QUANTILE REGRESSION
ROGER KOENKER
Abstract. Classical least squares regression may be viewed as a natural way of extending the
idea of estimating an unconditional mean parameter to the problem of estimating conditional
mean functions; the crucial link is the formulation of an optimization problem that encompasses
both problems. Likewise, quantile regression o ers an extension of univariate quantile estimation
to estimation of conditional quantile functions via an optimization of a piecewise linear objective
function in the residuals. Median regression minimizes the sum of absolute residuals, an idea
introduced by Boscovich in the 18th century.
The asymptotic theory of quantile regression closely parallels the theory of the univariate sample
quantiles. Computation of quantile regression estimators may be formulated as a linear program-
ming problem and e ciently solved by simplex or barrier methods. A close link to rank based
inference has been forged from the theory of the dual regression quantile process, or regression
rankscore process.
Quantile regression is a statistical technique intended to estimate, and conduct inference about,
conditional quantile functions. Just as classical linear regression methods based on minimizing sums
of squared residuals enable one to estimate models for conditional mean functions, quantile regression
methods offer a mechanism for estimating models for the conditional median function, and the full
range of other conditional quantile functions. By supplementing the estimation of conditional mean
functions with techniques for estimating an entire family of conditional quantile functions, quantile
regression is capable of providing a more complete statistical analysis of the stochastic relationships
among random variables.
Quantile regression has been used in a broad range of application settings. Reference growth
curves for childrens' height and weight have a long history in pediatric medicine; quantile regression
methods may be used to estimate upper and lower quantile reference curves as a function of age, sex,
and other covariates without imposing stringent parametric assumptions on the relationships among
these curves. Quantile regression methods have been widely used in economics to study determinents
of wages, discrimination effects, and trends in income inequality. Several recent studies have modeled
the performance of public school students on standardized exams as a function of socio-economic
characteristics like their parents' income and educational attainment, and policy variables like class
size, school expenditures, and teacher qualifications. It seems rather implausible that such covariate
effects should all act so as to shift the entire distribution of test results by a fixed amount. It is of
obvious interest to know whether policy interventions alter performance of the strongest students in
the same way that weaker students are affected. Such questions are naturally investigated within
the quantile regression framework.
In ecology, theory often suggests how observable covariates affect limiting sustainable population
sizes, and quantile regression has been used to directly estimate models for upper quantiles of the
conditional distribution rather than inferring such relationships from models based on conditional
central tendency. In survival analysis, and event history analysis more generally, there is often also a
Version: October 25, 2000. This article has been prepared for the statistics section of the International Encyclo-
pedia of the Social Sciences edited by Stephen Fienberg and Jay Kadane. The research was partially supported by
NSF grant SBR-9617206.
1
2
ROGER KOENKER
desire to focus attention on particular segments of the conditional distribution, for example survival
prospects of the oldest-old, without the imposition of global distributional assumptions.
1. Quantiles, Ranks and Optimization
We say that a student scores at the th quantile of a standardized exam if he performs better than
the proportion , and worse than the proportion 1 , , of the reference group of students. Thus,
half of the students perform better than the median student, and half perform worse. Similarly, the
quartiles divide the population into four segments with equal proportions of the population in each
segment. The quintiles divide the population into 5 equal segments; the deciles into 10 equal parts.
The quantile, or percentile, refers to the general case.
More formally, any real valued random variable, Y , may be characterized by its distribution
function,
F y = Prob Y y
while for any 0
1,
Q = inffy : F y g
is called the th quantile of X. The median, Q 1=2 , plays the central role. Like the distribution
function, the quantile function provides a complete characterization of the random variable, Y.
The quantiles may be formulated as the solution to a simple optimization problem. For any
0
1, define the piecewise linear check function", u = u ,I u 0 illustrated in Figure
1.
τ
τ−1
ρτ (u)
Figure 1. Quantile Regression Function
Minimizing the expectation of Y , with respect to yields solutions, ^ , the smallest of
which is Q defined above.
The sample analogue of Q , based on a random sample, fy1; :::; yng, of Y 's, is called the th
sample quantile, and may be found by solving,
min
2R
n
X
i=1
yi , ;
QUANTILE REGRESSION
3
While it is more common to define the sample quantiles in terms of the order statistics, y 1
y 2 ::: y n , constituting a sorted rearrangement of the original sample, their formulation as a
minimization problem has the advantage that it yields a natural generalization of the quantiles to
the regression context.
Just as the idea of estimating the unconditional mean, viewed as the minimizer,
^ = argmin 2jR
X
yi , 2
can be extended to estimation of the linear conditional mean function E Y jX = x = x
0
by solving,
^ = argmin
2jRp
X
yi ,x
0
i 2;
the linear conditional quantile function, QY jX = x = x
0
i , can be estimated by solving,
^ = argmin
2jRp
X
yi ,x
0
:
The median case,
= 1=2, which is equivalent to minimizing the sum of absolute values of the
residuals has a long history. In the mid-18th century Boscovich proposed estimating a bivariate
linear model for the ellipticity of the earth by minimizing the sum of absolute values of residuals
subject to the condition that the mean residual took the value zero. Subsequent work by Laplace
characterized Boscovich's estimate of the slope parameter as a weighted median and derived its as-
ymptotic distribution. F.Y. Edgeworth seems to have been the first to suggest a general formulation
of median regression involving a multivariate vector of explanatory variables, a technique he called
the plural median". The extension to quantiles other than the median was introduced in Koenker
and Bassett 1978 .
2. An Example
To illustrate the approach we may consider an analysis of a simple first order autoregressive model
for maximum daily temperature in Melbourne, Australia. The data are taken from Hyndman,
Bashtannyk, and Grunwald 1996 . In Figure 2 we provide a scatter plot of 10 years of daily
temperature data: today's maximum daily temperature is plotted against yesterday's maximum.
Our first observation from the plot is that there is a strong tendency for data to cluster along the
dashed 45 degree line implying that with high probability today's maximum is near yesterday's
maximum. But closer examination of the plot reveals that this impression is based primarily on the
left side of the plot where the central tendency of the scatter follows the 45 degree line very closely.
On the right side, however, corresponding to summer conditions, the pattern is more complicated.
There, it appears that either there is another hot day, falling again along the 45 degree line, or
there is a dramatic cooling off. But a mild cooling off appears to be more rare. In the language of
conditional densities, if today is hot, tomorrow's temperature appears to be bimodal with one mode
roughly centered at today's maximum, and the other mode centered at about 20 .
Several estimated quantile regression curves have been superimposed on the scatterplot. Each
curve is specified as a linear B-spline. Under winter conditions these curves are bunched around
the 45 degree line, however in the summer it appears that the upper quantile curves are bunched
around the 45 degree line and around 20 . In the intermediate temperatures the spacing of the
quantile curves is somewhat greater indicating lower probability of this temperature range. This
impression is strengthened by considering a sequence of density plots based on the quantile regression
estimates. Given a family of reasonably densely spaced estimated conditional quantile functions,
it is straightforward to estimate the conditional density of the response at various values of the
conditioning covariate. In Figure 3 we illustrate this approach with several of density estimates
based on the Melbourne data. Conditioning on a low previous day temperature we see a nice
4
ROGER KOENKER
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yesterday’s max temperature
today’s max temperature
10
15
20
25
30
35
40
10
20
30
40
Figure 2. Melbourne Maximum Daily Temperature: The plot illustrates 10 years
of daily maximum temperature data in degrees centigrade for Melbourne, Aus-
tralia as an AR 1 scatterplot. The data is scattered around the dashed 45
degree line suggesting that today is roughly similar to yesterday. Superimposed
on the scatterplot are estimated conditional quantile functions for the quantiles
2 f:05; :10; :::; :95g. Note that when yesterday's temperature is high the spac-
ing between adjacent quantile curves is narrower around the 45 degree line and at
about 20 degrees Centigrade than it is in the intermediate region. This suggests
bimodality of the conditional density in the summer.
QUANTILE REGRESSION
5
today’s max temperature
density
10
12
14
16
0.05
0.10
0.15
Yesterday’s Temp 11
today’s max temperature
density
12 14 16 18 20 22 24
0.02
0.06
0.10
0.14
Yesterday’s Temp 16
today’s max temperature
density
15
20
25
30
0.02
0.06
0.10
Yesterday’s Temp 21
today’s max temperature
density
15
20
25
30
35
0.01
0.03
0.05
Yesterday’s Temp 25
today’s max temperature
density
20
25
30
35
40
0.01
0.03
0.05
Yesterday’s Temp 30
today’s max temperature
density
20
25
30
35
40
0.01
0.03
0.05
Yesterday’s Temp 35
Figure 3. Conditional density estimates of today's maximum temperature for sev-
eral values of yesterday's maximum temperature based on the Melbourne data:
These density estimates are based on a kernel smoothing of the conditional quantile
estimates as illustrated in the previous figure using 99 distinct quantiles. Note that
temperature is bimodal when yesterday was hot.
unimodal conditional density for the following day's maximum temperature, but as the previous
day's temperature increases we see a tendency for the lower tail to lengthen and eventually we see
a clearly bimodal density. In this example, the classical regression assumption that the covariates
affect only the location of the response distribution, but not its scale or shape, is violated.
3. Interpretation of Quantile Regression
Least squares estimation of mean regression models asks the question, How does the conditional
mean of Y depend on the covariates X?" Quantile regression asks this question at each quantile of the
conditional distribution enabling one to obtain a more complete description of how the conditional
distribution of Y given X = x depends on x. Rather than assuming that covariates shift only the
location or scale of the conditional distribution, quantile regression methods enable one to explore
potential effects on the shape of the distribution as well. Thus, for example, the effect of a job-
training program on the length of participants' current unemployment spell might be to lengthen the
shortest spells while dramatically reducing the probability of very long spells. The mean treatment
6
ROGER KOENKER
effect in such circumstances might be small, but the treatment effect on the shape of the distribution
of unemployment durations could, nevertheless, be quite significant.
3.1. Quantile Treatment E ects. The simplest formulation of quantile regression is the two-
sample treatment-control model. In place of the classical Fisherian experimental design model in
which the treatment induces a simple location shift of the response distribution, Lehmann 1974
proposed the following general model of treatment response:
Suppose the treatment adds the amount x when the response of the untreated
subject would be x. Then the distribution G of the treatment responses is that of the
random variable X + X where X is distributed according to F."
Special cases obviously include the location shift model, X = 0, and the scale shift model,
X = 0X, but the general case is natural within the quantile regression paradigm.
Doksum 1974 shows that if x is defined as the horizontal distance" between F and G at x,
so
F x = G x + x
then x is uniquely defined and can be expressed as
x = G
,1 F x ,x:
Changing variables so = F x one may define the quantile treatment e ect,
= F
,1 = G
,1 ,F
,1 :
In the two sample setting this quantity is naturally estimable by
^ = ^G,1
n , ^F,1
m
where Gn and Fm denote the empirical distribution functions of the treatment and control observa-
tions, based on n and m observations respectively.
Formulating the quantile regression model for the binary treatment problem as,
QYi jDi = + Di
where Di denotes the treatment indicator, with Di = 1 indicating treatment, Di = 0, control, then
the quantile treatment effect can be estimated by solving,
^ ; ^
0
= argmin
n
X
i=1
yi , , Di :
The solution ^ ; ^
0
yields ^ = ^F
,1
n , corresponding to the control sample, and
^ = ^G,1
n , ^F,1
n :
Doksum suggests that one may interpret control subjects in terms of a latent characteristic: for
example in survival analysis applications, a control subject may be called frail if he is prone to die at
an early age, and robust if he is prone to die at an advanced age. This latent characteristic is thus
implicitly indexed by , the quantile of the survival distribution at which the subject would appear
if untreated, i.e., YijDi = 0 = : And the treatment, under the Lehmann model, is assumed to
alter the subjects control response, , making it + under the treatment. If the latent
characteristic, say, the propensity for longevity, were observable ex ante, then one could view the
treatment effect as an explicit interaction with this observable variable. In the absence of such
QUANTILE REGRESSION
7
an observable variable however, the quantile treatment effect may be regarded as a natural measure
of the treatment response.
It may be noted that the quantile treatment effect is intimately tied to the two-sample QQ-plot
which has a long history as a graphical diagnostic device. The function ^ x = G
,1
n Fm x ,x is
exactly what is plotted in the traditional two sample QQ-plot. If F and G are identical then the
function G
,1
n Fm x will lie along the 45 degree line; if they differ only by a location scale shift,
then G
,1
n Fm x will lie along another line with intercept and slope determined by the location and
scale shift, respectively. Quantile regression may be seen as a means of extending the two-sample
QQ plot and related methods to general regression settings with continuous covariates.
When the treatment variable takes more than two values, the Lehmann-Doksum quantile treat-
ment effect requires only minor reinterpretation. If the treatment variable is continuous as, for
example, in dose-response studies, then it is natural to consider the assumption that its effect is
linear, and write,
QYi jxi = + xi:
We assume thereby that the treatment effect, , of changing x from x0 to x0+1 is the same as the
treatment effect of an alteration of x from x1 to x1+1: Note that this notion of the quantile treatment
effect measures, for each , the change in the response required to stay on the th conditional quantile
function.
3.2. Transformation Equivariance of Quantile Regression. An important property of the
quantile regression model is that, for any monotone function, h ,
Qh T jx = h QT jx :
This follows immediately from observing that
Prob T tjx = Prob h T h t jx :
This equivariance to monotone transformations of the conditional quantile function is a crucial
feature, allowing one to decouple the potentially conflicting objectives of transformations of the re-
sponse variable. This equivariance property is in direct contrast to the inherent conflicts in estimating
transformation models for conditional mean relationships. Since, in general, E h T jx 6= h E Tjx
the transformation alters in a fundamental way what is being estimated in ordinary least squares
regression.
A particularly important application of this equivariance result, and one that has proven extremely
influential in the econometric application of quantile regression, involves censoring of the observed
response variable. The simplest model of censoring may be formulated as follows. Let y
i denote a
latent unobservable response assumed to be generated from the linear model
y
i = x
0
i + ui i = 1;::: ;n
with fuig iid from distribution function F. Due to censoring, the y
i 's are not observed directly, but
instead one observe
yi = maxf0; y
i g:
Powell 1986 noted that the equivariance of the quantiles to monotone transformations implied
that in this model the conditional quantile functions of the response depended only on the censoring
point, but were independent of F. Formally, the th conditional quantile function of the observed
response, yi; in this model may be expressed as
Qi jxi = maxf0; x
0
i + F
,1
u g
8
ROGER KOENKER
The parameters of the conditional quantile functions may now be estimated by solving
min
b
n
X
i=1
yi ,maxf0; x
0
ibg
where it is assumed that the design vectors xi, contain an intercept to absorb the additive effect of
F
,1
u : This model is computationally somewhat more demanding than conventional linear quantile
regression because it is non-linear in parameters.
3.3. Robustness. Robustness to distributional assumptions is an important consideration through-
out statistics, so it is important to emphasize that quantile regression inherits certain robustness
properties of the ordinary sample quantiles. The estimates and the associated inference apparatus
have an inherent distribution-free character since quantile estimation is influenced only by the local
behavior of the conditional distribution of the response near the specified quantile. Given a solution
^ , based on observations, fy; Xg, as long as one doesn't alter the sign of the residuals, any of
the y observations may be arbitrary altered without altering the initial solution. Only the signs of
the residuals matter in determining the quantile regression estimates, and thus outlying responses
influence the fit in so far as they are either above or below the fitted hyperplane, but how far above
or below is irrelevant.
While quantile regression estimates are inherently robust to contamination of the response ob-
servations, they can be quite sensitive to contamination of the design observations, fxig. Several
proposals have been made to ameliorate this effect.
4. Computational Aspects of Quantile Regression
Although it was recognized by a number of early authors, including Gauss, that solutions to
the median regression problem were characterized by an exact fit through p sample observations
when p linear parameters are estimated, no effective algorithm arose until the development of linear
programming in the 1940's. It was then quickly recognized that the median regression problem could
be formulated as a linear program, and the simplex method employed to solve it. The algorithm
of Barrodale and Roberts 1973 provided the first e cient implementation specifically designed for
median regression and is still widely used in statistical software. It can be concisely described as
follows. At each step, we have a trial set of p basic observations" whose exact fit may constitute a
solution. We compute the directional derivative of the objective function in each of the 2p directions
that correspond to removing one of the current basic observations, and taking either a positive
or negative step. If none of these directional derivatives are negative the solution has been found,
otherwise one chooses the most negative, the direction of steapest descent, and goes in that direction
until the objective function ceases to decrease. This one dimensional search can be formulated as
a problem of finding the solution to a scalar weighted quantile problem. Having chosen the step
length, we have in effect determined a new observation to enter the basic set, a simplex pivot occurs
to update the current solution, and the iteration continues.
This modified simplex strategy is highly effective on problems with a modest number of obser-
vations, achieving speeds comparable to the corresponding least squares solutions. But for larger
problems with, say n 100;000 observations, the simplex approach eventually becomes considerably
slower than least squares. For large problems recent development of interior point methods for lin-
ear programming problems are highly effective. Portnoy and Koenker 1997 describe an approach
that combines some statistical preprocessing with interior point methods and achieves comparable
performance to least squares solutions even in very large problems.
QUANTILE REGRESSION
9
An important feature of the linear programming formulation of quantile regression is that the
entire range of solutions for
2 0;1 can be e ciently computed by parametric programming.
At any solution ^ 0 there is an interval of 's over which this solution remains optimal, it is
straightforward to compute the endpoints of this interval, and thus one can solve iteratively for the
entire sample path ^ by making one simplex pivot at each of the endpoints of these intervals.
5. Statistical Inference for Quantile Regression
The asymptotic behavior of the quantile regression process f^ : 2 0;1 gclosely parallels the
theory of ordinary sample quantiles in the one sample problem. Koenker and Bassett 1978 show
that in the classical linear model,
yi = xi + ui
with ui iid from dfF; with density f u 0 on its support fuj0 F u 1g, the joint distribution
of
p
n ^n i , i m
i,1 is asymptotically normal with mean 0 and covariance matrix D
,1.
Here = + F
,1
u e1; e1 = 1;0;::: ;0
0
; x1i 1; n
,1 P
xix
0
i ! D; a positive definite matrix,
and
= !ij = minf i; jg , i j = f F
,1 i f F
,1 j :
When the response is conditionally independent over i, but not identically distributed, the as-
ymptotic covariance matrix of =
p
n ^ , is somewhat more complicated. Let
i = xi
denote the conditional quantile function of y given xi, and fi the corresponding conditional density,
and define,
Jn 1; 2 = minf 1; 2g , 1 2 n
,1
n
X
i=1
xix
0
i;
and
Hn = n
,1 X
xix
0
ifi i :
Under mild regularity conditions on the ffig's and fxig's, we have joint asymptotic normality for
vectors i ;::: ; m with mean zero and covariance matrix
Vn = Hn i
,1Jn i; j Hn j
,1 m
i=1:
An important link to the classical theory of rank tests was made by Gutenbrunner and Jure ckov
a
1992 , who showed that the rankscore functions of H
ajek and Sid
ak 1967 could be viewed as a
special case of a more general formulation for the linear quantile regression model. The formal dual
of the quantile regression linear programming problem may be expressed as,
maxfy
0
ajX
0
a = 1 ,t X
0
1; a 2 0;1ng:
The dual solution ^a reduces to the H
ajek and Sid
ak rankscores process when the design matrix,
X, takes the simple form of an n vector of ones. The regression rankscore process ^a behaves
asymptotically much like the classical univariate rankscore process, and thus offers a way to extend
many rank based inference procedures to the more general regression context.
10
ROGER KOENKER
6. Extensions and Future Developments
There is considerable scope for further development of quantile regression methods. Applications
to survival analysis and time-series modeling seem particularly attractive, where censoring and
recursive estimation pose, respectively, interesting challenges. For the classical latent variable form
of the binary response model where,
yi = I x
0
i + ui 0
and the median of ui conditional on xi is assumed to be zero for all i = 1; :::; n, Manski 1975
proposed an estimator solving,
max
jjbjj=1
X
yi ,1=2 I x
0
ib 0 :
This maximum score" estimator can be viewed as a median version of the general linear quantile
regression estimator for binary response,
min
jjbjj=1
X
yi ,I x
0
ib 0 :
In this formulation it is possible to estimate a family of quantile regression models and explore,
semi-parametrically, a full range of linear conditional quantile functions for the latent variable form
of the binary response model.
Koenker and Machado 1999 introduce inference methods closely related to classical goodness of
fit statistics based on the full quantile regression process. There have been several proposals dealing
with generalizations of quantile regression to nonparametric response functions involving both local
polynomial methods and splines. Extension of quantile regression methods to multivariate response
models is a particularly important challenge.
7. Conclusion
Classical least squares regression may be viewed as a natural way of extending the idea of esti-
mating an unconditional mean parameter to the problem of estimating conditional mean functions;
the crucial step is the formulation of an optimization problem that encompasses both problems.
Likewise, quantile regression offers an extension of univariate quantile estimation to estimation of
conditional quantile functions via an optimization of a piecewise linear objective function in the resid-
uals. Median regression minimizes the sum of absolute residuals, an idea introduced by Boscovich
in the 18th century.
The asymptotic theory of quantile regression closely parallels the theory of the univariate sample
quantiles; computation of quantile regression estimators may be formulated as a linear programming
problem and e ciently solved by simplex or barrier methods. A close link to rank based inference
has been forged from the theory of the dual regression quantile process, or regression rankscore
process.
Recent non-technical introductions to quantile regression are provided by Buchinsky 1998 and
Koenker 2001 . A more complete introduction will be provided in the forthcoming monograph
of Koenker 2002 . Most of the major statistical computing languages now include some capa-
bilities for quantile regression estimation and inference. Quantile regression packages are avail-
able for R and Splus from the R archives at http: lib.stat.cmu.edu R CRAN and Statlib at
http: lib.stat.cmu.edu S, respectively. Stata's central core provides quantile regression esti-
mation and inference functions. SAS offers some, rather limited, facilities for quantile regression.
QUANTILE REGRESSION
11
REFERENCES
Barrodale, I. and F.D.K. Roberts 1974 . Solution of an overdetermined system of equations
in the `1 norm, Communications ACM, 17, 319-320.
Buchinsky, M., 1998 , Recent Advances in Quantile Regression Models: A practical guide for
empirical research, J.Human Resources, 33, 88-126,
Doksum, K. 1974 Empirical probability plots and statistical inference for nonlinear models in the
two sample case, Annals of Statistics, 2, 267-77.
Gutenbrunner, C. and J. Jure ckov
a 1992 . Regression quantile and regression rank score
process in the linear model and derived statistics, Annals of Statistics 20, 305-330.
H
ajek , J. and Z. Sid
ak 1967 . Theory of Rank Tests, Academia: Prague.
Hyndman, R.J., D.M Bashtannyk, and G.K. Grunwald, 1996 Estimating and Visualizing
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Koenker, R. and G. Bassett 1978 . Regression quantiles, Econometrica, 46, 33-50.
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Koenker, R. 2002 . Quantile Regression, forthcoming.
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cisco.
Manski, C. 1985 Semiparametric analysis of discrete response: asymptotic properties of the
maximum score estimator, J. of Econometrics, 27, 313-34.
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Powell, J.L. 1986 Censored regression quantiles, J. of Econometrics, 32, 143-55.
