Quantile regression

What is Quantile regression?

Quantile regression generalises OLS by minimising the asymmetric check function ρ_τ(u) = u·(τ − I(u < 0)) instead of squared residuals. The result is the conditional τth quantile of Y as a linear function of X. τ = 0.5 ⇒ median regression (LAD); τ = 0.1 / 0.9 ⇒ tail behaviour.

Two advantages over OLS: robust to outliers in y (the check function is bounded-derivative), and informative under heteroscedasticity (different τ give different slopes when the conditional distribution's shape changes with X).

Different from robust OLS (M-estimators): quantile regression specifically targets a conditional quantile, not just downweighting outliers around a conditional mean. Both have outlier-robust properties; QR has the additional benefit of letting you sweep τ to see how the relationship changes across the distribution.

When should I use Quantile regression?

• Skewed / heavy-tailed response distributions.
• Heteroscedasticity where the conditional spread changes with the predictor.
• When you specifically care about the median (LAD regression) rather than the mean.
• Tail-focused questions ('how does X affect the worst-20% of the response distribution?').

What data does it need?

Response + predictors + intercept toggle + quantile τ + SE method (rank-inversion / iid / nid / bootstrap / kernel).

What does it report?

β + SE / t / p (when available) + 95% CI per coefficient + Koenker-Machado pseudo-R¹.

What does it assume?

• Independent observations.
• Conditional quantile is linear in the predictors (for the targeted τ).

Formula

How do I interpret the result?

Slopes can vary across τ — significant difference indicates location-shift isn't the whole story; the conditional distribution's shape is changing with the predictor.

See also

• Linear regression
• Yuen-Welch trimmed-means t-test

References

• Koenker (2005). Quantile Regression.