Quantile Regression

Quantile Regression estimates the effect of regressors on different quantiles of the distribution of $Y$ — not just the mean as in OLS. It reveals how $X$ affects the "low", "median" and "high" groups differently (e.g. an effect on low-income vs high-income individuals).

When to use

Use Quantile Regression when you care about heterogeneous effects across the distribution, or when $Y$ is skewed/has outliers so the OLS mean is unrepresentative. Median regression ($\tau=0.5$) is more robust to outliers than OLS.

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Model specification

The $\tau$-th conditional quantile of $Y$ given $X$:

$$Q_{\tau}(Y_i \mid X_i) = X_i \beta(\tau)$$

Estimated by minimizing the asymmetrically weighted absolute errors (check function $\rho_\tau$):

$$\hat{\beta}(\tau) = \arg\min_{\beta} \sum_{i} \rho_{\tau}\big(Y_i - X_i\beta\big), \quad \rho_\tau(u) = u\,(\tau - \mathbb{1}[u<0])$$

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Running in EcoLab

1. Modeling module → Quantile regression family → Quantile.
2. Select $Y$, the $X$ variables, and a list of quantiles $\tau$ (e.g. 0.1, 0.25, 0.5, 0.75, 0.9).
3. Run; read $\beta(\tau)$ per quantile + the quantile-process plot; bootstrap SE; export the replication code.

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Replication code

Stata

* --- Quantile Regression ---
* Simultaneous quantile regression with bootstrap SE
sqreg lnwage educ exper, quantiles(0.25 0.5 0.75) reps(100)

* View results for each quantile
estimates table, stats(N)

R

# --- Quantile Regression ---
library(quantreg)

# Estimate at multiple quantiles
fit <- rq(lnwage ~ educ + exper, tau = c(0.25, 0.5, 0.75),
          data = df)
summary(fit, se = "boot", R = 200)

# Quantile-process plot
plot(summary(rq(lnwage ~ educ + exper, tau = seq(0.05, 0.95, 0.05),
                data = df), se = "boot"))

Python

import statsmodels.api as sm
import pandas as pd

X = sm.add_constant(df[['educ', 'exper']])
y = df['lnwage']

# Estimate at multiple quantiles
quantiles = [0.25, 0.50, 0.75]
results = {}
for q in quantiles:
    model = sm.QuantReg(y, X)
    res = model.fit(q=q)
    results[q] = res
    print(f"\n=== Quantile {q} ===")
    print(res.summary())

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Limitations

• SE typically requires bootstrap; heavier computation than OLS.
• Interpreting many quantiles is more complex than a single mean coefficient.

Video tutorial

Video Tutorial: Running Quantile Regression in EcoLab

See also

• Panel Quantile (FE-QR) · OLS · Catalog