Emerging Trends in The Social and Behavioral Sciences

Quantile Regression Methods

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Quantile Regression Methods

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Quantile Regression Methods
BERND FITZENBERGER and RALF ANDREAS WILKE

Abstract
Quantile regression is emerging as a popular statistical approach, which complements the estimation of conditional mean models. While the latter only focuses on
one aspect of the conditional distribution of the dependent variable, the mean, quantile regression provides more detailed insights by modeling conditional quantiles.
Quantile regression can therefore detect whether the partial effect of a regressor on
the conditional quantiles is the same for all quantiles or differs across quantiles.
Quantile regression can provide evidence for a statistical relationship between two
variables even if the mean regression model does not.
We provide a short informal introduction into the principle of quantile regression
which includes an illustrative application from empirical labor market research. This
is followed by briefly sketching the underlying statistical model for linear quantile
regression based on a cross-section sample. We summarize various important
extensions of the model including the nonlinear quantile regression model, censored
quantile regression, and quantile regression for time-series data. We also discuss
a number of more recent extensions of the quantile regression model to censored
data, duration data, and endogeneity, and we describe how quantile regression can
be used for decomposition analysis. Finally, we identify several key issues, which
should be addressed by future research, and we provide an overview of quantile
regression implementations in major statistics software. Our treatment of the topic
is based on the perspective of applied researchers using quantile regression in their
empirical work.

INTRODUCTION
We consider the linear regression model
yi = xi ? + ui ,
with observations i = 1, … , n and xi = (1, x2i , … , xKi ) is 1xK, includes a constant, and ? is Kx1.y and the regressors (covariates) x are observed, the error
term u is not observed, and ? is to be estimated. The error term is assumed to
be zero in expectation given any value of the covariates, and it is independent
of the covariates. The common approach to estimate the parameters of such a
model is ordinary least squares (OLS), which estimates the conditional mean
Emerging Trends in the Social and Behavioral Sciences. Edited by Robert Scott and Stephen Kosslyn.
© 2015 John Wiley & Sons, Inc. ISBN 978-1-118-90077-2.

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EMERGING TRENDS IN THE SOCIAL AND BEHAVIORAL SCIENCES

function E(y|x). This is the average value of y given the observed covariates.
A single parameter ? k is therefore informative about the partial relationship
between the covariate xk (with some abuse of notation) and the average value
of y holding all other covariates constant. It is therefore an estimate of the
average effect in the population from which the observations i are randomly
sampled.
Quantile regression (QR) follows a somewhat different approach. Instead
of estimating the average population effect, it estimates the effect at conditional quantiles of y given x (y and x are random variables with observations
yi and xi ). This is the conditional quantile function
qy|x (?) ∶= x?? ,
for quantile ? ∈ (0, 1). Similar to OLS, the linear QR model assumes that the
conditional quantile function is linear in the parameters ? ? , but the parameters can vary in ?. A single parameter ? ?k is the change in the conditional
quantile of y in response to a 1 unit increase in xk holding all other covariates
constant. If we consider ? = 0.5, it is the change in the conditional median of y
given x due to a 1 unit increase in xk . QR is thus more informative than mean
regression models as it considers the entire distribution of the dependent
variable. Of course, there is nothing wrong with conditional mean models.
However, they only focus on one feature of the conditional distribution as a
function of covariates. As a matter of fact if the effect varies across quantiles
and even changes its sign, the mean model may suggest no effect of a covariate on the mean, but the QR would reveal a more complete picture with nonconstant effects across quantiles. Having not observed an effect on the mean
may lead applied researchers to the conclusion that the variable does not play
a role in the model, but this may not be true. A prominent example for application of QR in social sciences is wage regression where individual wages are
explained by a number of covariates. When QR is applied to these models, it
allows the effect of covariates to vary across quantiles. For example, an additional year of education may well have a different effect on lower (? small)
and higher (? large) quantiles of the conditional wage distribution.
As an example, we estimate a wage equation with the log(wage) as the
dependent variable and a number of independent variables with a sample of
369,389 full-time working employees in Germany in June 2004. The sample is
an extract from German administrative labor market data (IAB Employment
Sample 2004).
Figure 1 shows how the estimated QR coefficients vary across quantiles
and how they relate to the OLS estimates (dashed line). For example, QR
estimates the conditional quantile function of females being 40% lower than
that of males at the first decile (quantile ? = 0.1), while being only 15% lower
at the ninth decile (quantile ? = 0.9). The OLS estimate suggests that the

0.40

0.50
0.2

0.4
0.6
Quantile

0.8

0.4
0.6
Quantile

0.8

1

1 = If unemployed in past
0

0.2

0

0.2

0.4
0.6
Quantile

0.8

1

0

0.2

0.4
0.6
Quantile

0.8

1

(b)

0.00 0.01 0.02 0.03 0.04

Tenure (in years)

0.30

1 = University degree

0.20
0

(a)

(c)

3

1

−0.15 −0.10 −0.05 0.00 0.05

Gender (1 = female)

−0.50 −0.40 −0.30 −0.20 −0.10

Quantile Regression Methods

(d)

Figure 1 Estimated QR coefficients (solid line) with 95% confidence intervals
(gray area). It also contains the estimated coefficient of the mean OLS regression
(dashed line).

average wage for a female is around 25% lower than that for a male with
the same other characteristics. The results therefore suggest that gender
differences in wages may be smaller in higher paid jobs, in contrast to the
so-called glass ceiling hypothesis. However, these results should not be
overinterpreted because of the restrictive set of covariates used.
Roger Koenker, who is the key contributor to the foundational research
of QR, has written several seminal articles, his widely used 2005 text book
“Quantile Regression”, and various surveys on QR methods. His textbook
provides all key references up to the year 2004. Together with coauthors,
Roger Koenker has contributed many computational resources to the
open-source statistical package R. His work provides formal presentations
of the material, detailed examples, and an introduction to computer code.
FOUNDATIONAL RESEARCH
Roger Koenker and Gilbert Bassett introduced the linear QR model in 1978
in their seminal article in Econometrica as a generalization of the estimation
of an empirical sample quantile.

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EMERGING TRENDS IN THE SOCIAL AND BEHAVIORAL SCIENCES

In a sample of n observations {y1 , … , yn }, the ? − quantile [? ∈ (0, 1)] of y is
that value qy (?) for which at most a share of (1 − ?) ⋅ 100 % of the observations
lie above that value and at most a share of ? ⋅ 100 % of the observations lie
below that value; Thus qy (?) cuts the observations into the lowest ? ⋅ 100 %
of the observations and the highest (1 − ?) ⋅ 100 % of the observations. For
instance, the median corresponds to ? = 0.5, the first decile to ? = 0.1, and the
ninth decile to ? = 0.9. Quantiles are an alternative form to represent the distribution of a statistical variable, such that Fy (qy (?)) := ?, where Fy (.) is the
distribution function with Fy (y) : = P(Y ≤ y). Quantiles [formally the quantile
process as a function of ? ∈ (0, 1)] represent the possible nonunique inversion
of the distribution function.
The main insight to introduce QR is that the determination of an empirical
quantile qy (?) can be viewed as the outcome of the following minimization
exercise:
]
[
∑
∑
|yi − q| + (1 − ?)
|yi − q| .
qy (?) ∶= arg minq ?
i ∶yi > q

i ∶yi <q

Implicit in this formulation is the focus on absolute differences from the
location parameter q and the asymmetric weighting with ? and (1 − ?),
depending on whether an observation lies above or below q. This is the
famous check function depicted in Figure 2.
For linear QR, we make q a function of covariates xi and model the conditional quantile of the response variable y, given xi as a linear function of xi ? ? .
Thus, the determination of the linear QR amounts to the following minimization exercise:
[
]
∑
∑
|yi − xi ?| + (1 − ?)
|yi − xi ?| .
?? ∶= arg min? ?
i ∶yi > xi ?

i ∶yi < xi ?

Linear QR coefficients describe the change in the conditional quantile of
the response variable when a covariate changes by 1 unit. Analogous to a
sample quantile, the implied regression relationship is such that at most a
share of (1 − ?) ⋅ 100 % of the observations lie above the regression line and
at most a share of ? ⋅ 100 % of the observations lie below. The calculation of
the regression coefficients corresponds to a linear program, which implies
many properties of linear QRs. For instance, if the matrix of covariates has full
rank K, then there will be at least K observations i with linearly independent
vectors of covariates xi such that the deviation from the regression line is
exactly zero (yi = xi ? ? ). This is the so-called interpolation property.
The fact that the minimization problem cannot be solved by simple calculus
methods is no restriction to today’s computer power. To calculate linear QRs,
effective algorithms based on refinements of the simplex method to solve

Quantile Regression Methods

5

Check function
1.6
τ = 0.2
τ = 0.5
τ = 0.7

1.4

1.2

ρ

1

0.8

0.6

0.4

0.2

0
−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

a

Figure 2 Check function.

linear programs are available. Interior point methods together with preprocessing of the data provide effective alternatives for very large data sets.
The asymptotic variance–covariance matrix is in fact very similar to OLS
regression, provided the response variable follows a continuous distribution
around the true conditional quantiles of interest. Instead of the variance of the
error term (as in the OLS case), the asymptotic variance–covariance matrix
involves the density of the response variable at the conditional population
quantile. The statistical theory of QR also provides the joint distribution of
the coefficient estimates at different quantiles, where the covariance matrix
across quantiles has a similar structure as the variance–covariance matrix
at the individual quantiles. Assuming a constant conditional distribution of
the response variable around the conditional quantile allows one to estimate
constant conditional densities based on the estimated residuals around the
conditional QR (excluding the exact zeroes resulting from the interpolation
property). In the heteroscedasticity case (which is the case when QR is interesting, see next paragraph), it would be necessary to use observation-specific
density estimates, which would be computationally difficult and cumbersome. In his 2005 textbook, Roger Koenker discusses a simple and elegant
alternative based on estimating QR slightly above and slightly below the

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EMERGING TRENDS IN THE SOCIAL AND BEHAVIORAL SCIENCES

quantile ? of interest, and then uses the implied conditional quantiles
to obtain local density estimates. In practice, many researchers resort to
bootstrapping methods to obtain asymptotically heteroscedasticity robust
inference. A pairwise bootstrap of the estimation of QR at different quantiles
(one estimates the QR at all quantiles of interest for the same resample) automatically provides estimates of the covariance across the different quantiles.
Estimating linear QRs for various values of ? provides a parsimonious picture of how the conditional distribution of the response variable changes
with the covariates (see earlier example). In fact, due to the linear programming structure of the estimation problem, it is straightforward to calculate
the entire process of QRs as a function of ? in a given sample, because the QR
only change at finitely many values of ?. However, due to the combinatorical complexity of the problem, researchers rarely calculate the entire process
in practice. Instead, they report the QR coefficients at selected equispaced
quantiles, for example, for each decile (? = 0.1, 0.2, … , 0.9) or each percentile
(? = 0.01, 0.02, 0.03, … , 0.99). When slope coefficients change across quantiles
(which can be investigated by means of standard Wald tests), this is an indication of heteroscedasticity, that is, the conditional dispersion of the response
variable changes with the covariates. In many applications (e.g., the effect of
unions on wages), one would expect that the effect of a covariate changes
along the conditional distribution (e.g., the effect of unions may be stronger
in the lower part of the distribution than in the upper part of the distribution),
that is, heteroscedasticity (changing dispersion) is a meaningful finding (e.g.,
unions reduce the dispersion of wages by increasing wages in the lower part
of the distribution more strongly than in the upper part of the distribution).
An important advantage of QR compared with OLS regression relates to
the equivariance property of quantiles under strictly monotone transformations, for which the ?th quantile of the values of the function corresponds
to the function value evaluated at the ?th quantile of the original value. For
instance, if we know that the conditional median of the logarithm of wages
is a linear function of the covariates, we know also that the median of wages
in levels corresponds to the exponential function applied to this linear function, thus modeling log wages entails modeling wages in levels. It is well
known that this is not the case for OLS regression, because the expected value
of wages is not equal to the exponential function applied to the expected
value of log wages. The equivariance property of quantiles allows for more
general strictly monotonous functions, such as the Box–Cox transformation
defined for positive responses. However, computational issues may arise,
because the inverse of the Box–Cox transformation may not be strictly positive for all data points, as pointed out by the authors and Xuan Zhang in
2010.

Quantile Regression Methods

7

An apparent limitation of linear QR is the fact that nonparallel regression
lines at different quantiles are bound to cross somewhere, thus for some
values of the covariates, the predicted values at a higher quantile (e.g., the
median ? = 0.5) lies below the predicted value at a lower quantile (e.g., for
? = 0.49). At the average values of the covariates, the ordering of predicted
quantiles is preserved. One can use a sizeable incidence of quantile crossing
among the observed values of the covariates as an indication for the need
to respecify the model in a more flexible way, for example, by introducing
nonlinear terms or nonparametric components as covariates. Holger Dette
and Stanislav Volgushev (Victor Chernozhukov, Ivan Fernandez-Val, and
Alfred Galichon (2010) have also written on this issue) discuss rearrangement methods (these involve smoothing of the estimates building on
isotone regression techniques) to impose monotonicity of the predicted
quantiles (not necessarily for coefficient estimates, for which the problem
cannot be resolved). However, these methods may not resolve a problem of
misspecification. It has been our experience that quantile crossing should
be used as a guidance for misspecification of the model and that quantile
crossing is often not a serious problem, if one allows for a sufficiently flexible
specification (see section titled “Nonlinear Models”).
Table 1 in the appendix provides an overview of the implementation of
QR in various statistical software packages. The “quantreg” package in R,
developed by Roger Koenker, is most important for the dissemination of
state-of-the-art QR techniques.
Table 1
Summary of Functionality of Major Statistical Packages
for QR Analysis
Statistical package
TSP

Method

R

STATA

Linear QR

• (quantreg)

• (qreg, sqreg)

• (lad with option)

Nonlinear QR
Nonparametric QR
Censored QR
Bootstrap for QR

• (quantreg)
• (quantreg)
• (quantreg)
• (quantreg)

—
—
• (lad with option)
• (lad with option)

IV QR
Decomposition

—
—

—
—

• (Hansen)
—

Panel QR
Competing Risks QR

• (rqpd)
cmprskQR

—
—
• (clad, cqiv)
• (sqreg with
option)
• (cqiv, Hansen)
• (various ado
files)
—
—

• (Hunter,
quantreg)
• (Hunter)
—
—
• (quantreg)

—
—

—
—

•, implemented (requires additional package/command).

Matlab

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EMERGING TRENDS IN THE SOCIAL AND BEHAVIORAL SCIENCES

CUTTING-EDGE RESEARCH
QR is nowadays applied to a variety of more advanced models for
cross-section, time-series, and panel data. Extensions of linear QR have been
developed since the mid-1980, but most of this research was conducted
after the year 2000 and it is still gradually developing. A broader process
of knowledge transfer from method-based research into broader applied
research in economics, and other social sciences did not start before the
year 2005 but since then it is increasing in pace and still picking up. Being
economists, it seems to us that empirical research in biostatistics has started
to pick up QR methods, in particular in survival analysis taking account
of censoring. Here, we present an overview of important model extensions
and fields of applications of more advanced QR models. Roger Koenker
contributed to some of these extensions, and the references given to his work
cover most of these extensions.
NONLINEAR MODELS
The linear functional form of the conditional quantile function might be
too restrictive in an application. Roger Koenker outlines the estimation of
nonlinear quantile functions in his textbook. The QR model can be extended
to allow for nonlinear relationships, that is, qy|x (?) = g? (x) with g? being
some nonlinear function that satisfies some regularity conditions. In the
abovementioned minimization exercise, g? (x) replaces the predicted values
xi ? ? . Nonlinear, strictly monotonous transformations (such as the logarithm
or a Box–Cox transformation) of the response variable to achieve a linear QR
(as discussed above) are a special case. The parametric, nonlinear QR model
can be applied if g? is known subject to some unknown parameters. This is
analogous to nonlinear least squares regression.
A nonparametric QR on a set of continuous covariates can be applied if g? is
unknown. A local smoothing approach such as kernel smoothing can be used
to locally estimate the unknown g? , the estimates being subject to the curse
of dimensionality. (The convergence rate of the estimator becomes slower
when the number of covariates increases.) Estimating a weighted linear QR
just on an intercept (the residuals are weighted by kernel weights) for continuous covariates produces the QR alternative of a Nadaraya–Watson (local
constant) kernel regression. Local linear (local polynomial) QR can be estimated using a local linear (local polynomial) approximation in the weighted
QR. Note that using a local constant kernel regression with the same bandwidth for all quantiles resolves by construction the quantile crossing problem. The literature also involves semiparametric specifications with additive
nonparametric components estimated using backfitting techniques, involving an iterative procedure. During an iteration step, each component of the

Quantile Regression Methods

9

QR specification is estimated recursively based on the previous estimates of
all other components.
CENSORED QUANTILE REGRESSION
The QR method can also be applied to a censored regression. Here, the conditional quantile function for y corresponds either to the censoring value or it is
linear in the covariates. A prominent example from social sciences involves
labor supply, which is either zero or positive and many individuals supply
zero hours of work. Another example involves health expenditures, which
are either zero or positive. Models for such response variables can be estimated by censored quantile regressions (CQR). The interpretation of estimation results regarding the observed censored values requires the computation
of partial effects accounting for censoring.
We first consider the simple case of right censoring (e.g., top coding of
wages or right censoring of durations of ongoing spells), where for censored
observations we only know that the statistical variable of interest exceeds a
certain known threshold. If this threshold is constant for all observations and
the ?th quantile of the censored observations lies below the threshold, we
know that the ?th quantile of the censored observations corresponds to the
?th quantile of the uncensored observations. Here, the censored observations
correspond to the actual values of the variable of interest, if they are not censored, and to the censoring value, if they are censored, that is, yi = min(y∗i , c)
where yi is the observed censored value, y∗i is the true uncensored value, and
c is the censoring threshold.
Roger Koenker discusses three approaches to CQR in his 2008 article.
The first approach developed by James Powell (1986) involves the case of
fixed censoring where the censoring values may vary across observations,
but it is known for all observations. The estimator replicates the censoring
mechanism in the regression specification for the censored observation as
g? (xi ) = min(xi ? ? , ci ), where ci is the observation-specific, known censoring
value. This is a special case of nonlinear QR. The Powell estimator provides a semiparametric alternative to the standard Tobit estimator for the
censored regression model, which relies on the assumption of a normally
distributed error term and which is inconsistent under heteroscedasticity.
However, the calculation of the Powell estimator is difficult when there
is a lot of censoring. Various modifications of the estimator have been
suggested in the literature to overcome these difficulties. Two appealing
approaches have been suggested by Steve Portnoy (2003) and by Limin Peng
and Yijian Huang (2008). These involve regression versions of nonparametric estimators of distribution functions under independent censoring
(Kaplan–Meier and Nelson–Aalen estimator), where the censoring values

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EMERGING TRENDS IN THE SOCIAL AND BEHAVIORAL SCIENCES

are only known for those observations that are censored (the case of random
censoring).
DURATION MODELS
The response variable in duration analysis (or survival analysis) is time until
an event or failure occurs (single-risk model). QR is an attractive approach
to analyze the distribution of a duration as it can allow for different effects of
covariates on lower and higher quantiles of the conditional distribution. For
example, when the response variable is the duration of unemployment and
we want to study the effect of a training programme on the job-taking time of
unemployed people, we would expect that such a training programme at the
first instance increases shorter unemployment periods, because the unemployed are locked into the programme. For longer unemployment periods,
we would hope to find a shortening effect of the training programme once
the training is completed. Thus, it is conceivable that even the sign of the
estimated coefficient of training varies across quantiles. Standard parametric
and semiparametric duration models such as proportional hazard models
do not possess this degree of flexibility as they typically model the effect of
a covariate by one single parameter. While linear QR can be directly applied
to duration data, these data are often characterized by being censored. In
the presence of censoring a CQR can be estimated. Roger Koenker did some
pioneering work on QR for duration models, which is reviewed in his 2005
textbook, and his 2008 article on CQR has a focus on applications in duration
analysis. Our 2006 survey discusses the usefulness and the limitations of QR
for duration analysis in the presence of independent censoring.
TIME-SERIES MODELS
QR is also becoming increasingly popular for the empirical analysis with
time-series data. Roger Koenker and Zhijie Xiao consider a class of quantile
autoregression models where the covariates involve the lags of the response
variable in discrete time. Such QR time-series models allow for a systematic
influence of the lagged dependent variable on the location, scale, and shape
of the distribution of the response variable. For the analysis of univariate time
series, the models include the autoregressive model (both stationary processes and processes with unit roots) and the autoregressive conditional heteroscedasticity (ARCH) model. Such models allow for asymmetric dynamics
and local persistence in time series and thus may bridge the gap between stationary and integrated time-series processes. Such models have been applied
to macroeconomic time-series data and financial data. A standard generalized autoregressive conditional heteroscedasticity (GARCH) model implies

Quantile Regression Methods

11

a symmetric persistence in the second moment of a time series, irrespective of the direction of change. A quantile autoregression model allows for
asymmetric dynamics, implying different responses in the conditional scale
depending on whether there was a strong downside or upside movement
of the response variable in the recent past. Downside movements or upside
movements in the recent past may involve different degrees of persistence
(for instance, unit root behavior may only exist in some part of the conditional
distribution of the response variable). Such effects may prove important in
the analysis of financial data. Extensions of quantile autoregression models to
the analysis of multivariate time series are possible, including the estimation
of quantile vector autoregressions and quantile co-integrating regressions.
KEY ISSUES FOR FURTHER RESEARCH
We discuss some pertinent issues for further research that are related to
our own research. Due to space constraints, the choice has to be somewhat
eclectic.
CENSORED QUANTILE REGRESSION
We raise four issues: First, Roger Koenker describes the major computational
difficulties involved and discusses some practical solutions (see above).
Somewhat practical approaches exist for the CQR model under fixed and
random censoring, although there does not seem to exist a consensus in the
literature on what works best (correspondingly, popular software package
use different algorithms to calculate CQR, often with little justification
of the particular choice). Second, it should be noted that identification of
CQR is a tenuous issue, because the CQR line involves an extrapolation
based on functional form assumptions into the censored part of the data. A
more substantive analysis of this issue would be useful. Third, the random
censoring case assumes that the censoring values are independent of the
response variable (at least conditional on the covariates). CQR models for
random censoring are therefore not applicable to all empirical problems.
While there has been some work for CQR if observations are dependent,
there is still a gap to accommodate various forms of dependent censoring
in the CQR model. Existing studies impose stringent assumptions on the
regression models and are plagued by high computational costs due to
multiple step algorithms. Fourth, two-limit CQR allowing for censoring
both from above and below (in an analogy to the two-limit tobit model) is a
straightforward extension under fixed censoring, and similar algorithms can
be used. Under random censoring, at most one of the two censoring points
is observed (and neither one is observed for uncensored observations) and

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EMERGING TRENDS IN THE SOCIAL AND BEHAVIORAL SCIENCES

the rationale for the algorithms suggested by Steve Portnoy (2003) and by
Limin Peng and Yijian Huang (2008) breaks down (see above).
DURATION MODELS
So far, QR models for duration data have been mainly used for single-risk
models with independent right censoring. Here, the QR model estimates
the conditional quantile functions of the duration distribution conditional
on a time-invariant set of covariates. We think that there are a number of
extensions still to be analyzed. These include multiple spell QR duration
models (this issue is related to panel data applications, see in the following
section) and QR models that explicitly allow for unobserved heterogeneity
and time-varying covariates. Duration analysis based on modeling hazard
rates can take these issues into account, typically under the restrictive
proportional hazard assumption. Regarding the last point, QR can condition
on a predetermined time path of the covariates, but the analysis may quickly
involve a large number of coefficients to be estimated. Although applications
of QR for competing risks models with possibly dependent competing risks
do exist, the applications cannot estimate the duration distributions of the
separate competing risks (transitions to a certain destination state) but
rather estimate the cumulative incidence function of the separate risks. The
cumulative incidence—the duration of observed transitions into a certain
destination state—is a common descriptive tool (especially in Biostatistics),
but in the absence of knowledge about the dependence structure between
risks, it is difficult to infer the duration distributions of the competing risks,
which are often objects of prime interest in the social sciences.
DECOMPOSITION ANALYSIS AND UNCONDITIONAL QR
For a long time, a major restriction of QR in applied research was that
conditional quantiles do not aggregate directly to the unconditional distribution of the response variable. While the overall mean is the weighted
average of cell means (or the fitted value of an OLS regression at the sample
mean of the covariates), it is not possible to calculate the aggregate as
?th quantile cannot be calculated based on the conditional ?th quantiles.
Analogously, the estimated QR coefficient at a certain quantile does not
reflect the effect of a uniform shift of one covariate by 1 unit on the aggregate
quantile. Relating conditional and unconditional quantiles is important for
so-called Blinder–Oaxaca type decomposition analysis. An example is the
analysis of gender differences in the wage distribution, which may be due
to gender differences in the distribution of covariates (characteristics) and
gender differences in the QR coefficients. To distinguish the two effects,

Quantile Regression Methods

13

researchers estimate counterfactual wage distributions (e.g., the wage distribution resulting if females exhibited male characteristics and still female
coefficients applied). This counterfactual distribution can be calculated by
determining the conditional distribution function of the response variable
based on the full process of QR coefficients, explicitly aggregating the
conditional distribution functions (by the law of total probability) and then
inverting the aggregate distribution function. This approach is described
by Victor Chernozhukov, Ivan Fernandez-Val, and Blaise Melly, who also
suggest its extension to sequential decompositions of the contribution
of individual covariates to the differences in aggregate distributions. A
simple alternative approach for decomposition analysis would make use of
inverse probability weighting to calculate counterfactual wage distributions
while balancing characteristics between males and females. This approach
is very important in the literature on (unconditional) quantile treatment
effects under the conditional independence assumption. A third alternative
denoted as unconditional QR is introduced by Serjo Firpo, Nicole Fortin,
and Thomas Lemieux. The basic idea is to estimate a discrete choice model
of whether the individual response does not exceed the aggregate ?th
quantile. Based on the fitted probabilities and the density of the aggregate
distribution at the aggregate ?th quantile, it is possible to study the effect
of uniform shifts in covariates on the aggregate ?th quantile. The approach
can be used for decomposition analysis. Estimating a linear probability
model allows to implement a sequential decomposition. The literature is
missing a comprehensive comparison of the different approaches to estimate
counterfactual distributions. One approach requires the specification of
conditional QRs, while the other approach requires the specification of the
unconditional QR. Unconditional QR may prove useful in duration analysis.
This idea has not yet been explored.
ENDOGENEITY
A huge literature has emerged on the estimation of QR under endogeneity
of some of the covariates. Just as OLS, QR estimation is inconsistent in such a
situation, and various approaches using instrumental variables (in analogy
to the two-stage least squares estimator of models of the conditional mean)
have been explored, and most of them are described by Roger Koenker
in Chapter 8.8 of his 2005 textbook (including the following ones). For
the case with a discrete endogenous covariate and a discrete instrument,
Alberto Abadie, Josh Angrist, and Guido Imbens (2002) have developed a
weighted QR estimator, where the weights identify the compliers. Victor
Chernozhukov and Christian Hansen (2008) use the fact that when the
true QR coefficient of the endogenous covariate is known, the instrument

14

EMERGING TRENDS IN THE SOCIAL AND BEHAVIORAL SCIENCES

should not affect the response variable. Andrew Chesher develops a general
treatment of recursive structural equation models in a QR setting with
continuous response variables and continuous endogenous covariates. He
is concerned with the estimation of the causal effect of a change in the ? 2 th
quantile in the reduced form equation of an endogenous covariate on the
? 1 th quantile of the response variable. These effects can be estimated as functions of nonparametric derivative estimates, and they can be aggregated,
provided necessary support conditions hold, to more conventional causal
effects. Instead, many researchers, including Roger Koenker, suggest control
function approaches, where the residual of the reduced form regression of
the endogenous covariate is used as an additional covariate in the structural
equation of interest to control for the endogeneity. Our reading of the
active literature on these topics suggests that the appropriate estimation
approach for QR under endogeneity of some of the covariates may be very
specific to the application of interest, because most approaches can easily
become infeasible in realistic applications with more complex models. More
guidance is needed here.
PANEL DATA MODELS
Whereas fixed-effects OLS regressions for longitudinal data provides consistent coefficient estimates for time-varying covariates, fixed-effects QR
(i.e., QR with a dummy variable for each panel observation) suffers from
the incidental parameter problem and do not provide consistent coefficient
estimates for time-varying covariates. Furthermore, in an actual application
it should be made sufficiently clear what it means to assume a separate
individual-specific effect for each quantile of the conditional distribution.
Effectively, a fixed-effects QR models the conditional distribution of the
response variable around the individual-specific effect. Roger Koenker
(as discussed in Section 8.7.2 of his 2005 textbook) suggests a common
individual-specific effect for all quantiles, when the ∼number of panel observations is small, and to implement a shrinkage estimator using an l1-penalty
for the individual-specific effects. This way the estimation problem still
involves a linear program and the estimation may set individual-specific
effects to zero, depending on the size of the penalty. All individual-specific
effects are set to zero for a sufficiently large penalty, while a sufficiently
small penalty replicates fixed-effects QR. It remains an open question how to
choose the size of the penalty. Furthermore, econometricians are concerned
about the fact that the resulting coefficient estimates are not consistent for a
fixed number of time periods. As an alternative to fixed-effects estimation,
Jason Abrevaya and Christian Dahl suggest a correlated random-effects
model for QR on panel data but similar conceptual issues arise for this

Quantile Regression Methods

15

model. More methodological research is needed regarding the proper modeling choices of individual-specific effects in QR. Given the state of literature,
we advise applied researchers to clarify the role of individual-specific effects
in QR in the context of their substantive question of interest and to avoid
mechanical analogies to fixed-effects OLS estimation.
APPENDIX: SOFTWARE IMPLEMENTATIONS
See Table 1
Sources:
•

•
•

•

•

•
•
•

•

R package quantreg: http://cran.r-project.org/web/packages/
quantreg/quantreg.pdf
R package rqpd: http://rqpd.r-forge.r-project.org
R package cmprskQR: http://cran.r-project.org/web/packages/
cmprskQR/index.html
STATA package CQIV: http://ideas.repec.org/c/boc/bocode/s457478.
html
Stata codes for decomposition analysis: (i) Chernozhukov, FernandezVal, Melly: http://www.econ.brown.edu/fac/Blaise_Melly/code_
counter.html; (ii) Firpo, Fortin, Lemieux: http://faculty.arts.ubc.ca/
nfortin/datahead.html
TSP: http://www.tspintl.com
Matlab package: http://sites.stat.psu.edu/∼dhunter/code/qrmatlab/
Matlab function: https://www.mathworks.co.uk/matlabcentral/
fileexchange/32115-quantreg-m-quantile-regression
Matlab functions by C. Hansen: http://faculty.chicagobooth.edu/
christian.hansen/research/#Code
REFERENCES

Abadie, A., Angrist, J., & Imbens, G. (2002). Instrumental variables estimates of the
effect of subsidized training on the quantiles of trainee earnings. Econometrica,
70(1), 91–117.
Chernozhukov, V., Fernández-Val, I., & Galichon, A. (2010). Quantile and probability
curves without crossing. Econometrica, 78(3), 1093–1125.
Chernozhukov, V., & Hansen, C. (2008). Instrumental variable quantile regression:
A robust inference approach. Journal of Econometrics, 142(1), 379–398.
Peng, L., & Huang, Y. (2008). Survival analysis with quantile regression models. Journal of the American Statistical Association, 103, 637–649.

16

EMERGING TRENDS IN THE SOCIAL AND BEHAVIORAL SCIENCES

Portnoy, S. (2003). Censored regression quantiles. Journal of the American Statistical
Association, 98, 1001–1012.
Powell, J. L. (1986). Censored regression quantiles. Journal of Econometrics, 32(1),
143–155.

FURTHER READING
Abrevaya, J., & Dahl, C. M. (2008). The effects of birth inputs on birthweight. Journal
of Business & Economic Statistics (American Statistical Association), 26, 379–397.
Chernozhukov, V., Fernandez-Val, I., & Melly, B. (2013). Inference on counterfactual
distributions. Econometrica, 81, 2205–2268.
Dette, H., & Volgushev, S. (2008). Non-crossing non-parametric estimates of quantile
curves. Journal of the Royal Statistical Society: Series B, 70, 609–627.
Fitzenberger, B., & Wilke, R. A. (2006). Using quantile regression for duration analysis. Allgemeines Statistisches Archiv, 90, 105–120.
Fitzenberger, B., Wilke, R. A., & Zhang, X. (2010). Implementing Box–Cox quantile
regression. Econometric Reviews, 29(2), 158–181.
Firpo, S., Fortin, N. M., & Lemieux, T. (2009). Unconditional quantile regressions,
Econometrica, 77(3), 953–973.
Koenker, R. (2005). Quantile regression. Cambridge, England: Cambridge University
Press.
Koenker, R. (2008). Censored quantile regression redux. Journal of Statistical Software,
27, 1–24.
Koenker, R., & Bassett, G. (1978). Regression quantiles. Econometrica, 46, 33–50.
Koenker, R., & Xiao, Z. (2006). Quantile autoregression. Journal of the American Statistical Association, 101, 980–990.

BERND FITZENBERGER SHORT BIOGRAPHY
Bernd Fitzenberger has been a Full Professor of statistics and econometrics
at the University of Freiburg since 2007, where he teaches econometrics and
labor economics as well as an associate editor of “Empirical Economics”.
He previously held positions at the universities of Dresden, Mannheim, and
Frankfurt. He obtained his PhD in Economics, a Master of Science in Statistics
from Stanford University, and his diploma in Economics from the University
of Konstanz.
His research focuses on labor economics and microeconometric methods.
Currently, he works on the following topics: wage structure, unions/wage
bargaining, evaluation of active labor market policies, vocational training,
school-to-work transitions, employment of mothers after first birth, and
quantile regression. He is a research associate of the ZEW (Mannheim), a
research fellow at IZA (Bonn) and at ROA (Maastricht), a research affiliate
at the Institute for Fiscal Studies (London) as well as an editor of “Empirical

Quantile Regression Methods

17

Economics” and a member of the editorial board of the “Journal of the European Economic Association” and the “Journal for Labour Market Research.”
He has published around 45 per reviewed papers in international journals
in Economics and Statistics and in edited volumes. He has been co-editor of
six per reviewed special issues of international journals in Economics and
Statistics.
RALF ANDREAS WILKE SHORT BIOGRAPHY
Ralf Andreas Wilke is a Full Professor of Applied Econometrics at Copenhagen Business School since 2014. From 2001 until 2014 he was a Reader
at the Department of Economics and Related Studies at the University of
York. Previously, he was an Associate Professor at the University of Nottingham (2011) and he held Lectureships at the Universities of Nottingham
(2007–2011) and Leicester (2006–2007). He was a Research Economist at
the Centre for European Economic Research in Mannheim (2002–2006). He
obtained his PhD in Economics from the University of Dortmund in 2002.
He has a Research Master Degree from the Toulouse School of Economics
and he graduated from the University of Bonn.
He is a Research Professor at the Institute for Employment Research,
Nuremberg, since 2013, and a Research Fellow of the Centre of European
Economic Research, Mannheim, since 2009. He is an associate editor of
“Empirical Economics”.
He has published around 25 per reviewed papers in international journals
in Economics and Statistics.
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Repeated Cross-Sections in Survey Data (Methods), Henry E. Brady and
Richard Johnston
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Tamlin S. Conner and Matthias R. Mehl
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18

EMERGING TRENDS IN THE SOCIAL AND BEHAVIORAL SCIENCES

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Wiedermann
Translational Sociology (Sociology), Elaine Wethington



Quantile Regression Methods
BERND FITZENBERGER and RALF ANDREAS WILKE

Abstract
Quantile regression is emerging as a popular statistical approach, which complements the estimation of conditional mean models. While the latter only focuses on
one aspect of the conditional distribution of the dependent variable, the mean, quantile regression provides more detailed insights by modeling conditional quantiles.
Quantile regression can therefore detect whether the partial effect of a regressor on
the conditional quantiles is the same for all quantiles or differs across quantiles.
Quantile regression can provide evidence for a statistical relationship between two
variables even if the mean regression model does not.
We provide a short informal introduction into the principle of quantile regression
which includes an illustrative application from empirical labor market research. This
is followed by briefly sketching the underlying statistical model for linear quantile
regression based on a cross-section sample. We summarize various important
extensions of the model including the nonlinear quantile regression model, censored
quantile regression, and quantile regression for time-series data. We also discuss
a number of more recent extensions of the quantile regression model to censored
data, duration data, and endogeneity, and we describe how quantile regression can
be used for decomposition analysis. Finally, we identify several key issues, which
should be addressed by future research, and we provide an overview of quantile
regression implementations in major statistics software. Our treatment of the topic
is based on the perspective of applied researchers using quantile regression in their
empirical work.

INTRODUCTION
We consider the linear regression model
yi = xi 𝛽 + ui ,
with observations i = 1, … , n and xi = (1, x2i , … , xKi ) is 1xK, includes a constant, and 𝛽 is Kx1.y and the regressors (covariates) x are observed, the error
term u is not observed, and 𝛽 is to be estimated. The error term is assumed to
be zero in expectation given any value of the covariates, and it is independent
of the covariates. The common approach to estimate the parameters of such a
model is ordinary least squares (OLS), which estimates the conditional mean
Emerging Trends in the Social and Behavioral Sciences. Edited by Robert Scott and Stephen Kosslyn.
© 2015 John Wiley & Sons, Inc. ISBN 978-1-118-90077-2.

1

2

EMERGING TRENDS IN THE SOCIAL AND BEHAVIORAL SCIENCES

function E(y|x). This is the average value of y given the observed covariates.
A single parameter 𝛽 k is therefore informative about the partial relationship
between the covariate xk (with some abuse of notation) and the average value
of y holding all other covariates constant. It is therefore an estimate of the
average effect in the population from which the observations i are randomly
sampled.
Quantile regression (QR) follows a somewhat different approach. Instead
of estimating the average population effect, it estimates the effect at conditional quantiles of y given x (y and x are random variables with observations
yi and xi ). This is the conditional quantile function
qy|x (𝜏) ∶= x𝛽𝜏 ,
for quantile 𝜏 ∈ (0, 1). Similar to OLS, the linear QR model assumes that the
conditional quantile function is linear in the parameters 𝛽 𝜏 , but the parameters can vary in 𝜏. A single parameter 𝛽 𝜏k is the change in the conditional
quantile of y in response to a 1 unit increase in xk holding all other covariates
constant. If we consider 𝜏 = 0.5, it is the change in the conditional median of y
given x due to a 1 unit increase in xk . QR is thus more informative than mean
regression models as it considers the entire distribution of the dependent
variable. Of course, there is nothing wrong with conditional mean models.
However, they only focus on one feature of the conditional distribution as a
function of covariates. As a matter of fact if the effect varies across quantiles
and even changes its sign, the mean model may suggest no effect of a covariate on the mean, but the QR would reveal a more complete picture with nonconstant effects across quantiles. Having not observed an effect on the mean
may lead applied researchers to the conclusion that the variable does not play
a role in the model, but this may not be true. A prominent example for application of QR in social sciences is wage regression where individual wages are
explained by a number of covariates. When QR is applied to these models, it
allows the effect of covariates to vary across quantiles. For example, an additional year of education may well have a different effect on lower (𝜏 small)
and higher (𝜏 large) quantiles of the conditional wage distribution.
As an example, we estimate a wage equation with the log(wage) as the
dependent variable and a number of independent variables with a sample of
369,389 full-time working employees in Germany in June 2004. The sample is
an extract from German administrative labor market data (IAB Employment
Sample 2004).
Figure 1 shows how the estimated QR coefficients vary across quantiles
and how they relate to the OLS estimates (dashed line). For example, QR
estimates the conditional quantile function of females being 40% lower than
that of males at the first decile (quantile 𝜏 = 0.1), while being only 15% lower
at the ninth decile (quantile 𝜏 = 0.9). The OLS estimate suggests that the

0.40

0.50
0.2

0.4
0.6
Quantile

0.8

0.4
0.6
Quantile

0.8

1

1 = If unemployed in past
0

0.2

0

0.2

0.4
0.6
Quantile

0.8

1

0

0.2

0.4
0.6
Quantile

0.8

1

(b)

0.00 0.01 0.02 0.03 0.04

Tenure (in years)

0.30

1 = University degree

0.20
0

(a)

(c)

3

1

−0.15 −0.10 −0.05 0.00 0.05

Gender (1 = female)

−0.50 −0.40 −0.30 −0.20 −0.10

Quantile Regression Methods

(d)

Figure 1 Estimated QR coefficients (solid line) with 95% confidence intervals
(gray area). It also contains the estimated coefficient of the mean OLS regression
(dashed line).

average wage for a female is around 25% lower than that for a male with
the same other characteristics. The results therefore suggest that gender
differences in wages may be smaller in higher paid jobs, in contrast to the
so-called glass ceiling hypothesis. However, these results should not be
overinterpreted because of the restrictive set of covariates used.
Roger Koenker, who is the key contributor to the foundational research
of QR, has written several seminal articles, his widely used 2005 text book
“Quantile Regression”, and various surveys on QR methods. His textbook
provides all key references up to the year 2004. Together with coauthors,
Roger Koenker has contributed many computational resources to the
open-source statistical package R. His work provides formal presentations
of the material, detailed examples, and an introduction to computer code.
FOUNDATIONAL RESEARCH
Roger Koenker and Gilbert Bassett introduced the linear QR model in 1978
in their seminal article in Econometrica as a generalization of the estimation
of an empirical sample quantile.

4

EMERGING TRENDS IN THE SOCIAL AND BEHAVIORAL SCIENCES

In a sample of n observations {y1 , … , yn }, the 𝜏 − quantile [𝜏 ∈ (0, 1)] of y is
that value qy (𝜏) for which at most a share of (1 − 𝜏) ⋅ 100 % of the observations
lie above that value and at most a share of 𝜏 ⋅ 100 % of the observations lie
below that value; Thus qy (𝜏) cuts the observations into the lowest 𝜏 ⋅ 100 %
of the observations and the highest (1 − 𝜏) ⋅ 100 % of the observations. For
instance, the median corresponds to 𝜏 = 0.5, the first decile to 𝜏 = 0.1, and the
ninth decile to 𝜏 = 0.9. Quantiles are an alternative form to represent the distribution of a statistical variable, such that Fy (qy (𝜏)) := 𝜏, where Fy (.) is the
distribution function with Fy (y) : = P(Y ≤ y). Quantiles [formally the quantile
process as a function of 𝜏 ∈ (0, 1)] represent the possible nonunique inversion
of the distribution function.
The main insight to introduce QR is that the determination of an empirical
quantile qy (𝜏) can be viewed as the outcome of the following minimization
exercise:
]
[
∑
∑
|yi − q| + (1 − 𝜏)
|yi − q| .
qy (𝜏) ∶= arg minq 𝜏
i ∶yi > q

i ∶yi <q

Implicit in this formulation is the focus on absolute differences from the
location parameter q and the asymmetric weighting with 𝜏 and (1 − 𝜏),
depending on whether an observation lies above or below q. This is the
famous check function depicted in Figure 2.
For linear QR, we make q a function of covariates xi and model the conditional quantile of the response variable y, given xi as a linear function of xi 𝛽 𝜏 .
Thus, the determination of the linear QR amounts to the following minimization exercise:
[
]
∑
∑
|yi − xi 𝛽| + (1 − 𝜏)
|yi − xi 𝛽| .
𝛽𝜏 ∶= arg min𝛽 𝜏
i ∶yi > xi 𝛽

i ∶yi < xi 𝛽

Linear QR coefficients describe the change in the conditional quantile of
the response variable when a covariate changes by 1 unit. Analogous to a
sample quantile, the implied regression relationship is such that at most a
share of (1 − 𝜏) ⋅ 100 % of the observations lie above the regression line and
at most a share of 𝜏 ⋅ 100 % of the observations lie below. The calculation of
the regression coefficients corresponds to a linear program, which implies
many properties of linear QRs. For instance, if the matrix of covariates has full
rank K, then there will be at least K observations i with linearly independent
vectors of covariates xi such that the deviation from the regression line is
exactly zero (yi = xi 𝛽 𝜏 ). This is the so-called interpolation property.
The fact that the minimization problem cannot be solved by simple calculus
methods is no restriction to today’s computer power. To calculate linear QRs,
effective algorithms based on refinements of the simplex method to solve

Quantile Regression Methods

5

Check function
1.6
τ = 0.2
τ = 0.5
τ = 0.7

1.4

1.2

ρ

1

0.8

0.6

0.4

0.2

0
−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

a

Figure 2 Check function.

linear programs are available. Interior point methods together with preprocessing of the data provide effective alternatives for very large data sets.
The asymptotic variance–covariance matrix is in fact very similar to OLS
regression, provided the response variable follows a continuous distribution
around the true conditional quantiles of interest. Instead of the variance of the
error term (as in the OLS case), the asymptotic variance–covariance matrix
involves the density of the response variable at the conditional population
quantile. The statistical theory of QR also provides the joint distribution of
the coefficient estimates at different quantiles, where the covariance matrix
across quantiles has a similar structure as the variance–covariance matrix
at the individual quantiles. Assuming a constant conditional distribution of
the response variable around the conditional quantile allows one to estimate
constant conditional densities based on the estimated residuals around the
conditional QR (excluding the exact zeroes resulting from the interpolation
property). In the heteroscedasticity case (which is the case when QR is interesting, see next paragraph), it would be necessary to use observation-specific
density estimates, which would be computationally difficult and cumbersome. In his 2005 textbook, Roger Koenker discusses a simple and elegant
alternative based on estimating QR slightly above and slightly below the

6

EMERGING TRENDS IN THE SOCIAL AND BEHAVIORAL SCIENCES

quantile 𝜏 of interest, and then uses the implied conditional quantiles
to obtain local density estimates. In practice, many researchers resort to
bootstrapping methods to obtain asymptotically heteroscedasticity robust
inference. A pairwise bootstrap of the estimation of QR at different quantiles
(one estimates the QR at all quantiles of interest for the same resample) automatically provides estimates of the covariance across the different quantiles.
Estimating linear QRs for various values of 𝜏 provides a parsimonious picture of how the conditional distribution of the response variable changes
with the covariates (see earlier example). In fact, due to the linear programming structure of the estimation problem, it is straightforward to calculate
the entire process of QRs as a function of 𝜏 in a given sample, because the QR
only change at finitely many values of 𝜏. However, due to the combinatorical complexity of the problem, researchers rarely calculate the entire process
in practice. Instead, they report the QR coefficients at selected equispaced
quantiles, for example, for each decile (𝜏 = 0.1, 0.2, … , 0.9) or each percentile
(𝜏 = 0.01, 0.02, 0.03, … , 0.99). When slope coefficients change across quantiles
(which can be investigated by means of standard Wald tests), this is an indication of heteroscedasticity, that is, the conditional dispersion of the response
variable changes with the covariates. In many applications (e.g., the effect of
unions on wages), one would expect that the effect of a covariate changes
along the conditional distribution (e.g., the effect of unions may be stronger
in the lower part of the distribution than in the upper part of the distribution),
that is, heteroscedasticity (changing dispersion) is a meaningful finding (e.g.,
unions reduce the dispersion of wages by increasing wages in the lower part
of the distribution more strongly than in the upper part of the distribution).
An important advantage of QR compared with OLS regression relates to
the equivariance property of quantiles under strictly monotone transformations, for which the 𝜏th quantile of the values of the function corresponds
to the function value evaluated at the 𝜏th quantile of the original value. For
instance, if we know that the conditional median of the logarithm of wages
is a linear function of the covariates, we know also that the median of wages
in levels corresponds to the exponential function applied to this linear function, thus modeling log wages entails modeling wages in levels. It is well
known that this is not the case for OLS regression, because the expected value
of wages is not equal to the exponential function applied to the expected
value of log wages. The equivariance property of quantiles allows for more
general strictly monotonous functions, such as the Box–Cox transformation
defined for positive responses. However, computational issues may arise,
because the inverse of the Box–Cox transformation may not be strictly positive for all data points, as pointed out by the authors and Xuan Zhang in
2010.

Quantile Regression Methods

7

An apparent limitation of linear QR is the fact that nonparallel regression
lines at different quantiles are bound to cross somewhere, thus for some
values of the covariates, the predicted values at a higher quantile (e.g., the
median 𝜏 = 0.5) lies below the predicted value at a lower quantile (e.g., for
𝜏 = 0.49). At the average values of the covariates, the ordering of predicted
quantiles is preserved. One can use a sizeable incidence of quantile crossing
among the observed values of the covariates as an indication for the need
to respecify the model in a more flexible way, for example, by introducing
nonlinear terms or nonparametric components as covariates. Holger Dette
and Stanislav Volgushev (Victor Chernozhukov, Ivan Fernandez-Val, and
Alfred Galichon (2010) have also written on this issue) discuss rearrangement methods (these involve smoothing of the estimates building on
isotone regression techniques) to impose monotonicity of the predicted
quantiles (not necessarily for coefficient estimates, for which the problem
cannot be resolved). However, these methods may not resolve a problem of
misspecification. It has been our experience that quantile crossing should
be used as a guidance for misspecification of the model and that quantile
crossing is often not a serious problem, if one allows for a sufficiently flexible
specification (see section titled “Nonlinear Models”).
Table 1 in the appendix provides an overview of the implementation of
QR in various statistical software packages. The “quantreg” package in R,
developed by Roger Koenker, is most important for the dissemination of
state-of-the-art QR techniques.
Table 1
Summary of Functionality of Major Statistical Packages
for QR Analysis
Statistical package
TSP

Method

R

STATA

Linear QR

• (quantreg)

• (qreg, sqreg)

• (lad with option)

Nonlinear QR
Nonparametric QR
Censored QR
Bootstrap for QR

• (quantreg)
• (quantreg)
• (quantreg)
• (quantreg)

—
—
• (lad with option)
• (lad with option)

IV QR
Decomposition

—
—

—
—

• (Hansen)
—

Panel QR
Competing Risks QR

• (rqpd)
cmprskQR

—
—
• (clad, cqiv)
• (sqreg with
option)
• (cqiv, Hansen)
• (various ado
files)
—
—

• (Hunter,
quantreg)
• (Hunter)
—
—
• (quantreg)

—
—

—
—

•, implemented (requires additional package/command).

Matlab

8

EMERGING TRENDS IN THE SOCIAL AND BEHAVIORAL SCIENCES

CUTTING-EDGE RESEARCH
QR is nowadays applied to a variety of more advanced models for
cross-section, time-series, and panel data. Extensions of linear QR have been
developed since the mid-1980, but most of this research was conducted
after the year 2000 and it is still gradually developing. A broader process
of knowledge transfer from method-based research into broader applied
research in economics, and other social sciences did not start before the
year 2005 but since then it is increasing in pace and still picking up. Being
economists, it seems to us that empirical research in biostatistics has started
to pick up QR methods, in particular in survival analysis taking account
of censoring. Here, we present an overview of important model extensions
and fields of applications of more advanced QR models. Roger Koenker
contributed to some of these extensions, and the references given to his work
cover most of these extensions.
NONLINEAR MODELS
The linear functional form of the conditional quantile function might be
too restrictive in an application. Roger Koenker outlines the estimation of
nonlinear quantile functions in his textbook. The QR model can be extended
to allow for nonlinear relationships, that is, qy|x (𝜏) = g𝜏 (x) with g𝜏 being
some nonlinear function that satisfies some regularity conditions. In the
abovementioned minimization exercise, g𝜏 (x) replaces the predicted values
xi 𝛽 𝜏 . Nonlinear, strictly monotonous transformations (such as the logarithm
or a Box–Cox transformation) of the response variable to achieve a linear QR
(as discussed above) are a special case. The parametric, nonlinear QR model
can be applied if g𝜏 is known subject to some unknown parameters. This is
analogous to nonlinear least squares regression.
A nonparametric QR on a set of continuous covariates can be applied if g𝜏 is
unknown. A local smoothing approach such as kernel smoothing can be used
to locally estimate the unknown g𝜏 , the estimates being subject to the curse
of dimensionality. (The convergence rate of the estimator becomes slower
when the number of covariates increases.) Estimating a weighted linear QR
just on an intercept (the residuals are weighted by kernel weights) for continuous covariates produces the QR alternative of a Nadaraya–Watson (local
constant) kernel regression. Local linear (local polynomial) QR can be estimated using a local linear (local polynomial) approximation in the weighted
QR. Note that using a local constant kernel regression with the same bandwidth for all quantiles resolves by construction the quantile crossing problem. The literature also involves semiparametric specifications with additive
nonparametric components estimated using backfitting techniques, involving an iterative procedure. During an iteration step, each component of the

Quantile Regression Methods

9

QR specification is estimated recursively based on the previous estimates of
all other components.
CENSORED QUANTILE REGRESSION
The QR method can also be applied to a censored regression. Here, the conditional quantile function for y corresponds either to the censoring value or it is
linear in the covariates. A prominent example from social sciences involves
labor supply, which is either zero or positive and many individuals supply
zero hours of work. Another example involves health expenditures, which
are either zero or positive. Models for such response variables can be estimated by censored quantile regressions (CQR). The interpretation of estimation results regarding the observed censored values requires the computation
of partial effects accounting for censoring.
We first consider the simple case of right censoring (e.g., top coding of
wages or right censoring of durations of ongoing spells), where for censored
observations we only know that the statistical variable of interest exceeds a
certain known threshold. If this threshold is constant for all observations and
the 𝜏th quantile of the censored observations lies below the threshold, we
know that the 𝜏th quantile of the censored observations corresponds to the
𝜏th quantile of the uncensored observations. Here, the censored observations
correspond to the actual values of the variable of interest, if they are not censored, and to the censoring value, if they are censored, that is, yi = min(y∗i , c)
where yi is the observed censored value, y∗i is the true uncensored value, and
c is the censoring threshold.
Roger Koenker discusses three approaches to CQR in his 2008 article.
The first approach developed by James Powell (1986) involves the case of
fixed censoring where the censoring values may vary across observations,
but it is known for all observations. The estimator replicates the censoring
mechanism in the regression specification for the censored observation as
g𝜏 (xi ) = min(xi 𝛽 𝜏 , ci ), where ci is the observation-specific, known censoring
value. This is a special case of nonlinear QR. The Powell estimator provides a semiparametric alternative to the standard Tobit estimator for the
censored regression model, which relies on the assumption of a normally
distributed error term and which is inconsistent under heteroscedasticity.
However, the calculation of the Powell estimator is difficult when there
is a lot of censoring. Various modifications of the estimator have been
suggested in the literature to overcome these difficulties. Two appealing
approaches have been suggested by Steve Portnoy (2003) and by Limin Peng
and Yijian Huang (2008). These involve regression versions of nonparametric estimators of distribution functions under independent censoring
(Kaplan–Meier and Nelson–Aalen estimator), where the censoring values

10

EMERGING TRENDS IN THE SOCIAL AND BEHAVIORAL SCIENCES

are only known for those observations that are censored (the case of random
censoring).
DURATION MODELS
The response variable in duration analysis (or survival analysis) is time until
an event or failure occurs (single-risk model). QR is an attractive approach
to analyze the distribution of a duration as it can allow for different effects of
covariates on lower and higher quantiles of the conditional distribution. For
example, when the response variable is the duration of unemployment and
we want to study the effect of a training programme on the job-taking time of
unemployed people, we would expect that such a training programme at the
first instance increases shorter unemployment periods, because the unemployed are locked into the programme. For longer unemployment periods,
we would hope to find a shortening effect of the training programme once
the training is completed. Thus, it is conceivable that even the sign of the
estimated coefficient of training varies across quantiles. Standard parametric
and semiparametric duration models such as proportional hazard models
do not possess this degree of flexibility as they typically model the effect of
a covariate by one single parameter. While linear QR can be directly applied
to duration data, these data are often characterized by being censored. In
the presence of censoring a CQR can be estimated. Roger Koenker did some
pioneering work on QR for duration models, which is reviewed in his 2005
textbook, and his 2008 article on CQR has a focus on applications in duration
analysis. Our 2006 survey discusses the usefulness and the limitations of QR
for duration analysis in the presence of independent censoring.
TIME-SERIES MODELS
QR is also becoming increasingly popular for the empirical analysis with
time-series data. Roger Koenker and Zhijie Xiao consider a class of quantile
autoregression models where the covariates involve the lags of the response
variable in discrete time. Such QR time-series models allow for a systematic
influence of the lagged dependent variable on the location, scale, and shape
of the distribution of the response variable. For the analysis of univariate time
series, the models include the autoregressive model (both stationary processes and processes with unit roots) and the autoregressive conditional heteroscedasticity (ARCH) model. Such models allow for asymmetric dynamics
and local persistence in time series and thus may bridge the gap between stationary and integrated time-series processes. Such models have been applied
to macroeconomic time-series data and financial data. A standard generalized autoregressive conditional heteroscedasticity (GARCH) model implies

Quantile Regression Methods

11

a symmetric persistence in the second moment of a time series, irrespective of the direction of change. A quantile autoregression model allows for
asymmetric dynamics, implying different responses in the conditional scale
depending on whether there was a strong downside or upside movement
of the response variable in the recent past. Downside movements or upside
movements in the recent past may involve different degrees of persistence
(for instance, unit root behavior may only exist in some part of the conditional
distribution of the response variable). Such effects may prove important in
the analysis of financial data. Extensions of quantile autoregression models to
the analysis of multivariate time series are possible, including the estimation
of quantile vector autoregressions and quantile co-integrating regressions.
KEY ISSUES FOR FURTHER RESEARCH
We discuss some pertinent issues for further research that are related to
our own research. Due to space constraints, the choice has to be somewhat
eclectic.
CENSORED QUANTILE REGRESSION
We raise four issues: First, Roger Koenker describes the major computational
difficulties involved and discusses some practical solutions (see above).
Somewhat practical approaches exist for the CQR model under fixed and
random censoring, although there does not seem to exist a consensus in the
literature on what works best (correspondingly, popular software package
use different algorithms to calculate CQR, often with little justification
of the particular choice). Second, it should be noted that identification of
CQR is a tenuous issue, because the CQR line involves an extrapolation
based on functional form assumptions into the censored part of the data. A
more substantive analysis of this issue would be useful. Third, the random
censoring case assumes that the censoring values are independent of the
response variable (at least conditional on the covariates). CQR models for
random censoring are therefore not applicable to all empirical problems.
While there has been some work for CQR if observations are dependent,
there is still a gap to accommodate various forms of dependent censoring
in the CQR model. Existing studies impose stringent assumptions on the
regression models and are plagued by high computational costs due to
multiple step algorithms. Fourth, two-limit CQR allowing for censoring
both from above and below (in an analogy to the two-limit tobit model) is a
straightforward extension under fixed censoring, and similar algorithms can
be used. Under random censoring, at most one of the two censoring points
is observed (and neither one is observed for uncensored observations) and

12

EMERGING TRENDS IN THE SOCIAL AND BEHAVIORAL SCIENCES

the rationale for the algorithms suggested by Steve Portnoy (2003) and by
Limin Peng and Yijian Huang (2008) breaks down (see above).
DURATION MODELS
So far, QR models for duration data have been mainly used for single-risk
models with independent right censoring. Here, the QR model estimates
the conditional quantile functions of the duration distribution conditional
on a time-invariant set of covariates. We think that there are a number of
extensions still to be analyzed. These include multiple spell QR duration
models (this issue is related to panel data applications, see in the following
section) and QR models that explicitly allow for unobserved heterogeneity
and time-varying covariates. Duration analysis based on modeling hazard
rates can take these issues into account, typically under the restrictive
proportional hazard assumption. Regarding the last point, QR can condition
on a predetermined time path of the covariates, but the analysis may quickly
involve a large number of coefficients to be estimated. Although applications
of QR for competing risks models with possibly dependent competing risks
do exist, the applications cannot estimate the duration distributions of the
separate competing risks (transitions to a certain destination state) but
rather estimate the cumulative incidence function of the separate risks. The
cumulative incidence—the duration of observed transitions into a certain
destination state—is a common descriptive tool (especially in Biostatistics),
but in the absence of knowledge about the dependence structure between
risks, it is difficult to infer the duration distributions of the competing risks,
which are often objects of prime interest in the social sciences.
DECOMPOSITION ANALYSIS AND UNCONDITIONAL QR
For a long time, a major restriction of QR in applied research was that
conditional quantiles do not aggregate directly to the unconditional distribution of the response variable. While the overall mean is the weighted
average of cell means (or the fitted value of an OLS regression at the sample
mean of the covariates), it is not possible to calculate the aggregate as
𝜏th quantile cannot be calculated based on the conditional 𝜏th quantiles.
Analogously, the estimated QR coefficient at a certain quantile does not
reflect the effect of a uniform shift of one covariate by 1 unit on the aggregate
quantile. Relating conditional and unconditional quantiles is important for
so-called Blinder–Oaxaca type decomposition analysis. An example is the
analysis of gender differences in the wage distribution, which may be due
to gender differences in the distribution of covariates (characteristics) and
gender differences in the QR coefficients. To distinguish the two effects,

Quantile Regression Methods

13

researchers estimate counterfactual wage distributions (e.g., the wage distribution resulting if females exhibited male characteristics and still female
coefficients applied). This counterfactual distribution can be calculated by
determining the conditional distribution function of the response variable
based on the full process of QR coefficients, explicitly aggregating the
conditional distribution functions (by the law of total probability) and then
inverting the aggregate distribution function. This approach is described
by Victor Chernozhukov, Ivan Fernandez-Val, and Blaise Melly, who also
suggest its extension to sequential decompositions of the contribution
of individual covariates to the differences in aggregate distributions. A
simple alternative approach for decomposition analysis would make use of
inverse probability weighting to calculate counterfactual wage distributions
while balancing characteristics between males and females. This approach
is very important in the literature on (unconditional) quantile treatment
effects under the conditional independence assumption. A third alternative
denoted as unconditional QR is introduced by Serjo Firpo, Nicole Fortin,
and Thomas Lemieux. The basic idea is to estimate a discrete choice model
of whether the individual response does not exceed the aggregate 𝜏th
quantile. Based on the fitted probabilities and the density of the aggregate
distribution at the aggregate 𝜏th quantile, it is possible to study the effect
of uniform shifts in covariates on the aggregate 𝜏th quantile. The approach
can be used for decomposition analysis. Estimating a linear probability
model allows to implement a sequential decomposition. The literature is
missing a comprehensive comparison of the different approaches to estimate
counterfactual distributions. One approach requires the specification of
conditional QRs, while the other approach requires the specification of the
unconditional QR. Unconditional QR may prove useful in duration analysis.
This idea has not yet been explored.
ENDOGENEITY
A huge literature has emerged on the estimation of QR under endogeneity
of some of the covariates. Just as OLS, QR estimation is inconsistent in such a
situation, and various approaches using instrumental variables (in analogy
to the two-stage least squares estimator of models of the conditional mean)
have been explored, and most of them are described by Roger Koenker
in Chapter 8.8 of his 2005 textbook (including the following ones). For
the case with a discrete endogenous covariate and a discrete instrument,
Alberto Abadie, Josh Angrist, and Guido Imbens (2002) have developed a
weighted QR estimator, where the weights identify the compliers. Victor
Chernozhukov and Christian Hansen (2008) use the fact that when the
true QR coefficient of the endogenous covariate is known, the instrument

14

EMERGING TRENDS IN THE SOCIAL AND BEHAVIORAL SCIENCES

should not affect the response variable. Andrew Chesher develops a general
treatment of recursive structural equation models in a QR setting with
continuous response variables and continuous endogenous covariates. He
is concerned with the estimation of the causal effect of a change in the 𝜏 2 th
quantile in the reduced form equation of an endogenous covariate on the
𝜏 1 th quantile of the response variable. These effects can be estimated as functions of nonparametric derivative estimates, and they can be aggregated,
provided necessary support conditions hold, to more conventional causal
effects. Instead, many researchers, including Roger Koenker, suggest control
function approaches, where the residual of the reduced form regression of
the endogenous covariate is used as an additional covariate in the structural
equation of interest to control for the endogeneity. Our reading of the
active literature on these topics suggests that the appropriate estimation
approach for QR under endogeneity of some of the covariates may be very
specific to the application of interest, because most approaches can easily
become infeasible in realistic applications with more complex models. More
guidance is needed here.
PANEL DATA MODELS
Whereas fixed-effects OLS regressions for longitudinal data provides consistent coefficient estimates for time-varying covariates, fixed-effects QR
(i.e., QR with a dummy variable for each panel observation) suffers from
the incidental parameter problem and do not provide consistent coefficient
estimates for time-varying covariates. Furthermore, in an actual application
it should be made sufficiently clear what it means to assume a separate
individual-specific effect for each quantile of the conditional distribution.
Effectively, a fixed-effects QR models the conditional distribution of the
response variable around the individual-specific effect. Roger Koenker
(as discussed in Section 8.7.2 of his 2005 textbook) suggests a common
individual-specific effect for all quantiles, when the ∼number of panel observations is small, and to implement a shrinkage estimator using an l1-penalty
for the individual-specific effects. This way the estimation problem still
involves a linear program and the estimation may set individual-specific
effects to zero, depending on the size of the penalty. All individual-specific
effects are set to zero for a sufficiently large penalty, while a sufficiently
small penalty replicates fixed-effects QR. It remains an open question how to
choose the size of the penalty. Furthermore, econometricians are concerned
about the fact that the resulting coefficient estimates are not consistent for a
fixed number of time periods. As an alternative to fixed-effects estimation,
Jason Abrevaya and Christian Dahl suggest a correlated random-effects
model for QR on panel data but similar conceptual issues arise for this

Quantile Regression Methods

15

model. More methodological research is needed regarding the proper modeling choices of individual-specific effects in QR. Given the state of literature,
we advise applied researchers to clarify the role of individual-specific effects
in QR in the context of their substantive question of interest and to avoid
mechanical analogies to fixed-effects OLS estimation.
APPENDIX: SOFTWARE IMPLEMENTATIONS
See Table 1
Sources:
•

•
•

•

•

•
•
•

•

R package quantreg: http://cran.r-project.org/web/packages/
quantreg/quantreg.pdf
R package rqpd: http://rqpd.r-forge.r-project.org
R package cmprskQR: http://cran.r-project.org/web/packages/
cmprskQR/index.html
STATA package CQIV: http://ideas.repec.org/c/boc/bocode/s457478.
html
Stata codes for decomposition analysis: (i) Chernozhukov, FernandezVal, Melly: http://www.econ.brown.edu/fac/Blaise_Melly/code_
counter.html; (ii) Firpo, Fortin, Lemieux: http://faculty.arts.ubc.ca/
nfortin/datahead.html
TSP: http://www.tspintl.com
Matlab package: http://sites.stat.psu.edu/∼dhunter/code/qrmatlab/
Matlab function: https://www.mathworks.co.uk/matlabcentral/
fileexchange/32115-quantreg-m-quantile-regression
Matlab functions by C. Hansen: http://faculty.chicagobooth.edu/
christian.hansen/research/#Code
REFERENCES

Abadie, A., Angrist, J., & Imbens, G. (2002). Instrumental variables estimates of the
effect of subsidized training on the quantiles of trainee earnings. Econometrica,
70(1), 91–117.
Chernozhukov, V., Fernández-Val, I., & Galichon, A. (2010). Quantile and probability
curves without crossing. Econometrica, 78(3), 1093–1125.
Chernozhukov, V., & Hansen, C. (2008). Instrumental variable quantile regression:
A robust inference approach. Journal of Econometrics, 142(1), 379–398.
Peng, L., & Huang, Y. (2008). Survival analysis with quantile regression models. Journal of the American Statistical Association, 103, 637–649.

16

EMERGING TRENDS IN THE SOCIAL AND BEHAVIORAL SCIENCES

Portnoy, S. (2003). Censored regression quantiles. Journal of the American Statistical
Association, 98, 1001–1012.
Powell, J. L. (1986). Censored regression quantiles. Journal of Econometrics, 32(1),
143–155.

FURTHER READING
Abrevaya, J., & Dahl, C. M. (2008). The effects of birth inputs on birthweight. Journal
of Business & Economic Statistics (American Statistical Association), 26, 379–397.
Chernozhukov, V., Fernandez-Val, I., & Melly, B. (2013). Inference on counterfactual
distributions. Econometrica, 81, 2205–2268.
Dette, H., & Volgushev, S. (2008). Non-crossing non-parametric estimates of quantile
curves. Journal of the Royal Statistical Society: Series B, 70, 609–627.
Fitzenberger, B., & Wilke, R. A. (2006). Using quantile regression for duration analysis. Allgemeines Statistisches Archiv, 90, 105–120.
Fitzenberger, B., Wilke, R. A., & Zhang, X. (2010). Implementing Box–Cox quantile
regression. Econometric Reviews, 29(2), 158–181.
Firpo, S., Fortin, N. M., & Lemieux, T. (2009). Unconditional quantile regressions,
Econometrica, 77(3), 953–973.
Koenker, R. (2005). Quantile regression. Cambridge, England: Cambridge University
Press.
Koenker, R. (2008). Censored quantile regression redux. Journal of Statistical Software,
27, 1–24.
Koenker, R., & Bassett, G. (1978). Regression quantiles. Econometrica, 46, 33–50.
Koenker, R., & Xiao, Z. (2006). Quantile autoregression. Journal of the American Statistical Association, 101, 980–990.

BERND FITZENBERGER SHORT BIOGRAPHY
Bernd Fitzenberger has been a Full Professor of statistics and econometrics
at the University of Freiburg since 2007, where he teaches econometrics and
labor economics as well as an associate editor of “Empirical Economics”.
He previously held positions at the universities of Dresden, Mannheim, and
Frankfurt. He obtained his PhD in Economics, a Master of Science in Statistics
from Stanford University, and his diploma in Economics from the University
of Konstanz.
His research focuses on labor economics and microeconometric methods.
Currently, he works on the following topics: wage structure, unions/wage
bargaining, evaluation of active labor market policies, vocational training,
school-to-work transitions, employment of mothers after first birth, and
quantile regression. He is a research associate of the ZEW (Mannheim), a
research fellow at IZA (Bonn) and at ROA (Maastricht), a research affiliate
at the Institute for Fiscal Studies (London) as well as an editor of “Empirical

Quantile Regression Methods

17

Economics” and a member of the editorial board of the “Journal of the European Economic Association” and the “Journal for Labour Market Research.”
He has published around 45 per reviewed papers in international journals
in Economics and Statistics and in edited volumes. He has been co-editor of
six per reviewed special issues of international journals in Economics and
Statistics.
RALF ANDREAS WILKE SHORT BIOGRAPHY
Ralf Andreas Wilke is a Full Professor of Applied Econometrics at Copenhagen Business School since 2014. From 2001 until 2014 he was a Reader
at the Department of Economics and Related Studies at the University of
York. Previously, he was an Associate Professor at the University of Nottingham (2011) and he held Lectureships at the Universities of Nottingham
(2007–2011) and Leicester (2006–2007). He was a Research Economist at
the Centre for European Economic Research in Mannheim (2002–2006). He
obtained his PhD in Economics from the University of Dortmund in 2002.
He has a Research Master Degree from the Toulouse School of Economics
and he graduated from the University of Bonn.
He is a Research Professor at the Institute for Employment Research,
Nuremberg, since 2013, and a Research Fellow of the Centre of European
Economic Research, Mannheim, since 2009. He is an associate editor of
“Empirical Economics”.
He has published around 25 per reviewed papers in international journals
in Economics and Statistics.
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