quantile regression. the intuition hypothetical distributions
TRANSCRIPT
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Quantile Regression
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The intuition
Hypothetical Distributions
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The intuition
OLS Regression Results
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The intuition
Quantile Regression Results
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An alternative approach
Logistic Regression Models
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Examples
Five ideas from your (or your friends’) research where this approach might be useful.
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Some examples
Image from Koenker (http://www.econ.uiuc.edu/~roger/research/intro/jep.pdf)
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Some examples
Image from Koenker (http://www.econ.uiuc.edu/~roger/research/intro/jep.pdf)
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Image from Koenker (http://www.econ.uiuc.edu/~roger/research/intro/jep.pdf)
Some examples
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Image from Koenker (http://www.econ.uiuc.edu/~roger/research/intro/jep.pdf)
Some examples
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Some examples
Image from Bitler et al. 2006 AER paper.
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Image from Bitler et al. 2006 AER paper.
Some examples
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Some examples
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Some more examples
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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).
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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.
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Technical intuitions
Image from Pindyck and Rubinfield (Econometric models and economic forecasts)
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Formulae (OLS)
ikkii Xy
iii Xy 2)(
min
)()( 1 YXXX
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Formulae (LAD)
ikkii Xy
iii Xy
min
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Formulae (LAD vs OLS)
iii Xy
min
iii Xy 2)(
min
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Formulae (LAD at ≠.5)
ikkii Xy
iiii Xyiii
Xyiii XyXy 1
min
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Formulae (LAD at ≠.5)
ikkii Xy
iiii Xyiii
Xyiii XyXy 1
min
Negative residuals Positive residuals
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Technical (semi) intuitions
Image from Koenker (http://www.econ.uiuc.edu/~roger/research/intro/jep.pdf)
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Why we might care
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Why we might care
Skewed Distributions
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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)
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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
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ccccLAD