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Working Paper Series
INFERENCE
ON
Institute ot
QUANTILE REGRESSION PROCESS,
AN ALTERNATIVE
Victor Chernozhukov
Working Paper 02- 12
February 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.com/paper.taf '/abstract id^ xxxxx
DEWEY
2
f
Massachusetts Institute of Technology
Department of Economics
Working Paper Series
INFERENCE
ON
QUANTILE REGRESSION PROCESS,
AN ALTERNATIVE
Victor Chernozhukov
Working Paper 02-1
February 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
http://papers.ssni.com/paper.taf7abstract id= xxxxx
at
INFERENCE ON QUANTILE REGRESSION PROCESS,
AN ALTERNATIVE
VICTOR CHERNOZHUKOV
Abstract.
A
very simple and practical resampling test
inference based on Kmaladzation, as developed in in
alternative has competitive or better power, accurate size,
of
non-parametric sparsity and score functions.
series data.
effect of
It
is
offered as
an alternative to
Koenker and Xiao (2002a).
This
and does not require estimation
applies not only to iid but also time
Computational experiments and an empirical example that re-examines the
re-employment bonus on the unemployment duration support
this approach.
Key Words: bootstrap, subsampling, quantile regression, quantile regression process,
Kolmogorov-Smirnov
test,
unemployment duration
1.
Introduction
Inference in quantile regression models, pioneered by the classical work of Koenker
Bassett (1978),
crucial to a
is
and
wide range of economic analyses. For example, evaluation
of the distributional consequences of social programs requires inference concerning nature,
direction,
and quantity of the impact throughout the entire outcomes distribution. See
e.g.
Abadie (2002), Buchinsky (1994), Heckman and Smith (1997), Gutenbrunner and Jureckova
(1992), McFadden (1989), Koenker and Xiao (2002a), and Portnoy (2001). Just like in the
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.
It differs
however from the
early approaches by replacing the basic (indicator) regressors with general ones.
The main
difficulty associated
with such inference
tures, estimated nuisance parameter, or non-i.i.d.
is
the Durbin problem - the model's fea-
data induce parameter-dependent asymp-
1
jeopardizing distribution-free inference.
In a recent Econometrica paper Koenker
and Xiao (2001) proposed an ingenious and intricate theory, based on Khmaladze transformation, that purges the tests statistics from the non-distribution-free components, restoring
totics,
MIT working
paper, January 2002. Contact: 50
Memorial Drive, E52-262
F,
Cambridge
MA
02142,
VCHERN@MIT.EDU.
This project would have been possible without receiving help and encouragement from Takeshi
and Roger Koenker.
Hansen
for
I
would
like to
express
my
appreciation and gratitude to them.
I
also
Amemiya
thank Christian
very useful comments.
The problem gained importance
after a long series of
bibliography.
1
remarkable works by Durbin,
listed
in
the
VICTOR CHERNOZHUKOV
2
distribution-free inference.
dictable
component
The approach
uses recursive projections to annihilate the pre-
a martingale that limits to a standard Brownian
in the process, leaving
motion, thus overcoming the Durbin problem.
Here we suggest a simple resampling alternative that
complex Khmaladze transformation,
(ii)
metric nuisance and score functions,
(iii)
the sense that
does not require the estimation of the nonparahas an accurate size and the optimal power (in
has same power as the test with
its
does not require the somewhat
(i)
very competitive with Khamaldzation, (iv)
known
The
basic idea
The
process.
is
is
aimed
at substantively
extremely simple.
H
The key
correctly, regardless of
),
which makes
is
it
com-
a useful complement to
based on the quantile regression
statistic is
denoted H, under the null hypothesis. In
whether the null
As a
an appropriately re-centered quantile process.
result,
is
H
sampling purposes, we choose the subsample bootstrap,
cf.
we resampie
true or not,
as well as the entire null
law of the process are correctly estimated under local departures from the
(1999).
is
expanding the scope of empirical inference.
statistic has a limit distribution,
order to estimate
2
robust to dependent data, and (v)
is
putationally and practically attractive. Therefore, the approach
Khmaladzation and
value
critical
Politis,
null.
For
re-
Romano, and Wolf
which has computational, practical, and certain theoretical advantages over the
usual (unsmoothed) bootstrap for quantile regression,
Bickel (2000). Horowitz (1992)'s
mechanism
cf.
Buchinsky (1995) and Sakov and
smoothed bootstrap may
also be
an attractive resampling
for these tests.
The underlying principle differs from the conventional bootstrap tests for goodness of fit.
The conventional tests resampie from a probability model that is consistent with the null,
see e.g Romano (1988), Andrews (1997), and Abadie (2002). Although such approach is
potentially useful in quantile regression settings,
remains unknown, because the
its validity
quantile regression families estimated in emprical work are typically mis-specified and in-
complete probability models (regression quantile
and the
In
tail
what
and
id => and
we use P*
follows,
—
>
to denote
in distribution for
to denote (outer) probability,
weak convergence
random
The questions posed
in the
for
(1974),
(1989).
under P*.
fundamental econometric and
effect,
effect, cf.
Quantile regression
is
statistical literature are
(1999),
Koenker and Xiao (2002a).
tests
do not generally have
or,
Abadie (2002), Heckman and Smith (1997),
an important and practical tool
about such distributional phenomena.
Khmaladze
whether
a location-scale effect, or a general shape
Koenker and Machado
example, a stochastic dominance
and McFadden
and convergence
The Testing Problem
the treatment exhorts a pure location
Doksum
which possibly depends on n,
bounded functions
in a space of boi
vectors, respectively,
2.
effect, cf.
lines are not constrained to avoid crossing,
regression quantiles are not estimated).
this property, see
Koenker and Xiao.
for learning
INFERENCE ON QUANTJLE REGRESSION PROCESS
Suppose
Y
is
X
the outcome variable, and
FYlx
are regressors. Let
the conditional distribution function and the r-quantile of
Y
3
and Fy\ X (r) denote
The
given X.
basic conditional
quantile model takes the linear in parameters form:
F- A.(r)=X'/?n (r).
1
7 =
6 T, where
for all t
[e,
1
—
U~
e]
This
are quantiles of interest.
model
Y—
entire
shape of the conditional distribution and includes the
X'f3n (U), where
location-scale models as special cases.
Pn(r)
is
As
in
made dependent on
The
f/(0, 1).
To
a
is
random
coefficient
stated model allows regressors to affect the
classical linear location
power
facilitate the local
and
analysis, parameter
the sample size n.
Koenker and Xiao (2002a), we consider the following
#(t)/3„(t)-7-(t)
=
null hypothesis:
reT,
*(t).
(1)
where R{t) denotes a q x p matrix, q < p = dim(/3), r£l', and \£(t) denotes a known
^ T -» R9 We assume that functions R{t), ^(t), t(t), and /?o( T ) = lim n /?n (T)
function
:
.
are continuous in r.
The
tests will
be based on the Koenker-Bassett quantile regression process
(3n (-):
n
V
pn {r) = arg min
pT
(Y,
- Xtf)
r € T,
,
(=1
where p T (u)
=
u(t
— I(u <
Other estimators, such
0)).
as Chamberlain's
minimum
distance
or instrumental variable quantile regression estimators, can be considered as well, depending
on the problem.
We
will focus
on the basic inference process:
«n(T)=(5(T)A,(T)-f(T)-*(T))
and derived from
it
Kolmogorov-Smirnov and Smirnov
Sn =
where
||a||
definite
v
=
v/nsup||i; n (r)||v (T)
r€T
Example
1
The
in r.
(Location Hypothesis).
An
Sn = n
statistics
/
||u n
Sn = f{\Znv n (-)),
(r)||| (r) dr,
(3)
JJ
—^
Va'Va, the symmetric V(r)
symmetric matrix uniformly
,
(2)
)
V(t) uniformly
choice of
V
and
in r,
and V(t)
V
discussed in section
is
important hypothesis
is
a positive
3.
that of the classical
is
location-shift regression
F^x (T)=X'a + 1 -Fwhere
V
is
(a2, ...,a p )',
independent of X.
i
{r),
or
In this case, R(t)
= R =
[0
:
•
V,
/p -i] and r(r)
=
r
=
which asserts that the quantile regression slopes are constant, independent of
The component r can be estimated by any method
LAD, OLS, and others in Koenker and Xiao (2002a).
t.
Y = X'a + 7
consistent with the null, for
example
VICTOR CHERNOZHUKOV
4
Example
X
Hypothesis). The
2 (Location-scale
and
to affect only the location
F-^{T)
= X'a + X'j-F-
l
V is independent of X. In this case,
= 0. Estimates of a and 7 F 7 (t) can
1
\I/(t)
•
v
Example
D=
3 (Stochastic
/0_i(-)
is
Dominance
•
(t),
be obtained by using
see
/?i(-),
R =
OLS
[0
:
Ip-i],
and
projection of each
Koenker and Xiao (2002a).
Hypothesis). Suppose
The
V,
D =
test of stochastic
1
denotes receipt and
dominance, or whether
unambigiously beneficial, in the model
F^ x
dominance
involves the
= a + 7 Fy
r(r)
on the intercept
denotes non-receipt of a treatment.
the treatment
moments:
Y = X'a + X'-y
or
(T),
where
component of slopes vector
location- scale shift regression constraints
scale of Y, but not any other
(T)
=
D5(T)
+
X'd( T ),
null
5(t)
>
for all t
0,
1
G
versus the non-dominance alternative
5(t)
<
0,
for
some r €
T.
=
0,
In this case, the least favorable null involves t(t)
P(t)
=
R —
—[1,0...],
^(r)
=
0.
and
and one may use the one one-sided Kolmogorov-Smirnov or Smirnov
Sn = v ninfrST max( — 5n (r),0) and Sn = y/n J max( — 5n (r), 0)\\y {r) dr to test
(5(t),0(t)')',
statistics
/
||
the hypothesis.
We
will
maintain the following assumptions.
A.l (Y ,X
t
t
,t
<
n)
is
stationary and strongly mixing on probability space
A. 2 Law of (Yt,Xt,t <
(a) for
P ™\
n),
[
is
contigious to
a fixed continuous function p(r)
R(t)Pu(t)
(b) for a fixed
-
r(r)
=
*(t)
(a)
continuous function g(r)
Under any
local alternative,
7 —>
+ g(r),
R(r)p(T)-r(r)
A. 3
:
:
P'"l, 3
some
W
g(r)
and
=
mean Gaussian
and either
for
each n
for
each n
y/n
- Pn {-)) => &(•)•
^(T), where (6,p,?)
(J3 n {-)
on
p.
\/"(-R(')
_
are jointly
functions with nondegenerate covariance kernel,
Under the global alternative, A2(b), the same holds, except that the
needs not have the same distribution as in A3 (a).
e.g.
)-
= ^(T)+g(T).
A2(a),
(b)
As defined
Pn
p(r)/y/n, or.
7 —> Rq and
R(.)) => p(.), v/n(f(-) -r(-)) => -?(•)> jointly in
zero
(fi.3",
87 in van der Vaart (1998)
limit
(b, p,
<;)
INFERENCE ON QUANTILE REGRESSION PROCESS
A.l allows a wide variety of data processes:
sufficient
but
is
iid,
time
and panels.
Mixing
is
not necessary for consistency of subsampling. Stationarity can be replaced
by more general
Romano, and Wolf
stability conditions, see ch. 4 in Politis,
and A. 2(b) formulate a local and a global alternative.
is
series,
5
A. 3
is
(1999). A. 2(a)
very general condition, that
implied by a wide variety of conditions in the literature, most remarkable and general of
which are given in Portnoy (1991), who allows shape heteroscedasticity and dependent data.
Thus A. 3. substantively generalizes Koenker and Xiao's (2001) or Koenker and Machado's
(1999) conditions (local-to-location-scale assumption and
the hypotheses in Examples
Proposition
2,
sampling) which elegantly suit
but are restrictive and not necessary in Example
Under conditions Al, A2a, A3,
1.
1.
and
1
iid
Vnvn {-) =>«() =u(-) +
3.
in £°°(7)
d{-)+p(-),
MO
=
where u(t)
and d(r)
i?(r)'6(r)
=
{Po{ T )p{ T )
sn
2.
Under Al, A2b, A3, y/n(vn {-) -
d{T)
=
{(3o{t)p{t)
1.
<f(r)).
once g
statistics in (3)
The
+
And Sn
usual component u
iid
oo
if
=
u(-)
/(%/"<?(•)
components that
Under
the null,
p
=
0,
+ d(-). where u{r) = Z?(t)'6(t) and
+ Op {\)) —^ oo (which is true for
illustrate the
Durbin problem:
typically a Gaussian process with non-standard covariance
is
kernel, so its distribution can not
away by imposing
•
s = /M-)).
g{-)) => v(-)
—^
?( r ))
^ 0/
limit consists of three
The
=»
+
be feasibly simulated.
conditions. However, the
This problem
may be assumed
problem does not go away, once the data
is
a time series or a panel. In such setting, Koenker and Xiao's method unfortunately does
not apply in
2.
its
present form.
Component d
Koenker and Xiao
is
the Durbin
isolate
nonstandard covariance
component that
is
present because
R and r
are estimated.
d as a chief problem that makes the entire term v to have a
kernel.
They use Khmaladzation
to annihilate this
component.
3. Component p, which describes deviations from the null, determines the test's power.
As Koenker and Xiao show, the Khmaladzation inadvertently removes some portion of this
component as well. In fact they gave examples where p is removed completely, such as piece-
wise linear densities. Such densities are not commonplace, yet they can easily approximate
a well behaved density.
Nevertheless,
Koenker and Xiao show that Khmaladzation has
respectable power in most practical cases.
Khmaladzation requires estimation of several nonparametric nuisance functions - the
and various score functions, see Koenker and Xiao for details. The
sparsity functions
feasibility of this
may depend on
the procedure
not laborious. Otherwise, for instance in Example
functions
is
is
more
difficult
the underlying model. Under a location-scale shift model,
and has not been implemented nor had
its
3,
estimation of score
theory been established.
VICTOR CHERNOZHUKOV
6
we describe a simple approach that
In the next section,
very useful in practice, does
is
not erase components of p under any circumstances, and does not require nuisance function
estimation.
From a constructive point of view, the approach
of the Koenker and Xiao methods, which are brilliant
Rather, the approach
is
meant
not intended to be a critique
is
and useful
in
be a useful complement, aimed
to
many
conceivable cases.
at substantively
expanding
the scope of empirical inference.
Resampling Test and Its Implementation
3.
The
3.1.
Test.
The approach
is
v n {r)
based on the mimicking process v and
=
v n {r) -g{r),
Proposition 2. 1. Given A.l, A2a, A. 3
A. 3 v n (T)^v(-)=u(-)+d(-),
S n =^S
Under
when
local alternatives,
vn
=
{-)
Sn
=
=> v
statistic
Sn
f{v n (-)).
§ n => S. 2.
(-),
Given A.l, A2b,
/(«(•))
Sn
the statistic S n correctly mimics the null behavior of
This does not happen under global alternatives, but this
the null is false.
:
,
even
is
not
important.
we use
In what follows
v n (r) itself to estimate g(r),
and use the subsample bootstrap
consistently estimate the distribution of S, which equals that of
The
usual bootsrap will also work, but subsampling
grounds explained
(2000).
The
(1999),
in
is
S
to
under the null hypothesis.
preferable on both computational
Buchinsky (1995) and theoretical reasons given
in
Sakov and Bickel
4
subsample bootstrap, introduced by
basic idea of the
is
statistic
to
approximate the sampling distribution of a
Politis,
statistic
computed over smaller subsets of data. The resampling
Romano, and Wolf
based on the values of this
tests
based on subsampling
are done in three steps.
Step
1.
when Wt
For cases
such subsets
B n is
"n choose
=
(Yt,Xt )
6."
subsets of size b of the form {Wi,
each i-th subset,
i
< Bn
is iid,
construct
For cases when
...,
{W }
t
is
all
subsets of size
6.
The number
a time series, construct
of
B n = n— 6+1
Wi+b-i}- Compute the inference process v^ n ,i(')^
f° r
5
.
Denote by v n the inference process computed over the entire sample; and by
computed over the i-th subset of data:
i>f> in> i
the
inference process
Another attractive alternative
is
the smoothed bootstrap as in Horowitz (2001). Subsampling
is
used
for
pragmatic, computational reasons. In addition, Sakov and Bickel (2000) show that subsampling, combined
-2
with interpolation, yields the same minimax order of occupancy as smoothing once subsample size b oc n ' 5
.
A
smaller
2.5 in Politis,
number B n of randomly chosen subsets can
Romano, and Wolf (1999).
also
be used,
if
Bn
—> oo as n
—>
oo,
cf.
Section
INFERENCE ON QUANTILE REGRESSION PROCESS
and define § nj 6
S„ i6i2
for cases
,
=
7
= supv^lK^r) -
when Sn
is
and
->
)])<
H
=b
u n (r)||v(r) or S nbi
,
under
-
b,i{j)
un (r)|||,,dT,
Define
statistics, respectively.
x}.
- ff(-)ll = v^ x Op (l/y/n) -^4 0, including when
- <?() + (g(-) - v n (-))\\ = Vb\\v nAi {-) - g(-) + op (l)\\,
b -> oo, V^ll^nO)
y/b\\v n
\\v n
H{x) = Pr{S <
and
bii {-)
Therefore, the distribution of
i.
coincides with
Step
f° r instance
-
= Pr{S<x}
ff(r)=p(r)/ > /fi. Therefore
uniformly in
^n(
Kolomogorov-Smirnov or Smirnov
G{x)
As 6/n
—
f{vb[vt,, n ,i{')
7
(r, g)
§„,{,,,
can consistently estimate G, which
Thus, the following steps are
local alternatives.
clear.
Estimate G[x) by
2.
B„
Gn
,
b
= S- ^l{S n (M (r)<x}.
1
(x)
,
!=I
Step
3.
The
critical
value
is
obtained as the
cnJb {l-a)
The
size
a
Theorem
(i)
test rejects
Given A.l
1.
When
Under
where
(3
Under
//(x)
v
a)
-a)
= Pr{f{v
p
-^
is
is
A2b,
—> oo,n —»
Pn (5n >
a),
0, (f // is
-^ G-
a
(l
G
-
is
+p())
it
Bn —
oo,
cn
,
a).
c
oo,
>
6
_1
(l
i6
(l
—
(l
-
continuous at
Pn (Sn >
-
a)) -> a.
H~
l
(l
a)) -+
continuous at
G _1 (l —
Pn (Sn >
-
a),
a):
—
a):
/?,
-a)),
test is asymptotically
—* °°- the
critical values are
—
continuous at i/
//"'(I
if
c n> (,(l
x
at
c„, 6 (l
>
wften
a),
a)) -*
i/ie
—^
and Sn
oo:
1.
covariance function of
nondegenerate.
Thus the resampling
Op {\)
may be
known
power
and
j3
goes to one.
Sn — ^
unbiased and has the same power as the
critical value.
Furthermore
Under global
if
as p(r)
—>
useful to account for an error in
G~\(l — a) caused by
of standard error to
oo or
— oo,
alternatives, the estimated
oo for Kolmogorov-Smirnov and Smirnov
we added 1.69 times an estimate
conservative when B„ is small.
simulations,
when Sn >
and G(x) are absolutely continuous
and v
In practice
^
+ p{-)) >
corresponding test that uses a
f( v o(')
-
//"'(I
G n<b (-): 6
a-th quantile of
= G-^(l-a).
0, b
H
0. if
—
-^> //"'(l - a),
{-)
- a)
=
A2a, p
global alternative
c„, 6 (l
(iv)
-
—>
A. 3 as b/n
local alternative
c„, 6 (l
(iii)
-
the null is true,
c„, 6 (l
(ii)
the null hypothesis
1
G~\(l —
Bn
a), to
statistics.
being "small".
make
the test
In
more
VICTOR CHERNOZHUKOV
8
3.2.
Approximations.
It
may be sometime more
with the largest cell size 5n —>
Corollary
1.
as
n —>
practical to use a grid
Propositions 1 and 2 and Theorems
Estimation of V(r).
1
is
—
as
We
for estimating V*(r), uniformly consistently in t.
iid
n —»
oo.
we could
In order to increase the test's power
a (generalized) Andersen-Darling weight. In
general cases.
set
_1
,
many methods
samples, there are
are not aware of any results for
more
Note that we would need a Newey and West (1987) type estimator that
consistent uniformly in
Subsampling
7
and 2 are valid for piece-wise constant
V(t) =V*{t) =Var[u(r)]
which
in place of
oo.
approximations of the finite-sample processes, given that 5 n
3.3.
7n
is
t.
can be used to estimate V(t), even without assuming asymptotic
itself
integrability conditions. This
is
possible by using a percentile
method
in conjunction with
asymptotic normality.
Consider the truncated variance
Vk( t =
)
=
K(t)
=
Uj
1
*-
—
_1
Var[w K (r)]
p
1= i[Lj(t),Uj(t)]
,
where v k {t)
a large compact
is
We
a-quantile of Vj(t).
set.
=
v{t) x
E.g.,
I
km (v{t)),
L3 =
a-quantile of vj(t), and
can estimate V^(r) using Theorem 2 stated below.
Note that having estimated the truncated variance, we may stop there since
V£(r)
«
V*(t), and
Theorem
Second, using that v(t)
=
1
K,
applies to any positive definite symmetric V(t).
N(0, V*(r)), we can use the percentile method
estimate of diagonal elements of V*(r) based on V^-(t).
correlations,
for a large
we can then estimate
off-diagonal elements.
to obtain
an
Using symmetrically trimmed
In simulations
we simply used
un-truncated variances.
Theorem
2 provides the uniformly consistent in t estimates of
any truncated moments of
the process v n (r), including the trimmed correlations. This theorem
of the ingenious results of Politis, Romano, and Wolf (1999).
Let r
i—>
is
v(t) be an element of £°°(T), equipped with the sup norm,
of measurable Lipshitz functions
\\<p(v)
-
<p(v')\\
<p
:
<
£°°(T)
c
•
sup
—>
K
M.
||i,(t)
-
a direct consequence
and L(c,
k)
be a
class
that satisfy:
v'(t)1
\\ip(v)\\
<
k,
T
where c and k are suitably chosen positive constants. For probability laws
Q
and
Q', define
the bounded Lipshitz metric (which metrizes weak convergence) as
Pbl{Q,Q')
Useful examples of
vp
p
ip
include (p(v)
and we replace the indicator
=
=
sup\\EQ if-
EQ np\\.
w(t) to / a-('D(t)), where (v\,
...,v
p)
m = v™
1
x
...
x
1 K{t) (v(t)) by a smooth approximation / A( -)(u(t)) which
.
INFERENCE ON QUANTILE REGRESSION PROCESS
K,
vanishes outside compact set
for all r.
correlations. For example, if v(t)
Vax[v K (r)]
is
This defines
Ln^
Theorem
in £
oc
kinds of truncated
all
= E Lnb [v{r) 2 f K (v(t))} = B~ ^[(^(t) l
:
Under assumption A.1-A.3,
2
v n (r)) f K (v nAl (r)
-
vn (r))]
=1
denotes the subsampling (outer) law of v n ^j;(•)
2.
moments and
a scalar (for clarity sake), then
2
where
9
letting
L and Lq
—
v n {-) in £°°(7).
denote the laws ofv(-) and
vq(-)
(7), respectively,
p~
pBL{Ln,b'L) -2»
and L equals Lq under
t
£ 7) within
0,
local alternatives. In particular, for
functions (v
>->
v(T) m f K{r) (v(T)),
Ti(c,k),
sup
E L ^[v(r) m fKav(r))}
-
EL [v(r) m fK{r) (v{r))]
^0.
7£T
The
last
statement remains true even when /K (r) (-)
Choice of Block
replaced by l/r( T )(
is
Romano, and Wolf
(1999) various rules are suggested for choosing appropriate subsample size. Politis, Romano,
and Wolf (1999) focus on the calibration and minimum volatility methods. The calibration
method involves picking the optimal block size and appropriate critical values on the basis of
simulation experiments conducted with a model that approximates a situation at hand. The
minimum volatility method involves picking (or combining) among the block sizes that yield
more stable critical values. More detailed suggestions emerge from Sakov and Bickel (1999)
and Buchinsky (1995). Sakov and Bickel (2000) suggest that choosing b = hi 2 5 yields 7
the optimal minimax accuracy (in conjunction with extrapolation). Our own experiments
indicated that the constant k between 3 and 10 are attractive both computationally and
3.4.
Size. In Sakov and Bickel (1999) and in Politis,
'
qualitatively,
which well accorded with the results of Sakov and Bickel (1999)
for the
sample
median.
4.
A Computational Example
The computational experiment
that
we consider
is
that of Koenker and Xiao (2002a).
This allows us to compare the performance of the resampling
without prejudicing against the
Their result
is
for
latter.
Khamaladzation
Consider the location-shift hypothesis as in Example
the subsample bootstrap with replacement
replacement versions are asymptotically equivalent once b 2 /n
(1999).
test vs.
—
>
0.
However, the replacement and nonSee
e.g.
Politis,
Romano, and Wolf
VICTOR CHERNOZHUKOV
10
1.
The data
is
generated from the model:
Y = OL + pX + cT{X )-eu
i
i
i
=70+7!
o{Xi)
X{,
d ~N{Q,1), Xi ~iV(0,l),
a=
Under the
=
null hypothesis 71
0.
0,0
We
= l,7o = l-
examine the empirical rejection probabilities
for the
sample sizes and
we used the OLS estimate of /3 and 7 = [.05, .95]. When 71 =
the model is a
location-shift model, and the rejection rates yield the empirical sizes. When 71 7^
the
model is a heteroscedastic model, and the rejection rates give the empirical powers. Table
1 reports the results and compares them with Khmaladzation. Other details of the set up
are as those reported in Koenker and Xiao (2002b).
heteroscedasticity parameter 71
test for different choices of
the
.
In constructing
test,
Table 2 speaks for
From
ples.
serious
itself.
these results,
complement
The resampling
test
powerful and accurate even in small sam-
is
say that the resampling test emerges as a respectable,
fair to
it is
Khamaladzation method. The method
to the
is
also quite robust to
variation of subsample size - a wide variety of subsample sizes performs very well (including
when the resampling mechanism
small sub-samples
(6
=
5n 2
5
'
)
5.
To
illustrate the present
is
the n out of
n bootstrap), suggesting that even
are both computationally
An
and qualitatively
fairly
attractive.
Empirical Application
approach, we will re-analyze and expand on the main empirical
question considered in Koenker and Xiao (2002a).
The question concerns
the Pennsylvania
re-employment bonus experiment conducted by the U.S. Department of Labor, 8 which was
conducted
scheme
in the 1980's in order to test the incentive effects of
unemployment insurance
for
were randomly offered a cash bonus
and
if
an alternative compensation
(UI). In these controlled experiments,
if
UI claimants
they found a job within some prespecified of time
the job was retained for a specified duration.
The
goal was to evaluate the impact of
such a scheme on the unemployment duration.
As in Koenker and Xiao (2002a) we focus on the compensation schedule that includes a
lump-sum payment of six times the weekly unemployment benefit for claimants establishing
the reemployment within 12 weeks (in addition to the usual weekly benefits). The definition
of unemployment spell includes one waiting week, with the maximum of uninterrupted full
weekly benefits of
27.
The model under
consideration
is
the linear conditional quantile model for the logarithm
of duration:
QW)(T\X) =
There
see e.g.
is
a(r)
+
5(t)
D + X'P(r),
a significant empirical literature focusing on the analysis of this and other similar experiments,
Meyer
(1995)'s review.
INFERENCE ON QUANTILE REGRESSION PROCESS
where
is
T
is
D
the duration of unemployment,
is
1
the indicator of the bonus
offer,
and
I
X
a set of socio-demographic characteristics (age, gender, number of dependents, location
within the state, existence of recall expectations, and type of occupation). Further details
are given in
Koenker and
Bilias (2001).
The
estimate of
Quantile Treatment Effect for
is
<5(-)
plotted in Figure
1.
Unemployment Duration
0.4
Quantile Index
Figure
The
1
three basic hypotheses described in table 3 include:
T=
•
treatment
•
treatment affects only the location and scale of the outcome log(T),
•
treatment
constant across most of the distribution
effect is
unambiguously
effect is
beneficial: 5(t)
These hypotheses specialize Examples 1-3
implemented following section 3, and the
implemented
for
subsample
size of 3000.
<
for all r
to the present case.
results are given in
We
sample despite that n
designs
The
as, effectively,
first
=
—
[.15, .85]),
G 1.
The resampling test is
Table 3. The tests were
could not consider subsamples of smaller sizes
because they often yielded singular designs (many components of
taking on positive value with probability 2
(
10%). Thus,
we
X
are
dummy
variables
dealt with effectively a small
6384 (see Goldberger (1991) on characterizing close-to-singular
the small-sample designs).
two hypotheses are decisively
Koenker and Xiao (2002a). This
and powerful inferences even
tion of these hypotheses
is
is
rejected, strongly
supporting the conclusion of
notable given that the resampling tests provide accurate
in effectively small samples, as
we saw
in
Table
2.
The
rejec-
an additional strong evidence in favor of the quantile inference
VICTOR CHERNOZHUKOV
12
paradigm of Lehmann-Doskum, which emphasizes ubiquity of quantile
shift effects
and the
general impossibility of describing such treatment effects as merely shifting the location and
scale.
The hypothesis
statistic is 0,
of stochastic dominance, the third one,
while for rejection
it
is
necessary that
it
decisively supported.
is
exceeds the value of
The
2.6.
test
This
additional result complements the set of inferences given in Koenker and Xiao (2002a).
Thus the bonus
offer creates a first
order stochastic dominance effect on the unemployment
duration, supporting the efficacy of the program.
6.
A
Conclusion
simple and practical resampling test
offered as
is
technique, suggested in Koenker and Xiao (2002a).
(same power as the
test
with known
parametric nuisance functions.
It
critical value)
an alternative to the Khmaladzation
This alternative has optimal power
and does not require estimation of non-
applies both to iid and time series data.
Finite-sample
experiments provide a strong evidence in favor of this technique and an empirical illustration
illustrates its utility.
Appendix A
and
Proof of Proposition
1
Proof of Theorem
Part
in Politis,
1
2
result
(1999).
immediate from A.2-A.3.
There are few
Kolmogorov-Smirnov
give the proof for the
is
of the proof follows standard arguments for subsampling consistency, as
II
Romano, and Wolf
The
statistic.
details that
we have
to
fill
out before then.
Extensions to other statistics defined
in
We
the text are
straightforward.
To prove
I.
(i)- (iii),
GnAx) = B~
Gn
EG n A x =
)
in Politis,
Next
,
b
(x)
Pn(Sb
l
1
x).
l[
two
For
facts: fact 1,
V
K
G n ,b(x)
and write out
V 1/2 (t)
[sup
(V6K,6,i(r) - g(r))
+
-fb(g(r)
-
v n (r)))
I
iid case:
(1999),
by
Theorem
uniformly in
l
'2
(r)
G„A X
being a U-statistic of degree
)
3.2.1,
combined with
b;
< «],
and otherwise: by
contiguity, conclude
G n ,b(x)
—^
LLN
G(x).
i
(Vb(v n ,bAr) - g(r)) + Vb(g(r) - v„(r)))
<
-
|
]T sup|v 1/2 (r)(\/fc(^, b .,(r)-s(r)))|<
Romano, and Wolf
collect
1
J^
= B-
<
G n ,b(x)
define
|vi/*(t) (Vb(v n ,bAr)
-
g(r))
+
y/b(g(r)
The paradigm is formulated in a series of works by Lehmann
Machado (1999), and Koenker and Bilias (2001).
-
n/aT,
«„(r)))
(1974),
Doksum
(1974),
Koenker and
INFERENCE ON QUANTILE REGRESSION PROCESS
where A n
=
1/2
(r)i>(r)y- 1/2 (r)V and A„
sup, maxeig (v'-
eq-ty 10 on p.460 in
Amemiya
(1985).
<
l[Ai
where
l
n
=
v/l/A n and u n
10
(x/u n
= v A„,
Fact 2 follows from Fact
- Wn )] <
and
wn
is
<
l[A,
qn
=
max[|u„
—
1|, |/„
—
—^
1|]
Thus
0.
<
x]
sup r maxeig (v'" 1/2 (r)V(r)V>
1
and by ||A||-|NI <
l[Ai
<
{l/ln
||A
+
-uj[|
- 1/2
<
(r)) by
||.4||
+ |MI>
+ UI„)]
defined below.
By A2 and A3 and assumptions on V and V w n =
0,
=
13
wp—
>
1
sup_ y/b V
l{E n )
=
1,
rl/2
(r)(u n (r)
where E„
=
—
= Op (\/b/\/T) -^
p(r))
{v n ,q n
<
>
S} for any 5
0.
G n ,b(x - e)l{En < G n ,b{x)l{E n <
< G n ,i,(x) < G n ,b(x + e). Now pick
—
so that [x
e,x + e] are continuity points of G(x). For such small enough t G n ,b(x + c) —^ G(x — c),
e >
for c = e and c = — e, which implies G{x — e) — e < G nt b{x) < G(x + e) + e w.p. — 1. Since e and <5(e)
-1
can be set as small as we like, G n ,b{x) —^ G(x). Now note that x = G
(l — a) is a continuity point by
II.
Thus
G n ,b(x +
for small
enough
e
>
there
is
S
>
0,
so that
e)l(En) so that with probability tending to one:
by
fact 2:
G n ,b(x —
)
)
e)
,
>
assumption. Convergence of quantiles
is
implied by the convergence of distribution functions at continuity
points.
III. (iv) follows
from
Proof of Theorem 2
Romano, and Wolf
Lifshits (1982) or
I.
The proof
(1999).
II.
The
Davydov,
Lifshits,
of the first statement
class of
moment
is
and Smorodina (1998) by A3.
a direct corollary of Theorem 7.3.1
functions / defined prior to the proof
measurable, since these functions are a measurable function of a
random
subsampling truncated moments thus follows from the definition of
degenerate uniformly
in r,
it
pi,.
vector v„(r).
III. Finally,
is
in Politis,
clearly Borel
The convergence
because v(t)
is
of
non-
has continuous bounded density by Gaussianity, by A. 3, Ev(t) p 1 k ^^(v(t))
can be approximated arbitrarily well by Ev{r) p fK (-)(v{r))
for
some v
>->
f(T) p /K(-) 6 L(c',k') for
some
c',
/.'
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10
Let A be the
sup x x'Ax/x'x.
maximum
eigenvalue (characteristic root) of the symmetric matrix A.
Then A =
VICTOR CHERNOZHUKOV
14
(1976): "Kolmogorov-Smirnov tests when parameters are estimated," in Empirical distributions and
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m
m
m
Table
H=
Empirical Rejection Results for
1.
.5
Power
=
7
n= 100
n- 200
=
n =
72
7
=
.2
7i
level
Khmaladze Test
H= Bofmger
H=.6 Bofmger
Bofmger
Size
5%
Power
Size
=
-
7
5
=
7
=
.2
Power
Size
7i
=
7
-5
=
7=2
7i
=
-5
0.101
0.264
0.898
0.035
0.211
0.755
0.016
0.126
0.641
0.070
0.480
0.988
0.041
0.406
0.990
0.022
0.280
0.964
300
0.062
0.622
0.998
0.043
0.665
1.000
0.029
0.416
0.998
400
0.043
0.809
1.000
0.043
0.809
1.000
0.035
0.632
1.000
Notes: All results are from Koenker and Xiao (2002b). Symbol
relative to the
TABLE
tic), for
Bofmger
Empirical rejection results for
2.
various
K,
=
b
K
5%
H denotes different
bandwidth choices
rule.
resampling test (Smirnov Statis-
x n 2 / 5 using 250 bootstrap draws and 500 Repe,
titions.
Subsampling
'
rest
=
7=2
=
7=2
0.014
0.348
0.980
0.026
0.350
200
0.052
0.752
1.000
0.059
300
0.058
0.910
1.000
0.058
400
0.054
0.980
1.000
0.064
7i
=
-5
7
Boot strap Test (b=n)
Power
Size
100
7
=
n=
n=
n =
n
Subsampling Test (K = 10)
(K=5)
Power
Size
Power
Size
7i
=
-5
7
=
7
=
.2
7i
=
.5
0.954
0.022
0.316
0.968
0.728
1.000
0.038
0.728
1.000
0.924
1.000
0.074
0.918
1.000
0.978
1.000
0.056
0.970
1.000
maximal
sim.
0.009
s.e.
0.009
0.009
Notes: All results are reproducible and the programs are available from the author.
Table
b
=
3.
3000
(
The
test results for the
Hypothesis
Null
Location-shift
6(t)
Location-scale shift
S(t)
=S
=a+
Dominance
6{t)
<
Effect
re-employment bonus treatment, using
subsampling with replacement
)
Alternative
± 5
a + 7q(t)
S(T)
7a(r)
S(t)
-fi
3t:5{t) >
Smirnov
Statistic
5%
level critical value
Decision
Reject
2.46
1.31
2.47
1.30
Reject
0.00
4.59
Accept
Date Due
-yfet]o~>
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