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working paper
department
of
economics
Instrumental Variables Estimates of the
Effect of Subsidized Training
on the
Quantiles of Trainee Earnings
Alberto Abadie
Joshua Angrist
Guido Imbens
No. 99-16
October 1999
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
WORKING PAPER
DEPARTMENT
OF ECONOMICS
Instrumental Variables Estimates of the
Effect of Subsidized Training
on the
Quantiles of Trainee Earnings
Alberto Abadie
Joshua Angrist
Guido Imbens
No. 99-16
October 1999
MASSACHUSETTS
INSTITUTE OF
TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASS. 02142
PliACHuSETTS INSTITUTE
OF TECHNOLOGY
LIBRARIES
Instrumental Variables Estimates of the Effect of
Subsidized Training on the Quantiles of Trainee
Earnings*
— MIT
Alberto Abadie
— MIT and NBER
Guido Imbens — UCLA and NBER
Joshua Angrist
Revised: September 1999
Abstract
The
effect of
government programs on the distribution of participants' earnings
important for program evaluation and welfare comparisons. This paper reports
timates of the effects of
The estimation
uses a
gram impacts on the
JTPA
training programs
new instrumental
quantiles of
(QTE) estimator accommodates exogenous
sion
when
selection for treatment
is
variables.
covariates
linear
where the
empirical results
at low quantiles.
of earnings for
first
step
is
effects
and reduces to quantile
regres-
We
show that the JTPA program had the
men
QTE estimator can
JTPA
this
develop distribution theory
estimated nonparametrically.
Perhaps surprisingly, however,
pro-
This quantile treatment
programming problem, although
requires first-step estimation of a nuisance function.
for the case
method that measures
exogenously determined. The
be computed as the solution to a convex
es-
on the distribution of earnings.
variable (IV)
outcome
is
For women, the
largest proportional
impact
training raised the quantiles
only in the upper half of the trainee earnings distribution.
*We thank Moshe Buchinsky, Gary Chamberlain, Jinyong Hahn, Jerry Hausman, Whitney Newey,
Shlomo Yitzhaki, and seminar participants at Berkeley, MIT-Harvard, Penn, and the Econometric Society
Summer 1998 meetings for helpful comments and discussions. Thanks also go to Erik Beecroft at Abt
Associates for providing us with the National JTPA Study data and for helpful discussions. Abadie
acknowledges financial support from the Bank of Spain. Imbens acknowledges financial support from the
Sloan Foundation.
1.
Effects of
many
Introduction
economic variables on distributions of outcomes are of fundamental
A
areas of empirical economic research.
government programs
leading example
interest in
the question of
is
how
affect the distribution of participants' earnings, since the welfare
analysis of public policies involves distributions of outcomes. Policy-makers often hope that
subsidized training programs will reduce earnings inequality by raising the lower quantiles
of the earnings distribution
of
and thereby reducing poverty (Lalonde (1995), US Department
Labor (1995)). Another example from labor economics
distribution of earnings.
unionism
Fortin,
is
Freeman
One
is
the effect of union status on the
of the earliest studies of the distributional consequences of
(1980), while
more recent analyses include Card
and Lemieux (1996), who have asked whether changes
in
(1996),
and DiNardo,
union status can account
a significant fraction of increasing wage inequality in the 1980s.
for
Although the importance of distribution
effects is
widely acknowledged, most evaluation
research focuses on average outcomes, probably because the statistical techniques required
to estimate effects on
restrict
the
means
are easier to use.
Many
econometric models also implicitly
treatment effects to operate in the form of a simple "location shift"
mean
effect
captures the impact of treatment at
treatment on a distribution
and there
is
is
easy to assess
all
quantiles.
when treatment
perfect compliance with treatment assignment.
Of
status
,
in
which case
course, the impact of
is
randomly assigned
Randomization guarantees
that outcomes in the treatment group are directly comparable to outcomes in the control
group, so valid causal inferences can be obtained by simply comparing the treatment and
control distributions.
in
The problem
of
how
to
draw
randomized studies with non-compliance or
assignment
is
more
In this paper,
difficult,
inferences about distributional effects
in observational studies
however, and has received
we show how
less attention.
with non-random
1
to use a source of exogenous variation in treatment status
Rosenbaum and Rubin (1983), and
Manski
Heckman,
Smith
and
Clements
(1985).
(1994),
(1997), Imbens and Rubin
and
Abadie
discuss
(1999a)
effects on distributions. Manski (1994, 1997) develops estimators for
(1997),
bounds on quantiles.
'Discussions of average treatment effects include Rubin (1977),
Heckman and Robb
-
an instrumental variable - to estimate the
effect of
treatment on the quantiles of the
distribution of outcomes in non-randomized studies, or in situations where the offer of
treatment
is
randomized but treatment
(QTE) estimator
is
itself is
used here to estimate the
voluntary. This Quantile Treatment Effects
effect of training
on trainees served by the
Job Training Partnership Act (JTPA) of 1982, a large publicly-funded training program
designed to help economically disadvantaged individuals.
JTPA
The data come from the National
Study, a social experiment begun in the late early 1980s at 16 locations across the
JTPA
to evaluate the effects of
training. For this study,
JTPA
US
applicants were randomly
assigned to treatment and control groups. Individuals in the treatment group were offered
JTPA
training, while those in the control
group were excluded
Only 60 percent of the treatment group actually received
treatment assignment as an instrument
The treatment
subpopulation we
effects
in the control
of effects
affected
The
like
the
This terminology
JTPA, the
by an instrumental
Rubin
is
fact, in
(1996).
used because in randomized
the case of the
effects for
JTPA, where
who
(almost)
is
variable.
established by Imbens
approach to instrumental variables
and Angrist (1994) and Angrist, Imbens, and
Imbens and Rubin (1997) extended these
results to the identification of
hypotheses about distribution impacts such as stochastic dominance.
papers developed simple estimators or a scheme for estimating the
2
a
trials
whose treatment status
how
the effect of treatment on distributions, and Abadie (1999a) showed
quantiles.
for
compilers are also representative
cases, compilers are those
identification results underlying the compilers
first
we can use the
relevant subpopulation consists of people
group received treatment,
on the treated. 2 In other
(IV) models were
training, but
for treatment.
always comply with the treatment protocol. In
no one
a period of 18 months.
estimated using the framework developed here are valid
call compilers.
with partial compliance,
for
to test global
But neither
effect of
of these
treatment on
We focus here on conditional quantiles because quantiles provide useful summary
Angrist and Imbens (1991) discuss the relationship between instrumental variables and effects on the
treated.
in the
Orr, et
al.
(1996) and
Heckman, Smith, and Taber (1994) report average
JTPA. Heckman, Clements and Smith
using a non-IV framework.
(1997) estimate the distribution of
effects
JTPA
on the treated
treatment
effects
statistics for distributions,
and because quantile comparisons have been
recent discussions of changing
wage inequality
Murphy
(1994)).
and Buchinsky
(1992)
The paper
organized as follows.
is
discusses the identification problem.
(see, e.g.,
at the heart of
Chamberlain (1991), Katz and
Section 2 outlines the conceptual framework and
Section 3 presents the estimator, which allows for a
binary endogenous regressor (indicating exposure to treatment) and reduces to Koenker
and Bassett (1978) quantile regression when
selection for treatment
is
exogenous.
Like
quantile regression, the estimator developed here can be written as the solution to a convex
linear
programming (LP) problem, although implementation
estimation of a nuisance function in a
of the
QTE estimator requires
Finally, Section 4 discusses the estimates of
first step.
on the quantiles of trainee earnings. The estimates
effects of training
for
women show larger
proportional increases in earnings at lower quantiles of the trainee earnings distribution.
But the estimates
men
for
suggest the impact of training was largest in the upper half of
the distribution and not at lower quantiles as policy-makers perhaps would have wished.
Conceptual Framework
2.
The setup
outcome
is
as follows.
variable,
The data
consist of
Y, a binary treatment indicator D, and a binary instrument, Z. In the
case of subsidized training,
Y
is
earnings,
D
indicator of the randomized offer of training.
not everyone
n observations on a continuously distributed
who was
program participation, and
indicates
Z
offered training received
and
it
D
are not equal in the
status, say
a
D would indicate union status,
dummy
indicating individuals
and
who work
Z
As
in
Rubin
of causal effects,
Y
an
because
might be a
would be an instrument
in firms that
organizing campaigns (Lalonde, Marschke and Troske (1996)).
vector of covariates,
is
and because a few people who were not
offered training received services anyway. In a study of the effect of unions,
measure of wages,
JTPA
Z
We
for
union
were subject to union
also allow for
an r x
1
X.
(1974, 1977)
we
and our
earlier
work on instrumental variables estimation
define the causal effects of interest using potential
outcomes and
potential treatment status.
D,
Yd,
we
In particular,
outcomes indexed against
define potential
and potential treatment status indexed against Z,
D
z
.
Potential outcomes and
potential treatment status describe possibly counterfactual states of the world. Thus,
tells
us what value
take
if
had
D =
Z
D would take Z were equal to
if
were equal to
d.
The
0.
Similarly, Yd tells us
while
1,
Do
tells
us what value
D\
D would
what someone's outcome would be
if
they
objects of causal inference are features of the distribution of potential
outcomes, possibly restricted to particular subpopulations.
The observed treatment
status
is:
D = D + (A In other words,
if
Z=
1,
then D\
the observed outcome variable
is
observed, while
1
terfactual
causal inference
is
difficult is
outcomes as being defined
Z = 0,
then
D
observed. Likewise,
is
-(1-D).
for
(1)
that although
for everyone,
one potential outcome are ever observed
we think
of
all
possible coun-
only one potential treatment status and
any one person. 3
Principal Assumptions
2.1.
The
principal assumptions of the potential outcomes
Assumption
2.1:
For almost
all
values of
Independence; (Yi,Y ,Di,D
(i)
(ii)
)
is
jointly independent of
^ E[D
\X).
D
=
First-Stage: £[Di|X]
(iv)
MONOTONICITY: P(D >
The
1
\X)
framework
for
IV
are stated below:
X,
Non-Trivial Assignment.- P(Z = 1\X) e
(hi)
3
if
Z.
is:
Y = Y -D + Y
The reason why
Do)
Z
given
X.
(0, 1).
1.
idea of potential outcomes appears in labor economics in discussions of the effects of union status.
See, for example, Lewis' (1986) survey of research
on union
relative
wage
effects.
Assumption
ment status
subsumes two related requirements.
2.1(i)
First,
identify the causal effect of the instrument. This
is
comparisons by instru-
equivalent to instrument-
error independence in traditional simultaneous equations models.
comes are not directly affected by the instrument. This
Angrist, Imbens
and Rubin (1996)
how they
Assumption
JTPA
differ.
is
Second, potential out-
an exclusion
two requirements and
for additional discussion of these
2.1(i) is plausible
because of the randomly assigned
See
restriction.
(though not guaranteed) in the case of the
offer of
treatment.
Assumption 2.1(h) requires that the conditional distribution of the instrument not be
degenerate.
in
The
two other ways as
some
and treatment assignment
relationship between instruments
well.
correlation between
As
D
in simultaneous equations models,
and Z;
this
is
effect in
and
it is
D in one direction.
Monotonicity
assumption
for the
a
plausible in
most applications
JTPA, where
D =
It is
for (almost) everyone.
and coun-
on observed covariates, X. For example,
evaluation studies focus on estimating the difference between the average outcome
(which
of treatment (which
A
is
inference problem in evaluation research involves comparisons of observed
for the treated
4
2.1(iv) guarantees identification of
This monotonicity assumption means that the
terfactual outcomes, possibly after conditioning
many
Imbens
4
automatically satisfied by latent-index models for treatment assignment.
also a reasonable
The
Also,
any model with heterogeneous potential outcomes
that satisfies assumptions 2.1(i)-2.1(iii).
instrument can only affect
require that there be
stated in Assumption 2.1 (hi).
and Angrist (1994) have shown that Assumption
meaningful average treatment
we
restricted
is
latent-index
is
is
model
observed) and what this average would have been in the absence
counter- factual).
Outside of a randomized
trial,
the difference in
for participation is
D= 1{A
where Ao and Ai are parameters and 77
D\ — l{Ao + Ax > 77}, and either D\
is
t?
an error term that
> Do
everyone, then monotonicity holds for Z'
+ Z-A! -
or
=1—
Dq > D\
Z.
>
is
0}
independent of Z. Then
for everyone.
If
Ax
<
Do = l{Ao >
Dq > D\
so that
t]},
for
average outcomes by observed treatment status
E[Y \X,D =
1
1]
- E[YQ \X,D =
0]
is
typically a biased estimate of this effect:
= {E[Y \X,D =
1]
- E[YQ \X,D=
D=
1]
- E\Y
1
+ {E[Y
The
first
also
be written as E\Y\
is
term
in brackets
the bias term.
is
\X,
\X,
D=
1]}
0}}.
the average effect of the treatment on the treated, which can
— Yq\X,D =
1]
since expectation
is
a linear operator; the second
For example, comparisons of earnings by training status are biased
trainees are selected for training
if
on the basis of low earnings potential. This bias extends
to comparisons other than the mean.
For example, the relationship above holds
if
we
replace conditional expectations with conditional quantiles.
2.2.
An
Identification Using Instrumental Variables
instrumental variable solves the problem of identifying causal effects for a group of
individuals
whose treatment status
is
affected
The
by the instrument.
following result
(Imbens and Angrist (1994)) captures this idea formally:
Lemma
2.1:
Under Assumption
(and assuming that the relevant expectations are
2.1
Z = \\- E[Y\X, Z = 0]
= E[Y
E[D\X, Z = 1] - E[D\X, Z = 0]
E[Y\X,
1
E[YX
-Y \X,Di
to individuals for
>
D
]
called a Local
is
whom D\ > Dq
as compliers because in a
whatever their assignment. In other words, the
whose treatment status was changed
cannot be identified
(i.e.,
we never observe both D\ and Do
where Do
E[Y1
=
in the
randomized
who comply with
trial
We
with partial
the treatment protocol
experiment induced by
Z
.
Note that individuals
are compliers) because
any one person. Also note that
in the special case
for everyone,
-Y \X,D >D
l
Q]
= E[Y
-Y \X,D =
= E[Y
-Y \X,D =
l
l
refer
set of compliers is the set of individuals
we cannot name the people who
for
}.
1
Average Treatment Effect (LATE).
compliance, this group would consist of individuals
in this set
-Y \X,D >D
finite)
l
l]
l],
= E[Y
1
-Y \X,D
1
= l,Z=l]
D
LATE
so
The equivalence between
the effect of treatment on the treated.
is
compliers and effects on the treated in cases where
distributional characteristic
The compliers concept
and not
is
any
identically zero holds for
just means.
LATE
at the heart of the
is
Do
effects for
explanation for
how IV methods
work.
compliers are.
For these people,
Z —
Suppose
D, since
framework and provides a simple
initially
it
is
that
we could know who the
D > D
always true that
x
.
This
observation plus Assumption 2.1 leads to the following lemma:
Lemma
Given Assumption 2.1 and conditional on X, treatment
2.2:
(independent of the potential outcomes) for compliers: (Yi, Yo) -L D\X,
Proof: Assumptions
1,
D = 0. When D —
x
A
that (YU
2.1(i) says
1
D =
and
Lemma
consequence of
2.2
is
0,
D
Y
,
uD
)
±
assignment
is
X
course, as
compliers
is
it
stands,
Do]
(i.e.,
2.2 operational,
= P(Z —
This function
Lemma
2.3:
±
)
.
Z\X,D =
1
D
can be substituted for Z.
is
:
D=
2.2 is of
effect
0,D >
X
even though treatment
E[Y,
]
we
it
(Abadie, 1999b) Let
-
Y \X,D
for the
define the following function of
D-(l-Z)
(l-D)-Z
i-ttoPO
MX)
when
"identifies compliers" in
h(Y,D,X)
be
D,
same
Z
x
Do]
=
>
D
}.
(2)
'
To
{S}
'
D = Z,
otherwise k
is
negative.
the following average sense:
any integrable
p7p^~^)
individual).
and X:
real function of
Then, given Assumption 2.1,
E[h{Y,D,X)\D >
1
no practical use because the subpopulation of
1\X). Note that k equals one
useful because
D =
we do not observe D\ and Dq
_
where n (X)
DQ
ignorable
that in the subpopulation of compliers, comparisons
- E[Y\X
Lemma
not identified
make Lemma
>
x
is
not ignorable in the population:
E[Y\X,D = 1,D >
Of
D
Z\X, so (Yi,Y
means by treatment status estimate an average treatment
of
D,
status,
E[k h(Y, D,X)}.
(Y,D,X).
To
why
see
this
into three groups:
is
true, note that,
who have D\ >
compilers
= Dq =
and never-takers who have D\
E[h(y,D,X)\X,Di>D
by monotonicity, the population can be partitioned
0.
= p(D^D^\X~) E[h{Y D X)lX]
{
'
]
X, we have the
Z=
Z =
individuals with
all
D=
and
1
'
=D =
£[/i(F,D,X)|X,A
Likewise, those with
1,
Thus,
E[h{Y,D,X)\X~Di
Monotonicity means that
who have D\ = Dq =
Dq, always-takers
1]
= Z> =
1
P(A = D =
•
0]
•
P(D = D =
1
D =
and
1|X)
0\X)\.
must be never-takers.
must be always-takers. Since
Z
is
ignorable given
and never-takers as a function of
following expressions for always-takers
observed moments:
E[h(Y,D X)\X,Di
t
= A) =
E[h(Y,D,X)\X,D =
1
= E[h{Y,D,X)\X,D = 1,Z = 0]
D-(l-Z)
1
h(Y,D,X) X
E
P{D = 1\X,Z = 0)
1 " TTo(X)
l]
D = 0] =
E[h(Y, D,X)\X,
D=
0,
1
P(£ = 0|X,Z =
P{D\
= Dq =
An
and never-takers using P(D\
0\X)
= P(D =
implication of
condition involving
Abadie (1999b). 5 In the next
5
For example,
if
we
define
fi
(fi,
—
a)
>
Dq], and
identified
fi
E\Yq\D\
is
1
~ D)
-
TTo(X)
1)
WAX)
A'
is
Z—
1).
= D =
1\X)
Integrating over
X
= P(D = l\X,Z =
identified for compliers. This point
we show how Lemma
and
completes the argument.
that any parameter defined as the solution to a
section,
0)
2.3
is
moment
explored in detail in
can be used to develop an
and a as
same intercept that is
E[k (Y - m - aD) 2 }.
then,
0\X,
Lemma 2.3
(Y,D,X)
P
== 1]
Z given X can similarly be used to identify the proportions
Monotonicity and ignorability of
of always-takers
Z
=
argmin (ma) E[(Y
-
m - aD)
a — E\YX - Yq\D\ > Do], so
by conventional IV methods).
8
2
\Di
>
D
],
a is LATE (although fi
By Lemma 2.3, (/u,a) also
that
is
not the
minimizes
estimator for the causal effect of treatment on the quantiles of an outcome variable.
Quantile Treatment Effects
3.
The QTE Model
3.1.
The
QTE
linear
estimator
and additive
analysis
is
based on a model where the
is
when the treatment
model because the resulting estimator
when
regression
is
there
3.1:
For 6 £
Qg(Y\X,D,Di >
As a consequence
(0, 1),
there exist unique ag
of
D
)
JTPA
Lemma
2.2,
we
training changed the
G
R
and
(3 e
G
W such
that
= a e D + X%.
and Yq
ofY
given
(4)
X
for compliers.
median earnings
and
D
for compliers.
This
tells us, for
example,
of participants. Note, however, that
where average differences equal differences
in
may also be of
focus on the marginal distributions of potential outcomes because identification
social welfare
—
Yq requires
much
stronger assumptions and because economists
comparisons typically use differences in distributions and not the
distribution of differences for this purpose (see,
6
IV and ordinary
not the quantile of the difference (Yi— Yq). Although the latter
of the distribution of Y\
making
and quantile
the parameter of primary interest in this model, ag,
in contrast with average treatment effects,
is
)
denotes the 9-quantile
gives the difference in the ^-quantiles of Y\
interest,
QTE
of interest are defined as follows:
1
averages, ag
Koenker and Bassett (1978) quantile
simplifies to
Q e {Y\X, D, D > D
whether
The
with X, but we use an additive
no instrumenting. The relationship between
is
estimated.
is
(OLS).
The parameters
where
effect varies
is
therefore analogous to the relationship between conventional
least squares
Assumption
treatment and covariates
at each quantile, so that a single treatment effect
straightforward
regression
effect of
e.g.,
Atkinson (1970)). 6
Heckman, Smith and Clements (1997) discuss models where features of the distribution of the difference
— Yq) are identified. They note that this may be of interest for questions regarding the political
economy of social programs. If the ranking of individuals in the distribution of the outcome is preserved
(Y\
The model above
differs in
a number of ways from the model in the seminal papers
by Amemiya (1982) and Powell (1983), who used
least absolute deviations to estimate
a simultaneous equations system. Their approach begins with a traditional simultaneous
equations model, and
Rather, the idea
tailed.
is
not motivated by an attempt to characterize effects on distributions.
to improve
is
Most importantly,
of interest in the
function.
on 2SLS when the distributions of the error terms are longwith the parameters in equation
in contrast
Amemiya/Powell setup do
(4),
the parameters
not, in general, define a conditional quantile
7
The parameters
as (see Bassett
of the conditional quantile function in equation (4) can be expressed
and Koenker
{a e ,Pe)
where p e (X)
is
=
ai*g
(1982)):
mm
(a,/3)eM'-+ 1
i
the check function, defined as p e (A)
Therefore, using
Lemma
2.3,
{a e ,Pe)
=
ae and
minimand
fi e
=
(9
is
— 1{A <
A)],
0})
•
A
for
- aD -
(5).
globally convex in (ag,P e ) since
for compilers times
However, since k
is
some
negative
objective function turns out to be non-convex.
tion problems of this type (piecewise linear
do not ensure a global optimum
real A.
X'0)].
positive constant
(5)
it is
(see, e.g.,
when
D
A number
is
equal to the
{P{D\ > Do))-
lowing the analogy principle (Manski (1988)) a natural estimator of (ag,/3 g )
counterpart of
any
are identified as
argmin (Q/3)eMr+1 E[k p e {Y
This population objective function
check-function
E [Pe( Y ~ aD ~ x 'P)\ D >
is
Fol-
the sample
not equal to Z, the sample
of algorithms exist for minimiza-
and non-convex objective functions), but they
Charnes and Cooper (1957) or Fitzenberger
(1997a,b), for a discussion of a related censored quantile regression problem). Unlike the
under the treatment, then the estimator
impacts.
in this paper is informative about the distribution of treatment
King (1983) discusses horizontal equity concerns that require welfare analyses involving the joint
distribution of outcomes.
7
Amemiya and Powell papers comes from conditional median restrictions on the
However, a conditional median restriction on the reduced form does not imply that the
Identification in the
reduced form.
is a conditional median. In fact, for a binary endogenous regressor, conditional median
on the reduced form and structural equation are typically incompatible.
structural equation
restrictions
10
—
conventional quantile regression minimand, the sample analog of equation (5) does not have
a linear programming representation.
Now,
let
U=
(Y,D,X); applying the Law
of Iterated Expectations to equation (5),
we
obtain
{<*e,Po)
=
argmin (Q)/3)er+ i E[k v p g (Y
- aD -
X'j3)],
(6)
where
kv
u (U)
for
=
E[Z\U]
=
E[k\U\
= P(Z =
is
and
therefore non-negative.
Lemma
1\Y,
Under Assumption
3.1:
Do =
By
E[D-(l-Z)\U]
D =
1
—j
we show below, k u
= P{D > D
ku (U)
2.1,
First consider the product
monotonicity,
(i-D)-MU)
D, X). Although simple to derive,
Proof:
1.
—--^t
D-(1-MU))
1
of signal importance because, as
tation
is
=
D
—
(1
implies
= P(D(1 -
X
x
=
= P(A = D =
n v {U)
= E
1
D =
Z=
and
1\U)
l\U)
P{Z =
0\U)
^D
Q\D 1
P(Z = 0|A =
1
1
if
Hence:
1.
= P(D = D = 1\U)-P{Z =
- D) Z\U] = P(D =
a conditional probability
is
\U).
= P{D 1 = D Q = l\U)-P{Z =
Similarly, E[{1
second represen-
Z). This differs from zero only
D =
Z)
this
D =
l,U)
l,Ylt X)
0\X).
l\X). Therefore,
D(l-Z)
(l-D)Z
U
P(Z = 0\X)
P(Z = l\X)
- P(A = D = l\U) - P{D = D =
X
=
0\U)
==
P{D > D
X
\U).
a
A
consequence of this lemma
LP
is
that
it
is
possible to develop a
representation based on a sample analog of equation
11
(6)).
QTE
estimator with an
This can be thought of as
a scheme to "convexify" the sample analog of
(5).
The
resulting convex
QTE
estimator
minimizes a positively- weighted check-function minimand, with a global
minimum
be obtained as the solution to a linear programming problem
number
iterations.
This
is
in a finite
that can
of simplex
and Hahirs (1998) LP-type estimator
similar in spirit to Buchinsky
for
censored quantile regression.
An
interesting question
is
whether there
based on k u instead of the sample analog of
gies
The
distribution theory
is
non-parametrically in (X,
D)
(X).
7r
cells.
9
If
We
the
an estimator
can be shown that both strate-
it
In light of this result, the
and v Q (U) to construct an estimate of
developed assuming that
model consistently estimates
ear
In fact,
(5).
paper focuses on estimation by minimizing the sample analog of
quires first-step estimation of iro(X)
.
efficiency cost to using
produce estimators that are asymptotically equivalent. 8
rest of the
ku
any
is
X
is
k. u
(6).
This
re-
(U), denoted
discrete, so a saturated lin-
use series approximation to estimate v Q (U)
number
of terms in the series approximation
increases at an appropriate rate with the sample size, this procedure ensures that the esti-
mated conditional expectations converge
however,
it
may make
dimensionality of
when the
3.2.
X
is
to the true conditional expectations. In practice,
sense to use something less than a saturated model for
high.
We
the
discuss practical aspects of the estimation strategy further
results are presented in Section 4.
that
we have a random sample
(ao,f3 e )' for ag
and
j3 e
in equation (4).
{Y;,
If
D
z
,
Xi, Zi}™=1
.
Let
W
=
(1990).
Since k u
is
unknown, we estimate
See Newey (1994); the estimators
same functional.
(D,X')' and 6g
—
k u were known, the estimation problem would
reduce to a weighted quantile regression problem of the type discussed by
the
when
Estimation
Assume
8
X
this function
Newey and
nonparametrically in a
Powell
first
step
are asymptotically equivalent since they nonparametrically estimate
9
Buchinsky and Hahn (1998) similarly decompose the covariates in a censored quantile regression problem into a set of discrete variables and a set of continuous variables. They use non-parametric methods to
estimate the conditional probability of censoring in
cells
12
defined by the discrete variables.
and use the
fitted values k„([/i) in a
second step to construct the estimator:
n
%
=
1
- ]T 1{k„(^) >
argmin, GRr+1
0}
«„(^) p e (y, - W/5),
•
•
(7)
77,
First step estimation of
kv
is
carried out using non-parametric series regression.
increasing sequence of positive integers {A(fc)}^=1
(Y x
(
lj
,
...,Y
X(
K)). Assume
that
only takes on a
finite
=
Then, any random sample {Vi}"=1
{wi,...,wj}).
3
be indexed
X
and a positive integer K,
as {{Vi } i =1 }j =1
j
(X, D). In the
same
,
where {Vi j }
fashion, the
number
{(Z{,
let
For an
K
p (Y)
=
W
G
of values (so that
U )}f=1
V =
from
l
(Z,U) can
J
i
=1 are subsequences for distinct fixed values of
sample can be indexed as {{Vi^L^iLi, where
{l^,}£'= i are
subsequences for distinct fixed values of X. Now, a nonparametric power series estimator
v(U) of vq(U)
is
given by the Least Squares projection of {Z, }™ J=1 on {p K (Yi.)}™ 3=1 (this
amounts to non-parametric
Z
series regression of
Y
on
in each W-cell). Let u t
be the
fitted
values of such estimator for the observations in our sample. Consider the simple estimator
n(X)
of
ir
Z
(X) obtained by averaging
within
X. Our
cells of
first
step estimator of k u
is
given by:
^)~
~(TT\-1
K
1
3.3.
^-(1-%)
(l-A)-Pi
l-rr {Xl )
tt(^)
•
Distribution Theory
This subsection summarizes asymptotic results for the
QTE estimator.
Proofs are given in
the appendix.
Theorem
on W,
Y
3.1:
is
Under assumptions
and
and
3.1
if (i)
number
7r
(X)
is
i.i.d.; (ii)
bounded away from zero and one, and
of values; (iv) conditional on
W
,
eg is
conditional
differentiate at zero with density fee \w,Di>D {ty that
W;
(v)
kv
is
is
W
and D\ > Do
takes on a
is
continuously
bounded and bounded away from zero
bounded away from zero uniformly in
13
X
continuously distributed with bounded
density; the distribution function of eg conditional on
uniformly in
the data are
continuously distributed with support equal to a compact interval and density
bounded away from zero; (Hi)
finite
2.1
U
;
(vi)
for s equal to the
number of continuous
^ AA(0,O),
6e )
where
Y
derivatives in
Q = J-'ZJ"
1
ofu
K
n
2s
—
and
>
K
5
/n
—
Then, n l l 2 (6 e
P(D > D
1
]
\
andZ = E[W] with^ = K-m(U)+H(X)-{Z-7r
-MX))
(I
The asymptotic
Assumption
To produce an estimator
loss.
...
6"
~1 v ( Yj Wj
=
<pUQ h {-^~
...
.
a kernel function. Consider the following estimator of J:
J
1=1
i
robust to mis-specification
10
and
K
J=-Y
n
For
is
X
of the asymptotic variance matrix, let
(Y-W'8\
1
MS) = -h v {—h—)
is
in the /-cell of
X
,
v {Ui
)-^(8e )-W Wi
i
let
ni
Hi
=
k(K)
&
=
~ |> ~ Hn
=
)
In such a case, quantile regression estimates
3.1).
the best linear predictor under asymmetric
</?(•)
(1-D)-Z
(MX)) 2
variance formula provided by this theorem
of the functional form (in
where
2
—
m{U) = {e-l{Y-W6 9 < 0})-W,
(X)},
D-(l-Z)
H(X) = E m(U)
An
0.
>
= E[feg w>Dl>Do (0) WW'\D, > D
J
,
,
i
- Dt
•
(1
-
-
W& <
Zi)
l-7f(Xi)
K{Vi)
estimator of
(6
E
(1
-
0})
D
•
W
k
Z
t)
{1
(
~ Dk)
WW
'
Zil
Di
;i
"Z
- n(x )y
"
'
(1
*'
}
t
r
'
7?(Xi)
- 1{Y - w(6 e <
0})
•
Wi
+
H
t
{Z x
Srpfc)}.
can then be constructed as
n t—'
i=\
The
following theorem establishes the consistency of an asymptotic covariance matrix es-
timator.
10
Most
on quantile regression treats the linear model as a literal specification for condimodel can be viewed as an approximation. This interpretation is
discussed by Buchinsky (1991), Chamberlain (1991), Fitzenberger (1997), and Portnoy (1991).
of the literature
tional quantiles. Alternately, the linear
14
Theorem
Under the assumptions of Theorem
3.2:
for some neighborhood of
(Hi) ip(z)
\(p{z)
-
>
J (p(z) dz =
0,
(p(zo)\
0,
<C- \z-zQ
f€g \w,Di>D
1,
J
<
Thenn = J~ EJ1
\.
(t)
h
—
>
1
oo;
oo;
>
(ii)
first derivative;
C >
there exists
(iv)
—
nh A
0,
has bounded and continuous
(-)
tp(z)\dz
\z
and
3.1
such that
A Q.
Effects of Subsidized Training
4.
Background
4.1.
The JTPA began funding
training in October 1983,
and continued to fund
federally-
sponsored training programs into the late 1990s. The program included a number of parts
or "titles", the largest of
which
is
Title
II,
which supports training
JTPA Study
economically disadvantaged. At the time of the National
JTPA
Title II
programs were serving about
of roughly 1.6 billion dollars.
JTPA
1
judged to be
for those
in the early 1990s,
million participants a year, at an annual cost
services were delivered at 649 sites, also called Service
Delivery Areas (SDAs), located throughout the country.
Title II of the
ized evaluation.
JTPA
11
is
unusual in that
explicitly incorporated a
The National JTPA study
ever undertaken in the US.
participants at 16
it
The JTPA
SDAs. These
sites
is
mandate
for
random-
the largest randomized training evaluation
evaluation study collected data on about 20,000
were not a random sample of
and
all
SDAs;
rather, they
implement the experimental design,
were chosen
for diversity, willingness
and the
and composition of the experimental sample they could provide. Although
size
ability to
the non-random selection of sites raises issues of external validity (as in
als),
within
sites,
applicants were randomly selected for
JTPA
many
treatment.
The
clinical tri-
evaluation
sample includes applicants who applied between November 1987 and September 1989.
The
original study of the labor-market
persons for
whom
impact of Title
II services
continuous data on earnings (from either State unemployment insurance
(UI) records or two follow-up surveys) were available for at least 30
11
Other parts of the JTPA, such as Title
III
programs
for
workers
et al.
(1996),
Bloom,
et al.
(1997),
who
months
lost their jobs as
after
15
random
a consequence of
Background for this section
and the US Department of Labor (1999) website.
international competition, did not include a randomized evaluation.
from Orr,
was based on 15,981
is
drawn
assignment.
Although data are available on a range of labor market outcomes
sample, we focus on the
sum
of earnings in this
30-month period since
best measure of the program's lasting economic impact
at
months following
is
probably the
on participants. Individuals who
JTPA services
for period
their application (though they could participate in other
programs
were not offered treatment were generally excluded from receiving
of 18
this
for this
any time).
The JTPA was a complicated program
service providers included
ganizations,
community
sistance
colleges, State
skills,
strategies.
basic education, or both;
(OJT/JSA);
(iii)
other services that
and/or a combination of the
first
were recommended as part of the
two.
employment
The types
and private-sector training agencies.
grouped into three general service
occupational
that offered a wide range of services.
may have
SDA, the data
in the analysis
(i)
classroom training in
included probationary employment
JTPA
intake process, before
though individuals were assigned to treatment with
their
or-
on-the-job training and/or job search as-
For the National
JTPA
community
of services offered can be
These strategies are
(ii)
services,
JTPA
sample were
Study, service strategies
random assignment.
different probabilities
artificially
Al-
depending on
balanced to maintain a 2/1
treatment-control ratio at each location.
The JTPA
generally
offered services to a
deemed
number
eligible for training if
of different groups.
Title II applicants were
they faced one of a number of "barriers to em-
ployment". These included long-term use of welfare, being a high school dropout, 15 or
more recent weeks
ability,
of
unemployment, limited English
reading proficiency below 7th grade
barriers were
unemployment
spells
level, or
proficiency, physical or
an
arrest record.
and high-school dropout
gorized as being in one of five groups: adult men, adult
status. Applicants were cate-
women, female youth, male youth
men and women
because the samples are largest for these two groups. There are 6,102 adult
and 5,102 adult men with 30-month earnings data.
16
dis-
The most common
non- arrestees, and male youth arrestees. In this study we focus on adult
30- month earnings data
mental
women
with
4.2.
Average Effects
Using our
earlier notation,
having been enrolled for
offer of
Y
is
JTPA
30- month earnings,
services,
Z
and
D indicates those who were recorded as
Although the
indicates the offer of services.
treatment was randomly assigned, only about 60 percent of those offered training
actually received
JTPA
which randomized the
This
services.
is
JTPA
a consequence of the
offer of services early in
evaluation design,
the application process, but did not compel
those offered services to participate in training.
While
all
applicants indicated at least
some
interest in receiving
offered treatment were not necessarily notified immediately,
JTPA
services, those
and some time may have passed
before training could begin. In the meantime, applicants selected for treatment
found jobs, received services somewhere
else,
or simply lost interest.
Providers
may have
may
also
have had an incentive to delay the enrollment of applicants that they thought were unlikely to benefit
some
from treatment, while some SDAs were unable to find service providers
for
applicants. Also, on the other side of the randomization offer, a small proportion of
those selected for the control group (1.6 percent) received
imenters' attempt to prevent this.
12
Treatment status
is
JTPA
services despite the exper-
therefore likely to be correlated
with potential outcomes and cannot be treated as exogenous.
Although treatment status
framework appear to apply
itself
was not randomly assigned, the assumptions of our
in this case: the
randomized
been independent of potential outcomes, the
offer of
treatment
offer of
treatment
is
is
likely to
have
unlikely to have affected
outcomes through any mechanism other than treatment
itself,
and denial of
made treatment more
likely.
Moreover, because of the
randomization
is
not likely to have
very low probability of receiving
JTPA
services in the control
compilers in this case can also be interpreted as effects on those
Since training offers were randomized in the National
not required to identify training
12
Adult
men and women who were
services than were received
effects.
Even
JTPA
(Z
—
by
0) group, effects for
who were
treated.
Study, covariates (X) are
in experiments like this, however,
offered treatment ultimately received about 150
by the members of the control group.
17
services
more hours
it
is
of training
customary to control
for covariates to correct for
chance associations between
D
and
X
Moreover, in our setup, covariates can be used to describe earnings
in Orr, et al, 1996).
quantiles for compilers in population subgroups, since
we estimate Qg(Y\X, D, D\ >
We
and Hispanic applicants, a
therefore include as covariates
group dummies, and dummies
worked at
dummies
a
least 12
by
differed
holders),
AFDC
dummies
receipt (for
recommended
women) and whether the
is
carried out separately for
men and women
for
Most
more
The
I.
other)
and
There are more minority applicants than
be recommended
likely to
Average 30-month earnings
for
for
in the
program
rules, a relatively
applicants also have low previous
men were recommended
of the
OJT/JSA
services, while the
classroom training than
sample are about $19,000
employment
women were
OJT/JSA
for
low
men and
or other
$13,000
women.
As a benchmark
Table
II
reports
for the
OLS and
impact of training. 14 The
while the
OLS
first
column reports unadjusted trainee/non-trainee
differences,
estimates in column (2) are from a regression of the dependent variable on
training difference
are precisely
purposes of comparison with earlier analyses of the JTPA,
conventional instrumental variables (2SLS) estimates of the
the covariates and a training
13
OJT/JSA,
sex.
proportion of high school graduates.
services.
applicant
since previously reported results
in the general population and, not surprisingly given the
slightly
dummy
married applicants, 5 age-
service strategy (classroom,
Descriptive statistics are reported in Table
rates.
for
).
whether earnings data are from the second follow-up survey 13 In addition,
for
the analysis
for
GED
for black
D
weeks in the 12 months preceding random assignment. Also included are
for the original
dummy
dummies
graduates (including
for high-school
(as
is
dummy
$3,970 for
measured
for
(D). Without the use of covariates, the training/non-
men and
$2,133 for women. Trainee/nontrainee differences
both men and women.
OLS
estimates of training effects in
comes from a background survey conducted as part of the JTPA intake
is similar to that described in Appendix B of Orr et al. (1996), except
that we collapsed some categories and omitted SDA dummies because they had low explanatory power.
14
As with the quantile regression and QTE estimates discussed later, the standard errors in Table II
The
process.
covariate information
The
covariate
list
used here
are robust in the sense that they provide consistent estimates of the asymptotic variance of the estimators
under general misspecification.
18
models with covariates are similar to the differences without controls.
The reduced-form
Not
effects of
Z
surprisingly, since
the offer of treatment are reported in columns (3) and
was randomly assigned without conditioning on
estimates with and without covariates differ
are not directly comparable with the
OLS
little.
(4).
covariates, the
Note that the reduced form estimates
estimates since
many
and
(6) of
of those offered training
did not actually receive training.
The instrumental
variable estimates in columns (5)
ized offer of treatment (Z) as an instrument for
construct the estimates in columns (1) and
those offered training.
When
(2).
less
OLS
2SLS estimate
size of the
for
corresponding
men
OLS
is
among
$1,593
estimate.
This amounts to a 9 percent
estimate.
a 15 percent earnings increase for
similar to those reported in previous studies.
women. These
results are
15
Estimates of Quantile Treatment Effects
4.3.
Table
III reports
OLS and
conventional quantile regression estimates of the effect of train-
ing.
The
OLS
estimate of the training coefficient
tile
regression as was used to
$1,780 with a standard error of $532, not dra-
is
matically different from the corresponding
men and
use the random-
II
This corrects for non-participation
than half the
For women, however, the 2SLS estimate
same
in the
covariates are included, the
with a standard error of $895,
earnings increase for
D
Table
covariates are the
regression estimates
same
as those used to construct the estimates in Table
is
$3,754 for
show that the gap
in quantiles
proportionate terms) below the median than above
earnings
is
it.
$2,215 for women.
by trainee status
is
136 percent higher. For
dramatic, but
still
marked. Like the
women
OLS
is
The
The quan-
much
larger (in
For men, the .85 quantile of trainee
about 13 percent higher than the corresponding quantile
the .15 quantile
is less
men and
II.
for non-trainees, while
the difference in impact across quantiles
estimates
shown
in the table, the quantile
regression coefficients do not necessarily have a causal interpretation. Rather they provide
15
The 2SLS
et al.
(1996).
estimates in Table II are very close to those in Table 4.6 of the National
The
JTPA
study by Orr
estimates are not identical because the covariates are not identical. Percentage effects
were computed as the coefficient on training, divided by
and other covariates
set to
means
for the treated.
19
fitted values
with the training
dummy
set to zero
a descriptive comparison of earnings distributions
Implementation of the
QTE
estimator requires
oretical results in the previous section are
and non-trainees.
for trainees
step estimation of k u
first
based on non-parametric
The
.
the-
series estimation of
the conditional expectations in k„, but this leaves open a range of possibilities. Since the
elements of
X are discrete, non-parametric estimation of E[Z\X]
is
in principle straightfor-
ward. In practice, however, a fully saturated model leads to problems with small or missing
covariate
dom
is
We
cells.
assignment,
Z
therefore estimated Z£[.Z|X] using the restriction that because of ran-
and
X should be uncorrelated in large samples.
simply the empirical E[Z\. The expectation vq(U)
separate models for
A
value.
series
D = 0,
1.
= E[Z\Y,D,X]
QTE
but for
A
tiles
the
little
explanatory
approximation was used to estimate terms in Y. Selection of the order
The order
is
for
the same for both
16
magnitude though
earlier, for
men
the
women
less precisely
estimated than the 2SLS estimates in Table
the 2SLS estimates are not
2SLS estimates
much
smaller than
are considerably smaller than
particularly interesting finding for
men
is
that the
OLS
II.
estimates,
OLS.
QTE estimates
of effects
on quan-
exhibit a pattern very different from the quantile regression estimates. In particular,
QTE
estimates show no evidence of a change in the .15 or .25 quantile.
at low quantiles are substantially smaller
error) of the effect
on the
regression estimate
effect
on the
is
.15 quantile for
men
For example, the
is
men
is
estimates
QTE
esti-
estimate (standard
$121 (475), while the corresponding quan-
$1,187 (205). Similarly, the
.25 quantile for
The
than the corresponding quantile regression
mates, and they are small in absolute terms.
tile
was estimated using
estimates of the effect of training on median earnings, reported in Table IV, are
similar in
As noted
resulting estimate
Most X's were dropped because they had
the series approximation was guided by cross-validation.
values of D.
The
QTE estimate
(standard error) of the
$702 (670), while the corresponding quantile regression
lc
Hausman and Newey (1995) use a similar approach to dimension-reduction for non-parametric estimation of consumer demand equations. Given estimates of k„, we computed QTE coefficient estimates by
weighted quantile regression using the Barrodale-Roberts (1973) linear programming algorithm for quantile regression (see, e.g., Koenker and D'Orey (1987)). A biweight kernel was used for the estimation of
standard errors.
20
estimate
is
$2,510 (356). This suggests that quantile regression estimates of training effects
at low quantiles are especially distorted
by positive selection on earnings
potential. It seems
that training did not really change the lower deciles of the trainee earnings distribution for
men. In contrast with the
results at
low quantiles, however, the
male earnings above the median are large and
QTE estimates of effects on
statistically significant
(though
smaller
still
than the corresponding quantile regression estimates).
The
QTE
women show
estimates for
with the largest proportional
effects at
low quantiles. For example, training
women by
to raise the .15 quantile of earnings for
The
significant effects of training at every quantile,
is
estimated
$324 (175), an increase of 35 percent.
estimates also suggest training raises the .85 quantile by $1,900 (997), but this
increase of only 7 percent.
Most
of the
QTE
estimates for
women
proportional impact on the lower
course,
women's earnings are
tail of
an
are reasonably close to
the corresponding quantile regression estimates. Thus, whether or not training
as endogenous, the estimates support the notion that for
is
women
training
treated
is
had a bigger
the earnings distribution than the upper
tail.
especially low in this sample, so large proportional effects
Of
do
not translate into large dollar amounts. 17
Orr, et
al.
concluded that
(1996) reported effects by subgroups but found no clear patterns.
(p.
160) "the benefits of
of different types of
JTPA. Our
are broadly distributed across a wide variety
men and women." Heckman,
concluded that heterogeneity of impacts
the
JTPA
results
is
Smith, and Clements (1997) similarly
important but that most
women
benefited from
do not contradict these general conclusions, but they nevertheless
show more heterogeneity
in
program
effects
than
is
revealed by a simple analysis within
subgroups. In particular, our results strongly suggest that training for adult
much
larger proportional effect
upper
tail
17
(though the absolute
Interestingly,
QTE
They
on the lower
effect
tail
women had
of the earnings distribution than
on the lower
on the
tail is small).
estimates of the proportional effects of training on
tional quantile regression estimates not only because the training
impact
men
is
are smaller than conven-
lower, but also because the
is bigger.
This reflects the fact that the constant and covariate-effects estimated by QTE are
Qe{Yo\X, D\ > Do). This is bigger than the quantile regression intercept because of positive selection
male compilers.
constant
21
a
for
for
Perhaps most striking among our findings
is
not seem to have raised the lower quantiles of their earnings.
an
by program operators to target
effort
higher earnings potential.
in
any assessment using a
distribution
more
well
off,
it
may be
results in distributional changes that
does
because of
men
with
would be undesirable
social welfare function that weights the lower tail of the earnings
seems
One response
purpose of the
JTPA was to aid economically
likely that the lower quantiles are of particular
to this finding might be that few
JTPA
concern to
applicants were very
so that distributional effects within applicants are of less concern than the fact that
the program helped
many
applicants overall. However, the upper quantiles of earnings were
who
reasonably high for adult males
this
This
services at relatively easy-to-employ
heavily. Since the ostensible
disadvantaged workers,
policy makers.
The
men
the result that training for adult
upper
tail is
participated in the National
JTPA
Study. Increasing
therefore unlikely to have been a high priority.
Summary and Conclusions
5.
This paper reports estimates of the
ings for participants.
on quantiles. The
We
QTE
use a
effect of subsidized training
new estimator
on the quantiles of earn-
for the effect of a non-ignorable
estimator can be used to determine
how an
intervention affects
the distribution of any variable for individuals whose treatment status
binary instrument.
The estimator accommodates exogenous
conventional quantile regression
when the treatment
is
covariates
exogenous.
It
treatment
is
changed by a
and
collapses to
minimizes a convex
piecewise-linear objective function similar to that for conventional quantile regression,
can be computed as the solution to a linear programming problem after
tion of a nuisance function.
this first step is
The paper develops
estimated nonparametrically.
and
first-step estima-
distribution theory for the case where
QTE
estimates of the effect of training on
the quantiles of the earnings distribution suggest interesting and important differences in
program
effects at different quantiles,
women. These
the
JTPA
differences are large
and
differences in distributional
impact
for
men and
enough to potentially change the welfare analysis of
program.
22
Appendix
Proof of Theorem
3.1:
This proof largely follows that of Theorem
1 in
Buchinski and
Hahn
(1998). Consider,
='^2 gi (j,K)
Gn(T,K)
i
where
gi (r, k)
and
egi
\/n(Se
-
—
=
K{Ui)
{9
— W-5g. The
Y{
Now,
5e)-
function
define r„(r,K)
MLLUA
almost surely.
By
- n-
\{e ei
1'2
W[t)+ -
Gn (r, 1{kv >
= E[G n (T, k)}.
= _ n -V2 W
.
e+]
0}
Kv{Ui)
.
kv )
•
Note
+ (1 - 6)
is
- n~ 1 ' 2 W(r)- -
[{e ei
convex in r and
it
is
e«]},
minimized
at
rn
=
that,
{6
_
1{£0i
_ n -l/2
W
-
<
<
,
Q})
and Weierstrass domination,
(iv)
dE[g(T,K
l/
)
]
,
dr
d 2 E{g(r,
.
r=0 =
«„)]
7^
[t=o= n-
1
OTOT
_„-l/2
-n- 1 / 2
E[Wk v (U)
(9
l{e e
=
0})]
0,
WW'\D > D Q P(D 1 >
E[ftolW:Dl>Do (0)
X
]
£>„).
Then,
=
r n (r, k v )
=
E[f£ ^ WiDl>Do (0) WW|£>i > £>
feo\w,Di_>Do(fy are bounded away from zero, J
where J
o(l),
P{Pi > A))- Note
•
]
Cn (Ui)
+
-r'Jr
that by
(ii)
and since both k v and
non-singular. Define,
is
= n- x '2 {e -
<
l{eei
0})
•
W
it
n
Un(K)=Y,K(U )-UUi),
i
i=l
and
= k(^)
?„([/;, «, r)
Note that E\u n (K v )\
G n (j,n)
•
=
=
=
{9
0,
[(e ei
~ n~ 1 ' 2 W{t) + -
e+]
+
(1
-
6)
{(e gi
- n~
1
'2
W[t)~ -
e£]
+ r'U^)}.
then
r„(T,K )-l-(G7l (r,K)-rn (r,K^))
1/
r„(T, k„)
-
T'u) n (K)
+
(G„(r, k)
+ r'w„( K )
{r n (r, k„)
+ /£[«„(«„)]})
n
= r n (r, K„) -
r'w n («0
+ ^{PnC^i. K
'
T)
~ E
-
9 n( t/
'[/
-
K ^> T )]l
i=l
Lemma
A.l
:
w„(l{K y > 0} £„)
=n _1/2 ^Vi + oP (l)
with
i=i
23
£ty
=
and
2
£|M| <
oo.
Lemma
A. 2
Applying
:
Eti(ft,( t[..l^>0}-ic„,T)-%(C/,K„
Lemma
a given
r.
Let
rj
=J
n
2
Define A„(r)
convex in
r,
(
1
T~
7
0}
>
cj n (l{K l/
?n)'
•
J( T ~
7
Ku)
=
0}
k„).
•
=
ln)
T
is
=
Lemma
|r n (r)|
Therefore, by
>
E r+1
0}
by
Op(l)- So,
.
Op (l)
that:
jT _ T W "(1{^ >
'
K„)
0}
+ - 7^ J??n
=
^
r'Jr
+
.
o p (l). Since A n (r)
is
K(t)-\t'Jt -0,
k„)
=
-
Lemma 3
(t
in
-
ry
n )'J(r
-
?
?
J - - rfn Ji ln + rn (r)
Buchinsky and Hahn (1998), we have that r n
=
?7
n
+ o p (l).
A.l
-6 e )^N(O,J- ZJl
1
),
E[ipip'].
PROOF OF Lemma A.l
zero.
+
Then,
n 1 / 2 (6
£=
'
T
K„)
0}
•
G„(t, 1{k„
where
Note
>
r'w n (l{^
= G n (i~, l{«t/ > 0} K„) + r'o; n (l{K„ > 0} K„), then X n (r)
applying Pollard's convexity lemma (Pollard (1991)):
any compact subset of
with sup TgT
*Vt "
o
2
g
sup
where
o p (l).
A. 2:
Gn (T, l{Ku >
for
=
T)]}
1
This assumption
is
:
To prove
this
lemma we
use the assumption that k v
probably stronger than necessary but
it
is
bounded away from
allows us to ignore the trimming using
K
1 2
s
making the asymptotics easier. Assumption (vi) implies that,
((K/rij) / + K~ ) —>
—
=
almost surely for all j 6 {1, ..., J}. Therefore sup t/eW \v
op (l) (see, e.g., Newey (1997), Theorem
vq\
4). Since txq is bounded away from zero and one (by (iii)), then sup;y eW \k u — k u = o p (l). Since /c„ is
bounded away from zero, with probability approaching one the trimming is not binding and we can ignore
\{k v
>
0},
\
it
for
the asymptotics.
uj n
Let
7Tq
{K u )
be the population
^n
=
= -= y.m(Ui)
V n ~(
mean
-F=V m(/7i)
V™^
of
Z
V
-
7—-
1
for the /-cell of
i
•
1
V
TTOi-TTt
- 7r 0(A
X
7T
i )
and ?
its
(A i )
+ Rn
/
sample counterpart.
r
-(TTi-TToi)
(l-7Ti)-(l-7r0i)/
24
(l-A,)-^,
A, •(!-£*,)
tt'-tt'
(i_^).(i_4)
o,
note that
D ir {\-VM
l-D^-Vi,
111
fe
-9
n'
V
1
=
l
Lemma
\V
D
f& ~
,m
V^
< sup
Then, applying
(1-Tf')-
(1 -*{,),/
@ii
ii)
~
v Oii)
— Vq\
n
'fe
A, (%,
,
-^Oi,) ^
Op(l).
*W
V
Newey and McFadden
4.3 in
\
)
(l-5?')-(l
-^)J
(1994),
-l
I
l
lX n -TV
(i-A,)-^_
£
(!-£>)
D-(l-i/
i/
m([/)
(7T
(X))2
A.-(i-hh.)
\
(1-^.(1-4)
J
(1)
(l-D)-Z
)
m(C/)
A"
(l-Tro(X))'
D-(l-Z)
(i-MX)Y
(x)y
(7r
X
Therefore,
i(K v )
To
"
lf.
V^ n
+
^Vi/(X )-{^-7r
/
A-(l-gj)
(l-A)-gj
V
1-T0(^i)
TToCX)
i
(X )}+Op (l).
I
prove,
J_V- an A
A-(i-Pi)
(l-A)-gj
>™
Vn
"7=
^i
JT(
1
V
-
A
•
(1
-
Zi)
7T7T
l-TTo(Ai)
(1
- A)
~
7r
Zi
lv
(Ajj
+
,
Op(l)
/
notice that
A
V
t=l
•
(l
-
vj)
l-Tro(Xi)
(l
"
- A)
Pt
no(Xi)
J
-.
ni
l
- A,)
tt
25
(A%)
Pjj
So,
we
show that
just have to
for
each j €
{1,
J}
(l-A)-zv
Di. -(1-Vi.)
(
...,
i,=i
:^m(^). 1-
+0,(1
,(!)
l-Tro(^)
,
ttoP^.;
This will be done by checking assumptions 6.1 to 6.6 in Newey (1994). Assumptions 6.1 and 6.2 follow
from the conditions of the theorem (see Newey~(1994), page 1373). Assumption 6.3 holds with
d —
and ad = s. Assumption 6.4 holds for b(z) — and derivative equal to
directly
m(U)
5
Assumptions 6.5 and 6.6 follow from: (i) rij K~ 2s — 0; (ii)
/rij —
check Assumption 6.5 note that (vi) implies that s > 5/2, therefore
6.5 is also valid with d — 0). To check assumption 6.6 note that since
K
>
(almost surely). In particular, to
>
K K~ —
m(U)
i
D
1-D
-MX)
MX)
s
<
(note that Assumption
>
00,
then, there exists a sequence £ K such that
E
as
K
-
oo (see
Newey
D
m(U)
l
~£)-tKP
MX),
l-no(X)
(1994), page 1380 last paragraph.)
Di
(1
-
Vi)
(1
-
Di)
Tn^ mm \
V"
Now, applying the
(TT\
(-L
l
A-(l-^Oi)
-
I
-MX,)
(1
~
~ Di)
Z%
MXi)
)
Di
V
Proof of Lemma
lemma
A. 2
:
(1994),
-
VQi)
(1
- ZO
+
o p (l)
(1
7To(A'i
N
2=1
VQ
MXi)
l-A
result of the
Newey
Vi
x
and the
results in
MXi)
)
1
K (U)
^ + Op(l)
TTo(A'i)
- A)
holds.
Note that pn (Ui,l{K v > 0}-K v ,T)-pn (Ui,K
1/
,T)
=
(1{k„
> 0}-K„-n v )-Sn (Ui,T),
where
Sn (Ui,r)
so \Sn (Ui,r)\
E[n
9-[{e ei
+
(1-6).
< n- x l 2 \{\e 8i < n- l ' 2 \W[ T \}
\
\Sn (Ui,r\}
=E
<
n
~Fu
1'2
\
E[l{\e e
\
-n-" 2 W' T)+-el}
=
<
l
[(e
w - n"
/2
W!t)~ -
e,"]
+ r'UU),
\W[t\. Also,
jT^W'tW
w (n-W \W'r\)
1
\W'r\]
- Feolw (-7i-V 2 \W'r
-1/2
26
\W't\
2-E[feolw (0)-\W'T\
2
}<™
Then,
n (Ui,l{K
^p
=
v
>
0} -K V ,T)
-pn (Ui,K v ,T)
i=l
1
2
1
<
sup
-
-k„\
\k v
-
^n-
ueu
\Sn (Ui,T)\
=
o p (l).
i=l
Also, by cancellation of cross-product terms,
J2 E i(Pn(. U,^,r)f
=
^2p n (Ui,Ku,r) -E\p n {U,K v ,T)\
1=1
v»=l
<
£[l{|e fli
and the
result of the
lemma
Proof of Theorem
Consistency of
£
\W[t\
•
2
]
0,
3.2:
)\W,D 1
E[<Ph {6
< n-^l^/rl)
holds.
easy to prove and
is
|
>D
we
will focus
/*
1
=
]
i<P
=
on
J.
By
fv-
W'6f>\
{
h
and
(ii)
<
(hi), for
<
e*
eg:
dV
/y|w,J3 1 >A,(v)
J
fi(z)feo\W,D
i
1
>D (h-z)dz
=
fe e \W,D 1
>D o (0) + h
=
fe e \W, Dl
>D o {0) +
dz
Z-<p(z)'
dz
O(h).
(A.1)
Therefore,
limiE[<p h (6 e )\
a —>U
By
equation (A.l) and condition
lim
h —*0
E [k v
<p h
is
kv
,
</?(•)
and
n
h?
—
>
oo,
•
>d o (0)-
]
is
eventually bounded (in
X
P(D >
Do]
X
,
Do]
WW'\D > D
WW'\D >
[feelw Dl>Do (0)
>
1
P(D > D
l
]
D
)
).
=
0(l/h 2 ).
then
Notice that, since k„
X>„(0i)
Di
W are bounded,
^ 5>„(tfi)
i
,
x
var( Ku -^ h (6e)-WW')
Since
U ]W
3.1, E[<p h (6o)\W,Di
bounded, we have that
also
h —>Q
= E
Also, since
]
hm E [E [<p h {6 g )\W, D > D
=
WW'}
(6e)
W
1
Theorem
(iv) in
absolute value) by a constant. Since
W,D > D =
<p hi
is
<p hii
1 E [U\ w Dl>Do (0)
WiW[
(6 e )
,
bounded away from zero (uniformly
3e) WiWl -
K v {Ui)
<p hA
(8e)
WW'ID, >
D
]
P(D 1 >
D
(A.2)
).
in U),
W W[
{
i=l
n
< C sup
ueu
\k v
- kv
1
\
- V"
~
n
\\k v
fh
i(Se)
WiW-
=C
•
sup
ueu
27
|re„
-
1
k„\
-
~
"
V"^
i=l
•
fh,i( Se
Under the assumptions of Theorem
1
3.1, sup,y eW \k u
—
kv
0.
>
In addition,
n
~
-
—
\
1
-
I
2=1
i=l
C-^-h ±-h ¥e-8
*
e
\\.\m\
i=l
C
<
•
n 1 /2
||?e
-8 e
\\-
(n
1
/^ 2 )- =
1
o p (l).
As shown above,
'
i=i
Therefore,
=
£ f>„(0i) ¥>m&) WiW! lY,K v (Ui)
•
2
By
(i)
and
(iv), for
I Y,K v {Ui)
ip
=1
<p h>l
(6 e )
WiWi + o„(l).
(A.3)
2=1
some constant C
Ki (6e)
WiWi - K v {Ui)
<p hii
W W[
(6 e )
t
< C n 1 /2
•
\\6 e
-Se\\- {n
l'2
h 2 )- 1
=
o p {\).
2=1
(A.4)
Combining equations (A. 2), (A.3) and
(A.4),
we
get
J
28
A J.
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31
Table
I
Means and Standard Deviations
Assign] nent
Entire
Treatment
Control
Sample
A.
Difference
(t-stat.)
Men
Number
of observations
5,102
3,399
1,703
Baseline Characteristics
Age
32.91
[9.46]
High school or
GED
Married
Black
Hispanic
Worked
less
weeks
than 13
in past year
32.85
[9.46]
33.04
[9.45]
-.19
(-67)
.69
.69
.69
-.00
[.45]
[.45]
[.45]
(-.12)
.35
.36
.34
[.47]
[.47]
[.46]
.02
(1.64)
.25
.25
.25
.00
[.44]
[.44]
[.44]
(.04)
.10
.10
.09
.01
[.30]
[.30]
[.29]
(.70)
.40
.40
.40
.00
[.47]
[.47]
[.47]
(.56)
Experimental Characteristics
Second follow-up
Training
.29
.30
.28
[.46]
[.46]
[.45]
.42
.62
.01
[.49]
[.48]
[.11]
.02
(1.14)
.61
(70.34)
Service strategy:
Classroom training
OJT/JSA
Other
Outcome
30
.20
.21
.19
[.40]
[.41]
[.39]
.02
(1.73)
.50
.50
.50
.00
[.50]
[.50]
[.50]
(.07)
.29
.29
.31
[.46]
[.45]
[.46]
-.02
(-1.57)
variable:
month earnings
19,147
19,520
18,404
1,116
[19,540]
[19,912]
[18,760]
(1.96)
32
continued from previous page
Entire
Assigm nent
Treatment
Control
Sample
B.
Difference
(t-stat.)
Women
Number
of observations
6,102
4,088
2,014
Baseline Characteristics
Age
33.33
33.33
[9.78]
High school or
GED
Married
Black
Hispanic
Worked
less
weeks
than 13
in past year
AFDC
[9.77]
33.35
[9.81]
.72
.73
.70
[.43]
[.43]
[.44]
.22
.22
.21
[.40]
[.40]
[.39]
-.02
(-.09)
.03
(2.01)
.01
(1.55)
.26
.27
.26
.01
[.44]
[.44]
[.44]
(.95)
.12
.12
.12
-.00
[.32]
[.32]
[.33]
(-.89)
.52
.52
.52
-.00
[.47]
[.47]
[.47]
(-.08)
.31
.30
.31
[.46]
[.46]
[.46]
-.01
(-1.03)
Experimental Characteristics
Second follow-up
Training
.26
.26
.25
.01
[.44]
[.44]
[.43]
(.45)
.45
.66
.02
[.50]
[.47]
[.13]
.64
(80.24)
Service strategy:
Classroom training
OJT/JSA
Other
Outcome
.38
.38
.39
-.01
[.49]
[.49]
[.49]
(-37)
.37
.37
.38
-.01
[.48]
[.48]
[.49]
(-.85)
.24
.25
.23
[.43]
[.43]
[.42]
.02
(1.40)
variable:
30 month earnings
13,029
13,439
12,197
[13,415]
[13,614]
[12,964]
1,242
(3.46)
Note: The first three columns of the table report means and standard deviations (in brackets)
for the National JTPA Study 30-month earnings sample. The last column shows the difference in
means by assignment status and reports the t-statistic (in parenthesis) for the null hypothesis of
equality in means.
33
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^
Table III
Quantile Regression and OLS Estimates
Dependent
variable:
30-month earnings
OLS
A.
%
Impact of Training
High school or
GED
Black
Hispanic
Married
Worked
less
weeks
than 13
in past year
Constant
0.50
0.75
3,754
1,187
2,510
4,420
4,678
4,806
(536)
(205)
(356)
(651)
(937)
(1,055)
21.20
135.56
75.20
34.50
17.24
13.43
4,015
339
1,280
3,665
6,045
6,224
(571)
(186)
(305)
(618)
(1,029)
(1,170)
0.85
-2,354
-134
-500
-2,084
-3,576
-3,609
(626)
(194)
(324)
(684)
(1087)
(1,331)
251
91
278
925
-877
-85
(883)
(315)
(512)
(1,066)
(1,769)
(2,047)
6,546
587
1,964
7,113
10,073
11,062
(629)
(222)
(427)
(839)
(1,046)
(1,093)
-6,582
-1,090
-3,097
-7,610
-9,834
-9,951
(566)
(190)
(339)
(665)
(1,000)
(1,099)
9,811
-216
365
6,110
14,874
21,527
(1,541)
(468)
(765)
(1,403)
(2,134)
(3,896)
Women
Training
%
Impact of Training
High school or
GED
Black
2,215
367
1,013
2,707
2,729
2,058
(334)
(105)
(170)
(425)
(578)
(657)
18.46
60.76
44.42
32.25
14.47
8.09
6,373
3,442
166
681
2,514
5,778
(341)
(99)
(156)
(396)
(606)
(762)
-544
22
-60
-129
-866
-1,446
(397)
(115)
(188)
(451)
(679)
(869)
-1,151
-31
-222
-995
-1,620
-1,503
(488)
(130)
(194)
(546)
(911)
(992)
-667
-213
-392
-758
-1,048
-902
(436)
(127)
(209)
(522)
(785)
(970)
-5,313
-1,050
-3,240
-6,872
-7,670
-6,470
(370)
(137)
(289)
(522)
(672)
(787)
AFDC
-3,009
-398
-1,047
-3,389
-4,334
-3,875
(378)
(107)
(174)
(468)
(737)
(834)
Constant
10,361
649
2,633
8,417
16,498
20,689
(815)
(255)
(490)
(966)
(1,554)
(1,232)
Hispanic
Married
Worked
less
weeks
.
0.25
Men
Training
B.
Quantile
0.15
Note:
The
than 13
in past year
table reports
OLS and
quantile regression estimates of the effect of training on earnings.
The
specification used also includes indicators for service strategy recommended, age group and second
follow-up survey. Robust standard errors in parenthesis.
35
Table IV
Quantile Treatment Effects and 2SLS Estimates
Dependent
variable:
30-month earnings
2SLS
Quantile
0.15
A.
0.50
0.75
0.85
Men
Training
%
Impact of Training
High school or
GED
Black
Hispanic
Married
Worked
less
weeks
than 13
in past year
Constant
B.
0.25
1,593
—vzt
702
1,544
3,131
3,378
(895)
(475)
(670)
(1,073)
(1,376)
(1,811)
8.55
5.19
11.99
9.64
10.69
9.02
4,075
714
1,752
4,024
5,392
5,954
(573)
(429)
(644)
(940)
(1,441)
(1,783)
-2,349
-171
-377
-2,656
-4,182
-3,523
(625)
(439)
(626)
(1,136)
(1,587)
(1,867)
335
328
1,476
1,499
379
1,023
(888)
(757)
(1,128)
(1,390)
(2,294)
(2,427)
6,647
1,564
3,190
7,683
9,509
10,185
(627)
(596)
(865)
(1,202)
(1,430)
(1,525)
-6,575
-1,932
-4,195
-7,009
-9,289
-9,078
(567)
(442)
(664)
(1,040)
(1,420)
(1,596)
10,641
-134
1,049
7,689
14,901
22,412
(1,569)
(1,116)
(1,655)
(2,361)
(3,292)
(7,655)
Women
Training
%
Impact of Training
High school or
GED
1,780
324
680
1,742
1,984
1,900
(532)
(175)
(282)
(645)
(945)
(997)
14.60
35.47
23.14
18.37
10.06
7.39
3,470
262
768
2,955
5,518
5,905
(342)
(178)
(274)
(643)
(930)
(1026)
-123
-401
-1,423
-2,119
(204)
(318)
(724)
(949)
(1,196)
-554
Black
(397)
Hispanic
-1,145
-73
-138
-1,256
-1,762
-1,707
(488)
(217)
(315)
(854)
(1,188)
(1,172)
-652
-233
-532
-796
38
-109
(437)
(221)
(352)
(846)
(1,069)
(1,147)
-5,329
-1,320
-3,516
-6,524
-6,608
-5,698
(370)
(254)
(430)
(781)
(931)
(969)
AFDC
-2,997
-406
-1,240
-3,298
-3,790
-2,888
(378)
(189)
(301)
(743)
(1,014)
(1,083)
Constant
10,538
984
3,541
9,928
15,345
20,520
(828)
(547)
(837)
(1,696)
(2,387)
(1,687)
Married
Worked
less
weeks
than 13
in past year
table reports 2SLS and QTE estimates of the effect of training on earnings. Assignment
used as an instrument for training. The specification used also includes indicators for service
strategy recommended, age group and second follow-up survey. Robust standard errors in parenthesis.
Note:
The
status
is
36
DEC
2000
Date Due
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