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Quantile Regression
The intuition
Hypothetical Distributions
The intuition
OLS Regression Results
The intuition
Quantile Regression Results
An alternative approach
Logistic Regression Models
Examples
Five ideas from your (or your friends’) research where this approach might be useful.
Some examples
Image from Koenker (http://www.econ.uiuc.edu/~roger/research/intro/jep.pdf)
Some examples
Image from Koenker (http://www.econ.uiuc.edu/~roger/research/intro/jep.pdf)
Image from Koenker (http://www.econ.uiuc.edu/~roger/research/intro/jep.pdf)
Some examples
Image from Koenker (http://www.econ.uiuc.edu/~roger/research/intro/jep.pdf)
Some examples
Some examples
Image from Bitler et al. 2006 AER paper.
Image from Bitler et al. 2006 AER paper.
Some examples
Some examples
Some more examples
Some examples
Pronghorn densities (y) by shrub canopy cover (X) on n = 28 winter ranges (data from Cook and Irwin 1985) and 0.90, 0.75, 0.50, 0.25, and0.10 regression quantile estimates (solid lines) and least squares regression estimate (dashed line) for the model y = b0 + b1X +e .
(From Cade and Noon, 2003).
Some examplesQuantile regression was used to estimate changes in Lahontan cutthroat troutdensity (y) as a function of the ratio of stream width to depth (X) for 7 years and 13 streams in the eastern Lahontan basin of the western US. A scatterplot of n = 71 observations of stream width:depth and trout densities with 0.95, 0.75, 0.50, 0.25, and 0.05 quantile (solid lines) and least squares regression (dashed line) estimates for the model ln y = b0 + b1X +e. From Cade and Noon, 2003.
Technical intuitions
Image from Pindyck and Rubinfield (Econometric models and economic forecasts)
Formulae (OLS)
ikkii Xy
iii Xy 2)(
min
)()( 1 YXXX
Formulae (LAD)
ikkii Xy
iii Xy
min
Formulae (LAD vs OLS)
iii Xy
min
iii Xy 2)(
min
Formulae (LAD at ≠.5)
ikkii Xy
iiii Xyiii
Xyiii XyXy 1
min
Formulae (LAD at ≠.5)
ikkii Xy
iiii Xyiii
Xyiii XyXy 1
min
Negative residuals Positive residuals
Technical (semi) intuitions
Image from Koenker (http://www.econ.uiuc.edu/~roger/research/intro/jep.pdf)
Why we might care
Why we might care
Skewed Distributions
Issues• Small samples
– Guidelines: The 30 observations rule? (Chernozhukov)
• Suitable dependent variables– Does your metric make sense?
• Accessibility– (Relatively) new outside of economics– Solution: Find a friend in economics?– More difficult with thornier data (categorical DV’s, panel
data, etc)
Issues
• Cluster robust standard errors– Solutions:
• Bootstrapping se’s• Sandwich estimators (see stata code online)
• Thinking about effects– Effects on the distribution
– Rank preservation assumptions
• Distribution of Y not of X
11 )ˆ(ˆˆ)ˆ(ˆ
XFXXXXFXVc
ccccLAD
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