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)

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)()( 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

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min

Formulae (LAD at ≠.5)

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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

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