Panel Quantile Regression (FE-QR)

Panel Quantile (FE-QR) extends quantile regression to panel data with fixed effects by unit. It estimates effects at different quantiles while controlling for unobserved unit characteristics — combining the strengths of FE and quantile regression.

When to use

Use it when you have panel data and want heterogeneous effects across quantiles while removing individual effects (e.g. a policy's effect on firms at different productivity levels).

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Model specification

$$Q_{\tau}(Y_{it} \mid X_{it}, \alpha_i) = X_{it} \beta(\tau) + \alpha_i$$

where $\alpha_i$ is the unit fixed effect. Common methods: Koenker (penalized FE-QR), Canay (two-step), Machado–Santos Silva (MM-QR).

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Running in EcoLab

1. Modeling module → Quantile regression family → Panel Quantile.
2. Declare entity/time, $Y$, $X$, and a list of quantiles $\tau$.
3. Run; read $\beta(\tau)$ by quantile; bootstrap SE; export the replication code.

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Replication code

Stata

* --- Panel Quantile Regression (FE-QR) ---
* Requires: ssc install xtqreg
xtset id year

* Canay (2011) two-step FE quantile regression
xtqreg y x1 x2, i(id) quantile(0.5)

* Multiple quantiles
foreach q in 0.25 0.50 0.75 {
    xtqreg y x1 x2, i(id) quantile(`q')
}

R

# --- Panel Quantile Regression (Canay 2011 approach) ---
library(quantreg)

# Step 1: Estimate FE via within-estimator (mean regression)
library(plm)
fe_model <- plm(y ~ x1 + x2, data = pdata, model = "within")
alpha_i   <- fixef(fe_model)

# Step 2: Remove fixed effects from dependent variable
pdata$y_tilde <- pdata$y - alpha_i[as.character(pdata$id)]

# Step 3: Standard quantile regression on demeaned data
fit_qr <- rq(y_tilde ~ x1 + x2, data = pdata,
             tau = c(0.25, 0.5, 0.75))
summary(fit_qr, se = "boot", R = 200)

Python

import statsmodels.api as sm
import pandas as pd

# --- Panel Quantile Regression (Canay 2011) ---
# Step 1: Demean Y by removing individual FE (within mean)
group_mean = df.groupby('id')['y'].transform('mean')
df['y_tilde'] = df['y'] - group_mean

# Step 2: Run quantile regression on demeaned data
X = sm.add_constant(df[['x1', 'x2']])
quantiles = [0.25, 0.50, 0.75]

for q in quantiles:
    res = sm.QuantReg(df['y_tilde'], X).fit(q=q)
    print(f"\n=== Quantile {q} ===")
    print(res.summary())

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Limitations

• FE-QR estimation is not unique (multiple approaches, results may differ).
• Some methods need a large $T$; computation is heavy.

Video tutorial

Video Tutorial: Running Panel Quantile Regression in EcoLab

See also

• Quantile Regression · FEM/REM · Catalog