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Quantile Regression
ISQS 5349 – Regression Analysis
Spring 2014
Daniela Sanchez
March 13, 2014
What is Quantile Regression?
 A form of regression analysis designed to estimate models for
the conditional median or other conditional quantile
functions of the predictor variable (Y) against the covariates
(X’s).
 Different slopes/rates of change (β’s) for different quantiles
of the response variable (Y) distribution.
2
Background
 Boscovich proposed median regression in the 18th century.
 Laplace and Edgeworth further investigated that idea.
 Mosteller and Tukey (1977) first stated that functions could
be fitted to describe parts of the response variable (y)
distribution aside from simply the mean of the distribution.
 Quantile regression (other than median) is the work of Roger
Koenker and Gilbert Bassett (1978) – University of Illinois.
What is a Quantile?

OLS vs. Quantile Regression

(Hao and Naiman, 2007; Koenker, 2000)
OLS vs. Quantile Regression
Characteristics
OLS
Regression
Quantile
Regression
Assumed Distribution
for Errors
Normal
No Distribution
Assumption
Variance Assumption
Constant Variance
(Homoscedasticity)
Non-Constant Variance
(Heteroscedasticy)
Accomodated
Linearity Assumption
Mean is a linear function
of X
Quantile is a linear
function of X
Uncorrelated Errors
Assumption
Assumption is necessary
Assumption is necessary
(Cade and Noon, 2003; Hao and Naiman, 2007)
Quantile Regression
Graph adapted from Fitzenberger (2012)
Quantile Regression

Quantile Regression – March Madness Example

March Madness Example Continued
 Why Quantile Regression?
 Teams’ consistencies (different variances).
 Teams’ performance non-symmetric (non-normal distributions).
 Very high and low scoring games occur (outliers).
 Predictions for certain gambling opportunities may necessitate predictions
for parts of the score distribution aside from the mean.
 Caveats later controlled for:
 Positive/negative momentum (correlated/dependent errors).
 Single game scores for both teams usually similar (dependent errors).
March Madness Example Implementation
 Data on 2,940 games for 232 Division I NCAA teams
 199 quantiles calculated for each team
 Using past data, score predictions made for each pair of teams in
the tournament at each of the 199 quantiles
Note: this model assumes independence of errors which is unlikely in reality. More in-depth analysis using more
advanced statistical and quantile regression techniques and survival analysis are used in the paper to deal with such issues.
R-Code

Formula
Tau
Method
(He and Wei, 2005); Quantile Regression - R
•
•
•
•
•
•
“br” = simplex method – Barrodale and Roberts (1974)
“fn” = interior point method – Frisch-Newton (1997)
“pfn” = Frisch-Newton with pre-processing
“fnc” = enables linear inequality on fitted coefficient
“lasso” = penalized method using lasso penalty
“scad” = penalized method using Fan and Li’s smoothly clipped absolute deviation penalty
Comparison of More Common Algorithm Methods
“br”
• Default
• Good for up to
several thousand
observations
“fn”
• Good for a larger
problem
(He and Wei, 2005); Quantile Regression – R; Susmel
“pfn”
• Good for much
larger problems
• Similar to “fn” but
quicker
Methods of Calculating Standard Errors
Summary.rq(object, se=“ ”…) or Summary(object,se=“ ”…)
“iid”
• Direct
estimation /
sparsity
estimation
• Computes
estimate of
asymptotic
covariance
• iid errors
“rank”
• Inversion of
rank tests
• Default iid
errors but noniid can be
accommodated
• For non-iid,
option
iid=FALSE
“boot”
• Bootstrap
methods
• Pairwise
bootstrap (noniid allowed)
• Parzen, Wei, and
Ying (non-iid
allowed)
• Markov Chain
Marginal
Bootstrap
(MCMB)
For a discussion of the methods and their relative advantages/disadvantages see
http://www.econ.uiuc.edu/~roger/research/rqci/rqci.pdf
(He and Wei, 2005); Quantile Regression – R; Susmel
Other Quantile Regression Applications
 Applications
 Engineering: Building energy consumption vs. temperature/weather
and varying levels of end uses (NREL) - Henze et al. (2014)
 Upper and lower control limits desired
 Marketing: Tourist spending patterns vs. various spending stimuli (e.g.
length of stay, job type, gender, age, etc.) - Lew and Ng (2012)
 Market segmentation desired
 Accounting/Finance: - Earnings vs. firm size, financial leverage, and
R&D expenditures - Li and Wang (2011)
 Prior research inconclusive regarding effect of factors on earnings
On a Practical Note
 Is CEO total compensation associated with firm size?
 I examine CEO Total Compensation as a function of Total
Assets.
 Y = CEO Total Compensation S&P1500 firms
 X = Total Assets (size proxy)
 Merged 2012 data downloaded from COMPUSTAT and
EXECUCOMP.
 Total Compensation data is in thousands
 Total Assets data is in millions
Quantile Regression

(Koenker and Hallock, 2001)
Quantile Regression: tau = .50
Intercept
tau = .50
Centercept
tau = .50
• The intercept is a centercept and estimates the quantile
function of Total CEO Compensation conditional on
mean Total Assets at each particular quantile.
Interpreting Coefficients?
 The same way as
ordinary regression
coefficients.
 The total asset
quantile coefficients
are positively
associated with total
compensation.
Conclusions

References

Cade, B. S., & Noon, B. R. (2003). A gentle introduction to quantile regression for ecologists. Frontiers in Ecology and the Environment, 1(8),
412-420. http://www.fort.usgs.gov/products/publications/21137/21137.pdf

Fitzenberger, Bernd (2012). Quantile Regression. Universität Linz.
http://www.econ.jku.at/members%5CDerntl%5Cfiles%5CPHD%5CFitzenberger_QuantileRegression.pdf

Hao, L., & Naiman, D. Q. (2007). Quantile regression (No. 149). Sage. http://www.sagepub.com/upm-data/14855_Chapter3.pdf

He, X., & Wei, W. (2005). Tutorial on Quantile Regression. Cached page: http://webcache.googleusercontent.com/search?q=cache:IugoWaFOXoJ:epi.univparis1.fr/servlet/com.univ.collaboratif.utils.LectureFichiergw%3FID_FICHE%3D27872%26OBJET%3D0008%26ID_FICHIER%3D8337
9+&cd=1&hl=en&ct=clnk&gl=us

Koenker, R., & Bassett Jr, G. (1978). Regression quantiles. Econometrica: Journal of the Econometric Society, 33-50.

Koenker, R. W. (2000). Quantile Regression, article prepared for the statistics section of the International Encyclopedia of the Social
Sciences. University of Illinois: Urbana-Champaign, IL. http://www.econ.uiuc.edu/~roger/research/rq/rq.pdf

Koenker, R., & Hallock, K. (2001). Quantile regression. Journal of Economic Perspectives, 15(4), 143-156.
http://www.econ.uiuc.edu/~roger/research/rq/QRJEP.pdf

Koenker, R., & Bassett Jr, G. W. (2010). March Madness, Quantile Regression Bracketology, and the Hayek Hypothesis. Journal of Business &
Economic Statistics, 28(1). http://www.econ.uiuc.edu/~roger/research/bracketology/MM.pdf

Koenker, R. (2011). “Quantile Regression: A Gentle Introduction.” University of Illinois Urbana- Champaign.
http://www.econ.uiuc.edu/~roger/courses/RMetrics/L1.pdf

Quantile Regression – R Documentation for Package ‘quantreg’ version 4.30. http://svitsrv25.epfl.ch/Rdoc/library/quantreg/html/rq.html

Susmel, Rauli. “Lecture 10 Robust and Quantile Regression.” Bauer College of Business University of Houston.
http://www.bauer.uh.edu/rsusmel/phd/ec1-25.pdf
References for Noted Discipline-Specific Applications

Henze, G. P., Pless, S., Petersen, A., Long, N., & Scambos, A. T. (2014). Control Limits for Building Energy End
Use Based on Engineering Judgment, Frequency Analysis, and Quantile Regression.
http://www.nrel.gov/docs/fy14osti/60020.pdf

Lew, A. A., & Ng, P. T. (2012). Using quantile regression to understand visitor spending. Journal of Travel
Research, 51(3), 278-288. http://jtr.sagepub.com.lib-e2.lib.ttu.edu/content/51/3/278.full.pdf+html

Li, M., & Hwang, N. (2011). Effects of Firm Size, Financial Leverage and R&D Expenditures on Firm Earnings:
An Analysis Using Quantile Regression Approach. Abacus, 47(2), 182-204. doi:10.1111/j.14676281.2011.00338.x http://eds.a.ebscohost.com.lib-e2.lib.ttu.edu/ehost/pdfviewer/pdfviewer?sid=91bf3ebd6f4d-42dd-bb3b-e4818335144b%40sessionmgr4005&vid=2&hid=4110
Questions?
Thank You!