Fixed Effects Quantile Regression

Quick Reference

Class: panelbox.models.quantile.fixed_effects.FixedEffectsQuantile Import: from panelbox.models.quantile import FixedEffectsQuantile Stata equivalent: xtqreg y x1 x2, fe R equivalent: qrpanel::qregpd(y ~ x1 + x2, panel = "fd")

Overview

Fixed Effects Quantile Regression addresses a fundamental challenge: estimating quantile functions while controlling for unobserved entity-level heterogeneity. In standard panel models, fixed effects can be consistently estimated as \(N \to \infty\) with fixed \(T\). However, in quantile regression, the incidental parameters problem prevents consistent estimation of slope coefficients when fixed effects are included as separate parameters with short panels.

Koenker (2004) proposed an elegant solution: add an \(\ell_1\) (LASSO-type) penalty on the fixed effects to shrink them toward zero. The penalized objective function is:

\[\min_{\beta, \alpha_1, \ldots, \alpha_N} \sum_{\tau \in \mathcal{T}} \sum_{i=1}^{N} \sum_{t=1}^{T} \rho_\tau(y_{it} - X_{it}'\beta_\tau - \alpha_i) + \lambda \sum_{i=1}^{N} |\alpha_i|\]

The penalty parameter \(\lambda\) controls the shrinkage of fixed effects. PanelBox selects \(\lambda\) automatically via cross-validation (lambda_fe="auto") or accepts a user-specified value.

This approach produces consistent estimates of the slope coefficients \(\beta_\tau\) while preventing the fixed effects from dominating the estimation in short panels.

Quick Example

from panelbox.core.panel_data import PanelData
from panelbox.models.quantile import FixedEffectsQuantile

# Create PanelData object
panel_data = PanelData(data=df, entity_col="firm_id", time_col="year")

# Estimate at multiple quantiles with automatic lambda selection
model = FixedEffectsQuantile(
    data=panel_data,
    formula="investment ~ value + capital",
    tau=[0.25, 0.5, 0.75],
    lambda_fe="auto",
)
results = model.fit(method="L-BFGS-B", cv_folds=5)

When to Use

• Entity heterogeneity: unobserved entity characteristics correlated with regressors
• Short panels: \(T\) is small relative to \(N\) (the penalty handles the incidental parameters problem)
• Slope coefficient focus: primary interest is in \(\beta_\tau\), not the fixed effects themselves
• Distributional analysis with controls: study how covariate effects vary across quantiles while controlling for entities
• Comparison with Canay: when the location-shift assumption is suspect

Key Assumptions

• Linear conditional quantile: \(Q_\tau(y_{it}|X_{it}, \alpha_i) = X_{it}'\beta_\tau + \alpha_i\)
• Strict exogeneity: \(E[\rho_\tau'(y_{it} - X_{it}'\beta_\tau - \alpha_i) | X_i] = 0\)
• Additive fixed effects: entity effects enter additively
• Appropriate penalty: \(\lambda\) must balance bias and variance

Detailed Guide

The Incidental Parameters Problem

In quantile regression, the objective function is not differentiable, and the fixed effects \(\alpha_i\) are estimated with error of order \(O(1/T)\). Unlike OLS where the within-transformation eliminates fixed effects, no such transformation exists for quantile regression. The bias in \(\hat{\alpha}_i\) contaminates the slope estimates \(\hat{\beta}_\tau\).

The Koenker penalty addresses this by shrinking fixed effects:

• Small \(\lambda\): estimates close to unpenalized QR with individual dummies (biased slopes)
• Large \(\lambda\): all fixed effects shrunk to zero (equivalent to pooled QR)
• Optimal \(\lambda\): balance between controlling entity heterogeneity and minimizing bias

Data Preparation

FixedEffectsQuantile requires a PanelData object:

from panelbox.core.panel_data import PanelData

# From DataFrame
panel_data = PanelData(data=df, entity_col="id", time_col="year")

# The formula specifies the model
model = FixedEffectsQuantile(
    data=panel_data,
    formula="y ~ x1 + x2",
    tau=[0.1, 0.25, 0.5, 0.75, 0.9],
    lambda_fe="auto",        # automatic cross-validation
)

Lambda Selection

The penalty parameter \(\lambda\) is critical. PanelBox provides two approaches:

Automatic (Cross-Validation)Manual

# Entity-based K-fold CV preserves panel structure
model = FixedEffectsQuantile(
    data=panel_data,
    formula="y ~ x1 + x2",
    tau=0.5,
    lambda_fe="auto",
)
results = model.fit(cv_folds=5, verbose=True)

# Check selected lambda
print(f"Optimal lambda: {results.results[0.5].lambda_fe:.4f}")

# Specify lambda directly (useful for sensitivity analysis)
model = FixedEffectsQuantile(
    data=panel_data,
    formula="y ~ x1 + x2",
    tau=0.5,
    lambda_fe=1.5,
)
results = model.fit()

The cross-validation procedure (_select_lambda_cv) works as follows:

1. Compute \(\lambda_\text{max}\): the smallest \(\lambda\) that sets all fixed effects to zero
2. Create a log-spaced grid from \(0.001 \cdot \lambda_\text{max}\) to \(\lambda_\text{max}\) (20 values)
3. Perform entity-based K-fold CV, splitting by entities to preserve panel structure
4. Select \(\lambda\) minimizing average out-of-fold check loss

Estimation

results = model.fit(
    method="L-BFGS-B",   # optimization method (quasi-Newton)
    cv_folds=5,            # folds for lambda CV
    verbose=False,         # print progress
)

Interpreting Results

# Slope coefficients for each quantile
for tau in model.tau:
    r = results.results[tau]
    print(f"tau={tau:.2f}:")
    print(f"  Coefficients: {r.params}")
    print(f"  Std Errors:   {r.std_errors}")
    print(f"  Lambda used:  {r.lambda_fe}")

# Estimated fixed effects
fe = results.results[0.5].fixed_effects  # entity-level effects

# Many will be shrunk to zero (sparse solution)
n_nonzero = np.sum(np.abs(fe) > 1e-6)
print(f"Non-zero fixed effects: {n_nonzero}/{len(fe)}")

Shrinkage Path

Examine how coefficients change across \(\lambda\) values:

# Estimate at multiple lambda values
import numpy as np

lambdas = np.logspace(-2, 2, 20)
coefs_path = []

for lam in lambdas:
    model_temp = FixedEffectsQuantile(
        data=panel_data, formula="y ~ x1 + x2",
        tau=0.5, lambda_fe=lam
    )
    res = model_temp.fit()
    coefs_path.append(res.results[0.5].params)

# Plot shrinkage path
# Stable coefficients across lambda indicate robustness

Configuration Options

Parameter	Type	Default	Description
data	PanelData	required	Panel data object
formula	str	None	Model formula "y ~ x1 + x2"
tau	float/list	0.5	Quantile level(s) in \((0, 1)\)
lambda_fe	float/str	"auto"	Penalty: "auto" for CV or a positive float

Fit Parameters

Parameter	Type	Default	Description
method	str	"L-BFGS-B"	Optimization method
cv_folds	int	5	Number of CV folds for \(\lambda\) selection
verbose	bool	False	Print estimation progress

Result Attributes

Attribute	Description
params	Estimated slope coefficients \(\hat{\beta}_\tau\)
std_errors	Standard errors
fixed_effects	Estimated entity fixed effects \(\hat{\alpha}_i\)
lambda_fe	Penalty parameter used
converged	Optimization convergence flag

Diagnostics

Sensitivity to Lambda

# Compare estimates across lambda values to assess robustness
for lam in [0.1, 1.0, 10.0]:
    m = FixedEffectsQuantile(data=panel_data, formula="y ~ x1 + x2",
                              tau=0.5, lambda_fe=lam)
    r = m.fit()
    print(f"lambda={lam:5.1f}: beta = {r.results[0.5].params}")

Comparison with Canay Two-Step

from panelbox.models.quantile import CanayTwoStep

canay = CanayTwoStep(data=panel_data, formula="y ~ x1 + x2",
                      tau=[0.25, 0.5, 0.75])
canay_results = canay.fit()

# Large differences suggest location-shift assumption is violated
comparison = canay.compare_with_penalty_method(tau=0.5)

Tutorials

Tutorial	Description	Link
FE Quantile Regression	Penalty method with lambda selection
Lambda Selection	Cross-validation and shrinkage analysis

See Also

• Pooled Quantile Regression — baseline without fixed effects
• Canay Two-Step — faster alternative under location-shift assumption
• Location-Scale Model — non-crossing quantiles with FE support
• Non-Crossing Constraints — detect and fix quantile crossing
• Diagnostics — diagnostic tests for quantile models

References

• Koenker, R. (2004). Quantile regression for longitudinal data. Journal of Multivariate Analysis, 91(1), 74-89.
• Koenker, R. (2005). Quantile Regression. Cambridge University Press.
• Lamarche, C. (2010). Robust penalized quantile regression estimation for panel data. Journal of Econometrics, 157(2), 396-408.
• Galvao, A. F., & Montes-Rojas, G. V. (2010). Penalized quantile regression for dynamic panel data. Journal of Statistical Planning and Inference, 140(11), 3476-3497.