<-£)]
as
0,
l(£<v<Z)
Prob(v=-£) =
since
Finally,
^
M
-^
almost
P
surely
J as
-
—
£
theorem,
so by
the dominated convergence
J
b
I,>
>
E[lll{v£-£)S„-l(v:£-£)S„ll
(3
E[l(£<v<-£)IIS
11^]
—
Proof of Theorem
(A.l)
as
>
I
^
L
(1981)
w
(X)m(E)}.
random variable has
and
= JmCElf (E-ai X)dE
is
-E[m(E)s(E,X)|X].
Then by
E[m(E+a)
2
a
at
a =
+ l(v<-£)S
5(£,X)
3",
satisfying
i.e.
S =
7.2 of Ibragimov
0,
E[m(E+a)|X]
E[s(£,X)|X] = 0,
^(5) = l(v>-£)S
e
Lemma
d(X) = 3E[m(E+a) |X]/9a =
and
w
{X)E[m(E) |X]} = 0,
E[m(E)6(E,X)lX] = -Xl(v>-£){E[m(e)s(e,X)lX] +
Therefore,
By
bounded on a neighborhood of a =
|X]
differentiable in
mean-square.
finite
E[6(£,X)|X] = -X1(v>-£){E[s(e,X)|X] +
(A. 2)
=
Let
3.1:
3.1 this
and Hasminskii
]
Q.E.D.
6(c,X) = -X-1(v>-£)-<s(e,X) +
By Assumption
2
j3
S
S e
E[5(e,X)|X] =
for
0,
{X)E[m(e)^|X]} =
Then, by linearity of
3".
- t
w
t
€ J.
and
Lemma
Furthermore, by equation
E[5(e,X)m(E) X] =
|
-19-
J
0,
0.
A.l,
(3.1),
S -S = ^(6)
for any
E[S'e(5)] = E[S'E[6(c,X)|y,X]] = E[E[S' 5(e,X) y,X]]
|
= E[w (X)X'l(v>-£)m(c)6(e,X)] = E[w (X)X' l(v>-£)E[m(c)5(E,X) X]] =
|
Then, since
3"
is
the mean-square of objects of the form
orthogonal to the tangent
projection of
on the tangent
S
The following Lemma
Lemma
is
with
and
<
E[U(z)U(z)'
VR(^-Pq)
-^
Proof of
Lemma
iii)
the efficient score.
is
Q.E.D.
and
E[b(z)
]
If
<
Hi)
co;
^
—
^
i)
jS
ii)
;
there is
there
U(z)
-^
as
/3
qOig) +
-^
q(l3) = E[q(z.,(3)]
For
iv)
/3^;
((S-IBq)' Q((i-fSg)/2 +
Then
001/3-/3^11^);
N(0, Q~^E[U(z)U(z)' lQ~h.
r(z,/3)
Then by
is
nonsingular such that with probability one,
]
qCfB) =
nonsingular with
Q
S
follows that
the residual from the mean-square
p = argminS._M(z.,fi).
b(z)\\^-(3\\
[q(z,IS)-q(z.i3^)-U(z)'(!3-l3^)]/\\l3-l3^\\
there is
it
useful for the proof of Theorems 3.2 and 4.2.
\q(z,l3)-q(z,(3)\
E[U(z)] =
with
is
and hence
set,
Consider an estimator
A.2:
biz)
is
S
Therefore,
set.
G'(6),
0.
A.2:
=
By
ii)
and the triangle inequality,
< (b(z)+IIU(z)ll
|[q(z,p)-q(z,/3Q)-U(z)'(/3-/3Q)]/ll/3-/3Qll|
2
and the dominated convergence theorem,
E[r(z,/3)
]
—
>
).
as
/3
—
>
/3
.
The
conclusion then follows by Example 3.2.22 of Van der Vaart and Wellner (1996).
Proof of Theorem
for
c
:£
£,
q(e)
3.2:
is
r = min{-X'P,£}
Consider
linear in
c.
Then for
y
:£
I
£ 0,
and
y
r
= min{-v,£} ^
I.
- max{X'/3,--£} = y +r ^
that
(A. 3)
q(y-max{X'/3,-£})-q(y-max{v,-£}) - [q(y -max{X'/3,-£})-q(y -max{v,-£})]
=
l(y <0){ q(?)-q(r)
- [q(y +?)-q(y +r)]
-20-
}
Note that
£,
so
.
=
l(y <0){
IB,
=
l(y <0)q(0).
it
V
q(z,p)
is
= Ej"^q(z.,,p)Ai,
Q(/3)
continuous in
Also,
(i.
and
q(e)
11X11
llp-pil
It
follows by a standard uniform law of large numbers that
is
continuous in
p,
and that,
-^
sup„ „|Q(/3)-Q0)|
p€i3
consistency will follow from the standard argument
*
* _
„
[q(y -a)-q(y -a)]/(a-a)|
Assumption
3.2,
is
q(a,X)
is
at
a =
v,
uniquely minimized at
—OL
By
_ m(u)du.
bounded,
m(e)
almost
—
>
a,
-^ E[m(y -a)|X].
a =
q(a,X)
v.
is
It
—
*
>
E[q(y -a)-q(y -a)|X]/(a-a) =
q(a,X) =
follows that
E[m(y -a)|X].
a
decreasing in
for
a £
v,
Furthermore,
q(max{a,--£},X)
increasing in
a
-21-
~
m(y -a)
with derivative
a
increasing in
because
is
*
* ~
~
[q(y -a)-q(y -a)]/(a-a)
a
differentiable in
having a global minimum at
minimum
y
Therefore, by the dominated convergence theorem,
E[q(y -a)-q(e)|X]
V,
Q(|3)
p€jD
m(e)
^
|X)dy
p = argmin- ^Q(/3),
Since
0.
exists and
and by the fundamental theorem of calculus and
continuous almost everywhere, as
J'[J'^*~"m(u)du/(a-a)]f(y
= E[q(z,p)]
QO)
* ~
q(y -a)-q(y -a) = S
q('),
£ C,
if
*
* ^
By the definition of
surely.
e
£ Cw(X)|v-max{X'/3,-£}| < Cw(X)IIXII(IIPII + IIPqII) ^ Cw(X)IIXIl,
lq(z,/3)|
|q(z,p)-q(z,/3)|< Cw(X)
I
are Lipschitz in
max{v,--£}
respectively, so that
(A. 5)
.
does not
/3
= w(X)[q(y*-max{X'p,-£})-q(c)].
q(z,/3)
Note that
depend
q(e)
z = (y ,X),
follows that for
p = argmin^^^QO),
(A.4)
/3
and
l(y <0)q(0),
and adding terms to the objective function that do not depend on
change the minimum,
and
>
q(y-max{v,-£}),
q(y -max{v,-£y),
Then, since none of
on
q(r-r - [y +r-(y +r)])
for
a.
^
v.
Then, by
a
for
a £
also has a global
Therefore,
E[q(z,/3)|X] =
(A.6)
Now, for any
is
Prob(i4)
When
d(X)
global
minimum by
q{a,X)
0,
d =
it
{w(X)>0, v>-£,
d(X)
0,
>
and
X'|3
E[w(X)d(X)l(v>-£)(v-X' p)^] = 5'Q5
follows from the proof of Theorem 3.1 that
with derivative
d{X) - d q(a,X)/5a
so that
-d(X),
have a unique local minimum at
will
d
When
the previous argument.
a =
which
v,
max{X'/3,-£}
occurs,
v}.
?^
0,
>
E[m(y -a)|X]
2-
a = v
differentiable at
>
Also,
0.
>
> w(X)q(max{v,-n,X) = E[q{z,l3^)\X].
5 = /3-/3q ^ 0,
Assumption 3.2 implies that for
so that
/3,-£},X)
consider the event
P^
'^
/3
w(X}q(max{X'
is
|
a=v
a unique
= max{v,--£},
v
;*
2
so that
E[q(z,/3)|X] = w(X)q(max{X'|3,-£},X)
Then by
Prob(^)
>
=
Q{(3)
0,
= P(^)E[q(z,/3) U] + P(^^)E[q(z,/3) U^] =
E[q(z,/3)]
P(^)E[E[q(z,/3)|X]M] + P(^^)E[E[q(z,p)|X]M^]
)|X]M^] =
P(s4^)E[E[q(z,/3
We have now shown
Q(/3
Thus, the
).
w(X)q(max{v,-£},X) = E[q[z,(3^]\X].
>
that condition
i)
P(^)E[E[q(z,/3Q) X]
>
|
+
has a unique minimum at
QOq)
of
M]
Lemma
A.l holds.
Condition
ii)
/3
giving
follows by
*
equation
(A. 5).
Now, for condition
d -
let
iii),
{(y ,X):
v ^ -I
and
q{e)
is
*
continuously differentiable at
3.1
and
3.2.
derivative
at
On
a = max{v,--g>
q(y-max{X'
max{X'/3,-£}
d,
l(v>-£)X
at
/3
.
(3,-£})
is
and note
linear in
on
d,
-m(y-a).
/3
q(y-a)
Since
iv),
in a
is
Prob(4) =
by Assumptions
1
neighborhood of
/3
with
continuously differentiable
Then by the chain
differentiable with derivative
E[U(z)U(z)'] - E,
For condition
is
Also,
with derivative
-l(v>-£)m(E)X = U(z).
and
y -max{v,-£}},
on
rule,
d,
-m(y-max{v,-£})l{v>-£)X =
E[U(z)] = E[E[U(z)|X]] = -E[l(v>-£)E[m(E) X]X] =
|
condition
iii)
of
Lemma
A.l is satisfied.
note that as shown above
q(a,X)
is
differentiable with
*
derivative
3.1 that
-E[m(y -a)|X] = -E[m(E+v-a)
E[m(c+v-a)
|X]
is
|X].
differentiable in
-22-
Also,
a
it
with
follows as in the proof of Theorem
aE[m(c+v-a)|X]/aa = J^mlc+vjf (c+a|X)dc = Jm(c+v-a)f (e|X)dc
E
C
= E[m(c+v-a)s(c|X)|X],
s(g|X) =
where
(c |X)/f (e
f
SE[m{c+v-a) X]/9(x
that
X).
m(e)
Since
continuous almost everywhere,
is
continuous, so that
is
I
|
q(a,X)
follows
it
twice continuously
is
differentiable, and
q
(a,X) < Ca(X),
For notational convenience,
a(X) = E[
|
s(e X)
b = -I,
let
|
|
v = X'
X].
|
r
(3,
= max{v,b},
r = max{v,b},
and
*
*
on
suppress the
y
Then
.
a second order
mean-value expansion gives
E[q{z,/3)-q(z,/3Q)|X] = w(X)[q(r,X)-q(r,X)]
(A.7)
w(X)[q (r,X)(?-r) + q
(r,X)(?-r)^/2],
a
aa
where
r
lies
between
and
r
r.
Now, note that
r-r = l(v>b)l(v>b)(v-v) + l(v>b)l(v<b)(v-b) + l(v<b)l(v>b)(b-v)
= l(v>b)(v-v) + l(v>b)l(v<b)(v-b) + l(v<b)l(v>b)(b-v).
l(v>b)q (r,X) = l(v>b)q (v,X) = -l(v>b)E[m(E) |X] = 0,
Also,
b
and
so that for
v
between
v,
Iq
=
By
a
a
(r,X)(r-r)|
= |q (b,X)l(v>b)l(v<b)(v-b)
|
£
|q
(v,X) l(v>b)l(v<b) v-b
|
Ca(X)l(v>b)l(v£b)|v-v|^ < Ca(X)l(v>b)l{v<b)llXll^ll/3-/3
Prob(v=b) =
0,
l(v>b)l(v<b)
—»
convergence theorem and existence of
E[a(X)l(v>b)l(v£b)
11X11^1
-^
0.
w.p.l
as
E[a(X) 11X11
Therefore,
it
-23-
/3
2
]
—
>
/3
M b-v
11^.
so by the dominated
(as implied by
follows that as
|
p
—
>
Assumption
(3
3.1),
|
E[w(X)|q^(r,X)(?-r)|] < CE[a(X)l(v>b)l(v<b)IIXII^]ll/3-/3Qll^ =
(A.8)
0(11/3-/3^11^)
By similar reasoning,
(A.9)
E[w(X)|q
(A.IO)
E[w(X)lq
Also, as
/3
—
>
/3
E[w(X)q
(A.ll)
aa
(r,X)l(v>b)l(v<b)(v-b)^|] =
o(ll/3-/3_^ll^),
(r,X)l(v£b)l(v>b)(b-v)^|] =
0(11/3-/3^11^).
(r,X)
q
,
aa
—
>
so by the dominated convergence theorem,
d(X),
(r,X)l(v>b)(v-v)^] =
= (/3-/3q)'Q(/3-3q) +
O-^^)' E[w(X)q
aa
(?,X)l(v>b)XX' Kp-p^)
o(11P-/3qII^).
Taking expectations of equation
and applying the triangle inequality to equations
(A. 7)
(A.8) - (A.ll) then gives condition
The
iv).
from the
first conclusion then follows
*
conclusion to
Lemma
The second follows upon noting that
A.l.
Q = E[{d(X)/E[m(c)^|X]}d(X)l(v>-£)XX'] = E[SS']
Proof of Theorem 3.3:
l(X'/3 >
3.2,
(i
of
let
Note that
m(E)
if
-£)w(X)^m(y-X'/3)^XX'
and
is
w
Q.E.D.
].
differentiable then by Assumptions 3.1 and
l(X'/3 >
-£)w(X)m (y-X'/3)XX'
are continuous at
with probability one and dominated by functions with finite expectation.
Q
and
Q(z,/3)
Z
then follow by
Lemma
4.3 of
(X),
E =
and
E[{d(X)/E[m(G)^lX]}^E[m(c)^lX]l(v>-£)XX'] = E[SS'
w(X) =
if
Newey and McFadden
= w(X)XX'[q(y-max{X'/3,-£})-q(y-max{v,-£})],
(1994).
so that for
e
=
Consistency
In the other case,
(1,0,. ..,0)'
the first unit vector,
Q5 = Ij"^[Q(Zj,|3+ej6)+Q(z.,/3-e^5)-2Q(z.,p)]/(5^n).
Note that by equation
Then by
2.7.11 of
(A. 5),
IIQ(z,p)-Q(z,/3)ll
Van der Vart and Wellner
enough neighborhood of
p
for
Q(/3)
=
< Cw(X)llXll^llp-pil
(1996),
for any
E[Q(z.,/3)],
-24-
A
and
—
>
E[w(X)^IIXII^]
and
p,
/3
<
00.
in a small
^
suP|,^_pil^^lln"^^^I.^^{Q(z.J)-Q(z.,/3)-[Q(^)-Q0)]>ll
n
follows by
It
—
that
oo
>
= Q
Q^o = [Q(p+e,5)+Q{/3-e,5)-2Q(/§)]/5^
i
i
(A. 12)
Furthermore,
follows by equation
it
as defined in eq.
q(z,/3)
+ o
and the associated discussion that we can take
(A. 3)
Let
(A. 4).
(1).
p
= E[XX'q(z,/3)] = E[w(X)XX' q(max{X'
Q(/3)
for
1/2 2
5
/3,-£},X)].
= E[X.X
Q., (p)
in eq.
0)-Q.,
Q.,
finite by
M
.,
(/3
JKvJ
JK,
Noting that
that for
(A. 7)
and
r
It
follows by the
as defined there,
r
= E[w(X)X.X {q (r,X)(r-r) + q
JKCX
)
q(z,/3)].
"^
J
J"^
expansion
0.
(F,X)(?-r)^/2}].
OCOC
E[w(X)a(X)IIXll'^] < CE[w(X)a(X)^IIXII^] + CE[w(X)IIXII^] < CE[s(e X)^IIXII^] +
|
Assumption
3.1,
= E[w(X)d(X)l(v>-£)X
Jk
=
^r
XX'
.X,
k
J
Therefore, for
follows similarly to the proof of
it
(3-3
for
Lemma
A.l that for
]/2,
2
and noting that
,
iv)
C
ll^'+e 511
=0
2
(5
),
[Q .j^(/3+e^5)+Q .j^(p-e^5)-2Q .j^(p)]/5^
= [(^+e,5)'M., (^+e,6) + (j-e,5)'M., (^-e,5) - 2^'M., ^]/5^ + o
"
jk
1
= 2e'M.,
1
jk
e,
1
1
+ o
(1)
p
and
E[- |x] =
E
[•
|T,X] =
E
eq.
T =
[1{T)(
(A. 12).
^
l(y >0),
•
)
]
+ o
(1)
=
Q.,
^jk
p
(1)
p
+ o
p
(1).
Q.E.D.
^
^
^
note that E['] = E ['[T] = E [l(T)(-)]/P (T)
|X]/E [1(T)|X].
-25-
jk"
1
= E[w(X)d(X)l(v>-£)X.X,
J k
The conclusion then follows by
For truncated case and
jk
1
The following
Lemma
will be used i
i n
is
the proof of Theorem
Lemma
4.1:
If Assumptions 3A and
A.3:
are satisfied then
A.l
l(v^-k)S
e
3".
p
T^ =
Let
Proof:
s{c,X) = 1(T){s(e,X)-E [s(c,X) |T,X]}.
and
^ 0)
{y
E[S(e,X)|X] = E [S(e,X)|T,X] =
0.
E
Also,
s(e,X)
[11X11
= P
1
Note that
s(e,X)
(T)E[IIXII
]
= P*(T)E[IIXll^E[i(e,X)^|X]] = P*(T)E[IIXII^Var(s(£,X) X)] < P*{T)E[I1XII^E[s{e,X)^| X]] =
|
E
P*{T)E[IIXII^s(e,X)^] £
[IIXII^s(e,X)^] <
Consider
oo.
k
>
<
-v.
k
*
By the definition of
>
Let
0.
E
have
mic)
k,
nonzero on
will be
A(X) = E [(l(T),l(T^)m(E))'(l,m(E))|X].
[l(T'^)m(E) X] =
|
-E [l(T)m(£)|X],
Det(A(X)) = E [1(T )m{£)
£
-v.
c
2
[1(T )m{c)
X]
E
|
E [m(E)|X] =
we
0,
and hence
|X]E [1(T)|X] - E [1(T )m(£) X]E [l(T)m(£)|X]
|
|
-
Therefore,
Note that by
= E*[l(T^)m(£)^|X]E*[l(T)|X] + E*[l(T'')m(£) X]^
* ~
-1
D(X) = l{v^-k)E [s(£,X)(l,m(E)) X]A(X)
Let
k <
and the case where
M
,
|
>
(v
0,
< -k).
be some positive constant,
and
5(£,X) = -X-l(v<-^)-l(IID(X)ll£M)-[s(£,X) - D(X)(l(T),l(T^)m(£))'].
#
By the definition of
E
by
[m(E)^IIXIl^]
D(X),
<
oo
it
and
E
Hit
E [5(l,m(£))lX] =
follows that
[IIXII^s(£,X)^] <
oo.
By Lemma
0.
Also,
A. 5 of
E
[5' 5]
<
oo
Newey and Powell
*
(1990),
6
is
in the
tangent set
J
For
for the latent model.
5 e J,
let
^(5)
*
be the transformation from the latent to the observed data given by
By
Lemma
B.2 of
Newey
(1991),
the tangent set for the observed data
*
closure of
that
Also,
{S'(5)
1(T) -
E
:
6 e
J
[1(T)|T] =
e(s(E,x)) = s{e,X).
i?(6)
}.
Note also that 1(T
in
c
)
is
zero
the observed data, so that
Therefore,
by
S
in
in the
proof of
-26-
A.l.
the
mean square
e((l(T),l(T'^)m(£))) = 0.
= -X-s(£,X),
Lemma
is
the observed data, and
= l(v<-fc)-l(IID(X)llsM)-S
The conclusion then follows as
^{d) = 5-E [5|T].
Q.E.D.
Proof of Theorem
4.1:
Let
proof of Theorem
3.1,
E[6(E,X)(l,m(E)) X] =
X-w
S e
-
S
S.
5",
S = S
i.e.
X-w
-
t,
e
t
J.
By
Also, note that
3".
|
so that
Lemma
A. 3
^
^
it
then
E [1(T)S|X] =
0.
0,
^
E [1(T)S'{5(e,x)-E [6(e,X) |T]}]/P
^
S e
-
6(6) e
so that
oo,
the
in
(X)E [l(v>-ft)m(E) |X] = Xl(v>-fc)E [m(£)|X] =
^
E[S'E'(6)] =
6(6) = l(v>-A:)S
E[5(e,X)m(£) X] =
satisfying
6
E[ll5(c,X)ll^] <
l(v>-/t]m(c) = UT)\{v>-k)m{c),
A:,
Therefore,
(X)E [l(T)l(v>-/l)m(E)|X] =
Therefore, for any
c <
zero for
e(X-l(v>-£)-w (X)m(E)) =
follows that
and
|
m(c)
Note that by
5".
As shown
be as defined in equation (A.l).
5(c,X)
(T)
#
^
= E [1(T)S'5(£,x)]/P (T) - E[1(T)S'E [5(e,X) T]]/P (T)
|
= E [E [S'5(e,x)|X]]/P (T) = E[Xl(v>-^)E [m(E)6(E,X) |X]]/P (T) =
J
Then, since
is
the mean-square of objects of the form
orthogonal to the tangent
Proof of Theorem 4.2:
proof of Theorem
|q(z,/3)|
It
follows that
E
is
constant for
c
l(T'^)q(^).
3.1
q(z,/3)
that for
and
follows that
S
is
CE
It
follows as in the
(positive or negative),
|q(z,p)-q(z,/3)|< Cw(X)IIXII
[l(T^)|q(z,p)|] <
k
= w(X)[q(y-max{X'/3,-^l)-q(E)].
y
all
£ Cw(X)IIXII,
:^
it
Q.E.D.
set.
Let
6(6),
0.
[l(T^)w(X)IIXll]
-maxiX' p,-k}) £
k,
is
11^-/311.
Also, since
finite.
q(£)
1(T )q(y-max{X'/3,-£}) =
we have
Therefore,
E[q(z,p)] =
E
[l(T)q(z,/3)]/P (T) = E*[q(z,/3)]/P*(T) - E*[l(T^)q(z.p)]/P*(T)
= E*[q(z,/3)]/P*(T) - E*[l(T^)w(X){q(^) - q(E)}]/P*(T).
That
the
is,
E[q(z,/3)]
maximum
= C^E
[q(z,/3)]
+
C^
for constants
C
and
>
C
It
follows that
*
of
E[q(z,/3)]
will coincide with the
-27-
maximum
of
E
[q(z,/3)],
which has a
unique
is
maximum
at
(3
constant below
the properties
E[q(z,/3
+
)]
conditions
in the
Lemma
k,
has
E[q(z,/3)]
ii),
and
in
proof of Theorem
0(11/3-/3^11
3.1,
In particular,
Condition
A. 2.
of
iii)
so that the conclusion follows
Proof of Theorem
By equation
5.1:
p = p
the latent data at
m
E[q(z,/3)]
=
Thus,
3.1.
A. 2 also follows as
from the conclusion of
Follows by extending the results for the censored case to the
4.3:
truncated case analogously to the proof of Theorem 4.2.
d
inherits all
Theorem
Lemma
q(c)
Q.E.D.
Proof of Theorem
Theorem
[q(z,p)]
as in the proof of
),
Furthermore, since
3.1.
E
and so
k,
the censored case.
Lemma
of
iv)
proof of Theorem
in the
also linear below
is
it
0-/3 )'Q(/3-/3 )/2 +
i),
A. 2.
shown
as
,
5.1
so that
,
Also, by independence,
3.2.
E[w(X)l(v>-£)XX'
and
]
the conditional
Vn{^-J3
d
m
]
-^
= d(X)
m
second conclusion, note that for
moment
restriction
N(0,Q~ ZQ~
],
Y = w(X)l{v>-£)X
satisfied in
is
by the conclusion of
)
E[m{e-u
and
E = o-^E[w(X)^l(v>-£)XX'
Q.E.D.
2
m
)
|X]
=
2
cr
m
,
so that
giving the first conclusion.
and
U =
For the
l(v>-£)X,
(E[w(X)l(v>-£)XX' ])~^E[w(X)^l(v>-£)XX' l(E[w(X)l(v>-£)XX' ])"^-(E[l(v>-£)XX'
= (E[YU'])~^E[YY'](E[UY'])"^ - (E[UU'
Q
])"^
l)"'^
= (E[YU'])"^{E[YY'] - E[YU'](E[UU'])~^E[UY']}(E[UY'])~\
that
is
positive semi-definite by the matrix in the angle brackets being positive semi-
definite.
For the third conclusion, note that similar to the proof of Theorem 3.3,
a =
of
y.
^1=1
Q
X
—^
E[l(v>-£)] = a,
while
1
and
(Q_),
o
—
a«d
of
l(v.>--g)/n
>
a-d
a-d
m
follow as
a-d
m
the upper left element of
in
the proof of
Theorem
(a-d
Q.
3.3,
m5^)
is
Then
the upper left element
a-d
mm
—
^
a«d
and
except that here the sixth moments
are not needed to exist by virtue of only requiring convergence of the upper left
element of
Q^.
o
Q.E.D..
-28-
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