Wage inequality across quantiles (Quantile Regression)

This illustrates quantile regression: the return to education is heterogeneous between low- and high-wage groups — something OLS (mean only) cannot reveal. Figures are illustrative.

Summary: estimate $\beta(\tau)$ of educ at quantiles $\tau$ = 0.1, 0.25, 0.5, 0.75, 0.9 to see how education affects wages differently along the distribution.

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Step 1 — Ideation

• Question: does the return to education differ between high- and low-wage groups (glass-ceiling/floor effects)?

Step 2 — Literature Review

Wage-inequality literature; quantile effects of education.

Step 3 — Data Collection

Labor micro data (lnwage, educ, exper, controls) — as in the Mincer example.

Step 4 — Modeling

Choose the Quantile regression family → Quantile, with a list of $\tau$:

$$Q_{\tau}(\ln wage_i \mid X_i) = \beta_0(\tau) + \beta_1(\tau)\,educ_i + \dots$$

Illustrative results — educ coefficient by quantile (not real results):

Quantile $\tau$	$\hat{\beta}_1(\tau)$ (educ)
0.10	0.061
0.50	0.080
0.90	0.103

Sample interpretation: the return to education rises with the quantile (0.061 → 0.103) ⇒ education widens wage inequality (a larger effect for high earners). OLS gives a single mean (~0.08) that masks this difference.

Stata

* --- Wage inequality: Quantile Regression ---
sqreg lnwage educ exper female, quantiles(.10 .25 .50 .75 .90) reps(200)

* Compare coefficients across quantiles
estimates table, stats(N)

R

# --- Wage inequality: Quantile Regression ---
library(quantreg)

taus <- c(0.10, 0.25, 0.50, 0.75, 0.90)
fit  <- rq(lnwage ~ educ + exper + female, data = df, tau = taus)
summary(fit, se = "boot", R = 200)

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

Python

import statsmodels.api as sm
import pandas as pd
import matplotlib.pyplot as plt

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

# Estimate at multiple quantiles and collect educ coefficient
taus = [0.10, 0.25, 0.50, 0.75, 0.90]
coefs = []
for t in taus:
    res = sm.QuantReg(y, X).fit(q=t)
    coefs.append(res.params['educ'])
    print(f"\n=== Quantile {t} ===")
    print(res.summary())

# Plot the quantile-process for educ
plt.plot(taus, coefs, marker='o')
plt.xlabel('Quantile'); plt.ylabel('educ coefficient')
plt.title('Return to education across wage quantiles')
plt.show()

Step 5 — Reporting

Export a report + the quantile-process plot + replication code; SE by bootstrap.

Video tutorial

Video Tutorial: Running Quantile Regression in EcoLab

See also

• Quantile Regression · Panel FE-QR · Mincer example (OLS)