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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
 Different slopes/rates of change (β’s) for different quantiles
of the response variable (Y) distribution.
 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
Assumed Distribution
for Errors
No Distribution
Variance Assumption
Constant Variance
Non-Constant Variance
Linearity Assumption
Mean is a linear function
of X
Quantile is a linear
function of X
Uncorrelated Errors
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.
(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
• Default
• Good for up to
several thousand
• Good for a larger
(He and Wei, 2005); Quantile Regression – R; Susmel
• Good for much
larger problems
• Similar to “fn” but
Methods of Calculating Standard Errors
Summary.rq(object, se=“ ”…) or Summary(object,se=“ ”…)
• Direct
estimation /
• Computes
estimate of
• iid errors
• Inversion of
rank tests
• Default iid
errors but noniid can be
• For non-iid,
• Bootstrap
• Pairwise
bootstrap (noniid allowed)
• Parzen, Wei, and
Ying (non-iid
• Markov Chain
For a discussion of the methods and their relative advantages/disadvantages see
(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
 Y = CEO Total Compensation S&P1500 firms
 X = Total Assets (size proxy)
 Merged 2012 data downloaded from COMPUSTAT and
 Total Compensation data is in thousands
 Total Assets data is in millions
Quantile Regression
(Koenker and Hallock, 2001)
Quantile Regression: tau = .50
tau = .50
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
 The total asset
quantile coefficients
are positively
associated with total
Cade, B. S., & Noon, B. R. (2003). A gentle introduction to quantile regression for ecologists. Frontiers in Ecology and the Environment, 1(8),
Fitzenberger, Bernd (2012). Quantile Regression. Universität Linz.
Hao, L., & Naiman, D. Q. (2007). Quantile regression (No. 149). Sage.
He, X., & Wei, W. (2005). Tutorial on Quantile Regression. Cached page:
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.
Koenker, R., & Hallock, K. (2001). Quantile regression. Journal of Economic Perspectives, 15(4), 143-156.
Koenker, R., & Bassett Jr, G. W. (2010). March Madness, Quantile Regression Bracketology, and the Hayek Hypothesis. Journal of Business &
Economic Statistics, 28(1).
Koenker, R. (2011). “Quantile Regression: A Gentle Introduction.” University of Illinois Urbana- Champaign.
Quantile Regression – R Documentation for Package ‘quantreg’ version 4.30.
Susmel, Rauli. “Lecture 10 Robust and Quantile Regression.” Bauer College of Business University of Houston.
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.
Lew, A. A., & Ng, P. T. (2012). Using quantile regression to understand visitor spending. Journal of Travel
Research, 51(3), 278-288.
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
Thank You!