# Quantile Regression The intuition Hypothetical Distributions The intuition

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)

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)

Some examples Image from Bitler et al. 2006 AER paper.

Some examples Image from Bitler et al. 2006 AER paper.

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, and 0. 10 regression quantile estimates (solid lines) and least squares regression estimate (dashed line) for the model y = b 0 + b 1 X +e. (From Cade and Noon, 2003).

Some examples Quantile regression was used to estimate changes in Lahontan cutthroat trout density (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 = b 0 + b 1 X +e. From Cade and Noon, 2003.

Technical intuitions Image from Pindyck and Rubinfield (Econometric models and economic forecasts)

Formulae (OLS)

Formulae (LAD)

Formulae (LAD vs OLS)

Formulae (LAD at t≠. 5)

Formulae (LAD at t≠. 5) 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

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