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Quantile Regression
ISQS 5349 – Regression Analysis
Spring 2014
Laurie Corradino
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
but adjustments available
Assumption is necessary
but adjustments available
(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!