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31
O
I
I
Massachusetts Institute of Technology
Department of Economics
Working Paper Series
INFERENCE FOR DISTRIBUTIONAL EFFECTS USING
INSTRUMENTAL QUANTILE REGRESSION
MIT
MIT
Victor Chernozhukov,
Christian Hansen,
Working Paper 02-20
May 16,2002
Room
E52-251
50 Memorial Drive
Cambridge, MA 02142
This paper can be downloaded without charge from the
Social Science Research Network Paper Collection at
http://papers.ssrn.coin/paper.taf?abstract_id = 3 1 3426
:
1°
Massachusetts Institute of Technology
Department of Economics
Working Paper Series
INFERENCE FOR DISTRIBUTIONAL EFFECTS USING
INSTRUMENTAL QUANTILE REGRESSION
MIT
MIT
Victor Chernozhukov,
Christian Hansen,
Working Paper 02-20
May 6, 2002
1
Room
E52-251
50 Memorial Drive
Cambridge,
MA 02142
This paper can be downloaded without charge from the
Social Science Research Network Paper Collection at
http://papers.ssrn. com/paper. taf?abstract_id= xxxxx
INFERENCE FOR DISTRIBUTIONAL EFFECTS USING
INSTRUMENTAL QUANTILE REGRESSION
VICTOR CHERNOZHUKOV AND CHRISTIAN HANSEN
Abstract.
In this paper
we
describe
how
quantile regression can be used to evaluate
the impact of treatment on the entire distribution of outcomes, when the treatment
endogenous or selected
in relation to potential
outcomes.
We
is
describe an instrumental
variable quantile regression process and the set of inferences derived from
it,
on tests of distributional equality, non-constant treatment
dominance,
effects, conditional
focusing
and exogeneity. The inference, which is subject to the Durbin problem, is handled via a
method of score resampling. The approach is illustrated with a classical supply-demand
and a schooling example. Results from both models demonstrate substantial treatment
heterogeneity and serve to illustrate the rich variety of hypotheses that can be tested using
inference on the instrumental quantile regression process.
Key Words:
Quantile Regression, Instrumental Quantile Regression, Treatment Effects,
Endogeneity, Stochastic Dominance,
Hausman
Test,
Supply-Demand Equations, Returns
to Education
Address correspondence to V. Chernozhukov, Department of Economics, MIT, E52-262F, 50 Memorial
Drive, Cambridge,
of
MA 02142
meetings
in
is a provisional draft and an MIT Department
thank the participants and organizers of the EC 2001
(e-mail vchern@mit.edu.) This
Economics Working Paper.
March 2002.
We
Louvain-La-Neuve, where the research reported in this paper was presented.
Introduction
1.
In this paper
of treatment
we
on the
describe
how
quantile regression can be used to evaluate the impact
entire distribution of outcomes,
selected in relation to potential outcomes.
regression process
and the
We
when the treatment
set of inferences derived
from
it,
which
is
The paper
pling.
approach
is
self-selected or
focusing on tests of distribu-
tional equality, non-constant treatment effects, conditional dominance,
inference,
is
introduce an instrumental variable quantile
and exogeneity. The
is handled via a method of score resammethods and establishes their asymptotic validity. The
with a classical supply-demand and a schooling example.
subject to the Durbin problem,
describes these
illustrated
Inference about distributional outcomes
is
crucial in a
wide range of economic analy-
For example, formal evaluation of a social program requires inference concerning the
ses.
nature, direction,
and magnitude of the program's impact throughout the entire outcome
under alternative dis-
distribution, since evaluation involves integration of utility functions
tributions of outcomes. For further examples
and previous
literature, see
Atkinson (1970),
Abadie (2002), Foster and Shorrocks (1988), Heckman and Smith (1997), and McFadden
(1989). Just as in classical p-sample theory, e.g. Doksum (1974) and Shorack and Wellner
(1986), this kind of inference
is
based on the empirical quantile regression process, which
has recently been explored by Koenker and Xiao (2002).
The
goal of this paper
(IV-QR) process and a
is
to offer
an empirical instrumental variable quantile regression
set of inference tools derived
eliminates the endogeneity and selection bias
from
it.
Effectively, instrumentation
commonly occurring
in observational studies
and experiments with imperfect compliance. Thus, the IV-QR process allows us
to
measure
the exogenous treatment effect as in a fully controlled experiment, whereas the conventional
QR process
is
inherently biased.
Using the instrumental variable quantile regression process we describe and derive the
properties of a class of tests based
on
it
which allow us to examine:
The hypothesis
of distributional equality, or whether the treatment has a significant
The hypothesis
of non-constant or varying treatment effect, a fundamental hypothesis
1.
effect,
2.
of causal analysis,
3.
cf.
The
cf.
Heckman
(1990),
Doksum
(1974),
and Koenker and Xiao (2002),
hypothesis of conditional stochastic dominance, a fundamental hypothesis as well,
Abadie (2002) and McFadden (1989),
4.
The hypothesis
of exogeneity, or whether the treatment variable
essential hypothesis, e.g.
Hausman
(1978).
is
exogenous, another
A
difficulty
which
when implementing
arises
1
these tests
is
that some of
them
are subject
to the Durbin problem. That is,
duce parameter dependent asymptotics, endangering distribution-free inference.
the model's features or estimated nuisance parameters in-
A method
of score resampling, which bootstraps the scores or estimated influence functions without
estimates', is suggested for generating asymptotically valid critical values
recomputing the
A main
for these tests.
enables
fast, practical
other settings, and
its
advantage of this method
The method
implementation.
computational simplicity, which
is its
is
of independent interest in
immediate applicability to other problems
eral conditions given in this paper.
For example,
it
can be used
many
assured by the gen-
is
for conventional quantile
regression.
The
use of the approach
with random
ings.
is
illustrated through
demand
analyze the structure of
and
coefficients,
in the second,
We obtain clear evidence against
accepting the hypothesis of
first, we
demand system
two empirical examples. In the
within a simultaneous equations
for fish
we consider the
effect of schooling
on earn-
the exogeneity and constant effect hypotheses, while
order stochastic dominance in both of these examples.
first
Thus, our paper complements the previous work which has considered research in this
Hogg (1975), Abadie (1995) (2002), MaCurdy and Timmins
Hong and Tamer (2001), Abadie, Angrist, and Imbens (2001),
direction, including that of
McFadden
(1989),
Koenker and Xiao
(2002),
(1998),
Chernozhukov (2002) and
others.
This paper accompanies our previous paper, Chernozhukov and Hansen (2001), that
focused on modeling and identification of quantile treatment effects in the presence of endogeneity.
The
present paper goes further to establish the sampling properties of the entire
instrumental variable quantile regression process and of the inference processes and test
from
statistics derived
The remainder
It
also provides practical bootstrap tools to carry out the tests.
of the paper
the causal model and
examples developed
its
it.
sampling theory.
its
is
organized as follows. In the next section,
then presents the
IV-QR
2.
model
is
D
2
methods
it
we
will
use a con-
Potential or counterfactual real-valued outcomes
(D € D, a
subset of
ffif
),
and denoted
Y^
while potential
coined by Koenker and Xiao (2002) to emphasize Durbin's contribution to theory of
goodness-of-fit tests with estimated parameters
e.g.
process and
of the
and Section 6 concludes.
a simultaneous equations model. To describe
are indexed against treatment
See
5,
IV-QR
The use
The Causal Model
ventional potential outcomes framework.
The term was
process and develops
Section 4 develops inference procedures for the
are illustrated via two empirical examples in Section
following
briefly discuss
examples that underlie both the estimation and the empirical
in this paper. Section 3
presents practical inference for testing distributional hypotheses.
The
we
Heckman and Robb
and have an easy way to
(1986) and Imbens and Angrist (1994)).
refer to the problem.
treatment status
is
is
indexed against the instrument Z, and denoted
D=
an individual's outcome when
Z=
Dz
d and
Dz
For example, Yd
an individual's treatment status when
is
.
z.
The
d e D}, such as wages or demand, vary
Given the actual treatment D, the observed
potential or counterfactual outcomes [Yd,
across individuals or states of the world.
outcome
is
Y = YD
That
is,
.
only the D-th. component of {Yd,d e T>}
observed. Typically
is
D
is
selected in
relation to potential outcomes, inducing endogeneity or sample selectivity.
The
objective of our analysis
to learn
is
about the marginal distributions
the conditional quantiles of potential outcomes
or, equivalently,
Yd'.
Qyd |x(r),rGT,
where
T
is
a closed subinterval of
(0, 1).
Quantiles of potential outcomes are a primary input to decisions about the efficiency
of treatment
and
programs 3
social
potential outcomes
is
sample
.
The main
obstacle to learning about the quantiles of
misleading about the quantities in question, see
2.1.
The Model. The
(2001). This
model
e.g.
Heckman
it.
that affect the selection of
assumed that there
D by the individuals
is left
exists a vector of instrumental variables
D but do not affect the potential outcomes.
characteristics are denoted
and Hansen
Wald-type estimating equation and justifies a large variety
In this model, the selection of treatment
essentially unrestricted. It is
typically
(1990).
following model has been suggested in Chernozhukov
rationalizes a
of estimators based on
(Yo,D) are
selectivity or endogeneity-the observed
Z
Observed individual
by X.
The main restriction of the model is the similarity assumption. The similarity assumption
on the information that led to an individual's selection of treatment
the expectation of any function of the individual's rank does not vary across treat-
states that conditional
state,
ments. In other words, the selection presumes that the ex-ante expectation of a person's
"ability"
,
as
measured by the person's rank
sumption Al,
relative to people
in the distribution of unobservables,
vary across the treatments.
Assumption
1.
(IV-QR Model)
Al Potential Outcomes.
For almost every value of {X,Z)
Given
X = x,
for
some Ud
Yd = q{d,x, Ud
implying that q(d,x,r)
3
is
Ud
in As-
with the same observed characteristics (X,Z), does not
),
the r-th quantile of
This has been discussed, for example,
in
Abadie (2002).
3
Yd
.
~
U(0,
=
1),
{x,z),
A2
Selection. For unknown function
ment indexed by instrument status
and random vector V, the potential
given X = x, takes the form
S(-)
z,
treat-
Dz = 6{z,x,V).
A3 Independence.
Given X,
{Ud}
A4
Similarity. For each
independent of Z.
(V,X,Z)
d,d', given
Ud
A5 OBSERVABLES consist
is
is
equal in distribution to Ud'
of, for
-
U = UD
Y = q(D,X,U),
D = 5(Z,X,V),
X,Z
It is
important to note that the model
differs
Heckman and Angrist and Imben's LATE. The
from the conventional selection model of
differences mainly relate to the similarity
assumption and slightly different independence conditions.
An
extensive comparative dis-
cussion of the
IV-QR model
assumption
is
perhaps best highlighted in the demand and schooling examples described
in the next
two sections.
is
given in Chernozhukov and Hansen (2001).
The
discussion
is
The
role of each
elaborate since these examples underlie the
empirical analysis in this paper.
2.2.
Example: A Demand Model. The following is a general simultaneous equation
many classical structural models can be written as special cases. Consider the
model, and
following model
Yp = q (p, U)
Yp = p{p,z,U)
P £ {p q (p, Z,U) = p (p, It)}
i.
ii.
iii.
The map p
Yp
demand,
supply,
(2.1)
equilibrium.
:
random demand function, that is it is the demand when the price
the random supply function, that is the supply when the price is
p Hp. Additionally, Yp and Yp q(-), and p(-) depend on the covariates X, but this dependence
is suppressed. Random variable U is the level of the demand in the sense that
is
>-*
is
p. Likewise,
the
Yp
is
,
Q(P,
Demand
is
maximal when
the level of supply.
The
U—
1
U)
<
q{p, U')
when U < U'
and minimal when
r-quantile of the
U=
demand curve p
0,
holding p fixed. Likewise,
i-»
Yp
is
P^QY (T) = q(p,T).
the curve p ^ Yp
below the curve p
given by
p
Thus with probability
r,
lies
i->
Qyp (r).
It is
The
quantile treatment effect
—
q{p\r)
The
is
characterized by an elasticity
q{p,r)
by
or, if defined,
dlnq(p, r)/d\np.
depends on the state of the demand r (low or high) and may vary consid-
elasticity
erably with r.
when
For example, this variation could arise
the
number
and aggregation induces non-constant elasticity across the demand
summation of individual demand curves, holding the price fixed.
of buyers varies
levels as a process of
This example incorporates traditional models with additive errors
yp = q{p) + £,
Note that the model of demand
while in (2.2)
in
iii.
i.
is
more general
in that the price elasticity
As a
ii.
is
random,
(2.1) allows for general, non-parallel effects.
the equilibrium condition that generates endogeneity - the selection of
is
the actual price by the market depends on the potential
and
(2.2)
constant. In other words, (2.2) restricts the price effect to parallel shifts
it is
demand, while
Condition
in
= Q e {U).
where £
demand and supply outcomes
i.
result
P = 5(Z,V),
where
V
consists of U,
rium price
is
IX,
and other variables (including "sunspot"
Instrumental variables Z,
th quantile of the
demand
correlation between
2.3.
like
the equilib-
weather conditions and factor prices, that shift the supply
demand curve
Example:
— 0, 1.
Z
and
A Roy
The
function,
V
p
«->
q(p,r).
U
allow identification of the r
—
Perhaps remarkably, the model allows
to exist.
type Model. An individual considers two levels of schooling
outcome under each schooling level is given by
potential
{Yd ,d
The
if
not unique).
curve and do not affect the level of the
denoted d
variables,
=
0,l}.
individual selects his schooling level to maximize his expected utility:
D = arg
max \e
[
W (Yd
)
\X, Z,
de{ ° ,1}
V] }
'
(2-3)
=
max
arg
{E[W (q{d,X,Ud ))\X,Z,V]j
,
d
where
W
As a
is
the unobserved Bernoulli utility function, and
E is the
rational expectation.
result,
D = 5{Z,X,V)
where Z, X are observed,
unobserved variables that
V
is
affect
an error vector that depends on ranks {Ud } and other
the selection, and 5 is an unknown function.
5
The
similarity assumption imposes that
E
for
any function
[
W
(q(d,
Ud ))
X,
W, where E
is
\X,
Z,V]=E[W (q(d, X, [/„)) \X, Z, V]
(2.4)
the rational expectation (computed with respect to the
true probability law P). In other words, the decision maker's information does not allow
the objective discrimination of systematic variation of his ranks across the treatment states,
where the ranks are defined
the
same
X
relative to the observationally identical
and Z. In other words, the
on the information
similarity assumption
is
group of individuals with
nothing but a restriction
set of the individual.
Note that similarity
is
defined relative to the true expectation. In effect, (2.4) does not
an individual to have rational expectations while optimizing in (2.3). Indeed, we
could replace the expectation operator E in formula (2.3) by some subjective expectations
require
Eg
such that
D = arg max \eb
3.
3.1.
The
tions
[
W {q{d,X,U
\X,Z,V]
d ))
}.
The Instrumental Variable Quantile Regression Process
Principle.
We first
describe two important implications that arise from assump-
A1-A4.
Proposition
1
continuous and
(A1-A5 imply Wald IV). Suppose A1-A5 and
strictly increasing a.s.
Then for each t £
P[Y <q(D,X,T)\X,Z] =
T,
that r
i->
q(D,X,r)
is
(0, 1)
a.s.
(3.1)
Furthermore,
=
Equation
process t
f—y
Wald
(3.1) is a
q(d,x,r).
QY-q(D,x, T )\x,z{T)
style restriction that
a.s.
(3.2)
can be used to estimate the quantile
Identification of the quantile process in the population does not
require functional form assumptions. This result
is
reported in Chernozhukov and Hansen
(2001).
The main identification restriction (3.1) can be posed via (3.2) as an optimization problem, which we call the instrumental variable or inverse quantile regression for its "inverse"
relation to the (conventional) quantile regression of
the
IV-QR model and
Koenker and Bassett (1978). This
quantile regression together.
Recall from Koenker and Bassett (1978) that quantile regression
the best predictor of
Y
links
given
is
formulated as finding
X under the expected loss using the asymmetric least absolute
deviation criterion:
pT (u)
—
TU +
+
(1
— t)u~
.
In other words, assuming integrability, the r-th conditional quantile of
QY]x {r) = argmin
Note that median regression
Proposition
1
X
given
solves:
E[pT (Y -f(X))\X]
one important instance of quantile regression.
is
states that
Y
the r-th quantile of
is
random
variable
Y — q(D,X,r)
condi-
tional on (X, Z):
=
Thus, we
may
Proposition
QY-q{D,x,T)i T \X, Z)
a.s.
for each t.
pose the problem of finding a function q(d,
x, r) satisfying
equation (3.2) of
as the instrumental variable or inverse quantile regression:
1
Find a function q(x,
Y -q{X,D,r)
such that
d, r)
a solution to the quantile regression of
is
on(Z,X):
= arg mm EpT Y [(
q{D, X, r)
-
f{X, Z))\X, Z)
An
Instrumental Variable Quantile Regression Process. For estimation of the
rV-QR process, we focus on the basic linear model, which covers a wide range of applications:
3.2.
q{D,X,T)=D'a(T)+X'P(T),
where
is
D
is
an
I- vector
(3.3)
a fc-vector of (transformations) of covariates. The linear quantile model
special case of the
X
of treatment variables (possibly interacted with covariates) and
more general model presented
in
Assumption Al, and
is
is
obviously a
a basic model
of quantile regression research.
We
focus on a simple finite-sample analog of the the instrumental variable quantile re-
gression in the population. Define the weighted quantile regression objective function as
1
Q„(T,a,0,7)
^i(r)
Vi(r)
In principle,
-
= -J2pr(ri-Di<*-XiP-*i{'ryi)-Vi{T),
f
= $(t, Xi, Zi) is
= V(t, Xi, Zi) >
an
we may consider a
/-vector of instruments
is
(/
is
larger
number
=
,
of elements in vector $. However, this
\\x\\a
=
j|-y(a,
t)
as follows: for
&{t)
dim(Z?))
a weight function.
necessary as efficiency can instead be achieved by choosing
The IV-QR procedure
=
where
arg jnf
(/8(a,r),7(a,T))=
is
not
appropriately.
y/x'Ax
1]^,
arginf
7 )eBxS
(/?,
$ and V
such that
Q n (r, 0,^,7),
(3.4)
/
v
3 5^;
'
A and
where
are parameter sets, 3
"B
is
any
fixed
compact cube centered
at 0,
and
A
is
a
positive definite matrix.
The parameter estimates
are given by:
=
(fi(r)J(T))
(fi(r)^(a(T),r))
(3.6)
Let us denote
=
9{t)
(a(r)',/3(r)')'
There are three principal motivations
strictions
and 9{r)
=
(a(r)'^(r)')'-
for this estimator. First,
it
naturally links IV re-
and conventional quantile regression together by exploiting the principle described
in the previous section.
Second,
it is
computationally convenient, since the estimates can
be computed by implementing a series of ordinary quantile regressions (convex optimization problems) implying a need for a grid search only over the a-parameter. Third, it is
asymptotically equivalent to a
$
instruments
V
and weights
The instrumental
GMM estimator and achieves maximal efficiency by choosing
appropriately.
variable quantile regression process
§(-)
T
where
3.3.
is
a closed subinterval of
Assumptions.
set of
=
defined as
is
tGT),
(§{r),
(0, 1).
IV-QR
In order to obtain properties of the
process,
we
first
impose a
simple regularity conditions.
Assumption
Rl
2 (Conditions for Estimation). In addition to
Sampling.
(Y$,
Di, Xi, Z\) are iid defined on the probability space (Q,F,P) and
take values in a compact
set.
R2 Compactness. For all t, (a(T),/3(T)) G int A
R3 Full Rank and Continuity, a.s. sup yeR fY
d
d(a'
has
Ep
[l(y
<
p\ 7 ')
full
rank, and
is
Da xp+
'
'
x S,
\(
^y^^
continuous in (a,
ft,
7, r)
z,
x)
compact
—
>
V(r,
sets.
For
z,
x)
G
all r,
3",
$(t,
z,
functions
x)
3":
—
>
A
x
X ,D,z){y)
3 is compact
< K and
)
->
1,
$(r, x, z)
(t,z,x)
t-¥
G
Ax
3"
van der Vaart and Wellner (1996).
**( T >T
"B
uniformly in
(r, z,
/(r, z,x) are uniformly
77
and / are uniformly Holder in r: ||/(r',z,x) — /(r,z, x)|| < C|r
constants C > 0, and
< a < 1 are independent of (z,x,t, t').
in
:
x 3 x T.
$(t,z,x), V(t,z,x) g 5" and
functions in (z,x), C^-, 4 with the uniform smoothness order
See page 154
and convex.
^W
^( t - v
'
uniformly over
R4 Estimated Instruments and Weights. Wp
V(t,
suppose
(3.3),
>
x) over
smooth
dim(d, z,x)/2,
Condition Rl imposes
ables.
The compactness
relaxed. Condition
needed at
sampling and compactness on the support of the economic
hardly restrictive in micro-econometric applications, but
it
together with
Rl
imposes compactness on the parameter space. Such an assumption
parameter a(r) since the objective function
suffices for
is
asymptotic normality. The parametric identification condition
role of
R4
is
Chernozhukov
of independent interest. Essentially, this condition requires that
impacts the joint distribution of
R1-R4 may be
conditions
The
$
is
is
The
in R3
not convex in a.
similar in spirit to the nonparametric identification conditions obtained by
and Hansen (2001) and
the instrument
vari-
can be
rank condition in R3 implies global identification, and the continuity condition
full
is
R2
least for the
iid
is
refined at a cost of
(Y,
D)
many
at
relevant points. Clearly,
more complicated notation and
to allow possibly estimated instruments
needs to hold only for the non-discrete sub-component of
proof.
and weights. Smoothness
(d, x, z).
Condition
R4
in
R4
allows for a
wide variety of nonparametric and parametric estimators, as shown by Andrews (1994). See
also Andrews (1995), Newey (1990, 1997), and Newey and Powell (1990) for other examples
of such estimators.
3"
condition of
The smoothness
having a
finite
Estimation Theory. Theorem
3.4.
Theorem
(IV-QR
1
R4 can be
condition in
replaced by a
more general
L2(-P)-bracketing entropy integral.
describes the distribution of the
1
Process). Given Assumptions
1-2, for £j(r)
IV-QR
process.
=
Y{
-
TT')E^{r)^{T')'
— D^a^) + X[P{t)
=> b{r) in f°°{7),
uniformly in r, where
/J
and
b(-) is
(r,0(r))
= (r-l(e
i
(r)<O))
a mean zero Gaussian process with covariance function:
E
b{r)b{r')
=
J{r)-'S{ry)[J{T')-
1
]
',
where
J(t)
=E
Theorem
1
[f<T) (0\X,D,Z)V(T)[D'
simply states that
#(•) is
:
X'}}
,
S(t,t')
1
(Normality). For any
{^(e( Tj - 0( Tj )) }^
)
(min(r,r')
'.
approximately distributed as a continuous random
Gaussian function. This implies a variety of useful
Corollary
=
results.
finite collection of quantile indices {tj,j
A
1
n(o,
{
G J}
Jinr'sin^iJin)- ]'}^).
Corollary 2 (Efficient Weights and Instruments). When we choose the weights V*(t) =
/e(r) (0|X,Z), v(t) = ft{T) (0\D,X,Z), <F(r) = E[Dv(t)\X,Z]/V*(t), and^(r) = V*{r)[X'
$*(r)], the covariance function equals
:
E
b{T)b{r')'
=
(mm(r,r')
-
tt')
-
This choice of instruments and weights leads to an
[**(t)**(t')']
_1
-
procedure. This can be shown
efficient
by appealing to Chamer Iain's (1986) arguments. Regularity condition R4 allows use of a
wide variety of nonparametric estimators and parametric approximations of the optimal <3>
and V. For particular examples of such procedures, see Amemiya (1977), Andrews (1994,
1995), Newey (1990, 1997), and Newey and Powell (1990). An example of a simple and
practical strategy for empirical work is to construct $ as an OLS projection of D on X and
Z
(and possibly their powers) and set Vj
=
1.
Corollary 3 (Distribution-Free Limits). ForW(r)
= r(l-r) J(t)-
1
£?*(t)*(t)'[J(t)- 1 ]'
W(t)-*V^(0(t)-0(t))=»Bp (t),
where
Bp
is
a standard p- dimensional Brownian bridge
Bp
=
(p
I
+
k) with covariance
operator
E Bp (t)Bp (t') =
4.
-
(min(r,r')
tt')Ip
.
Inference
Several distributional hypotheses have been posed in the fundamental econometric and
statistical literature.
For example, an essential hypothesis
is
a pure location (constant treatment) effect or a general shape
whether the treatment exhibits
effect, e.g.
Doksum
(1974)
and
Koenker and Xiao (2002), or whether the treatment creates a stochastic dominance effects,
cf. Abadie (2002), Heckman and Smith (1997), and McFadden (1989). In structural and
causal empirical analysis, the hypothesis of endogeneity
Hausman
tests, cf.
Hausman
(1978).
In this section,
we
is
also fundamental, motivating
describe inference procedures to
test these hypotheses.
4.1.
The Inference Problem.
All of our hypotheses will be
embedded
in the following
null hypothesis:
B.(t)(6„(t)
-
r„(r))
=
0,
for
each r e 7,
r£f.
(4.1)
where R(t) denotes a known q x p matrix, q < p = dim(#) and
The parameter
#„(-) is made explicitly dependent on the sample size n to accomodate asymptotic analysis
under local alternatives.
10
The
tests will
will focus
be based on the instrumental variable quantile regression process, §().
on the basic inference process,
u„(T)=i?(r)(4(r)-rn (r)),
and
We
Sn =
form
statistics of the
(4.2)
it. In particular, we will be interand Smirnov-Cramer-Von-Misses (CM) statistics,
f(^/nvn (-)) derived from
ested in the Kolmogorov-Smirnov (KS)
which have
Sn =
respectively,
is
where
j|
v/nsup||?; n (T)|| A(T)
rST
=
a||^
Sn = n
,
%/a'Aa, the symmetric A(t)
a positive definite symmetric matrix uniformly in
in Sections 4.3
Example
1
and
|K(r)||? (r) dT,
(4.3)
J-J
t.
—
A(r) uniformly in
r,
and A(r)
The
choice of
A and A
is
discussed
>
4.5.
A
(Hypothesis of Equality of Distributions).
basic hypothesis
is
that the
treatment impacts the outcomes significantly:
a(r)
^
for
some r
in 7.
In this case,
R( T )
=R=
[1,0,
and
...]
r(r)
=
0.
The next example imposes a restrictive, yet simple mechanism through which the treatment may operate. This mechanism requires that the effect is constant across the distribution.
The
alternative
is
modern
causal models, see e.g.
with varying
Example
The
that the effect varies across quantiles.
of heterogeneous treatment effects
is
alternative hypothesis
of fundamental importance because
Heckman
(1990),
which were developed
it
motivates the
specifically to
cope
effects.
2 (Location- Shift or Constant Effect Hypothesis). The hypothesis of a
constant treatment effect
D
that the treatment
is
but not any other moments. That
affects only the location of
outcome Y,
is,
3a q(t)
:
=
a, for each
t£J.
In this case,
R( T )
=R=
Rr =
[1,0...]
and
r(r)
=r= («,/?)',
implying
a,
which asserts that the a(r)
is
constant across
by any method consistent with the
all
null, e.g. f
Example
3
(Dominance Hypothesis). The
treatment
is
unambiguously
=
r G T. The component r can be estimated
(a(^)'', $(\)')''
test of stochastic
beneficial, involves the
a(r)
>
0,
for all
li
dominance
r €
7
dominance, or whether the
null
versus the non-dominance alternative
<
a(r)
0,
for
some r £
T.
In this case, the least favorable null involves
=
R( T )
and one may use the one-sided
KS
B,
or
=
[1,0...]
and r(r)
CM statistics,
/
0,
Abadie (2002),
cf.
Sn — \/nsupmax(— a(-r),0), and Sn — n
=
||
max(— a(r),0)||^ (T) dr
to test the hypothesis.
Example 4 (Exogeneity Hypothesis).
In a basic model, the quantiles of potential or
X,
counterfactual outcomes, conditional on
are given by
Q Yd \x(T) = d'a(r)+X'f3(T).
Suppose that the treatment D is chosen independently of outcomes, that is D is independent
of {Ud}, conditional on X. Then the quantiles of realized outcome Y, conditional on D and
X,
are given
by
=
<?y|D x(T)
£'a(T)
;
Thus, in the absence of endogeneity,
(q:(t)',/3(t)')'
The
quantile regression without instrumenting.
QR estimates,
and
$(-),
+ *'/?(r).
can be estimated using the conventional
difference between
can be used to formulate a
Hausman
IV-QR
estimates, #(-),
test of the null hypothesis of
no endogeneity:
for each
q:(t)=i9(t) 1
and
i9(-)i is
the
QR
=
[1, 0, ...],
i?(r),
(4.4)
alternative of endogeneity states:
The Assumptions. We
will
Assumption 3 (Conditions
X
1.1 (Yj, Z?j,
n),
i,
P„\
(a) for
is
1
maintain the following technical assumptions.
for Inference).
contiguous to some
P
[
"), 5
(9ti{t)
-
r n (r))
=
:
).
The law of (Yt
g(r),
:
7 —>
W and
g{r)
= p(T)/y/n,
7 —¥ K9
R(r)(9(T)-r(T))=g(r).
is
Pn
and either
a fixed continuous function p(r)
(b) for a fixed continuous function g(r)
Contiguity
(4.5)
.
Zi) are iid on the probability space (O, F,
R(t)
5
plim
= 0(t).
and r{r)
3t£T:q(t)^i?(t)
4.2.
=
estimate of a(-) obtained without instrumenting. In this case,
R(t)
The
r in 7, where $(r)
defined as on p. 87 in van der Vaart (1998)
12
and
for each
n
or,
for each
n
,
D Z X
t,
t
,
t
,
t
<
Functions -R(r), g(r), p(r), and lim n
1.2
Under any
(a)
t.
local alternative, 12(a)
^(r(-)-r«(-))=*
jointly in £°°(T),
Under the
(b)
continuous in
r„(-r) are
where
b(-)
and
<*(-),
mean Gaussian
are jointly zero
<f(-)
same
global alternative, 12(b), the
functions,
holds, except that the limit
needs not have the same distribution as in 12(a).
(&(-), d(-))
Conditions 1.1(a) and 1.1(b) formulate a local and a global alternative.
Condition
1.2
and
requires that the estimates of 6(-)
guaranteed by Theorem
r(-) are
asymptotically Gaussian.
and asymptotic
In our examples, this
is
QR. Note
formulated so that other asymptotically Gaussian estimators of the
that 1.2
parameters of the
stated after
4.3.
is
IV-QR model
Assumption
1.
are permitted. Detailed discussion of this assumption
We
now prepared
are
Under Assumptions
=
=
0),
Pn (Sn >
and when
c(l
Under Assumptions
main
result.
1,
assume that
KS
or
CM statistics
in i°°{7)
(-)+p(-)),
d(r)). If vo(-) has non-generate covariance kernel,
c(l
-a))^a = JP(/(«d(-)) >
^
when
2,
-5- /?
= P(/(u
3:11 (b),
-A
oo,
alternative.
-
c(l
a)),
0),
and
(-)
Pn (Sn >
Note that
for the case of one-sided tests in
+P(0) >
c(l
-
a))
>
a.
3:I2(b),
c(l
2 states the limit distribution of the
and the global
is
we have for a < 1/2
- a))
Sn
Theorem
^S = f(v
the null is not true (p
Pn {Sn >
natives
—
7?(t)(6(t)
the null is true (p
and 3:I2(a)
3:11 (a),
1, 2,
Sn
2.
to state a
2 (Inference). For f denoting the two- and one- sided
where vq(t)
results for conventional
4.
Inference Theory.
Theorem
1
-
a)) -*
KS and
1.
CM
statistics
in the statement of
Example
3,
under local
Theorem
2
we
alter-
implicitly
the local or global alternatives to
the least favorable null violate the composite null.
As
it
stands,
Theorem
2 does not provide us with operational tests, since
we do not know
the critical value
c(l-a):P(/(«o(-))>c(l--a))
In general, one faces the Durbin problem
not-distribution free
and simulations are
= a.
when estimating c(l — a)
infeasible.
13
since the limit
is
generally
Examples
In several important cases, such as
equal to zero and so
is
and
1
not present. In these cases,
it is
the Durbin
3,
component d(-) is
an appropriate
possible by picking
weight matrix A(r) to achieve asymptotically distribution-free inference, at least under
iid
sampling.
Corollary 4 (Distribution-Free Inference). Suppose
[R(t)W{t)R{t)}~
-
||a( t )
||
A(t)
1
with
W(t)
=
in the definition of the
= A(r) + Op(l)
t(1
KS
d(-)
=
0.
If
and
CM
statistics
f
in (4-3),
=
we pick matrix A(r)
- r)J(r)- 1 £;[*(r)*(T)'][J(r)-
1
]'
to enter the
norm
and suppose we have
uniformly in r, then
/K(-)) => /(£,()),
where
Bq
is
the standard q-dimensional
Brownian bridge with covariance function:
EBq {r)Bq {T)' =
(min(r,r') -rr')/?
-
Examples 2 and 4, the transformation used in Corollary
4 will not provide distribution- free limits. There are several ways to proceed. One method is
a martinagale transformation using Khmaladzation, cf. Koenker and Xiao (2002). Another
method is a simple resampling with recentering the inference process around its sample realization, cf. Chernozhukov (2002) Simulation results in Chernozhukov (2002) suggest that
resampling has an accurate size and somewhat better power than Khmaladzation. In the
next section, we describe a different resampling method that delivers the same asymptotic
quality and is attractive computationally.
In other important cases, such as
.
4.4.
Inference by Resampling. The method of resampling we suggest
in this
not require the recomputation of the estimates over the resampling steps, which
laborious since the optimization problem requires
regressions for
many
values of
a and
r.
many computations
paper does
may be quite
of ordinary quantile
Instead we resample the linear approximation of the
empirical inference processes. In addition, to facilitate a feasible, practical implementation,
we employ the
m out of n bootstrap
(subsampling).
Suppose that we have a linear representation
v^n(-) - V^g(-)
where
Zj(-) is
1
6
for the inference process:
"
= ~r22 Zi ^ + °^( 1 )'
vn i=i
(
4 6)
-
defined below in Proposition 2.
Given a sample of the estimated scores (estimation
{zi(r),i
<n,r e
is
discussed below),
T},
consider the following steps.
The
a linear
full
bootstrap
is
hardly feasible in our second empirical example, even though
statistic.
14
we
are bootstrapping
Step
1.
Construct
The number
all
of such subsets
Denote by
<
subsets I of {j
Bn
n} of
"n choose
is
size
Denote such subsets as
b.
b."
computed over the
Vj b,n the inference process
< Bn
J,-,i
.
7
subset of data
j'-th
Ij,
t
Vj,b,n( T )
and define Sj >bjJl
=
f{Vb[vj
;
Sj^n
for cases
when 5„
is
=
b,n{-)])
2 *( T )>
as
SUpVb\\vjAn {T)\\y (T) Or Sj^n
Define for
S=
=
& /
\\vj,b,n( T ) III
M rfr
>
(CM)
f(vo(-))
= Pr{S <
T(x)
2.
l^
the Kolomogorov-Smimov (KS) or Smirnov-Cramer-Von-Misses
statistic, respectively.
Step
=
x}.
Estimate T(x) by
Bn
f
Step
3.
The
critical value is
M
size
a
= B"
1
£
obtained as the
C b ,n( l -<*)
The
(s)
=
l{5,"An( T )
1
'
when Sn >
Assumption 4 (Linear Representations).
Under any
V^ (£(-) - MO) =
-
r„(-))
jointly in £°°(7),
(b)
Under the
(&(-), d(-))
A
smaller
2.5 in Politis,
=
Ar
ff(0
where
_1
i\ n (-):
" «}-
1
a).
we make use of the following assumption.
In addition to 1.1 and 1.2
sums of
iid
-7=
zero vectors
-
£>(-A(0)*i(0 + <*.(!)
-^E*(-' r»(0)T«(0 + oPn (l)
b(-)
mean
T
1
l
—
q, jn (l
local alternative, 12(a), there exist
such that uniformly in t in
Vn(r(-)
quantile of
)
test rejects the null hypothesis
(a)
*>-
- Q = miiC T b,n(c) >
ffc^C 1
In obtaining the linear representation (4.6),
1.3
— a-th
<
and
d(-)
are jointly zero
global alternative, 12(b), the
same
=*
KO,
=* d(0,
mean Gaussian
functions,
holds, except that the limit
needs not have the same distribution as in 13(a).
number Bn of randomly chosen subsets can
Romano, and Wolf (1999). The subsampling
also be used,
is
if
the subsampling without and subsampling with replacement are equivalent
15
B„ —>
oo as
n —>
oo,
done with replacement. However,
wp —
>
1.
cf.
if
b
2
Section
/n —
0,
1.4
(a)
We
have the estimates r
and r
Zj(t,0(t))*j(t)
1-4
1-4
f(T))Tj(-r)
cZj(t,
that
take realizations in a Donsker class of functions with a constant envelope and
are uniformly consistent in r under the ^(-P) semimetric. 8
(b)
Wp-^
(c)
Functions
<C
1
for
each
and
Z,
i:
EPn U(T,On (T))fi{T)\ f= ^ = 0,EPn di(T,r n {T))fi(T)\ f= ^ = 0.
uniformly in
d, are L2(-P)-Lipschitz
[EP \\di(r,r) - di(T,r')fy
\\e'-6\\,
/2
<
r:
[i£j,||/j(-r,
6)
— ^(r,^')!! 2
1
]
C\\r'-r\\, uniformly in {0,6', r,r')
over compact sets.
Proposition 2 (Linear Representations). Under Assumption
= -7=J2 «*() + °p( 1
y/ng(-)
we have:
"
1
vW(-) -
3-4
(4-7)
),
where
Zi (-)
= R(.)
Thus, the estimate of
Zi(-) is
Zi(T)=R(r)
where J(t) and
-ff(r)
1
[J(-)- I«(-,e(-))*i(-)
-
1
ff(-)" ^(-,r-(-)T i
(-)]
•
given by
[jM-'Mr.flWlf.-W-ffM-^^fMjtilr)]
,
are any uniformly consistent estimates.
and r entering the null
the form defined above, and as-
In Assumption 4 condition 1.3 requires that the estimates of
hypotheses have asymptotically linear representation in
ymptotic normality applies to these estimates. Note that
asymptotically Gaussian estimators of the
and
1.4(c)
impose a smoothness needed
These condition are also
ence.
Condition 1.4(b)
is
IV-QR model
1.3 is
formulated so that other
are permitted. Conditions 1.4(a)
for developing the theory of the
satisfied in all of the
resampling
infer-
examples considered in this paper.
the condition of "orthogonality" which implies that the estimation of
weights $; and T, has no effect on the asymptotic distribution of the linear representation
in 1.3.
Proposition 3 in Appendix
ticular
the scores
1.
C
formally verifies that 1.3 and
Zi for
=
(r
1
li
is
not estimated,
(r,e(T))^ i (r)},
- l(Y < A«(r) + Xtffr))) ^(r) = V^t) [$*(•/-) ',*']'
t
Test of Constant Effect:
,
In this case, f (•)
= #(|)
is
an IV-QR estimate, and
defined above
*i(T)
8
=
Test of Equality of Distributions: Since r
where U{t,6{t))
li(-)
are satisfied for the par-
each of them.
zi (T)=R(r)[j(r)-
2.
1.4
implementations that we propose. Next, we briefly go through examples and state
/(W,r)
is
r(t)
consistent to
[/(tj-^mo-))*^) -
f(W,r) under
L 2 (P)
if
jur^u^u))^))]
sup T Vax [/(w,r)
16
-
/(w,r)l
.
-A
0.
for
Dominance
Test of
3.
= 0,
Since r
Effect:
the score
is
1
Z l (T)=B,(T)[j(rr h(T,e(T))^ l (T)},
Test of Exogeneity:
4-
defined in
Example
where dj(T,^(T))
the score
4,
Zi (r)
If
=
given by
is
X = (JD^)'
< X^(r))Ai,
l(Yi
J
Estimation of i/ and
estimated using conventional quantile regression as
is
[JW-^jfr.OWl^W - iKr)"^^^)]
R(t)
= (r -
r
matrices
is
t
,
= £/y| *(t?(t)'X)1X'.
ff(r)
The next Theorem
further discussed in section 4.5.2.
3 establishes the properties of the proposed resampling method.
Theorem 3 (Resampling Inference).
—^> H(t)
J(t) and H(r)
1.
When
uniformly in r over T. Then as b/n
the null is true (p
0^(1 - a)
2.
Under
w/iere
3.
Under
-2=>
local alternative
cnj.il
/3
Suppose Assumptions
- a)
= Pr(/(w
ifT
r-^l -
A2a
-*=>
(-)
— 0),
r-
!
Cn^l - a)
^
0^,
(l
-
a),
r_1
i/T
x
(l
(l
-
—
-5-
Pn (Sn > c^l - a))
->
—^
a.
r _1 (l —
a))
cn ib (l
that
a):
a))
continuous at
Pn (Sn >
(l
= Op (l),
is
>6
]
0, b
-
A 2b, Sn —^>
4. r(:r) zs absolutely continuous at
Pn (5„ > c
a),
(p
+ p(-)) >
global alternative
continuous at r~
is
—>
we have J(t)
—¥ oo, n —> oo
and
2-3,
-
a):
/?,
-a))oo:
>
wAen
i/ie
1.
covariance function of v
nonde-
is
generate a.e. in r.
Thus the resampling mechanism
consistently estimates the critical values,
and the
re-
sampling tests are asymptotically unbiased and have the same power as the corresponding
test in
Theorem
non-trivial
4.5.
2 that uses a
known
critical value.
power against the 1/^/n-local
Thus, the
and have
test are consistent
alternatives, as discussed after
Theorem
2.
Practical Implementation. Here we discuss practical implementation of the resam-
pling inference.
4.5.1.
Discretization. It
is
more
practical to use a grid
7n
in place of
7 with
the largest
size 5n —¥
as
n —\
Corollary
5.
Theorems 1-3 are valid for piece-wise constant approximations of
cell
oo.
sample processes using
7n
,
given that Sn
—>
as
17
n —>
oo.
the finite-
4.5.2.
Choice and Estimation of A(t). J(t) H{t). In order to increase the
power we
test's
could set
A*(T)
which
is
1
=Var[^(r)]-
a (generalized) Anderson-Darling weight.
for estimating A*(r),
An
= [fi*(T)]-
uniformly consistently in
9
1
,
In hd samples, there are
many methods
r.
obvious and uniformly consistent by 13-4 estimate of f2*(r).is given by:
n f—
i=l
Zi(T)=R(r) [j(;)- 1 I,(T,«(T))* i (r)-5(7)-
A
uniformly consistent estimate of J(t)
J(r)
=
1
<ij
(T,f(r))f i (r)]
.
given by Powell's (1986) estimator,
is
h (Yi - D'Mt) - X#(T))*i(T)[A,.Xi],
-Y,K
n
where Kh{x) = h~ l K(x/h) and K(-) is a compactly supported symmetric kernel with
two uniformly bounded derivatives, and h ~ Cn~ l b Estimates of
are only needed in
H
'
.
Examples 2 and
4.
In Example
and in Example
4,
a uniformly consistent estimate of H(r)
2,
is
given by Powell's estimator:
"
1
£=1
4.5.3.
Choice of the Block Size. In Politis,
Romano, and Wolf
(1999) various rules are sug-
minimum
gested for choosing an appropriate subsample size, including the calibration and
volatility
methods. The calibration method involves picking the optimal block size and ap-
propriate critical values on the basis of simulation experiments conducted with a model that
approximates the situation at hand.
combining)
among
The minimum
volatility
method
involves picking (or
the block sizes that yield the most stable critical values.
suggestions emerge from Sakov and Bickel (1999)
who
detailed
=
2 5
fen /
a related subsampling method that involves recomputing
yields optimal
performance
the estimates.
In our setting, this choice also appears reasonable.
for
More
suggest that choosing b
Our experiments and
those in Chernozhukov (2002) indicate that selecting k between 3 and 10 are attractive both
computationally and qualitatively.
This choice
from the
is
not readily suited to Example
interval T. Alternatively,
one
may
2, since
Var z,(j)
=
always simply use A(t)
18
0.
However, we can cut out
= /.
[J
— e, % + e]
Empirical Analysis
5.
This section contains results from two empirical examples that illustrate the use of the
The examples correspond to the models
example, we estimate the market demand for
inference procedures presented in this paper.
outlined in Sections 2.2 and 2.3. In the
we
fish,
and
5.1.
The Structure of Demand
in the second,
elasticities
first
consider the returns to schooling.
we present estimates of demand
demand, r, and assess the impact
for Fish. In this section,
which may potentially vary with the
level of
on the quantity distribution. The data contain observations on price and quantity
of price
of fresh whiting sold in the Fulton fish market in
from December
2,
1991 to
May
8,
New York
over the five
month period
These data were used previously in Graddy (1995)
the market and later in Angrist and Imbens (2000) to
1992.
to test for imperfect competition in
illustrate the use of the conventional
IV estimator as a weighted average of heterogeneous
demands. The price and quantity data are aggregated by day, with the price measured as
the average daily price for the dealer and the quantity as the total
The data
day.
also contain information
variables indicating weather conditions at sea,
demand
equation.
The
total
amount of fish
sold that
on the day of the week of each observation and
which are used as instruments to identify the
sample consists of 111 observations
for the
days in which the
market was open during the sample period.
The demand
function
we estimate
takes a standard Cobb-Douglas form:
Qin{Yp )\x(r)
Yp
=
A(r)
+ a(T)ln(p),
demand when price is set at p. The elasticity a(r) is allowed to vary
the quantiles, r, of the demand level. Note that this is a Cobb-Douglas version
demand model with random elasticity discussed in Section 2.2.
where
The
is
left
the
panel of Figure
1
provides the estimates of elasticities obtained by
across
of the
IV-QR
of
ln(F) on ln(P) using wind speed as the instrumental variable, while the right panel depicts
standard quantile regression (QR) estimates of the
region in each figure represents the
90%
effect of
confidence interval.
ln(P) on ln(Y).
The shaded
The estimated model does
not
include covariates, but the estimated elasticities are not sensitive to the inclusion of controls
for
days of the week or other covariates.
In general, the point estimates obtained through
corresponding
QR estimates.
The
IV-QR
differ substantially
the entire range of the quantity distribution and are uniformly small in magnitude.
QR estimates,
-2 at
the
from the
QR estimates appear to be approximately constant across
The
IV-
on the other hand, demonstrate a great deal of variability, ranging from near
low quantiles to -0.5 in the upper end of the distribution. Except at high quantiles,
IV-QR
estimates of the elasticities are uniformly greater in magnitude than the price
effects predicted
of price
by QR, demonstrating an upward bias induced by the joint determination
and quantity
in the market.
Also note that,
19
like
2SLS and OLS, the interpretation
TABLE
1
The
.
test results for
with replacement
Alternative Hypothesis
=
Effect, q(-)
while
(
subsampling
CM
Test Statistic
50
P-value for
<.01
Non-zero Effect
Fixed Elast. a(-)
=a
Random
Dominance
<
Non-Dominance
.50
= a Q1»()
Endogeneity
.18
a(-)
Exogeneity a(-)
of the
Equation, using b
)
Null Hypothesis
No
Demand
IV-QR and
QR
QR
.28
Elasticity
estimates are very different.
IV-QR
estimates a
demand model,
estimates the conditional quantiles of the equilibrium quantity as a function of
the equilibrium price.
Results from formal tests of the location-shift hypothesis and the endogeneity hypotheses
are contained in Table
downward
sloping
1.
We fail to reject the dominance hypothesis that a(-) <
demand
in view of the small
quantity, as expected there
is
=
Due
111.
weaker, giving p- values of only 18%.
at the
The
10%
level
5.2.
when we compare
=
level (recall that variances cancel
.25) of
demand
is
The Structure
and
demand. However, the
The hypothesis
we
each other for
overall results are
of constant elasticity
also rejected
is
the lower quartile elasticity with the median elasticity.
28%
overall results are weaker yielding only a
elasticity of
rest of the results require careful discussion
to the simultaneous determination of price
10%
lower quartile (t
tests) for the
indicating
clear evidence against the null of exogeneity. In particular,
reject the null of no-endogeneity at the
Hausman
The
at all quantiles.
sample n
0,
p-value.
These
results indicate that the
likely heterogeneous.
of the Returns to Schooling. As a further
illustration of the use
and inference methods presented in this paper, we use the data and
in Angrist and Krueger (1991) to estimate the effects of schooling
on earnings. In particular, we estimate linear conditional quantile models of the form
of the estimation
methodology employed
Qln(Ys )\x(r)
where S
as
is
reported years of schooling and
an instrument
We
for education.
= a{T)S + Xp(T),
X
is
a vector of covariates, using quarter of birth
10
focus on the specification used in Angrist and Krueger (1991) which includes state
11
of birth effects, year of birth effects, and a constant in the covariate vector.
we consider
consists of 329,509 males
The sample
from the 1980 U.S. Census who were born between
1930 and 1939 and have data on weekly wages, years of completed education, state of birth,
Specifically,
we use the
linear projection of S onto the covariates
X
and three dummies
for first
through
third quarter of birth, with fourth quarter as the excluded category, as the instrumental variable.
Note that the estimates of the schooling
coefficient are not sensitive to the specification of the
20
X
vector.
year of birth, and quarter of birth.
Appendix
The sample was
on the
selected based
criteria
found in
of Angrist and Krueger (1991).
1
QR estimates of the schooling coefficient are provided in Figure 2.
IV-QR and
region in each panel represents the
95%
The shaded
confidence interval. Both the quantile and inverse
quantile regression estimates suggest that the "returns to schooling" vary over the earnings
distribution.
The
first
constant treatment
row
of the treatment effects
do vary
12
statistically,
Table 2 reports the results from a
the
QR
in the
IV-QR
is
estimates.
and
The
While the
OLS
estimate.
demonstrated in the
clearly
is
test of the hypothesis of a
strongly rejected.
clustered around the
(solid line)
QR estimates
estimates, the
The shapes
all closely
estimates
which plots both the IV-QR
IV-QR
IV-QR
estimates which
most apparent
is
they are
lack of variability in the
2,
in
effect for
QR (dashed
variability
QR estimates
The
practical
first
panel of Figure
line) estimates.
Relative to the
appear to be approximately constant.
of the estimated treatment effects are also interesting.
The
QR
estimates
exhibit a distinct u-shape, implying higher returns to schooling to those in the tails of the
distribution than to those in the middle. However,
if
schooling
is
endogenous to the earnings
QTE and have no causal interpreconsistent for the QTE under endogeneity
equation, these estimates do not consistently estimate the
IV-QR
tation.
estimate, on the other hand, are
and show quite
the
IV-QR
different results
results
than those obtained through standard QR. In particular,
show returns to schooling of approximately 20% per year of additional
schooling at low quantiles in the earnings distribution which decrease as the quantile index
increases toward the middle of the distribution and then remain approximately constant at
levels
near the
QR
and
OLS
estimates.
This implies that the largest gains to additional
years of schooling accrue to those at the low end of the earnings distribution. This observation
is
consistent with the notion that people with high unobserved "ability"
,
measured as
the quantile index r, will generate high earnings regardless of their education
those with lower "ability" gain
The
third
more from the
training provided
row of Table 2 reports the results from the
would be expected,
it fails
level,
while
by formal education. 13
test of stochastic
to reject the null hypothesis of stochastic
dominance. As
dominance, confirming
our intuition that schooling weakly increases earnings across the distribution.
In the final row of Table
2,
we
test the endogeneity hypothesis.
The
test strongly rejects
the null hypothesis of no endogeneity, confirming the need to instrument for schooling in
the earnings equation. Note that this result contrasts the conclusion which could be drawn
from a Hausman
the
12
of
5%
level.
The test
14
for
test
OLS
the quantile regression estimates
is
not reported, but also strongly rejects the null hypothesis
effect.
"ability" is
used to characterize the unobserved component of earnings, which likely captures
elements of ability and motivation as well as noise.
The
estimates, which fails to reject the null at
Again, this confirms our intuition that endogeneity contaminates standard
a constant treatment
The term
based on the 2SLS and
test statistic is 3.64
and
is
distributed as a xi21
Table
The
2.
test results for
pling with replacement
Null Hypothesis
No
Effect.
Constant
a()
a(-)
Exogeneity a(-)
Equation, using b
=
P-value for
(
subsam-
KS
Test Statistic
<.01
Non-zero Effect
=a
1000
)
Alternative Hypothesis
=
Effect, a(-)
Dominance
Demand
Varying Effect
.03
<
Non-Dominance
.50
= a QR (-)
Endogeneity
.04
estimates of the returns to schooling and underscores the importance of accounting for this
endogeneity in estimation.
Overall, the results indicate that the effect of schooling
neous, with the largest returns accruing to those
distribution.
ses that
The example
on earnings
can be tested using the inference procedures presented in
effects
this paper.
The
results
which focus on a single feature of the out-
may fail to capture the full impact of the treatment and that examining
features may enhance our understanding of the economic relationships involved.
distribution
additional
6.
In this paper,
Conclusion
we described how instrumental
variable quantile regression can be used to
evaluate the impact of treatment on the entire distribution of outcomes
is
quite heteroge-
is
in the lower tail of the earnings
fall
also illustrates the variety of interesting distributional hypothe-
demonstrate that estimates of treatment
come
who
self-selected or selected in relation to potential
variable quantile regression process
and the
tests of distributional equality, non-constant
outcomes.
We
introduced an instrumental
set of inferences derived
treatment
when the treatment
from
effects, conditional
it,
focusing on
dominance, and
method
we demonstrated that the method is simple and computationally convenient and produces valid inference. The approach was illustrated through
two examples: estimation of the demand curve in a supply-demand system and estimation
exogeneity. Inference, which
is
subject to the Durbin problem, was handled via a
of score resampling. In the paper,
of the returns to schooling.
and exogeneity were
In both cases, the hypotheses of a constant treatment effect
rejected.
focus on a single feature of the
The
results suggest that estimates of treatment effects that
outcome distribution may
fail to
capture the
full
impact of
the treatment and serve to illustrate the variety of distributional hypotheses that can be
tested based
on the quantile regression process.
22
Appendix A. Proofs
We
use the following empirical processes in the sequel, for
f H+ En f(W)
For example,
and inner
/
if
is
=~J2 f(Wi)i
probabilities,
P* and P„
Gn f(W)
and —
means convergence
We
say that process
>
"with (inner) probablity going to 1."
if
n-Hx>
some pseudo-metric p on
>
and
15
1.
~~
^/(Wi))/=/- Outer
means
sup
77
|u„(0
>
—
in distribution.
>
0, there is 6
v n (l')\
€ £}
{/ t-> v„(l),l
>
<
rj)
Wp
is
—>
1
means
stochastically
:
e
p(l,l')<8
such that (£,/?)
£.,
Proof of Proposition
A.l.
will
for each e
limsup P*(
for
means: -4= I3"=i (/Wi)
.
are defined as in van der Vaart (1998). In this paper -^->
convergence in (outer) probability,
equi- continuous (s.e.) in £°°(L)
* Gnf(W) = "i £ (/W) - E f<Wi))
f
estimated function,
W = (Y, D, X, Z)
Conditioning on
P[Y<q[D,T]\Z =
bounded.
totally
is
X — x is suppressed.
For P-a.e. value z of
Z
z]
^P[q[D,UD ]<q{D,T]\Z = z]
= P[Ud <t\Z =
(3)
z],
f P[UD <t\Z
= IP
(4)
(
}
(5)
[US{ZyV)
= z,V = v]dP[V =
<t\Z = z,V=
t\Z
=
= P[U <t\Z =
z]
P[U <
z,
v]
v\Z
dP[V =
V = v] dP [V =
v\Z
=
z]
(A.l)
=
v\Z
z]
= z]
I
(7)
=
Equality (1)
T.
by Al and A5. Equality
is
by the similarity assumption A4:
(3) is
by
definition. Equality (4)
for each d, conditional
U&{z,v) equals in distribution to
Equality
t
(2)
by definition and equality
(6) is
q{d,r)
i->
continuous, since
is
(7) is
U
by A3. Note that equality
we assumed that r
i->
{q[D,UD < q[D,r]} by r
{q[D,UD < q[D,r]}
]
]
left-continuous
15
The proof
16
t
1-4
16
is
q[d, t]
1-4
q[d, r] non-decreasing
implies the event
{UD <
q(d,r)
on (0,1)
by A2. Equality
is
i-4
(2) is
immediate when
strictly increasing.
{UD <
for each d.
r}, since r
q[d, r]
On
is
To show
t} implies the event
the other hand, the
strictly-increasing
in (0, 1) for each d.
that given in Chernozhukov and Hansen (2001) and
is
said to be left-continuous
if
lim r
/
-
1T
23
q[d, r']
=
is
q [d,r].
(5) is
z)
.
holds more generally, simply note that for r € (0,1) the event
event
is
= v,X = x,Z =
on (V
given here for completeness.
and
Finally, since r
q[d, r] is strictly increasing, left-continuous,
>->
we have
= 0,
P[q[D,UD }=q[D,T}\Z=.z}
so that P-a.e.
P[Y <q[D,r}\Z}=P[Y <q[D,r}\Z}.
A.2.
tf
=
Proof of Theorem
(/3(r),0)
and
<p r
=
(u)
<
0)
f(W,a,ti,T)
/(W,a,tf,r)
where *(t)
where pr (w)
=
=
V(t)
(t
—
g{W,a,0,
r)
g(W, a, d,
t)
<
-
=
=
=
(A", *')',*(t)
l(u
proof
1. In the
(l(u
W
denotes (Y,D,X,Z).
Define for d
=
(,0,7)
and
r)
<p T
(Y - £>'a - A"/? - 4(r)' 7 )$(r),
¥> T
(F - D'a - X'0 - $(t)' 7 )*(t),
=
$(r,X,Z), §(r)
V(t)
H pT (Y - D'a - X'0 =pT (Y- D'a - X'0 -
•
(A",#(t))',$(t)
= $(X,Z);
4.(r)' 7 )t>(r),
*(t)' 7 ) V(r),
0))u. Let
Q„(a,tf,T)
=E„$0V, a, 0,t), Q(M,t)
=^5 (W,a,i?,r),
and
#(a,r)
=
(/3(a,r),
7 (a,r))
=
arg
#(a,r)
=
(/J(a,T),7(a,T))
=
arg
inf
Q„(a,i?),
inf
O
(a,$,r),
i?e3xS
=
«(t)
Step
1
arg inf
||
7 (a,T)||,
a*
(Identification) Here we show that 9(t)
=
arg inf ||7(a,r)||.
=
(a(T)' , 0(t)') uniquely solves the limit
optimization problem. Define
We know
11(9, t)
= EP
W
=
r)
by Proposition
1
[p T (Y
- D'a - X'0)9]
,
9t
=V
Q^T^Ep l<Pr(Y - D'a - X'flV]
that 6(t)
=
t
-
[X't
:
*J]'
,
(a(r)' ,0(t)') solves
II(0(t),t)=O.
Suppose there
exits another
in the
parameter set that solves this equation
n(0*(r),r)=O.
Then
for
any comformable non-zero vector A Taylor expansion gives
\'(TI(9*(t),t)-I1(9,t))=\'J(9 X (t),t)\*=0
for 0\(t)
is
on the
line
segment that connects 9*{t) and 9(t), and
A*
which yields a contradiction by the
full
=
(0*(r)-0(r))
rank assumption after setting A
24
=
A*.
So we have that, the true parameters
Etp r (Y
On
the other hand, by
(o:(t),/3(t)) solve the equation
- D'a{r) - X'0(r) -
R3 and by
=
$(r)'0)*(r)
0.
convexity in # of the limit optimization problem for each r and
a, {>(a,T) uniquely solves the equation:
Eip T (Y
- D'a - X'P(a,T) - $(T)'7(a,r))$(r) =
0.
We need to find a* (r) such that this equation holds and the norm of 7 (a, t) is as small as possible.
equal zero by Proposition 1. Thus a*(r) = a(r) is a
a* = q(t) makes the norm of 7(0:*, r) =
solution; by the preceding argument it is unique and (3(o*(t),t) — @{t).
Step 2 (Consistency) By Lemma 2
This implies by
Lemma
1 for
extremum
"
11
processes:
sup
\\'d(a,T)-'d(a,T)\\
"
(a,T)6^xT M
which
in
p,
-^0
<2„(a,tf,r)-Q(a,t?,T)
sup
(a,tf,r)6.Ax(2xS)xT
—
>
0,
(a.2)
turn implies
sup
ll7(a,r)|U-||7(o,r)|U
Ao,
{q,t)€Ax7
which by invoking
Lemma
1
again implies
sup
—
d(r)
tST
1
AO,
q(t)
'
which by equation (A.2) implies
for
^->0,
0(t)-/3(t)
sup
T6i
11
sup 7(6(t),t)reT
0-^0,
11
Step 3 (Asymptotics) By the computational properties
any a n (r) in a small ball at a T (r)
of quantile regression estimator #(a n ),
0(K/V^) = VnKf(W,a n (T)J(an (T),T)),T).
By lemma
2,
the following expansion of
r.h.s. is valid for
(A.3)
—
any a n (T) — a(r)
v^En /(W,an (r)^(a n (r)))=Gn /(W,an (r),^(a n (r),r)
>
>
uniformly
in r:
17
r)
+ VKEf(W,a n (T),V n (a n (T),T),T)
= G„/(W,a(T),1?(Q(T),T),T) +op (l)
+ y/EEf{W,an (T)J{a n (T),T),r)
Expanding the
last line further,
uniformly in r
0{K/y/K)
=
G„/(W,a(T),l?(T)) +Op(l)
+ (Mr) +
+
17
Note that by convention
(Ja (r)
oP (l))v^(^(a n (r),
r)
+ op (l))v^(a„(r) - o
in empirical process
theory
25
-
0(r))
t)),
(
Ef(W) means (Ef(W)) ,_>.
(A.5)
where
Mt) =
In other words for
WW) Ep br(y " D
any a n {r)
V^(^(a n ( r ),r) -
-
a(r) -^>
- J7
"
^
"
* (r)
'
7) * (T) W)=(o,0W)
uniformly in r
1
(t) J„(t)[1
+ 0p (l)]v/»T(au(T) - a(r)) + pp (l),
=- J7 (r)G„/(W,a T ,0 r
-0)
VrT(7(a„(r),r)
a{T)
^WGJ^a^W)
=-
tf(r))
'
- J7 (r) Ja (r)[l +
i.e
)
op (l)]VrT(a n (r)
-
a(r))
+ op (l),
where
[J/lW : Mt)']'
=
V«-
Center a shrinking closed ball at a(r) for each r and denote those balls
d(r)
Observe that by
Lemma
J7 (7-)Ja (T) and
yfc{&{ T )
where
=
arg
inf
1,
Jl7(an(r),r)|U.
2
N/ri||7(a„(r),r)|U
Since
=
B n (a(r)), wp —>
^4
have
-
o(t))
=
full
means that the plims
=
110,(1)
- Jt (t)J«(t)[1 + op (l)]v^Mr) -
rank, ^/n{a{r)
arginf
||
of the lhs
- a(r) =
P (1).
a(r))|| A ,
Hence by lemma
1
- J7 (r)G„/(W,a(r),0(r)) - J7 (t)Jq (t)mIL.
and rhs agree
in £°°(7).
Conclude that:
1
y/H(a(T)-a{T))
=
-(MtYMtYAJ^Mt))'
x (ja (r)'J7 (r)'^J7 (T))G„/(W,a(T),0(r))
=
0,(1)
and
1
i/n#(fi(T),T) -*(a(T),T))
= -J^ ^)]/1
a (r)(ja (r)'J7 (r)' J4J7 WJa(r))" Ja(r)'J7 (r)'ylJ7 (r)]
xG„/(W,a(r),0(r))-Op (l)
Now
note that due to simplifications because of invertibility of JQ J7 we have
1
^(7(a(r),r)-7(a(r)
)
r))^[-J7 (r)[/-JQ (r)(jQ (r)'J7 (r)')~ J7 (r)]Gn /(W,a(r),0(r))
=
x
Op (l)
Instead of working out the algebra to see a drastic simplification, using this fact and putting
(a„(T),i?(a„(T),T))
(A. 5)
=
we have uniformly
(&(t),0(q(t),t))
in
=
(a(r),/3(r),0
+
p (l/-v/n)j
back into the expansion
t
Gn f(W,a(T),0(r)) =
J(r)V^ (
26
f^lf^]
+ Ml)
J
Next by
Lemma
2
Gn f(W,a( T ),ti(T))
where G(r)
=> G(r) in £°°(T)
the Gaussian process with covariance function
is
=
S(t,t')
(min(r,r')
- tt')£$(t)$(t)'.
which yields the desired conclusion
*{%-.%)* war***.
A.3.
Proof of Theorem
2. Since
/
is
either
Kolmogorov-Smirnov or Cramer-von-Misses or one-
sided version of these statistics, Part 1 follows by continuous
mapping Theorem. Part
observing that
/(V»ff(-))
for
-^
= O p (l)
any tight element G„(-)
/(v^5(-)
We
x
>
once
That
3.11.4 in
/?
Vo(-)
> a
null is violated (once the
composite
is
of the specified sort,
is
absolutely continuous
follows from a generalized Andersen's
Lemma
Banach spaces,
for general
Lemma
van der Vaart and Wellner (1996).
A. 5.
Proof of Theorem
2.
Immediate from assumptions.
To
3.
simplify the presentation,
known. However, the case with the estimated matrices
proof Propostion
1 in
Chernozhukov (2002)
begin by showing Part
Step
oo,
has a nondegenerate covariance kernel.
Proof of Proposition
We
^
This follows by Theorem 11.1 in Davydov, Lifshits, and Smorodina (1998):
A.4.
in the
(?»())
also need that the distribution function of these limiting
the distribution of functional f(vo(-) +p(-)) where /
at
+
once the
in £°°(T) such /,
null is violated for one-sided tests).
statsistics is continuous.
=»
oo
2 follows by
I.
By assumption
1
is
we assume that A(r), J(t),H(t) are
straightforward by
e.g.
using the arguments
in part II of this proof.
using the following steps.
realizations of function
T
belongs to a Donsker set of functions.
We
H-»
will
z(W,r)
denote this set as
{£(W,T),reT,feE}
Consider the empirical process
(T,£WGn (£(r)),
which
is
Donsker by assumption with
limit
law denoted J{P). Consider also
its
subsample
tions
(r,0
^ GilM (£(r))
=
-^^ K(W ,t) - EZ{Wu
4
27
t))
,
j
=
l,...,Bn .
realiza-
G n (£(r)),
Let Jn (Pn) denote the sampling distribution of (t,£) h*
sampling distribution of (r,£)
i->
(£(f))-
Gj,j>, n
By Theorem
and
L b<n denote
let
7.4.1 in Politis,
the sub-
Romano, and Wolf
(1999)
-^
PL (Jn(Pn),L b>n )
where
By
-A
(A.6)
0,
denotes the Bounded-Lipschitz metric (Levy metric) that metrizes weak convergence.
pi,
Next
and pl (J(P),L Kn )
let
•
Jn{Pn,0 denote the sampling
•
L
•
J{P,Q denote
(A.6)
by „(£,)
(outer) law of r
denote the subsampling (outer) law of r
we have by
the limit law of r
definition of
maps t
This means that
i->
[Gj,&,n(£( r ))]i
pl
-A
,
since the projection
[G n (£(r))],
i->
[G„(f (r))],
i-»
sup[p L (Jn (Pn,0,L b n (0)}
Gn (£(r)).
h-»
G„(£(t)) are bounded uniform
-A
[PL (Jn {Pn ,z),L b n {z))]
,
-^
and sup[p L (J(P,€),L»,„(0)]
and
[pL
0,
in £ Lipschitz functionals of (£, r)
(J(P,z),L b<n (z))}
-A 0,
-*
(A.7)
provided
Jn(Pn,i)-> J{P,Z).
The
from the assumed in
last observtaion follows
1.4 (a)
Donskerness, assumed orthogonality
and continuity with respect to the Li(P) semi-metric by
1.4(a):
2
sup|£|k(£(r))-Gn (z(T))[
T
I
l£=l
For / denoting the two- and one-sided
•
Hn
4 SUp |Var[£(Wi5 r) -*(Wi, T)]
|
II
II
KS
T
and
I
CM
-
and
{Hn ,Hb n ) -2+
,
since the functionals / (G„ (£(•))) are
r
2.
is
(,
i
j(z(t))],
i7
definition of pi,
PL
Now we
0,
l£=Z
denote the (outer) distribution function of /[G„(z(r))],
Hb, n denote the subsampling distribution function of / [Gj
• r denote the distribution function of / [Goo(^(-))]i
(A.7)
-^
I
functionals on the empirical processes, let
•
By
1.4 (b),
and pL (H n ,H)
-^
0,
bounded uniform Lipschitz functionals of r
t->
Gn (£(•))-
need to convert this result into convergence of distribution functions at continuity points.
absolutely continuous
(
at
x
>
for
one sided
statistics) as
shown
in
the Proof of Theorem
Since the statistics are real-valued, convergence with respect to Levy metric
pointwise convergence at the continuity points x of T(x)
H
b
,
n (x)^T(x).
28
is
equivalent to
Step
Hb
t
II.
Now
note that
we
actually have the subsampling distribution Tb, n of /
but difference between T^n and
n,
Kn < f K M
/ [%>,„(-)] where
e.g.
when /
K„ =
is
KS
by invoking the condition
1
< Vb- sup
II
£„
Given the event
Hb, n (x
+
and then
1.4(b)- (c)
=
l(En )
where
1,
—>
Kn
,
>
for a small e
V"
|c||tf(r)
-
J?(r)||
+ <7||?(t) -
r(r)|||
/
>
5} for any 6
there
>
S
is
0.
0, Ht, tn {x
—
e)l(E n )
<
Ti) !Tl (x)l(En )
<
Hn ^{x + c)
I
-
(x
e)
<
—
T(x
—
c), for
r(x
-
-
e
>
e)
since e can be set as small as
we
<
Fb,ni x )
c
=
e
#ti,6(x
e).
= — e,
and c
< T bt7l (x) < T{x +
and T
+
e)
+
which implies
e
is
continuous at points of interest. This yields
III. Finally, convergence of quantiles is implied
by the convergence of distribution functions
1.
the conclusion
Step
+
e)l(En ) so that with probability tending to one:
have by Step
w.p.
[Gj, b ,n(-)}
1.3.
£„ = {#„ <
tffc.n
We
an d n °t
function:
sup \\Vb- [Ez{W, T )} z=i
->•
< /
(-)]
V
Thus wp
[i^,&,7i(-)]
be shown small. Indeed,
Ht, tn will
Ti, tn
(x)
——>
at continuity points. E.g. Politis,
Part 2 of Theorem 3
like
T(x).
Romano, and Wolf
immediate from Part
is
1
(1999).
and
contiguity.
Part 3 of Theorem 3 follows by steps that are identical to those in the proof of Part
that we have convergence of subsampling distribution
to
Tt,, n
some other
distribution T'
1,
^T
except
at the
— a) = O p (l) even if T' is not continuous at T' -1 (1 — a).
Indeed, we have c n (l — or ) < c„(l — a) < c„(l — a"), where a' and a" are picked such that T' is
/_1
_1
continuous at some finite r
(l — a') and r'
(l — a") (possible by tightness). We then have by
l
steps like in the proof of Part 1, c n (l - a ) -^ Y'~ (1 - a') and c n (l - a") -^ T'~ (1 - a").
continuity points.
By
tightness of T', c„(l
7
l
1
Part 4 has already been proved in the proof of Theorem
2.
Appendix B. Lemmas
Lemma
1
(Argmax Process).
Suppose that uniformly in
rr
in a
compact
set
H and for
a compact
setK
i.
Zn {ir)
is s.t.
ii.
Zoo{t!)
iii.
Q„(\-)
Zn(-)
Qn(Zn \n) > sup z£K Q n (z\n) -
= argsup z6A
—
-^
>
Qoo(-|-)
-
Qoo{z\-n)
is
en , e n
\ 0; Zn {n) = O p
(l) in e°°(H).
uniquely defined continuous process.
in£°°(K x U), where QooMO
Z«,(-) in e°°{U).
29
is
continuous.
Then
Proof. The
known
result is well
for the non-process case.
complicated than usual consistency arguments,
Suppose
We
that convergence in
first
iii.
Amemiya
cf.
The argument
(1985)
just slightly
is
holds uniformly in probability.
have
Q„(z|jr)-^Q„(z|7r)
where the convergence
—>
and uniformly
1,
Q n (Zn (n)\n) -
in
Pick any S
we can
7r:
c
(z, tt)
(B.l)
over compact sets. Uniformly in
e
=
pick c and
[Qoo(Zoo(t)K)
d
[ii]
e/3
> Q„(Z0O (7r)|7r) -
2e/3
>
- sup zeK ^ B
^ Qco{z)]
>
Q^Zn (n)\ir)
Hence wp ->
[iii]
> Q„(Zn (7r)|7r) -
inf ffGn
€ {c,d), d > c > O'wp
Q n {Zn {^)\^) > Qn{ZooW\n) — e/3 by definition,
Q n (Zco{ir)\ir) > Q„(Zoo(jr)|7r) -e/3 by (B.l).
[i]
>
1
Q^Z^Mln) - e.
Let {B(7r),7r 6 II} be a collection of balls with diameter S
0.
Then
Zoo(7r).
>
>
uniform in
is
e/3 by (B.l),
Q„{Z„(n)\n)
e
more
.
>
0,
each centered at
by assumption
ii,
and
for
any
so that for
P,(ee (c,c'))>l-e.
It
now
follows
wp becoming
Qoo(Zn (n)\n) >
— e,
greater than 1
-
Qoo(^cx)(7r)|7r)
uniformly in
Qoo(^oo(w)|7r)
+
7r
sup
Qcx>(^|7r)
z£K\B(ir)
Thus wp becoming greater than
1
=
sup
Qoo(z).
z£K\B{k)
— e,
sup
n .(7r)
||-Z
-
Zoo(7r)||
<
S.
7ren
But e
arbitrary, so the preceding display occurs with probability converging to one.
is
Lemma
I.
2 (Stochastic Expansions). Under assumption R1-R4,
Uniformly in
(q,/3,
j,t) in (i x
S
x 9 x T)
E„[$(W,a,/?,7,T)]^£[ff(W,a,/?,7,T)].
II.
Uniformly in r in
7
Gn f(W,a(T)J(T),j( T ),T) = Gn f(W,a(T),p(T),Q,T)+op (l),
for any (a(r),/?(r), 7(7-))
—
»
(a(r),/3(r),0) uniformly in7. Furthermore
Gn f(W,a(T),p(r),0,T)
where
Proof.
We
first
We
a Gaussian process with covariance function S(t,t') defined in
is
first
show
show that the
0i= {n=
is
G
=> G(t) in £°°(T),
II.
Denote
7r
=
(a,
/?,
7) and
II
=A
x
"B
x S where S
7
is
1.
a closed ball at
class of functions
{*,¥,ir,T)t-Kf>T (y-D'a-X'0-*(X,Z)''Y)*[X,Z),
Donsker, where
is
Theorem
defined in R4.
30
tt
en,$ gj,$6?}
0.
The bracketing number
of
7
by Cor 2.7.4
in
van der Vaart and Wellner (1996)
dim ''''A
\ogNu {e,7,L 2 {P)) =
some
for
5'
<
Thus 7
0.
)=0\\
0\\e
Donsker with a constant envelope. By Cor
is
staisfies
'-2+5'
2.7.4 in
van der Vaart and
Wellner (1996) the log of bracketing number of
X=
{(*,7t)k> {D'a-X'0-${X,Z)''y),
tt
€!!,$€?}
satisfies
log
N
.
[
]
/l^^\ =0 A
[e
)
6"
<
for
some
R3
the log of bracketing
2 +'"
= oi-
(e,X,L2 (P))
Exploiting the monotonicity and boundedness of indicator function and assumption
0.
V=
number
of
<D'a + X'P + $(X,Zy 1 ), jr€n,§€?}
{($,*).- 1(Y
satisfies
N
log
.
[
V
as well. Therefore
Class !H
is
is
Donsker since
]
(e,V,L 2 (P))
it
= 0i-
has a constant envelope by
Rl and R4.
formed by taking products and sums of Donsker classes
3,V, and
T={t4t}
:
H = 7-3 -V-7,
r
which
is
uniformly Lipshitz over (7 x
(1996) "X
is
Now we show
V), and
by Theorem 2.10.6
in
van der Vaart and Wellner
(II) using the established Donskerness. Define the process
=
h
This process
is
(*,*,'ir,T)t-tGn ipT (Y-D'a-X'P-$(X,Z)''y)y{X,Z).
Donsker (asymptotically Gaussian). Therefore the process
T^Gn
is
7x
Donsker.
also Dosnker
by
linearity in t
ip T
(Y-
D'ol{t)
- X'P(t))V{t, X, Z).
and by the uniform Holder property of
T^{a(T)',P(T)'MT,X,Z),*{T,X,Z))
in
t with respect to the supremum norm, by
verify
(ii)
(i)
R3 and R4.
(
To check the Donskerness,
the definition of stochastic equicontinuity in r with respect to the
finite-dimensional asymptotic normality by Linderberg-Levy
Gn <pT (Y - D'air) - X'P(t))^!(t,X,Z)
CLT.
)
L 2 {P)
it is
easy to
semi-metric and
Thus we have
^ G(t),
where G(r) has covariance function S(t,t').
in r,
—
and $(•)
we have by R3 and R4:
Since $(•)
>
$(•)
—
>
$(•) uniformly over
6n
=
compacts and
sup p{h(T),h{r))
31
-A
0,
tt(t)
—
7t(t)
—
>
uniformly
where p
is
the L2(P) semimetric on
JC:
= Var [<pT (Y - D'a - X'0 - $(X, Z)' 7 )$(X, Z)
p(h, h)
-<p f (Y- D'a - X'P - $(X, Z)' 7)§(X, Z)]
,
so that
sup \\Gn<p T (Y
- D'a{r) - X'$(t) -
$(r, X, Z)' 7 (t))$ (r,X,Z)
-G„ip r (Y - D'a{ T - X'0(t))*{t,X,Z)J
)
<
\\Gn<Pf(y-Ifa-X'P-*{X,Z)'i)ii(X,Z)
sup
p(h,h)<6 n
"
- Gn <p T (Y - D'a - X'P - *{X, ZYi)9(X, Z)\ -*+
as Sn ->
is
by stochastic equicontinuity of h
h->
G„ys T (Y
-
-D'a
- X'
- $(X, Z)' 7 )#(X, Z)
(which
a part of being Donsker).
Having shown
II,
9=
a simple way to
show
{(*,V,a,0,7,T)
^
I. is
to note that functions
p T (Y - D'a -
X'/3
- *(X,Z)'j)V(X,Z)}}
are uniformly Lipschitz over
(JxJx^lxSxgxT)
which by Theorem 2.10.6
in
van der Vaart and Wellner (1996) and assumption Rl means that
Donsker. From this we have a uniform
sup
||En
p r (y
3>
is
- D'a - X'0 - $(X,Z)'j)V(X,Z)
- EpT (Y - D'a - X'p - $(X, Z)'-y)V{X, Z)
and by uniform consistency of
9
LLN
and
V
Ao
and R4 we have
\\Ep T {Y-D'a-X'l3-${T,X,zy-y)V{T,X,Z)\.
sup
(o,)9,7,T)e(4x3xSxT}
l<J>=<J>,V=V
"
- Ep T {Y - D'a- X'0 -${T,X,Z)'f)V{T,X,Z)\\
-^
0.
Appendix C. Verification of Linear Representations
Proposition
3.
The conditions
and
1.3
1. 4
are verified for the proposed implementation in
Examples
1-4 under conditions R.1-R4-
Proof.
1
In
Example
by contiguity of
Pn
1, in
the test of equality of distributions,
relative to
P
under which Theorem
1
Z4 (T)=fl(T)[j(T)- / i
1
1.3 is satisfied for #(•)
was proven. Since
r
=
by Theorem
0,
(T,^(T))* i (T)],
where
h(T,9(T))
=
(t
- l(Y <
t
DMt) + X'tP(T))) ,%(r) = V^^dr)' ,X'}'
32
(C.l)
Condition
1.4(a)
Condition
1.4. (b)
Lemma
checked in the proof of
is
holds by Proposition
and since
1
2 in
$,- is
Appendix B,
cf.
the class of functions "K.
a function of (Xi, Z;) only. Condition
1.4(c)
holds by the bounded density condition R3.
In
Example
2, in
the test of constant effect,
=
r(-)
Q{\)
is
an IV-QR estimate. Thus
for
/*(•)
defined in (C.l)
zt(r)
[JM-^MMJiMt) - Jii^kdAi^id))]
R(r)
= k{h,0{i))^i(l)-
di(r,r(T))T,-(r)
i.e.
=
Thus
.
hold by the preceding argument.
I.3-I.4
Example 3, the test of Stochastic Dominance,
Example 1, for l defined in (C.l)
In
r
is
also
known, so the situation
is like
that in
t
Zi (T)=R(T)[j(T)so 1.3
Y
and
1.4
li (T,0(T))V i (T)],
are verifed.
In
Example
on
D,X,
e.g.
1
the test of no-endogeneity, the estimate of r(r)
4, in
denoted as
i?(r).
given by the ordinary
is
QR of
In this case under conditions R.1-R.4, the estimator $(r) satisfies 13,
Portnoy (1991):
n
v^ (*() -
*»(-))
di(T,*{T))
Thus the
score
is
=
R(t)
conditions 1.3 and 1.4 for
2 checks
1.4. (a)
=
(r
-
l(Yi
1
'2
J2 *(.*»()) + oVn (1),
< X[d{r))Xiz
X =
t
(Dj.Af)'
given by
Zi (r)
The
= Hty'n-
Z;
[JM-'iitT.OWl^W-ifW-^tr^W]
are checked above.
(t, #(t))\I'j (t)
(put Xi in place of $* and
7=
of the quantile regression coefficient $(r), so
As
for dt (r,
0). Note that Edi(r,
1.4. (b)
is
satisfied,
and
i?(r))
1.4. (c)
.
tf ),
=
the proof of Lemma
from the definition
holds by the bounded
density condition R.3.
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New York.
VAN der Vaart, A. W. (1998): Asymptotic statistics. Cambridge University Press, Cambridge.
van der Vaart, A. W., AND J. A. Wellner (1996): Weak convergence and empirical processes. SpringerVerlag, New York.
Wiley
&
Sons
Inc.,
J.
IQR: Treatment Elfect
0.2
FIGURE
1.
0.4
0.6
The sample
while the quantile index
0.2
0.8
size
is
is
111.
QR:
Price Effect
0.4
0.6
0.6
Coefficient estimates are on the vertical axis,
on the horizontal
axis.
The shaded region is the 90%
The first panel contains
confidence band estimated using robust standard errors.
demand obtained through instrumental variables
The second panel presents estimates of the effect of ln(P) on
estimates of the price elasticity of
quantile regression.
ln(Q) obtained through standard quantile regression. Estimates were computed
r£
[.05, .95].
for
OR: Schooling
IQR: Schooling
0.2
FIGURE
2.
0.4
0.6
The sample
0.E
size is 329,509.
Coefficient estimates are
on the
verti-
The shaded region is
band estimated using robust standard errors. The first panel
cal axis, while the quantile index
is
on the horizontal
axis.
the 95% confidence
contains estimates of the returns to schooling obtained through instrumental variables quantile regression. The second panel presents estimates of the effect of years
of schooling on earnings obtained through standard quantile regression. For comparison, the dashed line in the
first
panel plots the schooling coefficient estimated
through standard quantile regression. All estimates were computed
for
r €
[.05, .95].
SAB
2003
Date Due
Lib-26-67
MIT LIBRARIES
3 9080 02246 2425
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