Quantile Regression

Definition

A regression framework introduced by Koenker and Bassett (1978) that models the conditional quantiles of a response variable rather than the conditional mean. By minimising a weighted sum of absolute residuals, quantile regression estimates the relationship between covariates and any quantile of interest (e.g., the 10th percentile or the 90th). This is particularly valuable when the effects of covariates differ across the distribution — for example, the factors that affect wages may differ for low earners versus high earners. Quantile regression is robust to outliers and does not require assumptions about the error distribution.

Why It Matters

Ordinary regression describes the average relationship, which can mask heterogeneous effects across the distribution. Quantile regression reveals how predictors influence outcomes at every point — from the lowest to the highest — enabling richer policy analysis and more nuanced scientific conclusions.

Example

A labour economist studying returns to education finds that each additional year of schooling raises the median wage by $3,000 but raises the 90th percentile wage by $7,000. Quantile regression uncovers this heterogeneity, which a mean regression would obscure entirely.

Related Terms

• Quantile
• Ordinary Least Squares (OLS)
• Robust Standard Errors

Software Notes

• SPSS: Not built-in; use the QUANTREG R extension or manual computation
• R: rq() from the quantreg package; e.g., rq(y ~ x, tau = c(0.10, 0.50, 0.90))
• Stata: qreg y x, quantile(0.5) for median regression; sqreg for simultaneous quantiles