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Massachusetts Institute of Technology
Department of Economics
Working Paper Series
SIMPLE 3-STEP
REGRESSION
CENSORED QUANTILE
AND EXTRAMARITAL AFFAIRS
Victor Chemozhukov,
Han Hong,
MIT
Princeton University
Working Paper
01 -20
March 2001
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.ssm.com/paper.taf7abstract id=272499
MASSACHUSEnSINSIItUit
^OFT£ CHNOLOGY
AUG
Paug
1 5 2001
LIBRARIES
Massachusetts Institute of Technology
Department of Econonnics
Working Paper Series
SIMPLE 3-STEP
CENSORED QUANTILE
REGRESSION AND EXTRAMARITAL AFFAIRS
Victor Chemozhukov,
Han Hong,
MIT
Princeton University
Working Paper
March
01 -20
2001
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.ssm.com/paper.taf7abstract id=272499
SIMPLE 3-STEP CENSORED QUANTILE REGRESSION
AND EXTRAMARITAL AFFAIRS
VICTOR CHERNOZHUKOV AND HAN HONG
Abstract. This paper suggests simple
sion
3
and 4-step estimators
for
censored quantile regres-
models with an envelope or a separation restriction on the censoring probability. The
estimators
aire
theoretically attractive (asymptotically as efficient as the celebrated Powell's
censored least absolute deviation estimator). At the same time, they are conceptually simple
and have
or
trivial
models with
computational expenses. TheyLaxe, especially useful in samples of small
many
regressors, with desirable finite
sample properties and small
size
The
bias.
envelope restriction costs a small reduction of generality relative to the canonical censored
regression quantile model, yet its
also
main
plausible features remain intact.
The estimator can
be used to estimate a large class of traditional models, including normal Amemiya-Tobin
model and many accelerated
failure
and proportional hazard models. The main empirical
example involves a very large data-set on extramarital
estimate
45% — 90%
conditional quantiles.
affairs,
with high
68%
censoring.
We
Effects of covajiates are not representable as
women, with fewer children, and higher status, tend to engage
more than their opposites, especially at the extremes. Marriage
location-shifts. Less religious
into the matters relatively
longevity effect
is
and negative
at high quantiles.
Ed-
We
also
on the Stanford heart transplant data.
We
positive at moderately high quantiles
ucation and marriage happiness effects
aire
negative, especially at the extremes.
briefly consider the survival quantile regression
estimate the age and prior surgery effects across survival quantiles.
L Introduction
In statistics, biostatistics,
and econometrics, there has been a great deal of attention given
to censored data. This paper analyzes censored quantile regression models with
known
cen-
soring points, suggesting a very simple 3-step estimation procedure with congenial features.
This simplicity
is
achieved through the structured envelope and separation restrictions on the
censoring probability, yielding an easily implementable and well-behaved technique. These
restrictions preserve the plausible semi-parametric, distribution-free
tures of the model.
We
illustrate the
and heteroscedastic
procedure in two examples, an extramarital
fea-
affairs
example and the Stanford heart transplant data with complete censoring times.
The paper
first
is
organized as follows.
We
first
evaluate the censored quantile regression model,
proposed by Powell (1986), as a generalization of the Lehman-Doksum p-sample quantile
Key words and phrases. Median Regression, Quantile Regression, Fixed Censoring, Robustness, AccelerTime Model, Proportional Hazard model. Classification, Discriminant Analysis.
Chernozhukov is at Department of Economics, MIT, vchern@mit.edu; Hong is at Department of Economics,
Princeton and UCL, Louvain-la-Neuve, doubleh@princeton.edu
We thank Takeshi Amemiya, Roger Koenker, and Stephen Portnoy for very constructive, critical comments.
We also thank Joshua Angrist, Jerry Hausman, Jinyong Hahn, Sendhil MuUainathan, Whitney Newey for very
helpful conversations. Of course, the usual disclaimers apply.
ated Failure
.
1
VICTOR CHERNOZHUKOV AND HAN HONG
2
treatment problem for the case of censored data and general treatment.
CQR models are semi-
parametric, with distribution-free character, and equivariance to monotone transformations.
They embody
rich information
covariates, all
summarized
about
shifts in location, scale,
Ensembles of quantile regression
as the quantile treatment effects.
curves provide a more complete, often
more
and other moments induced by
interesting picture of relationships in the data
than conventional mean or median regressions alone. Graphically a spectacular technique,
they rouse curiosity, and engage attention.
CQR models compare favorably to the well-known
Amemiya-Tobin, Cox, Buckley-James, and other approaches, as they permit distributionforms of heteroscedasticity, including scale and non-scale
free specifications as well as rich
forms.
We briefly discuss available estimators for
the cases of fixed censoring. This discussion
motivates the k-step estimators on congeniality grounds, including implementation ease, good
performance
in small
samples and models of many continuous or discrete regressors, and cases
These
of very high censoring.
and simulations. The k-step
CQR
a constructive, robust, and well-behaved method to estimate the
traditional
Amemiya-Tobin, accelerated
for conditional quantile
tile
models as well as the
models had been recognized early
Effects. Although the
in the 19th century, only
Lehmann
in the past century did an extensive research of the subject begin.
Doksum
offers
and many Cox models.
failure,
Censored Quantile Regression and Quantile Treatment
need
CQR
Censored Regression Quantile Models
2.
2.1.
which
qualities especially pertain the empirical examples,
follow after the discussion of theoretical properties
(1974) and
(1974) formulated the theory of quantile-quantile plots, posing a p-sample quan-
treatment problem and arguing that location-shift models are insufficient to summarize
ubiquitous quantile shift
effects.
Koenker and Bassett (1978) introduced quantile regression
(QR) estimators that have evolved
into a popular
approach to data analysis. Another early
work of high merit by Hogg (1975) suggested instrumental variable estimators and gave a
first
empirical illustration of the conditional quantile model's breadth.
inal
works also lay the foundation of the
initial
A number
of sem-
Amemiya
(1981),
developments, including
Chaudhuri (1991),Chaudhuri, Doksum, and Samarov (1997), Powell (1986), Koenker and
Portnoy (1987), Portnoy (1991), Jureckova and Prochazka (1994), Newey and Powell (1990),
Buchinsky and Hahn (1998) Koenker and Machado (1999), Khan and Powell (2001), as well
as
many
QR is
others.
central to a substantial
number
of empirical studies, as
Koenker and
Hallock (2001) recently reviewed.'
Our
target
covariates
X
is
the conditional quantile function of the dependent real variable
in R*^,
Qy\x- Qy\x
is
The
Qy\x
is
given
the inverse of the conditional distribution function Fy\x-
QY\x{r)
therefore
Y
= mUv
:
Fyixiv)
>
r};
a complete description of the stochastic relation of
model of Qy\x is of fundamental importance. It
appealing and simple, incorporating classical linear model and
linear
'Koenker and Gelling (2001)'s introduction to quantile regression
is
is
F
to
X.
convenient, conceptually
linear location-scale
very worthy.
models
CENSORED QUANTILE REGRESSION
3-STEP
3
as important special cases,
Qy^^{r)=
We
X
assume that
X'Pir).
includes a constant and note that
(1)
may
it
incorporate a wide array of
polynomial and alternative transformations of the observable covariates. This model could
be specified
a particular quantile, say median, or for an ensemble of quantile curves,
for
providing a more complete, global description of conditional distribution.
shall stress en-route. In sequel
QR
model
Both
we assume that
(1) applies to
the quantiles of interest. Linear
a natural generalization of the classical location-shift model:
(1) is
Y = X'a + u,
where u
shift
independent of
is
or
local,
models and global models each have their own virtues, which we
single quantile restriction
X
,
(2)
with distribution function F, and of the linear location-scale
model
Y = X'a + XV,
>
where X'j
Indeed, in the
0.
= a + ^F~^{u).
the second /5(t)
for all T,
QR
whence
model
(3)
As
Thus,
is
=a+
F~^{u)ei, where ei
=
(1,0,
...)',
and
(2) implies that all the slope coefficients are the
in
same
monotone in the quantile index r. The general
make such restrictions. Thus, QR models ascribe a
implies that /3(t) are
(1), local or global, does not
rich role to covariates
effects.
case /3(t)
first
(3)
X, allowing them
to exhort location, scale, kurtosis,
and
quantile-shift
mean models, the QR models are distribution free.
a way to extend the Lehmann-Doksum p-sample quantile
the case with conditional
Another principal view of QR
is
as
treatment problem to the regression setting(Koenker and Gelling (2001)). In the two-sample
Lehmann
setting,
(1974)
and Doksum (1974) arrived
where the value of the untreated
is y,
F
Lehmann
F(y)
= G{y +
=
Evaluating A(y) at a quantile, y
shift
vs.
the familiar distance from the 45 degree line in the quantile-quantile plot of
is
=
a scale-shift form
Sq, for all r, or
(5o(t)
=
effect
could take a simple location-
So-\-SiF~^{t).
More
treatment can affect such features as skewness, kurtosis, and other moments,
by the quantile
shift effect S{-).
firstly,
=
generally, the
all
summarized
This 2-sample formulation leads to the linear model
(1),
+
^(^)^i5 where Di is the treatment indicator.
^"U''")
we may view linear QR models X'P(t) as a generalization of the p-sample
noting that Qy\d,{t)
Thus,
effect:
= G-\t) - F-\t);
G~^{t). For example, the quantile treatment
form 6{t)
and
= G-\F{y))-y.
F^'(t), one obtains the quantile treatment
6{t)
G
A(y)) or
A{y)
F~^{t)
y has the untreated
is defined by the random variable Y + A{Y).
defined the treatment effect A(y) as the horizontal distance between
in (y,p) coordinates
In fact, 5{t)
Suppose
the treatment adds amount A(y). If
distribution F, then the treated distribution, G,
Thus,
at the following formulation.
problem to the case of continuous or polychotomous treatment, represented by covariates
-^(j)i
J
>
2,
as in dose-response studies. In this case, P{j){t) can
be interpreted as a partial
derivative with respect to change in the treatment, or, equivalently, a quantile treatment
VICTOR CHERNOZHUKOV AND HAN HONG
4
of changing the treatment
effect
is
=
X(_,)
xq to xq
+
Secondly, introduction of covariates
1-
a pertinent way to control for observed heterogeneity. Regardless of whether covariates
bear causal or control meanings, we refer to
/0(t) as the quantile
X
X
treatment or the quantile
shift effects.
Equivariance to monotone transformations
as
els,
emphasized
state a slightly
Y
a useful property of quantile regression mod-
Chaudhuri, Doksum, and Samarov (1997), and Koenker
in Powell (1991),
and Gelling (2001)
is
and transformation models.
in connection to censoring, survival,
more general form. For a given measurable transformation Tz{Y)
and other variables Z,
it is
We
shall
of variable
obvious that
QT-AY)\x.zir) =Tz{Qy\x.z{t)), since
(4)
P[Y <QnxAr)\X,Z] = P[Tz{Y) <UQy^^Ar))\X,Z].
For example,
Y =
if
we estimate a
linear
model X'P{t)
for the
log(r), as in accelerated failure time models, then
logarithm of survival time T,
Qt{t\X) equals exp{X'/3{T)). This
property helps interpret and communicate data-analytic findings; and
conditional
mean models. Koenker and
it is
not shared by the
Gelling (2001) contains a lucid illustration in the
case of quantile regression survival analysis.
Transformation equivariance naturally leads to models of censored data. Assume that the
latent variable
and we
Y*
is
by the observable, possibly random, censoring points
left-censored^
Ci,
collect
X„
Y^=Y*yC^,
Assume Y*
is
= 1{Y^Q).
5^
conditionally independent of the censoring point
P
(y*
<
y\X,,
C) =P{Y* <
y\X^)
(5)
d,
that
is,
for all
y G M:
so that
,
^^^
Qy'Ix.c^^X'(3{t).
The
conditional independence assumption
sumption of independence between
example, for example, there
is
(YJ,
i,
is realistic
in
many
realistic
The assumption,
(but clearly not
because we know the transplant and the
possible to impute
them
example, the censoring point
of censoring
ies.
is
very
common
(see
is,
all)
that censoring points are
situations.
last follow
known
for
For example, in the analysis of
we can compute
up dates
for
each
i.
all
censoring points
In other situations,
Powell (1986) for discussion).^ In the extramarital affairs
naturally, zero, or "fixed", for all observations.
in social,
Conditioning on Cj, assumption
censored
as-
X^) and the censoring variables Ct. (In the Stanford
post-transplant survival times in the Stanford data-set,
it is
than the frequently made
a significant correlation between the "ace" and "surgery"
variables with the censoring times).
all
more
is
(6)
This type
psychometric, technometric, and econometric stud-
and transformation equivariance
yield the following
QR model:
QY\x,c,{r)
right-censoring
is
handled by reversing the
=
X'l3{T)VC,.
(7)
sign.
Here we do not consider the cases where the censoring points are unobserved. There
on median and M-estimation under random censoring - see
e.g.
Yang (1997) and Zhou
is
a growing hterature
(1992).
CENSORED QUANTILE REGRESSION
3-STEP
Early formulations of
go back to
(7)
Amemiya
Gaussian errors and fixed censoring point at
This model states that (5y|x(l/2)
of error Y^
-
V
X'/3
consistent estimators.
fact that
considered
0:
= X'p V
u~Af(0,a);
\inot;
(8)
and assumes a parametric form of distribution
0.
Tobin (1958)
Historically,
who
(1973) and Tobin (1958)
= X'P + u, \^X'p + u>0&ndO,
Y^
5
.
formulated
first
and Amemiya (1973) provided and
(8)
expenditure was nonnegative and frequently assumed value
specifically Tobin's example,
such censoring formulation
is
may be hard
median model Qy|x(l/2)
if
conditional on
expenditure
is
X
mean
to interpret as a
= X'pvO
for the observed
probability of "censoring"
recurrently criticized, since the
In his path-breaking work, Powell (1984) was
We
expenditure
median
first
stress
of a "latent variable," the conditional
an excellent one. Indeed,
is
greater than 1/2, then conditional
is
zero. Otherwise, the conditional
for the
many examples,
In
0.
negative values are sometimes meaningless or given imaginary interpretations.
that although x'P
justified
Analyzing expenditure on durable goods, Tobin accounted
is
median of
(reasonably approximated as) X'I3.
median and quantile regression
to stress
we refer to this model
the Amemiya- Tobin model.
in the distribution-free, fixed censoring world. For this reason,
the Powell
CQR
Having model
summarizes the
When
is
model, and to
(7) in
its
normal form
as to
mind, we can speak of the quantile treatment/shift
effect of
as to
effects. I3{t)
now
a treatment on both the censored and uncensored latent variable.
the latent variable has real meaning, such as the survival time, the interpretation
the same as discussed earlier, except that censoring
treatment
effects for all quantiles of interest.
the expenditure or extramarital examples,
When
it is
may
prevent identification of the
the latent variable
is
imaginary, as in
important to indicate the relevance of
/3(r).
Define the censored quantile treatment/ shift effect as a partial derivative
Sj{t,X,c)
=
"i? [{x
=
6j{t,x, c) characterizes the
(ej is
a zero vector with
1
eff'ect
- Qy\xA'^)
Qy|x+e,i,,c(-r)
+
ejvypir) V c
a
- x'/3(r) V c)]
/v
l(x'/3(r)>c)/?(r)
of changes in components of x keeping everything else fixed
in the j-th position).
/3(t) in
turn characterizes the treatment
effect 6{t,x,c).
As the
class of
CQR
models inherits the qualities of the uncensored
distribution- free character, talent of heteroscedasticity, they
Amemiya- Tobin models,
QR
models, such as
compare favorably to the normal
distribution-free homoskedastic accelerated failure models,
and Cox
proportional hazard models. For example, an accelerated failure time models are of the form
\og{T)
= a + XL^e + u,
(9)
where u has an unknown distribution F, X^-[ represents covariates without the
is
a location-shift model with quantile treatment effect given by
hazard Cox models are special cases of
(9), for
9.
Many
intercept. This
useful proportional
example, one with Weibull baseline hazard
model. More generally. Cox models can be written as In Ao(T')
= a + X'_i6 + au,
for
unknown
VICTOR CHERNOZHUKOV AND HAN HONG
6
Gumbel variate u. Vector 6 summarizes the
Cox model is a location-shift one up to a
view, one or the other model may be preferred
integrated baseline hazard function Ao and a
relevant quantile treatment effects, so that the
Prom a constructive point
depending on the context. However, when
transformation.
desired, the
CQR approach
Motivation
2.2.
is
of
heteroscedasticity
and distribution
free spirit are
valuable.
for k-step Estimators. Sample regression quantiles
may be
defined in
The most widely used method, due to Koenker and Bassett(1978), is ingeniously
we have n observations {Yi^Xi}. In the no-covariates case, the sample r-th
several ways.
simple. Suppose
quantile /3{t)
generated by solving the problem (Ferguson (1967)):
is
T.PriY^-P),
Ti^
where Pt{x)
=
{t
-
l{x
<
0))x.
Koenker and Bassett (1978) extended
this concept to the
regression setting as follows:
n
T.pr{y^-^i(^)-
g^,
They
also
(10)
and Powell (1984) and Portnoy (1991) developed the asymptotics. The asymp-
totic distribution parallels that of ordinary
sample quantiles. Also, as Koenker and Bassett
(1978) elaborated, the sample regression quantiles inherit good robustness and equivariance
features of the ordinary sample quantiles. Robustness
is
a strong motivation for quantile
re-
gression even in canonical, seemingly normal, homoskedastic cases, as heavy-tailed admixture
known
or outliers are
spoilers of
For brevity assume for
now
Ci
many
=
classical procedures.
0, Vi.
form with the semi-linear one {X'/3 V
In the censored
model
(7),
replacement of the linear
0),
n
leads to the celebrated Powell (1984), Powell (1986) estimator. Powell (1984) established the
asymptotic normality of /3p(T) and developed an inference theory, paving the way and setting
forth a standard for future work.
Despite the intuitive appeal, this method has not become popular in empirical research
due to
known computational
well
its
difficulty.^
See, for instance, Fitzenberger (1997b),
Fitzenberger (1997a) for remarkable research as well as extensive simulations.
Buchinsky
(1994) and Fitzenberger (1997a) designed ingenious computational algorithms, which Fitzen-
berger (1997a) recommends for low degrees of censoring, while admitting that "all practical
algorithms perform quite poorly
We
know
when a
or
Cox approaches found hundreds
For example, see Fitzenberger (1997a),
n
=
p. 15. In
100, in several important designs (e.g.
5%
to
censoring
of about three or four applications of censored
Amemiya-Tobin
from
lot of
37%
for various algorithms.
is
present" ? His conclusion
median regression
of applications,
the case of
50%
if
is
well
in econometrics. In contrast,
not thousands.
censoring, one regressor, and small sample
A, B), the frequency of computing the Powell estimator ranged
For some of the other designs results were better - convergence
3-
STEP CENSORED QUANTILE REGRESSION
7
substantiated by a very extensive Monte-Carlo experiment with tens of different practical designs. All
experiments involved only one regressor
complications. In
many
empirical applications, the censoring
in the heart dataset of section (3.6.2) - 37%.
and
respectively.
3,
and
is
The number
quite heavy and dimensional-
degree of censoring
(4),
theory
statistic
is
at
hand particularly
68%;
for the
mat-
the design of both theoretically
also implementable, practically attractive estimators. It
makes the problem
is
of regressors in these two datasets
This serves as one strong but not the only motivation
an important goal of
ter herein. Arguably,
elegant
Increase in dimension leads to further
For example, in the affairs example of section
ity is also high.
are 9
(!).
that requirement that
is
challenging.
In part motivated by such limitations, recent remarkable
work by Buchinsky and Hahn
and Khan and Powell (2001) suggested a number of alternative estimators. Buchinsky
and Hahn (1998) proposed to first estimate the propensity score h{Xi) = P {8i = l\Xi) by
(1998)
a nonparametric kernel regression, then select a set where
sample,
those observations
i.e.
and then use a transformed
to use any of the three
i
(iii)
:
where the conditional quantile
QR on the selected sample.
methods
>
h{Xi)
line
Khan and
is
1
—
t} of the whole
above zero,
X[P (t) >
0,
Powell (2001) also proposed
to perform the first stage selection:
estimators of the regression quantile,
h[Xi),
{i
(i)
maximum
score
nonparametric kernel propensity score estimator
(ii)
nonparametric locally linear conditional quantile estimator of Chaudhuri (1991)
(denoted q{Xi)). In the second step, they obtain the estimator by running a weighted
min^ Er=i AzPr
{Y,
- X[P) where
A,
e.g.
= K{h
(X,)
- (1 - r) - c)
or
A
{q {X,)
-
c), etc.
QR:
The
two-stage estimators are somewhat less efficient than the Powell estimator due to smoothing
and trimming.
Ideologically, these estimators share the ideas
behind the construction of the
Powell (1986) estimator (especially the estimating equation version of
imposed simultaneity to obtain
The suggested
first
stages are extremely attractive, but are only practical in low dimensions,
and have slow convergence
This
is
rates.
Local kernel smoothers apply to (sufficiently) continuous
will
Of
very confounding.
^/n- estimates could
^
when
ss 1.
Thus such asymptotics
.2
the bandwidths are carefully chosen]
sets for all of the first stage estimates.
estimators in our examples and
many
is
As a
other
cells.
observations per
void here.
is
many
(sufficiently) discrete
an asymptotic theory suggests that the
course,
be obtained by averaging within the
be on average roughly 6800/(8^)
69/(15 X 2^)
except that Powell
his single step estimator.
variables only, whereas a lot of applications, including ours, have
covariates.
it),
In the affairs example, there
cell;
in the heart
The computational burden
example [especially
very substantial in high dimensions, large data-
result,
we simply cannot use any of the
available
applications due to 1) heavy censoring, 2)
real-life
high dimensionality, 3) very small or large sample, and 3) polychotomous nature of numerous
regressors.
From a
As our
interest concerns
many
constructive angle, however,
we
quantiles; these settings are even
more confounding.
strongly stress that the mentioned estimators can be
potentially fruitful in a lot of conceivable cases.
In an unpublished report that precedes this one,
we suggested a number
of flexible para-
metric techniques that exploit the global approximation and classification ideas, that lead
frequencies ranged from
many
30%
to %70.
These results were obtained
regressors and larger n, the results can be expected to
for the case of
be worse.
one regressor.
In case of
VICTOR CHERNOZHUKOV AND HAN HONG
8
to consistent
and asymptotically normal estimators that are as
The
Powell estimator.
benchmark
we think is an esThe current approach
present paper extracts and further develops what
most implementable and usable part of our previous research.
sential,
is
efficient as the
based on the structured modeling restrictions that we put on the censoring probability.
These restrictions cause a reduction
many Cox models, and
incorporate, for example, Amemiya-Tobin,
models
as very
As a
free character.
offers
it
not only an
efficient, practical
for heteroscedas-
an easily computable (comparable to
result,
linear least squares), well-behaved, robust estimator
properties,
accelerated failure time
important special cases while at the same time allowing
and distribution-
ticity
but only a small one, since they allow to
in generality,
is
available.
way
Due
to the
good robustness
models, but also a good way to estimate very important traditional models.
computable,
easily
its finite
The procedure:
3.1.
sample properties can be studied in
Simple 3-step and k-step
3.
CQR
to estimate the general Powell
CQR
Because
it
is
fine details.
Estimators
This section describes the steps of the estimator and briefly sketches
the basic ideas behind them.
The Steps.
3.1.1.
Step
where
Si
is
Estimate a parametric classification (probability) modeler
1.
Xi indicates desired transforms of Xi and Ci.
r-l-c}, where c is strictly between
and r and
the indicator of not-censoring.
Next, select the sample Jq
=
{i
:
not too small (practical choice
Step
Obtain the
2.
p{X['^)
>
—
1
discussed in the appendix)
is
estimator
initial (inefficient)
/3o
.
(t)
by running the standard QR:
mmY, Pr{Y^-X',p).
(11)
ieJo
Next
as
select Ji
=
{i
:
X'^Po{t)
>
Cj
+
(5„}.
is
(5„
n —> DO (The formal condition on {5n}
is
a small positive number going to zero slowly
that y/n x ^„
therefore consistently selects the largest subset of
i
—>
oo and 5„
such that XIP{t)
>
\
0.).
This step
Ci, building
up the
efficiency of the next step.
Step
Denote
3.
Run
QR
(11)
with Ji
in place of Jq.
this 3-step estimator pi (r).
Step
4. (Optional) Further repeat step 3
^/3/-i(t)
>
a+
<5„} in
place of Jo.[/
In step 4 each repetition involves selecting Jj
Pi{t) from (11) using the sample
Made
.Jj.
=
=
{i
once or
2,3,
:
finite times,
X'-Pj-i{t)
>
Ci
+ 5n}, and
is
{i
:
then obtaining
(3j (r).
index can be relaxed by allowing the binary model to
have plausible stochastic complexity as measured by the uniform covering or bracketing entropy. The
the single index
=
...].
Denote the k-step estimators as
for brevity of proofs, the restriction of single
using sample Jj
particularly suited here from a practical point of view.
class of
CENSORED QUANTILE REGRESSION
3-STEP
What
3.1.2.
example,
its
we may
In the step 1
details are as follows.
extreme value, linear (polynomial) or any other model that
logit, probit,
Xi denotes any desired transform of Xi. For example, Xi
well.
{5i,Xi}
and
Some
in the steps?
is
9
may
use, for
the data
fits
consist oi
Xi,Ci
squares (Remark 4 allows for power series and regression spline approximations). In
an inconsistent estimator of the propensity score
general, this gives
but the inconsistency
not important as long as the misspecification
is
Indeed, the goal of the step 1
observations where h{Xi)
>
1
—
is
to select
some and not
where the quantile
r, i.e.
is
not too severe.
necessarily the largest subset of
line XI/3{t)
is
above
zero, so as to
obtain an initial consistent but inefficient estimator Po{t). This task can be carried out
for
example, p{Xl'yQ)
—
a lower bound on h{Xi):
c is
a.s.
=
(70
plim-y)
and
EXiX'-l{i G Jo)
it is
is
nontrivial,
meaning that the selected
a weak requirement since p()
is
Thus the envelope
function.
p{Xl^o)-c<h{X^),
(12)
set Jq
sufficiently rich -
is
restriction
is
not required to be a distribution
is
and not very
intuitive
Figure
(cf.
1
The above
may be
it
restriction in
Theorem
See the subsections below for formal details and further
for motivation).
discussion of this assumption.
but
restrictive,
replaced by an even weaker condition - the separating hyperplane
1
matrix
asymptotically invertible. Greater c and better model p{-) simplify the
This
selection task.
if,
Appendix B contains
construction assumes that the estimator
details for practical implementation.
7
reasonable and converges to a value
is
and the model p{Xlj). For example, 7
may be defined by minimizing Y17=\i^i " Pi^llWi >" which case under standard conditions
> 7o that solves min-y E[h{Xi) —p{X['-f)]^. Alternatively, quasi-ML methods can be used
7o
70 that minimizes a sensible distance between h{Xi)
—
and
will
be equivalent to weighted least-squares.
polynomial
logistic
model and estimated
Another attractive choice
tive
is
is
it
In our empirical section
using conditional
the Fisher-Rao discriminant analysis.
justifiable as follows (treat Ci as part of
has density or p.m.f. ^i(x) and Xj{(5j
P{S^
=
1\X^
=
=
X
or set C,
qi
=
means and
LQDA
P{5i
=
=
1)
1
—
go-
=
The discriminant prospec-
for brevity). If Xi|{(5i
x)
=
——
—
+
qog2{x)
Approximating gi{x) and
g-zix)
by normality with
Amemiya
(1985), p. 282).
Other forms of
Efron (1975), Press and Wilson (1978),
simulated and real-
LDA
tion
classifier
becomes
(because
it
different
variances, leads to the classical logistic linear-quadratic discriminant analysis,
(see e.g.
(cf.
1}
-,
gi
and
^2 could
of their concrete problems, but even normality assumption has been
results
=
0} - go{x), then by the Bayes's rule:
9191(2;)
where
we employed a
MLE.
life
Amemiya and
in
view
to produce
good
be chosen
known
Powell (1983)). Using both
examples, the studies found a surprisingly good performance of the
even when the regressors were binary! Naturally, when the normal approximarealistic,
discriminant analysis does
approximates the unconditional MLE).
In the terminology of
modern
classification analysis.
much
better than conditional approaches
LQDA
was among the top 3
classifiers for
VICTOR CHERNOZHUKOV AND HAN HONG
10
11 out of 22 datasets in the impressive Statlog project(see Michie
out-competed many sophisticated
classifiers.
The
and Taylor(ed) (1994)) and
Statlog project tested 23 algorithms on 22
commercially important problems (Michie and Taylor(ed) (1994)). For excellent
large-scale,
further refinements and generalizations of the discriminant approach see the literature cited
in the next subsection.
Formally we shall not confine ourselves to a particular estimator, instead an assumption
such as (12) or
generalization will be required to hold.
its
Under conditions
to
be stated,
(r)
/3i
is
asymptotically normal with variance equal to
that of the Powell estimator. Thus, starting with a "good" subset of observations, only two
QR suffice
recomputations of
also asymptotically
to obtain the Powell-efficient estimator. Estimators
asymptotic rationale for considering the fourth step?
efficiency structure of the Powell's
one step estimator
computing
QR
is
/3/
(t)
has the
more
efficient
3
than the selector
a unique property of the Powell
is
iterations
on step 4 are somewhat analogous to those
markable ILPA algorithm that Buchinsky (1994) designed
going beyond the third or fourth step
tion computational reasons), based
given the
2 estimator
the
is
and our 4-or-more step estimators.
single-step estimator
The
(r) are
in the sense that the selector in step
This efficiency structure
I3\{t).
>
For /
has y/n rate of convergence and Powell's variance, which
$q{t) used in
j3i
normal with variance equal to that of the Powell estimator. What
for the
not desirable on
is
in the pioneering
and
re-
Powell problem.^ However,
statistical
grounds (not to men-
on our Monte- Carlo experience. Our
result
show that
step only two recomputations of quantile regression lead to an
first classification
estimator (relative to the Powell estimator). (Regarding computational aspects, the
efficient
faster interior point algorithms,
Koenker and Portnoy
may be
(1987),
preferred to linear
programming.)
In summary, the estimation procedure has two very distinct features: a very simple, para-
metric classification
and the additional 3rd and the optional further
first step,
The
step.
estimators are as efficient as the Powell estimator.
3.2.
The Model Beneath: Normative and Agnostic
CQR model
in (7), together with the "nontrivial envelope" restriction (5),
as a model. This additional
restrictive
Set Cj
Xi^ Yi
is
normal
assumption
is
a
can be thought of
How
critical ingredient to yield the simplicity.
is it?
=
to simplify notation.
A
popular normal model assumes that conditional on
Then propensity
conditionally homoscedastic normal.
c.d.f.
A
cf).
significantly
simply assuming that 4>{x'^q)
h{x)^
Prospectives. The canonical
where
e.g.
x
=
(x,x'^,
—
...).
more general
c
is
CQR
score h[x)
4>{x'a), for the
is
model can be immediately obtained by
a nontrivial envelope of an
unknown propensity
score
Such assumption imposes neither normality nor conditional
homoscedasticity nor a location-scale sub-model. Similarly,
if
the
benchmark model
is,
say,
the Weibull proportional hazard model, as in the section about the heart example, then
'^see
Buchinsky (1994) or Fitzenberger (1997b))
then proceed with iterative
to Powell's estimator
The convergence
is
linecir
details.
not guaranteed and can be quite
to a local
The
programming computations
optimum does not
raure;
basic idea
until
is
to start at a value /?(r), say
convergence
is
reached.
see Fitzenberger (1997b)
and
0,
and
The convergence
earlier discussions.
lead to a consistent estimator-cf. Powell (1984), Powell (1986).
3-STEP
CENSORED QUANTILE REGRESSION
11
Y
Prob
/
t
of
UncensoKo Conditional
QuantHeFunctiqrt
/
'
/Propensity
/
,.'
/
Nol-
Score
/
Estimated
'
QF in step 2
^K^
Censoring Point
/
'
/
/X
//
sr
S2'
1
1
Estimated (Linear) "Envelope"
/
in
Step
1
1
1
y
/
/
Figure
How
1.
it
The
works.
solid line depicts the conditional
The propensity
quantile function and the propensity score.
equals to
1
—t
X
for the value of
s.t.
d
=
and below -
if
X'P{t) equals the censoring point
for
X
s.t.
sample
is
0,
>
Xi/3(r)
s.t.
of the uncensored
QR.
>
X/3{t)
and selecting
"envelope"
is
not
Next Step
0.
— r
1
Once a
0.
the conditional quantile function
0,
all
ideal,
i,
s.t.
fits
Xi G
linear
but this
the
irrelevant, since
is
selecting a sub-set of
QR
Note that the
[51,51'].
which
line,
is
it
z s.t.
acts as a
X[I3{t)
used to select
>
all
Xi £ [52,52']. The Third Step uses the selected sample,
which asymptotically gets close to the
Note that h{X) can cross the
In dimensions greater than
1
1,
—r
X
defined by the hyperplane X'P{t)
in
XP{t) <
model can be estimated by the usual
good separating hyperplane
:
above
It is
0.
Initial step involves fitting an "envelope" of the propensity
score,
i
score
conditional quantile line
W^
.
Thus,
in
ideal:
line only
G
=
M'',
0,
i
:
Xi £
[0,52'].
once at X'P{t)
=
0.
the crossing points are
which have a zero span
W^ the crossing points form a singularity as
well.
1-tau
VICTOR CHERNOZHUKOV AND HAN HONG
12
h{x)
=
Gumbel
g{x'a) for the
—
assuming that g{x'jo)
More
generally,
c
is
Theorem
c.d.f.
A much
g.
more
CQR
flexible
model
is
obtained by
a nontrivial envelope of the propensity score h{x).
1
replaces the intuitive envelope restriction by a
separation restriction, which requires that once pix'-jo)
above a threshold
is
c,
much weaker
h{x) > 1 — r.
This assumption allows the envelope to be an incorrect lower bound of the propensity score,
but only requires that
does a good job at selecting a correct subset of observations. Figure
it
quantile function, the further
is
1
For well behaved models, the further away from zero the conditional
illustrates the situation.
away from
1
—
r
the propensity score function, the easier
is
it
to carry out the classification. Classification/discrimination problems of this kind can be
subjected to the standard or more modern classification analysis
Breiman, Friedman,
(e.g.
Olshen, and Stone (1984), LeBlanc and Tibshirani (1996), Ripley (1996), Vapnik (2000)).
Therefore, in principle,
are available.
The
We
many
is
first classification
step
confine our discussion to the simplest, most familiar methods.
justification outlined
alternative
elaborate, structured strategies for the
above
is
based on a
An
asymptotic point of view.
classical,
to view such a procedure as a shrinkage
method where the
first
We
of approximating a propensity score envelope for smaller variance.
step trades bias
also argued
above
that due to the special nature of the problem the bias itself can be quite small. Thus, the
may be
shrinkage aspect
of primary importance in moderate-sized samples. This
is
confirmed
by the computational experiments discussed next.
More
generally,
from a
real-life (agnostic) perspective, either
models are approximations.
other, once
There
both are reasonable and
is
the linear quantile or envelope
no reason to believe one
more
is
we hope
flexible models. Therefore,
correct than the
this
paper
off"ers
an
organized way of thinking about and building such models.
In summary,
strictive
it
should be said the model studied here
than the canonical Powell
CQR model.
is,
many
other prospectives.
re-
Yet despite being more restrictive, the em-
CQR
phasized model leaves the general, plausible features of the
congenial from
somewhat more
of course,
The estimator
easily
is
intact.
This model
is
also
computable and applicable
to such examples as extramarital data (high censoring, very large sample,
many
categorical
regressors) or Stanford heart data (small sample, high censoring, categorical regressors). It
does well in Monte-Carlo expriments and very sensibly in
real-life
examples.
estimator will help procreate the presently scarce applications of the
B
discusses other practical aspects of estimation
and
Large Sample Properties. The following assumptions are made
tions (1), (5)-(7). For u^{t) = Y* - X-/3(t) and all TjJ < J of interest
1.
(i)
{{X^,Y*,Ci)}
a'^e
i.i.d;
Ui{T)
u near
zero.
Ht^[t)
X,
is
rj
£
(0, t).
equa-
zero,
is
and continuous, uniformly
0.
The support of
compact. Xi includes a constant.
= E fy^^i^T.^{Q\Xi)XiX[l[h{Xi) >
constant
in addition to
Ui{T) has the unique r-th conditional quantile at
the distribution of Xi,
(ii)
Appendix
has density /y^(^)(u|Xj), which
bounded uniformly in Xi, from above, away from
in
believe this
inference.
3.3.
Assumption
We
CQR models.
(1
—
r) 4-
77]
is
positive definite, for a fixed
CENSORED QUANTILE REGRESSION
3-STEP
(iii)
The pair of
model p and trimming constant
the binary
or a separating hyperplane of the propensity score: 3c
>
p(X-7)
-
(1
+
r)
c implies
=
for any 7 in a small neigborhood of jo
P{X-a >
is
a uniformly
Lipshitz in
and Xi denoting Xi and Xi,
/3{t)
Theorem
v)
Under
1.
>
same
if
holds
quence
(5„
1,
=
where Ao(t)
t(1
any other consistent
and
4.
l/Sol"^)
—
tion of several estimators for
Ti,
i
>
0,t;
€
t)
+
a.s.,
in v, for
-^ N
a
in
>
—
(1
t)
+ c}
is
an open neighborhood 0/70 or
cxd
estimator /3o(t)
—
is
H^\r,)Ao{T„Tj)HQ'{T,), where Ao(r,r')
is
—
1
Furthermore, the
r}).
used in step
2,
provided se-
Furthermore, the joint asymptotic distribu-
0.
>
and Sn i
{0,H^Hr)Ao{r)H^' {r))
— T)E{XiXll{h{Xi) >
< J
w
(0, t),
respectively.
initial
/3('7")|/(5„
a nontrivial envelope
plimj.; EXiX'^l{p{X'^jQ)
the stated assumptions, as 6^ x y/n -^
v^(/3/(r)-/3(r))
for finite /
-
(1
form
c
and continuous.
invertible. p{-) is strictly increasing
(iv)
>
h[Xi)
13
asymptotically normal, with covariance given by
=
[(r
A
r')
- Tr']E{Xa^h{X^) >
(1
-
r)
V
-
(1
r')).
Thus the second part
of the theorem allows for a wide range of alternative initial estimators
Mr).
Remark
literature
1.
and
Remark
2.
Our proof
is
uses an approach that
therefore both short
Assumptions
(i)
and
is
distinctly simpler than those in the cited
and straightforward to understand.
(ii)
are standard.
Assumption
"probability" model p(x'7o) to be moderately misspecified.
controlled by the constant
have discussed.
It
c.
Assumption
requires very
(iii)
(iii)
allows the parametric
The degree
of misspecification
rationalizes the parametric first step, as
weak smoothness conditions on the "probability model"
is
we
p(-):
continuity and strict monotonicity.
Remark
3.
No assumptions
made about
The trimming
are
limit 70 or of $q{t) to ^{t).
the rate of convergence of
device
is
to its probability
7
designed to eliminate the bias and
stochastic equicontinuity eliminates the impact of the variance of the preliminary steps. As-
sumption
in
a
(iv) requires, for
example, the distribution of
in the vicinity of 79.
The
X[a
to respond smoothly to changes
Lipshitz condition can be replaced by the weaker Holder
continuity.
Remark
4-
It is
useful to take account of the structural risk sourced by the complexity of
the envelope or separation models ip{x)
= p(i(x)'7) — c that
is
increasing with n, e.g.a;„(x)'7„
may be
a power series or regression polynomial spline
as X
p(in(x)'7„) converges uniformly in C""(X) to a fixed function, r
i->
latter property
in
may depend on
(1998) and references therein. For brevity
methods of these
The
result
is
preserved as long
>
dim(x)/2.
The
many of which are discussed
Newey (1997),Chen and Shen
particular estimation methods,
Stone (1985), Cox (1988), Andrews and
particular
series.
references.
r = {x^
Whang
(1990),
we do not reproduce the
Note that the resulting
i(<pix)
> c),ipee,ce
regularity conditions
class of "envelope"
[o, i]}
and
models
VICTOR CHERNOZHUKOV AND HAN HONG
14
is
Donsker as long as
>
smoothness order r
£°°(X), uniformly in
as that for
^=
is
a compact set in C"^(X) of boundedly differentiable functions of
This
c.
true since
is
{x h^ ip{x)-c,
E 0,
c/j
c
€
bracketing entropy integral for
V
is
finite
w.r.t to the
of
V
and Wellner
example
Hence
(1996).
and Donsker property holds
Furthermore, Donsker property
envelope.
number
for J^ are given in
(1998) or Corollary 2.7.4. in van der Vaart
ip
is
sup-norm
of the
is
in
same order
by monotonicity of the indicator function and
1]}
[0,
Lipshitz in
c) is
bracketing
1/2 (-P)
The bracketing numbers
Lipshitz property.
>
dim(a;)/2 and P{y:>{X)
when
preserved
V
is
19.9 in
van der Vaart
>
dim(2;)/2, the
if
in
r
view of a constant
random
multiplied by a
variable with a constant envelope, as required in the proof.
3.4.
Remarks on Robustness. The approach
erties,
because
it
allows outliers
regression estimator of
and heavy
pursued here enjoys certain robustness prop-
tails for
Koenker and Bassett (1978), used
arbitrary perturbations of
Y
above and below the
in steps 2
Y
is
heavy-tailed
(e.g.
example the
in the extramarital
and
3, is
This property
fitted line.
that of the ordinary sample quantiles and can be also advantageous
of
The
the dependent variable.
is
when
quantile
stable under
inherited from
the distribution
index
Hill estimate of the tail
implies thick tails). In such cases least-square based methods, such as Buckley- James,
The
suffer a great deal.
step
first
as long as the censoring indicators
MLE
is
also robust in typical implementations.
5i
are unaffected by perturbations in Yj, the conditional
estimates are stable. Discriminant analysis will enjoy the
values of
8i
may
change, there have to be sufficiently
many changes
For example,
same property. Even
[0(n)] to distort the
if
first
the
step
significantly.
3.5.
Remarks on Computation. Computational expense of the estimator
linear least squares
and appears to be of orders
faster
2
implemented
and higher
all
by, for
and coxph modules
They show
same order
as for linear least squares.
1)
Step
is
due to
based on the interior point and preprocess-
that both practical and theoretical computational times are of the
QR
software for
R
and
S-|-
environments
is
available
http://www.econ.uiuc.edu/ ~roger/research/rq/rq.html. Other
able software includes
3.6.
S-I-).
involve quantile regression with a very small computational expense
ing ideas.
statlib or
in
example, meiximizing a convex, smooth quasi-likelihood. Steps
Portnoy and Koenker (1997), whose algorithm
from
comparable to
than computing other survival/censored
regression estimators (based on our comparisons with survreg
1 is easily
is
QR modules
in
STATA, SAS,
avail-
Xplore.
Finite-Sample Properties and the Stanford Example. This subsection presents
a graphical bivariate simulation example, 2) a (standard) test of sensibility of our method
on the well-known Stanford heart data-set.
3.6.1.
A
tration,
Simulation Example and Concordance Diagnostics. For the sake of clear visual
we consider the case
examples intend to
tics.
clarify the
median regression
,i
in three simple bivariate
examples. These
workings of the method. They also help devise simple diagnos-
In each of the three models, there
The data {Y*,Xi)
i.i.d.
of
illus-
< n = 200
is
a fixed censoring mechanism: YJ
are generated as follows:
and mutually independent, u
~
A''(0, 3), /3(^)
=
Xi
(0, 1)'
=
(l,Xj)',
and
Xi
= Y*l {Y* > 0).
~ A''(l, 1), w, are
3-STEP
• B.
Y* =X'p{^)+u,
Y* = X'p{h) + Xu,
.
Y*
A.
.
By
C.
=
X'/3(^)
15
+ u/X.
construction, X'f3{^)
B and C
model, while
CENSORED QUANTILE REGRESSION
is
the conditional median function.
Model A
a standard normal
is
42%, 38% ,and 38% of the 200
are heteroscedastic normal models.
observations are censored in the respective data sets, which isn't atypical for a lot of empirical
applications.
The rows correspond
Figure 2 illustrates our procedure.
plots the pre-censored data {Y*,Xi),
The data clouds
x'l3{^).
A
logistic probability
using
5i
=
>
{Yi
1
0)
,Xi
and the
first
column
shows the true regression function
solid line
are representative of the models' nature.
model ^(^",'7)
is
estimated by maximizing a quasi-likelihood function,
= {Xi,Xf,Xfy.
In the second column, "+" points represent {Yi,Xi)
selected by the logistic envelope using
p (X-j) >
80%
>
of the observations with p{Xl'y)
The second
of the sample.
The
to models A-C.
1
—
J
1
—
+ c.
J
c
?=:
0.1
The
are selected.
chosen such that about
is
"o" points depict the rest
step regression quantile estimator, $o{^)^
the Koenker-Bassett procedure to the selected sample.
The
is
resulting
obtained by applying
fit
x'Po (r)
is
shown by
the dashed line and contrasted with the solid line of the true regression function x'/3o(^)-
The "+" points now present (yj,Xi) selected
using Xl$o{^) > Sn, where Sn ~ 0.7 is such that 90% of the observations with X'^Poi^) >
are selected. The "o" points represent the rest of observations. The third step estimator,
The
/3i(|),
column
third
illustrates the third step.
uses the selected sample, and the
fit
x'pi{^)
is
shown by the dashed
line.
These figures are representative of the principle, and they seem to accord well with the
asymptotic results described
earlier.
The
an incorrect model
logistic classifier, albeit
h{x), does well in separating out a good subset of observations - those with Xi
initial inefficient
estimator Po{^)
is
fairly reasonable.
step that separates a larger subset of
Monte-Carlo results discussed
Finally, the last
column
p{x'j) vs.
0.
The
serves as a classifier for the next
It
good obervations. Thus, the
a progressively larger sample and pools
>
for
last step
estimator uses
This agrees well with the
itself closer to the truth.
in the appendix.
plots simple diagnostics that
—
we found
The column
helpful.
C, (circles) and x'Pi{t)
—
plots
the logistic
fit
The
to account for the concordance of different classifiers in choosing the trimming
idea
is
quantile
fits x' Pq{t)
constants and deciding on the sensibility of the estimates.
separate out a set of observations for which Xj'/3(t)
classifier,
the
first
>
Each of the
Ci or h{Xi)
>
1
Ci (triangles).
classifiers tries to
— t. As
a conservative
step p{x''y), with the addition of trimming, should be seen as one selecting
a smaller set of observations.
The subsequent
classifiers
should confirm
all
or almost
all
of
this initial set.
The
classifier
concordance
is
presented by the placement of points in
diff'erent
quadrants of
the plots A.4-C.4. For example, on the plot B.4, a large proportion of observations for which
p{X-^)
>
1
—
T
is
confirmed by the quantile
classifiers, as
seen in the upper-right quadrant.
However, those points in the right-bottom corner, are dis-confirmed. Nonetheless, most of
these are
trimmed out by the trimming hurdle
c,
^(^^'7)
>
1
—
r
-I-
c.
Hence the discordance
VICTOR CHERNOZHUKOV AND HAN HONG
16
"from-the-right"
is
reduced or eliminated by such trimming. The left-bottom corners of A.4-
B.4 represent the points agreeably disqualified by both the logistic and quantile
The
upper-left corners represent the discordance of the logistic
In principle, this type of dissonance
the-left".
right"
,
since quantile classifier
observations. However,
the hurdle
(5„.
we
is
find
meant
it
classifiers.
classifier
"from-
not as pernicious as the one "from-the-
to asymptotically separate a larger, better subset of
prudent to reduce discordance "from-the-left" by adding
For example, on the plot A. 4 the
than a subset of
in selecting a superset, rather
is
and quantile
initial quantile classifier is overly optimistic
{i
:
X^ >
the discordance, and helps select a more appropriate
and the quantile models are approximate models of
set.
0}.
The trimming
factor 6n reduces
In practice, since both the envelope
reality, it
appears prudent to reduce the
discordance "from-the-left" as well.
Overall,
if
the concordance plots reveal a very drastic disagreement,
sensible warning to revise the
all at
it
trimming constants, envelope models, or the models
in question
once.
A.1.
Pro-consorod Data
A. 2.
1 at
and Znd Stops
A. 3. 3-rd Stop
.
.
Pro-censorod Data
1.2.
l9t
and 2nd Stops
B.3. 3-rd Stop
Concordan
A.4. Classlr.
.-^
J.
y
y
V
4. Cla»«if.
Concordonco
^•^'
rod Data
C.2.
1 9t
and 2nd Stops
Figure
3.6.2.
should serve as a
C-3. 3-rd Step
C4.
Clasair.
J
Concordance
2.
The Stanford Example. The section considers a well-known Stanford heart transplant
who received heart
transplants, 45 had died by the closing date and are uncensored. Thus 35% of the observations
are censored. The censoring times for the post-transplant survival are random observable.
data
We
set.
The
dataset
is
built into
S-l-
shall contrast our results with the
(heart).
Out
of the 69 patients
well-known and thorough study of Aitkin, Laird, and
Francis (1983), and follow their definition of variables:
• (log) survival time,
dependent variable: the log of the difference between the death and
the transplant time. Censoring times are log differences between the last follow-up and the
transplant dates.
•
Age: Age of the patient at the time of acceptance into the program.
3-STEP
•
Acc: Years since January
CENSORED QUANTILE REGRESSION
1,
17
1967 to acceptance into the program. This regressor
may
be seen as representing a technological progress.
• Surgery: indicator of a
We
estimate the
CQR
previous open heart surgery.
model:
Qy\x,C
where
C
=
{r)
+ X'O (r)) A
(a (r)
denotes the observable random censoring time.
analysis studied in Aitkin, Laird,
and Francis (1983)
=
QY\Xfi{r)
where F~^
(r)
is
{a
is
C,
(13)
A benchmark model
from survival
the accelerated failure time model:
+ aF-'[T) + X'e)^C,
Vr,
(14)
the inverse of (a) the extreme value (Gumbel) or (b) the standard normal
distribution function.
baseline hazard and
Model
model
(b)
thus a proportional hazard (PH) model with Weibull
(a) is
- a non-PH
AFT
seemed
Aitkin, Laird, and Francis (1983), these
model.
to
fit
Among many models
best. For comparisons,
considered in
we
shall use the
estimates of ^ in model (14) reported in Aitkin, Laird, and Francis (1983), table
way
discussed, another
to robustly estimate 9 in
model
(14)
is
5.
As we
any 6{t), say median,
to take
or average over several quantile regression estimates 9[t) with different indices t.
Because there are only few un-censored observations on patients with prior surgery, we
could estimate the
To
CQR
model with the surgery regressor only up to the 50%th
discuss higher quantiles,
we
also estimated the
Figure 4 presents the results.
The
first
and of slope
intercept coefficients, a{T),
four plots [read by rows]
confidence intervals.
The bottom
The shaded
The dotted
model divided by the shape
estimates of quantile shift effect,
.1
to
for
.5,
areas represent the pointwise
lines present the Aitkin, Laird,
Francis (1983) estimates of quantile treatment/shift effects,
PH
show the estimates of
three figures plot the quantile coefficient estimates for
the model with age and acc as regressors.
hazards (coefficients of
percentile.
the surgery regressor.
r ranging from
coefficients, 9{t), for
the model with age, acc and surgery as regressors.
80%
CQR model without
9, in
the model (14) with Weibull
coefficient).
the normal version of (14).
^, in
and
The dashed
lines are the
Note that the
lines are
horizontal since location-shift models impose constant treatment effects across quantiles.
Comparing the quantile shift estimates 9{t) in the CQR model with those in the models
(14), 9, across r, we observe much qualitative and quantitative similarity. At the median, for
example, the technological
effects (acc) are
both small and
insignificant.
The
effects of
age
are both negative and significant, and the effects of previous surgery are both positive and
significant. In other words, prior surgery
survival.
Overall,
it is
an encouraging
and younger age seem
fact that the 3-step
to prolong the post-transplant
CQR
estimator produces results
which compare well with a well-known, high-quality study.
Furthermore, as the
CQR
model
(15) nests
model
(17) as a special case,
we may confirm
the general validity of the models considered in Aitkin, Laird, and Francis (1983), at least
concerning the quantile treatment
to be constant across
all
effects.
estimated quantiles.
1967 to acceptance of the program
we should
In particular, the effects of prior surgery appear
is
The
effect of
the time between January
small and insignificant at
point out that the age effects differ across quantiles.
all
quantiles.
1,
However,
For low survival quantiles
,
VICTOR CHERNOZHUKOV AND HAN HONG
18
the age treatment effects are positive, small, and statistically insignificant. For middle and
higher survival quantiles the age
efi"ects
are negative and significant. Yet this variation does
not appear to (statistically) negate the location-shift models of Aitkin, Laird, and Francis
(1983).
Of course
these findings warrant further careful examination of the age effects once
more data becomes publicly
available.
conclusions are well sustained
bottom row
agreement of
columns of the second row
Ci
—
the-left"
X[p\
,
Note that
of these quantitative and qualitative
all
the surgery indicator
omitted from regression. See the
is
in Figure 4.
Finally, the
fit
when
(r).
almost
classifiers
in figure 4.
seems to be
We
fairly high, as
plotted the logistic
fit
p
presented in the last two
(X'^j) against the quantile
There are only a few discordant observations "from-the-right" or "fromall
of which are eliminated by trimming.
Intercept
^
^^^
0.3
0.2
O.n
0.4
0-5
0-6
0.1
0-2
.,.-
.-r
.
.
^--o
0.3
0.4
0.5
0.6
Ouantlle
Ouantlte
concord. 0.2 quantile
concord 0.4 quanti e
0,2
0.1
0,e
0.8
1.0
O.I
0.2
logistic clBsslller
0.6
O.B
logistic classtrie
age
Intercept
?-->,_
""1^ ^
^
;S
-*<
i.
0.2
Figure
4.
0.4
0.6
3.
Determinants of Extramarital Affairs: A
Extramarital
affairs,
an important
social
(1991),
DeLamater
CQR
Analysis
phenomenon, received much attention by an-
thropologists, psychologists, evolutionary biologists, sociologists,
Reiss, Anderson,
'"T
''I
(1981), Fair (1978), Miller
and Klein
and economists. See Cronk
(1981),
and Sponaugle (1980) and many references
South and Lloyd (1995),
therein.
spective analysis of the Redbook data-set on extramarital affairs.
We
We
present a retro-
shall
mainly contrast
our analysis with both the data and the model-analytic findings of Fair (1978). Fair (1978)
3-STEP
CENSORED QUANTILE REGRESSION
19
presents a utility-based optimization model of the time spent in the affair (aifair intensity)
as determined by preference for diversity, value of goods
labor and non-labor income,
4.1.
consumed
in
and outside marriage,
and time already spent with the spouse and paramour.
Data. The dataset has been
collected by the Redbook magazine. Fair (1978)
mater (1981) describe the collection procedures as well as the place of
only few similar data-sets.
The
"censoring".
with his
We
define
statistical
all
women, of which
This presents a very high degree of
the variables as in Fair (1978) in order to facilitate comparisons
and model-analytic
• Intensity of Affaire,
affairs.
among
this data-set
data-set covers 6388 first-time married
68.5% reported to have had no extramarital
and DeLa-
dependent
findings:
variable, defined as the
number
of different partners outside
marriage times the approximate number of relationships with each partner, divided by the
number
of years in marriage.
For 68.5% of respondents,
of respondents, the density function
it
is
For the rest
equal to zero.
sketched by the histogram and a kernel estimator in
is
figure 4.
A-ffsIr
Intensitv
Figure
4.
Simple histograms of the following regressors are given in figure
• Marriage Rating:
5:
respondents' rating of their marriage, on the scale from
1
to 5.
• Age, Years Married, No. of Children
respondents' rating of their religiosity, on the scale from
• Religiousity:
• Education: 9.0, 12.0, 14.0:
college graduate,
• Occupation,
6:
3:
to 4.
grade school, high school, and some college; 16.0, 17.0, 20.0:
some graduate
and advanced degree.
school,
Husband's occupation: Hollingshead's socio-economic status of occupation:
professional with advanced degree,
artist, etc.
1
managerial, administrative, business,
5:
white collar (administrative,
clerical),
2:
3:
teacher,
blue colar (farming, factory),
1:
student.
4.2.
Models. The
CQR
model assumes the following form (with C,
QYlx{T)
That
is,
=
{air)
+
is
appealing, as
we have
0):
X'_,e{T))yO.
the conditional quantile function of the affair level
functional form
=
discussed.
We
is
(15)
either zero or linear.
also consider a standard
This
normal
VICTOR CHERNOZHUKOV AND HAN HONG
20
h/Torrlago Flatir
Mo.
IVIsirri^d
chillc:
2
S
.
i
i
I
C^ccupatlc
Ftellglovj3lty
1
J3t»nrici*^
Oco.
I
i
[l
s
r^
t
»
1
f
p
1
i
1
6
1
i
Figure
5.
model:
QY\x{r)
where #~^(r)
is
=
{a
+ a<^-\T)+X'_,e)VO,
Vr,
(16)
the inverse of the standard normal distribution. Another benchmark model
is
the accelerated failure time model (for exp(y)) from survival analysis:
QY\x{r)
where
F
9 in this
will
is
=
{a
+ (jF'\T) + X'_,e)WO,
unspecified distribution function.
It is
Vr,
(17)
easy to estimate the quantile shift effects
model by taking or averaging any of the estimates of 6(t)
in the
CQR model.
This
not be necessary, since neither this nor the normal model are supported by the data.
Estimation and Model Comparisons. To construct the initial envelope/classifier
we examined the pairwise-plots of Y vs X. Many of covariates appeared to be associated
with higher dispersion of Y, which lead us to consider a number of polynomial powers in
4.3.
the logistic model
p{x'"f):
x consisted of xu^, x?
appeared to significantly improve the
the large sample
of the envelope
size.
was
Due
to
negligible.
heavy censoring
18,
which
is
results are
=
The trimming constant
it
-y
that
c
was
set to c si
.1
was not possible to estimate
(Q(r),0(T)')', t G {.4,
The dashed
normal model
(16)
according to the
was estimated by conditional MLE.
all
quantile coefficients 6{t). Iden-
This condition
.5.
summarized graphically
confidence intervals.
effects in the
X(j)X(j)
plausible in view of
depended on the nondegeneracy of the selected design matrix.
estimates oi (3{t)
90%
and certain interactions
Dimension of i was
prevented considering quantiles lower than
The
x?-,,
Sensitivity of the final estimates to further increasing the complexity
rule described in the appendix,
tification
fit.
^,
...,
in Figure 6.
.9},
The
solid line
denotes the 3-step
and the shaded region depicts the pointwise
line presents the
MLE estimates
6 of the quantile shift
obtained by Fair (1978).
6{t) significantly vary across quantiles, especially at higher ones. This presents an evident
violation of homoskedasticity assumption. Therefore, either
model
(16) or (17) are strongly
CENSORED QUANTILE REGRESSION
3-STEP
unsupportive of the data.
estimates of the normal model.
of both normality
heavy
(e.g.
interesting to briefly
It is still
It is
well
known
see Goldberger (1983),
we have both.
seem to correspond
for five out of eight variables, the estimates 6j
estimates 6j{t),t
It is
extreme quantile
to fairly
For the marriage longevity variable, on the other hand, the estimate
.9.
away from any of
T, so 6j,
understandably, can not even match the direction of the quantile shift
Furthermore,
djir).
Thus,
ks
This finding
an empirical illustration to the
CQR
of the
is
raw
Finally, the last
fairly
is
Figure
is
flexibility
-
is
discordant "from-the-left",
eliminated by the additional trimming. Discordance "from-theit is
Analysis. The most exciting matter
Religiosity effect
would have been the
on the breadth and
small proportion of observations
mildly present, and part of
effects 6{t) (cf.
it
in Figure 6 presents the concordance plots. Overall, the concordance
a major chunk of which
riglit" is also
earlier discussion
^
model.
appears to be good. Only
4.4.
dj
effect, since
can hardly be given any meaning in the present setting.
case that 9{t)
all r.
changes across
in several cases sign of ^j(t)
the normal model (16) or location-shift model (17) were adequate,
9 for
Hurd
interesting that
Oj is far
if
ML
of
that the estimates are not robust to violations
(1979) for proofs and simulation studies. In our example,
~
comment on the behavior
and homoskedasticity ~
tails)
21
is
also eliminated by trimming.
the interpretation of the estimated quantile shift
6).
quantiles of affair intensity
and especially
sociologists emphasized, the institutions
and norms of
expectedly negative at
strong at very high quantiles.
As
all
Judeo-Christian doctrine have had a major influence on American families, endorsing a pro-
and strictly-within-the-marriage orientation towards sexuality
creational
(1981)). This thesis
is
Education quantile
shift effects,
on the other hand, are more engaging.
negative and strongly negative at high quantiles.
weakly correlated
(e.g.
DeLamater
well supported by the data.
(.14),
but since we condition on
of this and other factors.
Education
The
Note that education and
religiosity,
effects are
religiosity are
the education effects are net
effects are inexplicable within the Fair's
model, yet
they appear to have a clear meaning in view of the relational (rather than recreational)
perspectives towards a paramour
among the more intelligent, educated, and not
DeLamater (1981)).
necessarily
religious individuals (Reiss (1980),
The
quantile treatment effects for age are nonpositive across
effects are negative at the
all
presented quantiles. Age
middle quantile and strongly negative at high quantiles.
means that the younger respondents
are
more
likely to
engage in an
affair, especially in
This
very
intense ones, holding everything else fixed. In Fair's analytic model, diversity considerations
("variety
is
a spice of
life")
the life-cycle considerations.
are incorporated in the utility but does not shed
On
much
light
on
the other hand, the elaboration by DeLamater (1981) on
the life-cycle perspective dynamics, within the social institutions and norms, conforms the
present data-analytic findings.
Women
engage
in affairs, especially
One view
affair
with an occupation of higher socio-economic status are relatively more
is
more intense
ones.
likely to
Explanations for this are to be looked
at.
that such status creates an interactional advantage, increasing the hazard of an
and subsequent marital dissolution (South and Lloyd
1995). Fair's analytic
model does
^
VICTOR CHERNOZHUKOV AND HAN HONG
22
Age
Marriage Rating
Intercept
—
o
d
c
5
=
O
d
o
O
"':
d
0.6
0.4
0.6
0.8
0.8
Quantile Index
Quantite Index
Years Married
No. children
Religiousity
o
o
°
°
**"«—»-o—et
r
o
o
%5;
o
'^%
\
0.4
0.8
0.6
1.0
0.6
0.4
0.8
Quantile Index
Quantile Index
Education
Occupation
Husband's Occ.
0.4
0.6
0.8
0.4
Quantite Index
concordance:
.6
concordance:
quantile
.8
0.2
0.4
0.6
1.
.9
quantile
(0
a
u
«
o
o
n
3
0.8
concordance:
quantile
S
c
0.6
1.0
Quantile Index
1
i
0.0
0.8
0.6
0.4
Quantile Index
c
o
0.S
3.0
logistic classifier
02
0.4
0.6
o
0.8
3.0
0.4
0-2
logistic classifier
0.8
0.6
logistic classifier
Fi(3URE 6.
not necessarily yield predictions about the direction of the status effect (because he treats
the status as proxy for labor income).
To the extent that higher
status
is
associated with
non-labor income, or to the degree that income effects dominate substitution
model may predict a positive
effect.
Husband's occupational status has very small positive or negligible
quantiles, except at very
insignificant at .95).
effects, Fair's
extreme ones, where
it
effect across
becomes very negative
Besides an (anecdotal) explanation that good
men
almost
all
is
positive but
pick
good wives,
(it
3-STEP
women may
CENSORED QUANTILE REGRESSION
value statusful husbands and pursue affairs,
of marital dissolution optimally low.
On
in
at all, optimally to
keep the risk
the other hand, Fair's analytic model predicts the
positive effects of a husband's status (income)
consumed
if
23
on the
affair level, as
a higher value of goods
marriage causes wives to substitute labor activities for time spent with family
and paramour. The Fair model, however, ignores the negative value of the dissolution option;
which
is
real as other studies point out
model explains only the middle
results suggest that Fair's
the high quantiles.
(South and Lloyd (1995)). Our quantile regression
quantiles, but does not apply to
seems plausible that incorporation of the dissolution
It
risk in Fair's
model, more along the lines of the Becker (1968) crime and punishment model, can make
conform to the present findings. Finally,
statusful
also not implausible that
more
intelligent or
is
slightly positive at
.6-. 8
quantiles and strongly negative
Fair, using behavioral considerations, postulates that
marriage longevity
positively relate to diversity quest, leading to an increased affair level.
not entirely clear
fact that only
why
the effect
is
effect is the
with the "cheating ability".
detection ability
marriage longevity
effect of
at high quantiles.
may
is
husbands can have better unobserved detection rates and the observed
interaction of this hidden
The
it
it
very negative at high quantiles. This
However,
may
it is
relate to the
married and undivorced respondents were selected to the sample, so that the
marriage longevity correlates with the marriage match quality and thus has a deleterious
effect
on
affairs.
Such an outcome would be a
alternatives) theory.
Our
clear prediction of the search (for spousal
finding thus partially disconfirms both the analytic
and
statistical
predictions of Fair (1978) derived from the normal model.
5.
The
This paper had three goals.
spectives on the global
CQR
Discussion
was to
first
offer
both theoretical and empirical per-
models as a useful way to approximate the quantile
The treatment variety
assumed by many commonly used models. In
ment/shift effects in censored regression settings.
is
simple location-shift effects
the
richer
treat-
than the
CQR
models,
the covariates and the treatment can affect such features as scale, skewness, kurtosis,
generally, the entire
The second
serious social
goal
shape of conditional distribution.
was to
offer
an empirical
phenomenon - extramarital
ogy, psychology,
or,
CQR
affairs.
and economics of marriage and
analysis of the determinants of a very
This
family.
is
an important topic within
To our
regret,
sociol-
we found no previous
implementable estimators that can be used in the settings of heavy censoring, many poly-
chotomous or continuous
prevail in
many
regressors,
and large or small samples.
areas of applied statistics.
mentable, well-behaved estimator.
Such data
sets
seem
to
This justified the pursuit of a practical, imple-
The suggested estimator can be used to robustly estimate
many traditional models. Studying this estimator in large
the global
CQR models
and
samples, in the well-known Stanford heart transplant data-set, and developing the
finite
as well as
diagnostic tools, was the third goal.
VICTOR CHERNOZHUKOV AND HAN HONG
24
Appendix
K
Below, C, const, and
we
are generic positive constants. For notation sake,
set
d
=
General case
0.
is
very
similar.
Proof of Theorem
Appendix A.
Part
consider
1. First
ffo
Qn
Vi„{z)
=
[z,
Z
= 4=
1)
— X[z/y/n) —
y/n[pT(ei
=
Rescaled statistic Z°
(t).
>
V'i„(z)l[p(X.'7)
and
PrCft)]
—
y/Ti{po (r)
1
-
T
1.
minimizes
{t))
+
where
c],
=Yi — Xl0 (r) = max (a;, —X[f3 (r)) The
e,
claim
.
is,
for finite
k:
Qoc
1
—
2,
+
T
and
c].
J
since
invertible
is
Qoo
by conditions
and
(ii)
(iv).
by the convexity theorem (Knight (1999), Theorem
LLN, CLT, and some standard
calculations, so
Qn
For any fixed
(a)
\l
— 7o| <
VC
Theorem
is
<5}
subgraph
2 of
{Qn
2,
and 5
>
class) in
(2,7)
— EQn
(z,
Type
(1994)
(b)
(a) is verified
70)
I
(i)
T with
random
in
But
if
7
=
70, (18) follows
by
-^
for
any fixed
2.
(19)
stochastically equicontinuous in 7,
is
where G
=
{-y
:
functions (bounded variation functions of a single index, a
class
^=
{x
>->
l\p{x~f)
>
1
—
r
+ c],
7 £ 5}
,
hence by
entropy condition with a constant envelope. This property
satisfies Pollard's
it
and continuous
are convex, finite,
(18) implies
also see Pollard (1991)).
5,
€ 5}
(2,7) ,7
variable Vin(2), Vin(z)
® J^,
by Theorem 3
in
Andrews
(1994).^,
|Vin(2)| has a constant envelope;
\V,n{z)\
Hence
(18)
remains only to verify that
it
Andrews (1994) include the
Andrews
by assumption
— Qn
7)
small. Indeed,
is
retained by the product of
since
(2,
Qn and Qoo
Since
—J~^W = Op (1),
uniquely minimized at
is
where
W = N{0,A),J = Ef^iO\X,)XiX',l[p{X',yo) > 1-t+c],A = T{l-T)EX,X',l\p{X'no) >
= W'z+\z'jz,
(z)
j<k),
j<fc)^(Qoc(2,),
{Qn{zjn),
by Theorem
in
1
Andrews
<
2\X,z\
<
const
(20)
(1994).
Space Q with pseudo-metric
p(7i,72)
=
sup
E\v,n(z) X [l{p(X'ni)
< C X \p{X'a, >p''[l-r +
c])
>
-
1
-
r
+
P(X.'7i
c}
>
-
l{p(A':72)
p-'[l
-
r
>
1
- r + c}]f
+ c],Xa2 >p-'[l-T + c])
(21)
+ P{X'n2 >p"'[l-r +
c])
-P{Xhi
>p-'[l-r-|-c],A:;72
> p"'[l -
r
+ c])
|
<
®We
const
X
72
—
71
note that Andrews allows for triangular sequences, which allows the r.v Vin{z) to be indexed by n.
Otherwise, we would have to refer to van der Vaaxt and Wellner (1996) 2.11.3 for dealing with functional
classes indexed
sense that 2
is
by
Note that convexity allows us to treat Vin{z) as
in the above arguments.
n.
fixed
r.v.
and not a function of
z, in
the
.
CENSORED QUANTILE REGRESSION
3-STEP
is
bounded, where the
totally
inequality follows from (20), and the second one follows from assumption
first
by bounding each of the two terms,
(iv)
example;
for
-r + c]) -P(X'7i
|p(.Y;7i >p"'[1
25
>P"'ll -T^-c\,X['y2 >p~'[l-r +
c])
|
<
|p(a';7i
>p-'[l-r +
|p(a:,'7i
>p"'[l-r + c])
-P{x[i, >p-'[l-r + c],A-:7i+A':(72-7i) >
c])
"
p"'[l
^
+ c])
|
<
-p(a','7i -
>
A' x
H^, -71II2
-
p"'[l
+
r
c])
|
< C^i\ where \X[
and
(a)
(71
72II2,
—
72)
<
|
—
A'||7i
sup
has a compact support X.
72II2, since A,-
(b) together implies that
Andrews
e.g.
(
- Qn
\Qn (2,7)
(1994) equations 3.34 and 3.36
- EQn
(2,70)
+ EQn
(2,7)
):
=
(z,7o)
|
Op
(
1)
.
l7-70|->0
Thus, to complete the proof of (19),
remains only to show that
it
\EQn
We
will
show that
=
for Si(7,7o)
EQn
(2,7)
>
l[p(A,'7)
- EQn
- EQn
7)
(2,
(2,70)
-
1
+
t
(2,
(2)
-
c]
= -^[{r -
7
close
>
—
,
i.
(1
-
1
r
70)^=^
-
+
r)
c]:
= Op(7 -
e,V^}]
enough to
which implies
+
=
A,'/3(t)
>
v"
V^V^n(z),
>
and
(i)
—
(1
r)
+
c
(iii).
Hence
>
small,
for v' ,v"
a.s.
,
+
p(A,'7)
by assumption
c,
(23)
^VU^)
Then
70.
70),
=
^[\A^V/„(2)l(A.'^(r)
since P[e,
[the
>
(22)
Hence, as 7 gets close to 70
E[^^V/„(2)s,(7,7o)|A,]
0]. A\so
Op (1)
(2) 6.(7,
?;=(2){A,'2
,
for all
=
-
=
where
-
l[p(A,'7o)
U=^ = V^EV.n
\[u < 0]}Xlz] + v^[
—
<
< Xlz/^/n)]. Set
l(ei
[l(ei
rii{z)
0)
—
implies h{Xi) > (1
r) + v' and so does p(A,'7o)
necessarily implies h{Xi) > (1 — r) + v'
Si(7,7o) /
Write y/nVin
70) 1^^
<
0|A,, A',/3(r)
>
t;"]
=
r
[
if
Ai/3(r)
>
v")\X,] X s,(7,7o)
>
0, €,
=
0,
uniformly
= max (-u^, — A','/? (r))
m
(24)
t,
has r-th conditional quantile at
E[V^v-U^)s,('rno)\Xi]
second
=
£[^^V/:(2)l(X.'^(r)
=
O[/„(0|A,)2'AiX;2l(A;/3(r)
line follows
>
^")1A.] X s,(7,7o)
(25)
>
v")] x 5,(7,70), uniformly in
by the standard calculations]. Therefore by assumptions
(iii)
i
and Lipshitz condition
(iv):
i?i^[^/^K/,;(2)s.(7,7o)|A.]
(24)
and
Part
is
(26)
2.
identical.
imphes
Proof
Rescaled
is
to
show the
for part 2
in part
statistic
Zl
is
The proof
result for /3i(r) with Po{t) as the selector.
similar to that of part
1,
for Pi{t),I
>
1
so only important differences are given. Notation
I.
=
y/ni^jii
(r)
—
/3(t)) minimizes
"
1
Qn{z,Po{.T),5n)
Consider ^„ as a parameter sequence.
= -yr^Y Vin{z)\[X[Po{T) >
Proceed identically 35
> 1 - t - c] by 1[X%{t) > 5n\, ^\p{X[^) >
N{0,Hg^AoH^^), Ho, Aq. It remains to show
1[p(A:7)
1
Qn{zJ{T),5n) - Qn{z,l3{T),0)
-
r
-^
,
:
5n\.
in part 1
up
-
[X[P{t)
0,
c]
by
for
1
to equation (19), only replacing
any fixed
(a) For any fixed z, {Qn {z,/3,5) — EQn {z,f3,S) (/3,5) G B x V} is
where B = {(3 \l3 - ^(r)| < C"}, V = {5
< 5 < C"} and C',C" >
in Andrews (1994) (VC subgraph classes) contain T = {x ^ l[i'/? >
:
(26)
(22).
It suffices
undefined here
=0(£[s,(7, 70)]) =0(7-70).
>
0]
and
W,
J,AhyW =
(27)
2.
stochastically equicontinuous in (3,6,
axe small. Indeed,
6],
{(3,6)
6
S
Type
I
functions
x V}. Hence
T
has
VICTOR CHERNOZHUKOV AND HAN HONG
26
a
finite
uniform covering entropy integral and a constant envelope,
This property
in
Andrews
retained by the product of space
T
S
P
x
is
totally
by Theorem
verified
is
entropy condition.
satisfies Pollard'
i.e.
®T,
with random variable V,n(2): ^in{z)
(1994), since |V',n(2)| has a constant envelope, (a)
Space
(b)
is
1
by Theorem 3
Andrews
in
(1994).
bounded under the L2 pseudometric:
f£|V'.„(z)x
p((/3i,5i),(/32,<52))ssup
[l{A','/?i
>
-
5i}
If.Y.'ft
>
^
H
^2}]
^
(28)
<
where
X
const
||;92
from (20) and from assumption
(28) follows
—
/3i
II,
+
x
const
||52—
(5i||.,,
analogously to the proof of (21), treating 5 as shifting
(iv),
the intercept parameter.
along with
(b),
(a),
implies
(
e.g.
Andrews (1994) eq
(3.34)
and
(3.36)
):
Qn{z,P,K) - Qn{z,l3{T),Q)-EQn{z,P,5n) + £Q„ (z, ^(t)
sup
,
Op(l).
0)
|/3-/3(r)|-tO
Thus, to complete the proof of
(27),
it
remains to show that
|£Q4z,/3,5„)-£Q„(2,/?(r),0)|^^^^ =Op(l).
=
(29)
By assumption on the sequence 5n, w.p. —> 1, Po{t) is
inside the ball with radius n'Sn, centered at P{t), where k' >
is small. By the compactness assumption on
{t))\ < ^6n- So set /3 inside this ball. Then
Xi, k' can be chosen so that w.p. — 1, sup^g-^ \^'{Po{t) —
implies x' P < Sn- Thus
for small enough k chosen as such, x' (3 > S„ implies x' /3 (t) > 0, and x' (i {t) <
uniformly
in
i,
7^
necessarily
implies
X,'/3(r)
a.s..
Hence,
>
Si{j3, P{r))
Let 5,(/3,/3(r))
\{X[P >
Sn)
-
l(.Y,'/3(r)
>
0).
)•
E[V^Vt{z)s,{l3,0{T))\X,]
smce P[e,
uniformly
<
in
0|X,-,
i
X,/3(r)
>
=
O]
r.
= E[J^VU{z)\{X[P{t) >
Also E[y/fiVZiz)s,{l3,p{T))\X.]
=
i?[^^C(z)l(A':/3(r)
=
O[fu{0\X,)z'XiXlzl{Xll3{T)
[the last equality again follows
and Lipshitz condition
>
0)|A',]
>
0)\X,] x s,(0,0iT))
=
0,
(30)
=
x s.(/3,^(r))
^^^^
x
0)]
by the standard
s.(/?,/9(r)),
Therefore by assumptions
calculations].
(iii)
(iv)
EE[V^V::{z)sdP,l3iT))\X,]
and (31) give (29).
Part 3. Finally, to show
=
0[Es.{l3,l3(r))]
= O{0 -
P{t))
+ 0{
<5„).
(32)
(30)
before by defining variable
over z
parts
=
1
(zi,
and
i
2,
<
J),
joint convergence of Zn(r,)
Z^
—
<
=
y/n{(3{Tj)
noting that
sum
of
B.
s.e.
— i3{tj)),
i
< J for
either step, proceed as
J) as minimizing the objective function: 671(2)
where Qn are objective functions defined as
Appendix
Inference:
(Zniri),i
processes
is s.e.
(see e.g.
in
part
1
or part
2.
=
X]i<j Qn{zi)
Repeat the arguments
in
Knight (1999)).
Practical Details of Estimation and Inference
Because of distributional equivalence with the Powell
dures developed by Powell apply without modifications.
CQR estimator,
The bootstrap
all
of the inference proce-
inference of Bilias, Chen,
and Ying
(2000) can be extremely useful in practice. All inference procedures are as for the standard quantile regression
procedure with the only difference being that the selected (rather than complete) sample
QR software need not make any
last step QR routine are all valid.
user of the standard
produced by the
fit
is
used. Therefore, a
modification - the standard errors, confidence intervals
Model p() and Trimming Constant: Parametric model p in step 1 should be reasonably good. Goodness of
all the standard statistical packages and are easy to carry out. It is always possible
checks are supplied in
to achieve a reasonably good
fit
using a series approximation of probability model.
not very small, although the Monte-Csirlo work shows that c
=
is
check sensitivity of the estimates Po{t) to increasing the constant
The theory
requires c to be
also quite sensible. In practice,
c.
one should
Also, c shouldn't be too big, for
we may
3-STEP
The
select very small Jo-
CENSORED QUANTILE REGRESSION
better the parametric
the smaller c can be
fit is,
set,
27
improving efficiency of the
initial
p(X'^) > 1 — r + c}
/3o- A sensible rule for choosing c is to compare the size of the selected sample J(c) = {i
for c =
and other values. Choosing c = q-th quantile of all p(A','7) s.t. > 1 — r, appears to be sound, as
it gives a control of percentage of observations from J(0) can be thrown out:#J(c)/#J(0) = (1 — g)%. This
rule, with q = 10%, was employed in simulations. We also set 5n corresponding to q = 5%.
:
Appendix
Finally, table
I
C.
Monte-Carlo Experiments
reports the result of a small
monte
standard location-scale model with an error term
(l,A',
by
e,
1
{A',
=
:
:
=
Y'
||A',||
Ui X (l
+
X',P
<
+
The
2}.
0.5X]j = i
censoring point
is
-0.75.
We
ei.
is
experiments.'"
The model we
consider was a
scale, for
i^i ~^-^'))' "^^^
We
used
X
and X^
little effect
parameter vector
^"^^^
in the
logit,
chosen at
(1, 1, .5,
~
A^ (0, 25),
— 1, — .5, .25), and
parametric propensity score regressions.
on the performance of the estimators. Therefore,
them
=
We
the
experimented
probit and linear models. However, the type of probability
Note that due
in table
I
we only report the
to the heteroscedasticity error structure, even the probit
not consistent with the true propensity score.
step estimators, and compare
is
Xi
distributions, truncated
error term has the multiplicative heteroscedasticity structure; for u,
results for the probit selection model.
model
cEirlo
by a linear-quadratic heteroscedcistic
draw Xi 6 R^ from independent standard normal
with different probability models including
model used has very
hit
to the Buchinsky
We
report the initial step, the
first
step and the third
and Hahn (1998) estimator and the Powell estimator.
Notably, the results from the monte carlo simulation show that for sample size 100, the step-3 (1=1)
estimator outperforms
all
other estimators in terms of root
deviation(MAE), both mean and median
increases both root
median
bicises.
mean square
error
biases, are
and mean absolute
Overall the step 5 estimator
estimator and the Powell estimator.
a large sample, n
=
The
still
mean square errors(RMSE).
Its
mean
absolute
comparable to other estimators. Iterating to step 5(1=5)
deviation, at the benefit of reducing both
mean and
compares favorably to both the Buchinsky and Hahn (1998)
initial inefficient
estimator (1=0) does fairly poorly, as expected. In
400, the 3 estimator and the Buchinsky and
and are both more favorable than other estimators
Hahn
in all dimensions.
(1998) estimator perform equally well
To conclude, the 3-step estimator does
better in a small sample, and very well in a large sample.
References
Aitkin, M., N. Laird, and B. Francis (1983): "Reanalysis of the Stanford Heart Transplant Data (in
Applications)," Journal of the American Statistical Association, 78(382), 264-274.
Amemiya, T. (1973): "Regression analysis when the dependent variable is truncated normal," Econometrica,
41, 997-1016.
"Two Stage Least Absolute Deviations Estimators," Econometrica, 50, 689-711.
Advanced Econometrics. Harvard University Press.
Amemiya, T., and J. L. Powell (1983): "A comparison of the logit model and normal discriminant analysis when the independent vaxiables are binary," in Studies in econometrics, tim,e series, and multivariate
statistics, pp. 3-30. Academic Press, New York.
Andrews, D. (1994): "Empirical Process Methods in Econometrics," in Handbook of Econometrics, Vol. 4,
ed. by R. Engle, and D. McFadden. North Holland.
Andrews, D. W. K., and Y.-J. Whang (1990): "Additive interactive regression models: circumvention of
the curse of dimensionality," Econometric Theory, 6(4), 466-479.
Becker, G. S. (1968): "Crime and Punishment: An Economic Approach," The Journal of Political Economy,
(1981):
(1985):
76(2), 169-217.
AND Z. Ying (2000): "Simple resampling methods for censored regression quantiles,"
Econometrics, 99(2), 373-386.
Breiman, L., J. H. Friedman, R. A. Olshen, AND C. J. Stone (1984): Classification and regression trees.
BiLIAS, Y., S. Chen,
J.
Wadsworth Advanced Books and Software, Belmont, CA.
Buchinsky, M. (1994): "Changes in U.S. wage structure
1963-87:
An
application of quantile regression,"
Econometrica, 62, 405-458.
"The
set
up
of our simulation
is
similar to those of Buchinsky
and Hahn
(1998).
VICTOR CHERNOZHUKOV AND HAN HONG
28
BucHiNSKY, M., AND
J.
Hahn
"An Alternative Estimator
(1998):
for the
Censored Regression Model,"
Econometrica, 66, 653-671.
Chaudhuri, p.
Ann.
tation,"
Chaudhuri,
K. Doksum,
P.,
and
X.,
local
Bahadur represen-
760-777.
and
A.
Samarov
(1997):
"On average
derivative quantile regression," Ann.
715-744.
Statist., 25(2),
Chen,
"Nonparametric estimates of regression quantiles and their
(1991):
Statist., 19(2),
X. Shen (1998):
"Sieve
extremum estimates
for
weakly dependent data," Econometrica,
66(2), 289-314.
Cox, D. D.
(1988):
"Approximation of
least squares regression
on nested subspaces," Ann.
Statist., 16(2),
713-732.
Cronk, L, (1991): "Human Behavioral Ecology," Annual Review of Anthropology, 20, 25-53.
DeLamater, J. (1981): "The Social Control of Sexuality," Annual Review of Sociology, 7, 263-290.
Doksum, K. (1974): "Empirical probability plots and statistical inference for nonlinear models in
the two-
Ann. Statist., 2, 267-277.
Efron, B. (1975): "The efficiency of logistic regression compared to normal discriminant analysis," J. Amer.
Statist. Assoc, 70, 892-898.
Fair, R. C. (1978): "Theory of Extramarital Affairs," Journal of Political Economy, 86(1), 45-61.
Ferguson, T. S. (1967): Mathematical statistics: A decision theoretic approach. Academic Press, New York,
Probability and Mathematical Statistics, Vol. 1.
FiTZENBERGER, B. (1997a): "Computational Aspects of Censored Quantile Regression," in Dodge Y. ed,
Proceedings of The 3rd International Conference on Statistical Data Analysis based on the Ll-Norm and
Related Methods, vol. 31, pp. 171-186. Institute of Mathematical Statistics Lecture Notes-Monograph Series,
Haywaxd, California.
(1997b); "A guide to censored quantile regressions," in Robust inference v. 14, Hanbook of Statistics,
sample
case,"
Amsterdam.
"Abnormal Selection Bias," in Studies in Econometrics, Time
by T. A. S. Karlin, and L. Goodman. New York: Academic Press.
pp. 405-437. North-Holland,
GOLDBERGER, A.
(1983):
variate Statistics, ed.
Hogg, R. V.
Series,
and Multi-
"Estimates of Percentile Regression Lines Using Salary Data," Journal of the American
(1975):
Statistical Association, 70(349), 56-59.
Kurd, M.
(1979):
"Estimation in Truncated Samples
When
There
is
Heteroscedasticity," Journal of Econo-
metrics, 11, 247-258.
JURECKOVA,
J.,
AND
Khan,
S.,
and
to appear in
Knight, K.
J.
J.
Prochazka
B.
(1994):
"Regression Quantiles and
Trimmed Least Squares
in Nonliear
of Nonparametric Statistics, 3, 201-222.
L. Powell (2001): "2-Step Estimation of Semiparametric Censored Regression Models,"
Regression Model,"
J.
Econometrics.
"Epi-convergence in distribution and stochastic equi-semicontinuity," Department of
(1999):
Statistics, University of Toronto.
KOENKER,
KOENKER,
R.,
AND G. S. Bassett (1978):
AND O. Gelling
"Regression Quantiles," Econometrica, 46, 33-50.
longevity:
A Quan"Reappraising
Medfly
appear,
also
at
Amer.
Statist.
Assoc,
to
http: //www.econ.uiuc .edu/"roger /research/home .html.
Koenker, R., AND K. Hallock (2001):
"Quantile Regression:
An Introduction," mimeo,
http://wuw.econ .uiuc edu/'roger /re search/home. html.
Koenker, R., AND J. A. F. Machado (1999): "Goodness of fit and related inference processes for quantile
regression," J. Amer. Statist. Assoc, 94(448), 1296-1310.
Koenker, R., and S. Portnoy (1987): "L-estimation for linear models," J. Amer. Statist. Assoc, 82(399),
tile
R.,
Regression
Survival
Analysis,"
(2001):
J.
.
851-857.
LeBlanc, M., and R. TlBSHlRANl
(1996): "Combining estimates in regression and classification," J. Amer.
Assoc, 91(436), 1641-1650.
Lehmann, E. L. (1974); Nonparametrics: statistical methods based on ranks. Holden-Day Inc., San Francisco,
Calif., With the special assistance of H. J. M. d'Abrera, Holden-Day Series in Probability and Statistics.
MiCHIE, S., AND Taylor(ed) (1994): Machine Learning, Neural, and Statistical Classifcation. Ellis Horwood,
Gregory Piatetsky-Shapiro.
Miller, B. C, and D. M. Klein (1981); "A Survey of Recent Marriage and Family Texts (in Survey
Essays)," Contemporary Sociology, 10(1), 8-21.
Newey, W., AND J. Powell (1990); "Efficient Estimation of Linear and Type I Censored Regression Models
under Conditional Quantile Restrictions," Econometric Theory, 6, 295-317.
Statist.
3-STEP
Newey, W. K.
(1997):
CENSORED QUANTILE REGRESSION
"Convergence rates and asymptotic normality
29
for series estimators," J.
Econometrics,
79(1), 147-168.
Pollard, D.
7,
(1991): "Asymptotics for Least Absolute Deviation Regression Estimator," Econometric Theory,
186-199.
PORTNOY,
S. (1991):
"Asymptotic behavior of regression quantiles
in nonstationeiry,
dependent
cases,"
J.
Multivariate Anal, 38(1), 100-113.
PORTNOY,
S.,
AND
R.
KOENKER
(1997):
"The Gaussian Hare and the Laplacian Tortoise,"
Statistical Science,
12, 279-300.
Powell,
J. (1984): "Least Absolute Deviations Estimation for the Censored Regression Model," Journal of
Econometrics, pp. 303-325.
Powell, J. L. (1986): "Censored Regression Quantiles," Journal of Econometrics, 32, 143-155.
(1991):
"Estimation of monotonic regression models under quaptile restrictions," in Nonparametric
in econometrics and statistics (Durham, NC, 1988), pp. 357-384. Cambridge
and semiparametric methods
Univ. Press, Cambridge.
and
S. Wilson (1978): "Choosing Between Logistic Regression and Discriminant Analysis,"
~
Assoc, 73(364), 699-705.
Reiss, L (1980): Family Systems in America. Holt, Reinhart and Winston, New York.
Reiss, 1., R. Anderson, and G. Sponaugle (1980): "A multivariate model of the determinants of extramarital sexual permissiveness," Journal of Marriage and Family, 42, 395-410.
Ripley, B. D. (1996): Pattern recognition and neural networks. Cambridge University Press, Cambridge.
South, S. J., and K. M. Lloyd (1995): "Spousal Alternatives and Marital Dissolution," American Socio-
Press,
J.
S. J.,
Amer.
logical
Statist.
Review, 60(1), 21-35.
Stone, C. J. (1985): "Additive regression and other nonparametric models," Ann. Statist., 13(2), 689-705.
Tobin, J. (1958): "Estimation of Relationships for Limited Dependent Variables," Econometrica, 26, 24-36.
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.
Vapnik, V. N. (2000): The nature of statistical learning theory. Springer- Verlag, New York, second edn.
Yang, S. (1997): "Extended weighted log-rank estimating functions in censored regression," J. Amer. Statist.
Assoc, 92(439), 977-984.
Zhou, M. (1992): "M-Estimation in Censored Linear Models," Biometrika, 79(4), 837-841.
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