Home > QUANTILE REGRESSION

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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.

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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.

�� ��−1 ���� (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 , ;

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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

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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.

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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

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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

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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

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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.

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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.

Page 10 |

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.

Page 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

Conditional Densities, J. of Comp. and Graphical Stat. 5, 315-36.

Koenker, R. and G. Bassett 1978 . Regression quantiles, Econometrica, 46, 33-50. Koenker, R. 2001 . Quantile Regression, J. of Economics Perspectives, forthcoming. Koenker, R. 2002 . Quantile Regression, forthcoming. Lehmann, E. 1974 Nonparametrics: Statistical Methods Based on Ranks, Holden-Day: San Fran-

cisco.

Manski, C. 1985 Semiparametric analysis of discrete response: asymptotic properties of the

maximum score estimator, J. of Econometrics, 27, 313-34.

Portnoy, S. and R. Koenker 1997 . The Gaussian Hare and the Laplacian Tortoise: Com-

putability of Squared-error vs. Absolute-error Estimators, with discussion, Statistical Science, 12, 279-300.

Powell, J.L. 1986 Censored regression quantiles, J. of Econometrics, 32, 143-55.

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