Quantile Treatment Effects¶

Quick Reference

Class: panelbox.models.quantile.treatment_effects.QuantileTreatmentEffects Import: from panelbox.models.quantile import QuantileTreatmentEffects Stata equivalent: qte y, treatment(d) quantiles(0.25 0.5 0.75) R equivalent: qte::qte(), DRDID::drdid()

Overview¶

Quantile Treatment Effects (QTE) extend traditional Average Treatment Effects (ATE) by estimating how a treatment affects different parts of the outcome distribution. While ATE answers "what is the average effect?", QTE answers "who benefits most and who benefits least?"

For a binary treatment \(D \in \{0, 1\}\), the QTE at quantile \(\tau\) is:

\[QTE(\tau) = Q_{Y(1)}(\tau) - Q_{Y(0)}(\tau)\]

where \(Y(1)\) and \(Y(0)\) are potential outcomes under treatment and control.

This reveals crucial heterogeneity that mean-based methods miss. For example, a job training program might:

Increase wages by 5% at the 10^th percentile (helps the lowest earners most)
Increase wages by 2% at the 90^th percentile (modest effect for high earners)
Show an ATE of 3% that masks this distributional heterogeneity

PanelBox implements four QTE estimation methods, each suited to different identification strategies.

Quick Example¶

import numpy as np
from panelbox.models.quantile import QuantileTreatmentEffects

qte = QuantileTreatmentEffects(
    data=df,
    outcome="wage",
    treatment="trained",
    covariates=["experience", "education"],
    entity_col="worker_id",
    time_col="year",
)

results = qte.estimate_qte(
    tau=[0.1, 0.25, 0.5, 0.75, 0.9],
    method="standard",
)

When to Use¶

Heterogeneous effects: treatment effects differ across the outcome distribution
Policy targeting: identify who benefits most from an intervention
Inequality analysis: understand how a policy affects distributional spread
Welfare analysis: evaluate if a program compresses or expands the distribution
Program evaluation: go beyond mean impact to characterize the full effect profile

Key Assumptions

Assumptions depend on the chosen method:

Standard QTE: conditional independence (selection on observables), common support
DiD QTE: parallel trends in quantiles (pre-treatment trends are common)
CiC: monotonicity of outcomes in unobservables, time invariance of unobservables distribution
Unconditional QTE: correct specification of the RIF regression

Detailed Guide¶

Conditional vs Unconditional QTE¶

Two distinct concepts:

Type	Definition	Interpretation
Conditional QTE	Effect at quantiles of \(Y\\|X\)	Effect for individuals at a given covariate-conditional quantile
Unconditional QTE	Effect at quantiles of \(Y\)	Effect at a given point in the marginal distribution

The unconditional QTE is often more policy-relevant: it tells us "what happens at the 10^th percentile of the wage distribution?" rather than "what happens for the lowest earner conditional on their education level?"

Method 1: Standard QTE¶

Estimates QTE by comparing conditional quantile regressions for treated and control groups:

results = qte.estimate_qte(
    tau=[0.1, 0.25, 0.5, 0.75, 0.9],
    method="standard",
)

# Access results
for tau in [0.1, 0.25, 0.5, 0.75, 0.9]:
    qte_est = results.qte_results[tau]
    print(f"tau={tau:.2f}: QTE = {qte_est['qte']:.4f} "
          f"(SE = {qte_est['se']:.4f})")

When: cross-sectional setting with good control variables (selection on observables).

Method 2: Unconditional QTE via RIF Regression¶

Uses the Recentered Influence Function (Firpo, 2007) to estimate marginal quantile effects:

results_unconditional = qte.estimate_qte(
    tau=[0.1, 0.25, 0.5, 0.75, 0.9],
    method="unconditional",
)

The RIF approach:

Compute the RIF for each quantile: \(RIF(y; Q_\tau) = Q_\tau + \frac{\tau - \mathbb{1}\{y \leq Q_\tau\}}{f_Y(Q_\tau)}\)
Regress RIF on treatment and covariates
The treatment coefficient is the unconditional QTE

When: policy-relevant marginal effects are needed.

Method 3: Difference-in-Differences QTE¶

For panel data with treatment and control groups observed over time:

results_did = qte.estimate_qte(
    tau=[0.1, 0.25, 0.5, 0.75, 0.9],
    method="did",
)

The DiD QTE extends the parallel trends assumption to quantiles:

\[QTE_{DiD}(\tau) = [Q_{Y,1}^{treat}(\tau) - Q_{Y,0}^{treat}(\tau)] - [Q_{Y,1}^{control}(\tau) - Q_{Y,0}^{control}(\tau)]\]

When: panel data with pre/post treatment periods and a control group.

Method 4: Changes-in-Changes (Athey & Imbens 2006)¶

A nonlinear generalization of DiD that relaxes parallel trends:

results_cic = qte.estimate_qte(
    tau=[0.1, 0.25, 0.5, 0.75, 0.9],
    method="cic",
)

The CiC method assumes:

Outcomes are monotone functions of a scalar unobservable: \(Y = h(U, T)\)
The distribution of \(U\) is time-invariant within groups
The production function \(h\) satisfies \(\partial h / \partial U > 0\)

When: parallel trends in quantiles is doubtful but monotonicity is reasonable.

Comparing Methods¶

import pandas as pd

tau_grid = [0.1, 0.25, 0.5, 0.75, 0.9]

comparison = pd.DataFrame({
    "tau": tau_grid,
    "Standard": [qte.estimate_qte(tau=t, method="standard")
                  .qte_results[t]["qte"] for t in tau_grid],
    "Unconditional": [qte.estimate_qte(tau=t, method="unconditional")
                       .qte_results[t]["qte"] for t in tau_grid],
})
print(comparison)

Interpreting Results¶

The QTEResult object provides:

# QTE at each quantile
results.qte_results[0.5]["qte"]     # point estimate
results.qte_results[0.5]["se"]      # standard error
results.qte_results[0.5]["ci"]      # confidence interval

# Overall heterogeneity
qte_values = [results.qte_results[t]["qte"] for t in tau_grid]
heterogeneity = np.std(qte_values)
print(f"QTE heterogeneity (std): {heterogeneity:.4f}")

Interpretation guide:

Pattern	Interpretation
QTE constant across \(\tau\)	Homogeneous effect (ATE suffices)
QTE increasing in \(\tau\)	Larger effects at the top of the distribution
QTE decreasing in \(\tau\)	Larger effects at the bottom (pro-poor)
QTE positive at low \(\tau\), negative at high \(\tau\)	Compresses the distribution
QTE negative at low \(\tau\), positive at high \(\tau\)	Expands the distribution

Configuration Options¶

Parameter	Type	Default	Description
`data`	DataFrame/PanelData	required	Data (panel or cross-sectional)
`outcome`	str	required	Outcome variable name
`treatment`	str	required	Binary treatment variable name
`covariates`	list	`None`	Control variable names
`entity_col`	str	`None`	Entity identifier (for panel data)
`time_col`	str	`None`	Time identifier (for panel data)

Estimation Parameters¶

Parameter	Type	Default	Description
`tau`	float/array	`0.5`	Quantile level(s) in \((0, 1)\)
`method`	str	`"standard"`	Method: `"standard"`, `"unconditional"`, `"did"`, `"cic"`

Method Selection Guide¶

Scenario	Recommended Method
Cross-sectional with controls	`"standard"`
Policy-relevant marginal effects	`"unconditional"`
Panel with common quantile trends	`"did"`
Panel without parallel trends	`"cic"`
Need non-crossing guarantee	Use Location-Scale instead

Diagnostics¶

Testing for Heterogeneous Effects¶

# Compare QTE across quantiles
tau_grid = [0.1, 0.25, 0.5, 0.75, 0.9]
qte_values = [results.qte_results[t]["qte"] for t in tau_grid]

# If QTE is approximately constant, ATE may suffice
from scipy import stats
# Rough test: are QTE values significantly different from their mean?
qte_mean = np.mean(qte_values)
qte_se = [results.qte_results[t]["se"] for t in tau_grid]
print(f"Mean QTE: {qte_mean:.4f}")
print(f"QTE range: [{min(qte_values):.4f}, {max(qte_values):.4f}]")

Checking for Crossing¶

from panelbox.models.quantile import QuantileMonotonicity

# If QTE curves cross, apply monotonicity correction
# (relevant when covariates are included)

Sensitivity Analysis¶

# Try different specifications
specs = [
    ["experience"],
    ["experience", "education"],
    ["experience", "education", "age"],
]

for covs in specs:
    qte_temp = QuantileTreatmentEffects(
        data=df, outcome="wage", treatment="trained",
        covariates=covs, entity_col="id", time_col="year"
    )
    r = qte_temp.estimate_qte(tau=0.5, method="standard")
    print(f"Covariates {covs}: QTE(0.5) = {r.qte_results[0.5]['qte']:.4f}")

Tutorials¶

Tutorial	Description	Link
QTE Basics	Standard and unconditional QTE estimation
QTE with DiD	Panel DiD and Changes-in-Changes methods

References¶

Firpo, S. (2007). Efficient semiparametric estimation of quantile treatment effects. Econometrica, 75(1), 259-276.
Firpo, S., Fortin, N. M., & Lemieux, T. (2009). Unconditional quantile regressions. Econometrica, 77(3), 953-973.
Athey, S., & Imbens, G. W. (2006). Identification and inference in nonlinear difference-in-differences models. Econometrica, 74(2), 431-497.
Callaway, B., & Li, T. (2019). Quantile treatment effects in difference in differences models with panel data. Quantitative Economics, 10(4), 1579-1618.
Chernozhukov, V., & Hansen, C. (2005). An IV model of quantile treatment effects. Econometrica, 73(1), 245-261.