Quantile Regression

Summary

Quantile regression models the effect of covariates on different parts of the outcome distribution, not just the mean. It reveals whether treatment compresses or expands the distribution — information invisible to standard (mean) regression.

Motivation

• 95% of applied econometrics focuses on averages, but distributions matter
• Wage inequality: upper quantiles rising, lower quantiles falling
• A training program might raise average wages but only help those at the top

The Quantile Regression Model

The $\tau$-th conditional quantile:

$$Q_{\tau }(Y_i|X_i)=X_i^{\prime }\beta (\tau )$$

Estimated by minimizing:

$$\underset{b}{\min }\sum _i{\rho }_{\tau }(Y_i-X_i^{\prime }b)$$

where ${\rho }_{\tau }(u)=u(\tau -1(u<0))$ is the check function.

Quantile Treatment Effects (QTE)

For binary treatment $D_i$:

$$QTE(\tau )=Q_{\tau }(Y_{1i})-Q_{\tau }(Y_{0i})$$

QTE ≠ Effect on Individuals at Quantile τ

The QTE compares the τ-th quantile of the treated distribution with the τ-th quantile of the untreated distribution. The people at quantile τ may be different individuals in each group.

The Approximation Property

Just as regression approximates the CEF, quantile regression approximates the conditional quantile function — even when the linear model is misspecified, it provides a useful weighted average of quantile partial effects.

See Also

• Regression and the CEF — mean regression as the baseline to compare against
• Local Average Treatment Effects — LATE vs QTE: both are local/distributional, not ATE
• Bayesian Linear Regression — Bayesian quantile regression uses asymmetric Laplace likelihood
• Discrete Choice Models — another approach to modeling non-mean outcomes
• Mostly Harmless Econometrics - Overview