MethodAtlas
Model-BasedModern

Quantile Treatment Effects (QTE)

Estimates how treatment shifts the entire outcome distribution, revealing heterogeneous effects across quantiles that average effects conceal.

Quick Reference

When to Use
When you suspect the treatment effect varies across the outcome distribution and average effects mask important heterogeneity.
Key Assumption
For conditional QTE: correct specification of the conditional quantile function. For unconditional QTE: the recentered influence function correctly linearizes the quantile functional.
Common Mistake
Interpreting conditional quantile regression coefficients as effects on unconditional quantiles. Use RIF regression for unconditional quantile effects.
Estimated Time
3 hours

One-Line Implementation

Stata: sqreg y treatment x1 x2, q(.1 .25 .5 .75 .9) reps(500)
R: rq(y ~ treatment + x1 + x2, tau = c(0.1, 0.25, 0.5, 0.75, 0.9), data = df)
Python: QuantReg(df['y'], df[['const','treatment','x1','x2']]).fit(q=0.5)

Download Full Analysis Code

Complete scripts with diagnostics, robustness checks, and result export.

Motivating Example

A state government introduces a subsidized job training program for unemployed workers. A randomized evaluation finds that the program increases average earnings by $1,200 per year -- a statistically significant and seemingly encouraging result. Policymakers celebrate.

But a closer look at the data reveals something the average conceals. Workers at the bottom of the earnings distribution -- those who struggled most before the program -- saw their earnings increase by $3,500. Workers at the top of the distribution, however, saw their earnings decrease by $800, possibly because they were diverted from better opportunities into the standardized program. The average of $1,200 is a weighted blend of a large positive effect at the bottom and a modest negative effect at the top. No single individual experienced the "average" effect.

This scenario is exactly the pattern studied by (Bitler et al., 2006), who analyzed the Connecticut Jobs First welfare reform experiment and found that mean impacts masked dramatic distributional heterogeneity. The program raised earnings for women at the lower quantiles and lowered them at the upper quantiles. A policymaker who saw only the mean effect would have drawn the wrong conclusions about who benefits and who is harmed.

solve this problem by estimating how the treatment shifts each quantile of the outcome distribution separately. Instead of asking "what is the average effect?", QTE asks "what is the effect at the 10th percentile? The 25th? The median? The 90th?" The result is a quantile process -- a curve showing how the treatment effect varies across the entire distribution. When this curve is flat, the average treatment effect tells the whole story. When it is not flat, the average is misleading, and QTE reveals the heterogeneity that average effects conceal.

A. Overview

What Quantile Treatment Effects Do

Standard regression methods -- OLS, difference-in-differences, IV -- estimate the effect of a treatment on the mean of the outcome distribution. This quantity is the (ATE). But the mean can hide as much as it reveals. A treatment that helps the poor and hurts the rich, a drug that cures some patients and harms others, a policy that compresses or widens the wage distribution -- all of these produce the same ATE if the gains and losses happen to average out.

Quantile treatment effects estimate the impact of a treatment at every point of the outcome distribution. The key object is the quantile process:

QTE(τ)=Qτ(Y(1))−Qτ(Y(0)),τ∈(0,1)\text{QTE}(\tau) = Q_\tau(Y(1)) - Q_\tau(Y(0)), \quad \tau \in (0, 1)

where Qτ(Y(d))Q_\tau(Y(d)) is the τ\tau-th quantile of the potential outcome distribution under treatment status dd. At τ=0.5\tau = 0.5, this is the effect on the median. At τ=0.1\tau = 0.1, it is the effect on the 10th percentile -- the bottom of the distribution. At τ=0.9\tau = 0.9, it is the effect on the 90th percentile -- the top.

Conditional vs. Unconditional QTE

There is a critical distinction between two types of quantile effects that researchers frequently confuse:

Conditional quantile effects (Koenker and Bassett, 1978) estimate Qτ(Y∣X=x)Q_\tau(Y | X = x) -- the τ\tau-th quantile of YY conditional on covariates. Standard targets this object. The treatment coefficient in a conditional quantile regression measures how treatment shifts the conditional quantile function at a given point in the covariate space.

(Firpo, Fortin & Lemieux, 2009) estimate the effect on the τ\tau-th quantile of the marginal (population) distribution of YY, integrating over the covariate distribution. These are the quantities that matter for policy -- "did the program raise the 10th percentile of wages in the population?" -- but standard quantile regression does not estimate them. is needed instead.

When to Use QTE

  • You suspect the treatment effect varies across the outcome distribution -- e.g., a training program helps low-earners more than high-earners
  • You care about distributional outcomes like inequality, poverty rates, or tail behavior
  • You want to know whether a treatment compresses or widens the outcome distribution
  • The average effect is small or zero, but you suspect offsetting effects at different parts of the distribution

When NOT to Use QTE

  • You are interested in who is affected (by observed characteristics) -- use subgroup analysis or estimation instead
  • Your outcome is binary or count-valued -- quantile regression is designed for continuous outcomes
  • You have very small sample sizes -- quantile regression at extreme quantiles requires substantial data in the tails
  • The treatment is endogenous and you lack instruments -- standard QR does not solve endogeneity (use IV-QR: (Chernozhukov & Hansen, 2005))

Common Confusions

When to Use

  1. The treatment effect likely varies across the distribution. Programs targeting disadvantaged populations, policies affecting wages or income, interventions with heterogeneous compliance -- these naturally produce distributional effects.

  2. You care about inequality or distributional outcomes. Does a minimum wage increase compress the wage distribution? Does a tax policy reduce the gap between the 90th and 10th percentile? QTE directly answers these questions.

  3. The average effect is zero or small but may mask offsetting effects. A drug with zero average effect might cure some patients and harm others. QTE reveals this pattern.

  4. You want a complete picture of the treatment. Reporting the full quantile process provides much richer information than a single ATE, especially for policy evaluation (Bitler et al., 2006).

Do NOT Use QTE When:

  1. Your outcome is discrete or binary. Quantile regression is designed for continuous outcomes. For binary outcomes, use subgroup analysis. For count outcomes, methods are limited.

  2. You want heterogeneity by observed characteristics. QTE reveals heterogeneity across the outcome distribution, not across subgroups. For subgroup effects, use interaction terms, , or causal forests.

  3. You have very small samples. Extreme quantiles (0.05, 0.95) require substantial observations in the tails. With n=100n = 100, the 5th percentile is based on only 5 observations.

  4. Endogeneity is a concern and you lack instruments. Standard quantile regression does not solve endogeneity. Use the IV-QR approach of (Chernozhukov & Hansen, 2005) if you have valid instruments.

Connection to Other Methods

  • OLS Regression: OLS estimates the effect on the conditional mean. QTE estimates effects on conditional (or unconditional) quantiles. When the treatment effect is constant (homogeneous), OLS and QTE at every quantile give the same answer.

  • Causal Forests: Causal forests estimate heterogeneous treatment effects by observed covariates (CATE). QTE estimates heterogeneity across the outcome distribution. These are complementary -- CATE tells you who is helped; QTE tells you where in the distribution the shift occurs.

  • Experimental Design: Randomization solves the selection problem for QTE just as it does for ATE. With experimental data, quantile regression identifies causal QTE without additional assumptions beyond randomization.

  • IV / 2SLS: Standard QR does not handle endogeneity. (Chernozhukov & Hansen, 2005) develop IV methods for quantile regression. The instrumental variable quantile regression (IVQR) model identifies quantile treatment effects when treatment is endogenous.

B. Identification

For quantile treatment effects to have a causal interpretation, specific assumptions must hold depending on the research design and the type of QTE being estimated.

Assumption 1: Unconfoundedness (for observational data)

Plain language: Conditional on observed covariates XX, treatment assignment is independent of potential outcomes at each quantile. This condition is the same as in OLS, but applied quantile by quantile.

Formally: Qτ(Y(d)∣X,D)=Qτ(Y(d)∣X)Q_\tau(Y(d) | X, D) = Q_\tau(Y(d) | X) for d∈{0,1}d \in \{0, 1\} and all τ∈(0,1)\tau \in (0, 1).

Under experimental data (randomization), this assumption holds by design. Under observational data, it requires the same selection-on-observables argument as for the ATE.

Assumption 2: Correct Specification of the Conditional Quantile Function

Plain language: The linear quantile regression model correctly specifies how covariates shift the conditional quantile function. Unlike OLS, where linearity is guaranteed to provide the best linear approximation to the CEF, quantile regression is only consistent for the conditional quantile function if the model is correctly specified.

Formally: Qτ(Y∣X)=X′β(τ)Q_\tau(Y | X) = X'\beta(\tau) for each τ\tau.

(Angrist et al., 2006) show that under misspecification, quantile regression coefficients can still be interpreted as weighted averages of conditional quantile effects, but the weights depend on the specification and may not correspond to the desired estimand.

Assumption 3: Rank Invariance (for individual-level interpretation)

Plain language: Treatment does not change an individual's rank in the outcome distribution. The person at the 30th percentile without treatment would also be at the 30th percentile with treatment (just at a different level). This allows interpreting QTE as individual-level treatment effects.

Formally: FY(1)(Yi(1))=FY(0)(Yi(0))F_{Y(1)}(Y_i(1)) = F_{Y(0)}(Y_i(0)) for all ii.

Rank invariance is a very strong assumption. It is plausible when treatment is a small perturbation (e.g., a modest wage subsidy) but implausible when treatment fundamentally reshuffles outcomes (e.g., a job training program that transforms some workers' career trajectories while leaving others unchanged). Without rank invariance, QTE identifies distributional effects but not individual-level effects.

C. Visual Intuition

Adjust the treatment effects at different quantiles to see how the QTE curve departs from the constant ATE. When the treatment compresses the distribution (helping the bottom more than the top), the QTE curve slopes downward and the ATE masks important heterogeneity.

Interactive Simulation

Quantile Treatment Effects vs. Constant ATE

DGP: Y(0) ~ N(50, 100). Treatment effect varies linearly from 3.0 at the bottom to 0.5 at the top. True ATE = 1.8. N = 500.

-3.4-1.30.93.05.27.30.100.250.500.750.90Quantile (τ)Treatment Effect
OLS / ATEQTE curve95% CI band

Estimation Results

Estimatorβ̂SE95% CIBias
OLS / ATEclosest1.9780.893[0.23, 3.73]+0.228
QTE at median3.0361.209[0.67, 5.41]+1.286
True β1.750———
500

Total observations (half treated, half control)

3.00

Treatment effect at the lowest quantile

0.50

Treatment effect at the highest quantile

2.00

Additional measurement noise

Why the difference?

The treatment effect varies substantially across quantiles: 3.0 at the bottom vs. 0.5 at the top. The QTE curve slopes downward, revealing heterogeneity that the single ATE (β̂ = 1.98) conceals. The treatment compresses the distribution by helping those at the bottom more than those at the top. Policymakers who rely on the ATE alone would miss this redistributive pattern.

D. Mathematical Derivation

The Quantile Regression Estimator

Don't worry about the notation yet — here's what this means in words: Quantile regression estimates conditional quantile functions by minimizing an asymmetric loss function. The estimator is consistent for the linear conditional quantile model under correct specification.

Setup. We want to estimate Qτ(Y∣X)=X′β(τ)Q_\tau(Y | X) = X'\beta(\tau) for a given τ∈(0,1)\tau \in (0, 1).

Step 1: The check function. Define the check function (also called the pinball loss):

ρτ(u)=u⋅(τ−1(u<0))={τ⋅uif u≥0(τ−1)⋅uif u<0\rho_\tau(u) = u \cdot (\tau - \mathbb{1}(u < 0)) = \begin{cases} \tau \cdot u & \text{if } u \geq 0 \\ (\tau - 1) \cdot u & \text{if } u < 0 \end{cases}

This function penalizes positive residuals by τ\tau and negative residuals by (1−τ)(1 - \tau). At τ=0.5\tau = 0.5, it reduces to the absolute value (median regression). At τ=0.9\tau = 0.9, it penalizes under-predictions (positive residuals) nine times more heavily than over-predictions.

Step 2: The optimization problem. The quantile regression estimator β^(τ)\hat{\beta}(\tau) solves:

β^(τ)=arg⁡min⁡β∑i=1nρτ(Yi−Xi′β)\hat{\beta}(\tau) = \arg\min_{\beta} \sum_{i=1}^{n} \rho_\tau(Y_i - X_i'\beta)

This minimization is a linear programming problem and can be solved efficiently using simplex or interior point methods (Koenker & Bassett, 1978).

Step 3: Asymptotic distribution. Under regularity conditions:

n(β^(τ)−β(τ))→dN(0,τ(1−τ)fY∣X(Qτ(Y∣X))2(X′X)−1)\sqrt{n}(\hat{\beta}(\tau) - \beta(\tau)) \xrightarrow{d} N\left(0, \frac{\tau(1-\tau)}{f_{Y|X}(Q_\tau(Y|X))^2} (X'X)^{-1}\right)

where fY∣Xf_{Y|X} is the conditional density of YY given XX evaluated at the τ\tau-th quantile. The "sparsity" function 1/fY∣X(Qτ)1/f_{Y|X}(Q_\tau) replaces the variance σ2\sigma^2 that appears in OLS asymptotics. Estimation of fY∣Xf_{Y|X} at the quantile is a nuisance -- bootstrap standard errors are generally preferred in practice.

Step 4: Comparison with OLS. OLS minimizes ∑i(Yi−Xi′β)2\sum_i (Y_i - X_i'\beta)^2, giving the conditional mean E[Y∣X]E[Y|X]. Quantile regression minimizes ∑iρτ(Yi−Xi′β)\sum_i \rho_\tau(Y_i - X_i'\beta), giving the conditional τ\tau-th quantile. The key difference: OLS produces a single β\beta vector; quantile regression produces a function β(τ)\beta(\tau) that varies with τ\tau.

RIF Regression for Unconditional Quantile Effects

Don't worry about the notation yet — here's what this means in words: RIF regression linearizes the quantile functional so that OLS on the transformed outcome recovers the unconditional quantile effect.

The problem. Standard quantile regression estimates the conditional quantile effect: how XX shifts Qτ(Y∣X)Q_\tau(Y | X). But policymakers typically want the unconditional quantile effect: how XX shifts Qτ(Y)Q_\tau(Y), the τ\tau-th quantile of the population distribution. These are not the same -- the law of iterated expectations does not hold for quantiles: EX[Qτ(Y∣X)]≠Qτ(Y)E_X[Q_\tau(Y|X)] \neq Q_\tau(Y).

Step 1: The influence function. The influence function of the τ\tau-th quantile functional is:

IF(Y;Qτ,FY)=τ−1(Y≤Qτ)fY(Qτ)IF(Y; Q_\tau, F_Y) = \frac{\tau - \mathbb{1}(Y \leq Q_\tau)}{f_Y(Q_\tau)}

where fY(Qτ)f_Y(Q_\tau) is the density of YY evaluated at QτQ_\tau.

Step 2: Recentering. The RIF adds back the quantile itself (Firpo et al., 2009):

RIF(Y;Qτ)=Qτ+τ−1(Y≤Qτ)fY(Qτ)RIF(Y; Q_\tau) = Q_\tau + \frac{\tau - \mathbb{1}(Y \leq Q_\tau)}{f_Y(Q_\tau)}

The key property: E[RIF(Y;Qτ)]=QτE[RIF(Y; Q_\tau)] = Q_\tau. This property means that the RIF is a transformation of YY whose expectation equals the distributional statistic of interest.

Step 3: OLS on the RIF. Regressing RIF(Y;Qτ)RIF(Y; Q_\tau) on XX by OLS gives:

E[RIF(Y;Qτ)∣X]≈X′γ(τ)E[RIF(Y; Q_\tau) | X] \approx X'\gamma(\tau)

The coefficients γ(τ)\gamma(\tau) have the interpretation of unconditional quantile partial effects -- the marginal effect of a small change in XX on the τ\tau-th quantile of the unconditional distribution of YY.

Step 4: Implementation. In practice:

  1. Estimate Q^τ\hat{Q}_\tau as the sample τ\tau-th quantile
  2. Estimate f^Y(Q^τ)\hat{f}_Y(\hat{Q}_\tau) using kernel density estimation
  3. Compute RIF^i=Q^τ+(τ−1(Yi≤Q^τ))/f^Y(Q^τ)\widehat{RIF}_i = \hat{Q}_\tau + (\tau - \mathbb{1}(Y_i \leq \hat{Q}_\tau)) / \hat{f}_Y(\hat{Q}_\tau)
  4. Run OLS of RIF^\widehat{RIF} on XX to get γ^(τ)\hat{\gamma}(\tau)

E. Implementation

Step-by-Step Workflow

  1. Estimate OLS first. Always report the ATE as a benchmark. The ATE is the effect on the conditional mean and serves as the reference for the quantile analysis.

  2. Run conditional quantile regression at the standard quantiles {0.10,0.25,0.50,0.75,0.90}\{0.10, 0.25, 0.50, 0.75, 0.90\}. Use bootstrap standard errors (not asymptotic) -- they are more reliable in practice.

  3. Plot the quantile process. Estimate at a fine grid (every 5th percentile from 0.05 to 0.95) and plot the treatment coefficient with 95% confidence bands. Overlay the OLS estimate as a horizontal dashed line. If the QTE curve lies entirely within the OLS confidence interval, there is no evidence of distributional heterogeneity.

  4. Test for heterogeneity. Use the Wald test from simultaneous quantile regression (sqreg in Stata) to test H0:βtreatment(0.10)=βtreatment(0.25)=⋯=βtreatment(0.90)H_0: \beta_{\text{treatment}}(0.10) = \beta_{\text{treatment}}(0.25) = \cdots = \beta_{\text{treatment}}(0.90).

  5. Estimate unconditional quantile effects if the policy question concerns population quantiles. Use RIF regression (Firpo-Fortin-Lemieux). Compare conditional and unconditional estimates -- large differences indicate that covariates importantly reshape the mapping from conditional to unconditional quantiles.

  6. Check for quantile crossing. Verify that fitted quantile lines do not cross: Q^0.10(Y∣X)<Q^0.50(Y∣X)<Q^0.90(Y∣X)\hat{Q}_{0.10}(Y|X) < \hat{Q}_{0.50}(Y|X) < \hat{Q}_{0.90}(Y|X) for all observations. Crossing indicates model misspecification.

  7. Report the interquantile range effect. The difference QTE(0.90)−QTE(0.10)\text{QTE}(0.90) - \text{QTE}(0.10) tells you whether the treatment compresses (negative) or widens (positive) the distribution.

Standard Errors and Inference

  • Bootstrap SEs are preferred for conditional quantile regression. The asymptotic variance involves the conditional density fY∣Xf_{Y|X} at the quantile, which is difficult to estimate reliably.
  • For simultaneous quantile regression (sqreg in Stata), the bootstrap is performed jointly across quantiles, enabling valid cross-quantile tests.
  • For RIF regression, standard OLS standard errors are valid because the RIF-transformed outcome is treated as the dependent variable in an OLS regression. Heteroscedasticity-robust SEs are recommended.
  • For extreme quantiles (τ<0.05\tau < 0.05 or τ>0.95\tau > 0.95), inference is unreliable in moderate samples. Report these with appropriate caveats or omit them.
library(quantreg)

# Quantile regression at multiple quantiles
taus <- c(0.10, 0.25, 0.50, 0.75, 0.90)
qr_fits <- lapply(taus, function(tau) {
rq(y ~ treatment + x1 + x2, tau = tau, data = df)
})

# Summary with bootstrap SEs
lapply(qr_fits, function(fit) summary(fit, se = "boot", R = 500))

# Quantile process plot
qr_full <- rq(y ~ treatment + x1 + x2, tau = seq(0.05, 0.95, 0.05), data = df)
plot(summary(qr_full, se = "boot"))

F. Diagnostics

F.1 Quantile Crossing

If the estimated conditional quantile at τ1\tau_1 exceeds the estimated conditional quantile at τ2>τ1\tau_2 > \tau_1 for some observations, the model is misspecified. This crossing means the linear model for Qτ(Y∣X)=X′β(τ)Q_\tau(Y|X) = X'\beta(\tau) implies a negative conditional density for some covariate values, which is impossible.

What to do: Check the fraction of observations with crossings. A small fraction (< 5%) is common and not alarming. Extensive crossing suggests that a linear specification is inadequate -- consider adding interactions, polynomial terms, or using a more flexible model.

F.2 Heterogeneity Test

Test whether the treatment effect is constant across quantiles. If the Wald test from sqreg fails to reject H0:β(τ1)=β(τ2)=⋯=β(τK)H_0: \beta(\tau_1) = \beta(\tau_2) = \cdots = \beta(\tau_K), there is no statistical evidence that QTE adds information beyond the ATE.

F.3 Goodness of Fit

There is no R2R^2 analog for quantile regression with a universally agreed interpretation. The pseudo-R2R^2 proposed by Koenker and Machado (1999) -- 1−V^(τ)/V~(τ)1 - \hat{V}(\tau) / \tilde{V}(\tau), where V^\hat{V} is the minimized check function and V~\tilde{V} is the check function from the intercept-only model -- can be reported but should not be over-interpreted.

F.4 Sensitivity to Bandwidth (RIF Regression)

RIF regression requires a kernel density estimate f^Y(Qτ)\hat{f}_Y(Q_\tau). The choice of bandwidth affects the RIF values and hence the estimated unconditional quantile effects. Report sensitivity to different bandwidth choices (e.g., Silverman's rule of thumb, oversmoothed bandwidth, and half the default bandwidth).

Reading the Quantile Process

The quantile process plot is the primary tool for communicating QTE results. It shows the treatment coefficient β^treatment(τ)\hat{\beta}_{\text{treatment}}(\tau) on the vertical axis and the quantile τ\tau on the horizontal axis, with a 95% confidence band.

Key patterns to look for:

  • Flat line: The treatment effect is constant across the distribution. The ATE tells the whole story. QTE adds no information.
  • Downward slope: The treatment helps the bottom of the distribution more than the top. The treatment compresses the distribution (reduces inequality).
  • Upward slope: The treatment helps the top more than the bottom. The treatment widens the distribution (increases inequality).
  • Zero crossing: The treatment helps some quantiles and hurts others. The ATE may be near zero despite large effects at specific quantiles.
  • U-shape or inverted-U: The treatment has complex distributional effects, with the middle of the distribution responding differently from the tails.

Reporting Conventions

  1. Always report the ATE (OLS) alongside the QTE. The ATE is the benchmark. Readers need to see whether QTE reveals additional heterogeneity.

  2. Report QTE at standard quantiles in a table: {0.10,0.25,0.50,0.75,0.90}\{0.10, 0.25, 0.50, 0.75, 0.90\} with standard errors and confidence intervals.

  3. Include the quantile process plot. The quantile process plot is the most informative visual. Show the OLS estimate as a horizontal dashed line with its confidence band for comparison.

  4. Report the interquantile range effect: QTE(0.90) - QTE(0.10). This quantity measures the treatment's effect on the spread of the distribution.

  5. Report the Wald test for equality of treatment effects across quantiles. If you fail to reject, acknowledge that QTE does not provide statistically significant evidence of distributional heterogeneity.

G. What Can Go Wrong

Assumption Failure Demo

Conditional vs. Unconditional Quantile Confusion

Researcher uses RIF regression to estimate unconditional quantile effects of a minimum wage increase on the population wage distribution

The minimum wage increase raised the 10th percentile of the unconditional wage distribution by $0.85/hour (SE = 0.22, p < 0.001) and had no significant effect on the 90th percentile ($0.03, p = 0.88). The wage distribution compressed.

Assumption Failure Demo

Crossing Quantile Lines Signal Misspecification

Linear quantile regression applied to log wages, where the conditional distribution is approximately symmetric and the linear model fits well across quantiles

Fitted quantile lines are well-separated: Q(0.10) < Q(0.50) < Q(0.90) for all observations. Zero crossings detected. The linear conditional quantile model is adequate.

Assumption Failure Demo

Extreme Quantile Instability

QTE estimated at tau in {0.10, 0.25, 0.50, 0.75, 0.90} with n = 5,000 observations. Each quantile has hundreds of observations in its neighborhood, and bootstrap confidence intervals are reasonably tight.

QTE at tau = 0.10: $2,100 (SE = 480). QTE at tau = 0.90: -$350 (SE = 520). Confidence intervals are informative and allow meaningful comparisons across quantiles.

Assumption Failure Demo

Rank Invariance Assumed Without Justification

Researcher reports QTE as distributional shifts without claiming they represent individual-level effects. States: 'The 10th percentile of earnings increased by $3,500, but we cannot determine whether this reflects the effect on specific individuals or a reshuffling of ranks.'

The interpretation is honest about what QTE can and cannot identify. The distributional finding is policy-relevant (the bottom of the distribution improved) regardless of whether rank invariance holds.

H. Practice

H.1 Concept Checks

Concept Check

A researcher estimates a quantile regression of wages on a job training program indicator and finds: QTE(0.10) = \$3,500 (p < 0.001), QTE(0.50) = \$1,200 (p = 0.02), QTE(0.90) = -\$800 (p = 0.15). The OLS estimate is \$1,100 (p = 0.03). What does this pattern tell you?

Concept Check

A labor economist reports: 'We estimated the effect of unionization on wages using quantile regression at tau = 0.10, 0.25, 0.50, 0.75, and 0.90. The coefficients show that unions raise wages more at the bottom of the wage distribution, consistent with wage compression.' A critic responds: 'Your conditional quantile regression does not tell you about the unconditional wage distribution.' Is the critic correct?

Concept Check

You run simultaneous quantile regression (sqreg) and test the null hypothesis that the treatment coefficient is equal across the five quantiles {0.10, 0.25, 0.50, 0.75, 0.90}. The Wald test gives chi-squared = 5.2 with 4 degrees of freedom (p = 0.27). What should you conclude?

H.2 Guided Exercise

Guided Exercise

Interpreting QTE from a Job Training Program Evaluation

You evaluate a randomized job training program using data from 2,000 participants (1,000 treated, 1,000 control). The outcome is annual earnings ($). You estimate both OLS and conditional quantile regression with bootstrap SEs (500 replications). Your output: Method | tau | Coeff | Boot SE | 95% CI | p-value OLS (mean) | -- | \$1,150 | \$420 | [\$327, \$1,973] | 0.006 QR | 0.10 | \$3,200 | \$710 | [\$1,808, \$4,592] | < 0.001 QR | 0.25 | \$2,050 | \$530 | [\$1,011, \$3,089] | < 0.001 QR | 0.50 | \$1,100 | \$480 | [\$159, \$2,041] | 0.022 QR | 0.75 | \$350 | \$620 | [-\$865, \$1,565] | 0.573 QR | 0.90 | -\$650 | \$880 | [-\$2,375, \$1,075] | 0.460 Wald test for equality of treatment across quantiles: chi2(4) = 14.8, p = 0.005. Quantile crossing: 0 of 2,000 observations show Q(0.10) > Q(0.50); 0 show Q(0.50) > Q(0.90). You also estimate RIF regression for unconditional quantile effects: tau | UQE | SE | p-value 0.10 | \$2,800 | \$650 | < 0.001 0.50 | \$1,050 | \$440 | 0.017 0.90 | -\$400 | \$820 | 0.625

What does the OLS estimate tell you about the average effect of the program? Is it significant?

Describe the pattern of QTE across quantiles. What does this pattern mean substantively?

Is there statistically significant evidence that the treatment effect varies across quantiles? How do you know?

Calculate the interquantile range (IQR) treatment effect. Does the program compress or widen the earnings distribution?

Compare the conditional QTE and unconditional QTE (RIF) at tau = 0.10. Why do they differ? Which is more relevant for policy?

H.3 Error Detective

Error Detective

Read the analysis below carefully and identify the errors.

A management researcher studies whether CEO succession (external vs. internal hire) affects firm performance differently across the performance distribution. She estimates quantile regression of ROA on an external_CEO indicator and controls (firm size, industry dummies, leverage) using panel data with 3,200 firm-year observations and 640 unique firms. She includes firm fixed effects as dummy variables in the quantile regression. She reports: qreg roa external_ceo firm_size leverage i.industry i.firm_id, quantile(0.10) qreg roa external_ceo firm_size leverage i.industry i.firm_id, quantile(0.50) qreg roa external_ceo firm_size leverage i.industry i.firm_id, quantile(0.90) Results: "External CEOs reduce ROA at the 10th percentile by 2.1 pp (p = 0.03), have no effect at the median (0.3 pp, p = 0.72), and increase ROA at the 90th percentile by 1.8 pp (p = 0.04). This shows that external CEOs increase performance inequality within firms." She uses asymptotic standard errors and does not report quantile crossing diagnostics.

Select all errors you can find:

Error Detective

Read the analysis below carefully and identify the errors.

A health economist evaluates a randomized preventive care program using an RCT with 800 participants (400 treated, 400 control). The outcome is annual healthcare expenditure. She reports quantile treatment effects at nine quantiles from 0.05 to 0.95. Her results: tau | QTE | SE | p-value 0.05 | -$12,400 | $8,200 | 0.13 0.10 | -$2,100 | $680 | 0.002 0.25 | -$450 | $210 | 0.03 0.50 | -$180 | $150 | 0.23 0.75 | $50 | $320 | 0.88 0.90 | $1,200 | $890 | 0.18 0.95 | $8,900 | $6,100 | 0.14 She concludes: "The program dramatically reduces healthcare costs for the sickest patients (tau = 0.05: $12,400 reduction) but increases costs for the healthiest (tau = 0.95: $8,900 increase). The program should be targeted to high-cost patients." She does not report OLS, does not test for equality across quantiles, and does not discuss rank invariance.

Select all errors you can find:

H.4 You Are the Referee

Referee Exercise

Read the paper summary below and write a brief referee critique (2-3 sentences) of the identification strategy.

Paper Summary

The authors study whether a minimum wage increase affected the earnings distribution differently across quantiles. Using state-level panel data from 2010-2020 (25 states, quarterly observations), they estimate conditional quantile regressions of log earnings on a minimum wage indicator, controlling for state unemployment rate, industry composition, and state and quarter fixed effects. They report that the minimum wage increase raised earnings significantly at the 10th percentile (\$0.45, p = 0.002) and 25th percentile (\$0.28, p = 0.01), had no effect at the median (\$0.05, p = 0.62), and reduced earnings at the 90th percentile (-\$0.18, p = 0.04). They conclude that the policy compressed the unconditional earnings distribution and recommend the policy as an inequality-reducing tool.

Key Table

VariableCoefficientSEp-value
Min wage x tau=0.100.4500.1450.002
Min wage x tau=0.250.2800.1100.011
Min wage x tau=0.500.0500.0980.620
Min wage x tau=0.75-0.0800.1250.522
Min wage x tau=0.90-0.1800.0880.041
State FEYes
Quarter FEYes
N (state-quarters)2,500

Authors' Identification Claim

The authors claim that controlling for state and quarter fixed effects and observable confounders isolates the causal effect of the minimum wage on the earnings distribution at each quantile.

I. Swap-In: When to Use Something Else

  • OLS Regression: Estimates the conditional mean effect. QTE at τ=0.50\tau = 0.50 estimates the conditional median effect. If the outcome distribution is symmetric, these are similar.

  • Causal Forests: Estimates heterogeneous treatment effects across the covariate space. Complementary to QTE: causal forests find subgroups with different effects; QTE finds distributional shifts.

  • Matching: Can be combined with QTE by matching on propensity scores and then estimating quantile effects within the matched sample.

  • Experimental Design: Randomization is the cleanest setting for QTE. With experimental data, the QTE at each quantile has a causal interpretation without functional form assumptions (Bitler et al., 2006).

J. Reviewer Checklist

Critical Reading Checklist

  1. Is the research question about distributional effects, or would the ATE suffice?
  2. Is it clear whether conditional or unconditional QTE is being estimated? Is the correct estimator used for the research question?
  3. Is the ATE (OLS) reported alongside the QTE for comparison?
  4. Is the quantile process plot provided, with confidence bands and OLS reference line?
  5. Are bootstrap standard errors used for conditional QTE (not asymptotic)?
  6. Is quantile crossing checked and reported?
  7. Is the heterogeneity test (Wald test for equality across quantiles) reported?
  8. Are extreme quantiles handled appropriately? Is the reported quantile range justified?
  9. Is rank invariance discussed if the results are interpreted as individual-level effects?
  10. If using RIF regression, is sensitivity to the bandwidth choice reported?

Paper Library

Foundational (4)

Koenker, R. & Bassett, G. (1978). Regression Quantiles.

EconometricaDOI: 10.2307/1913643

The foundational paper introducing quantile regression. Proposes estimating conditional quantile functions by minimizing an asymmetric absolute loss (check function), generalizing least absolute deviations to arbitrary quantiles. Establishes asymptotic theory and demonstrates robustness to outliers and heteroscedasticity relative to OLS.

Firpo, S., Fortin, N. M., & Lemieux, T. (2009). Unconditional Quantile Regressions.

EconometricaDOI: 10.3982/ECTA6822

Introduces the recentered influence function (RIF) regression for estimating unconditional quantile effects. Shows that standard quantile regression estimates conditional quantile effects that do not aggregate to unconditional effects. RIF regression transforms the outcome variable so that OLS on the transformed outcome recovers the effect of covariates on unconditional quantiles. The key innovation enabling policy-relevant distributional analysis.

Chernozhukov, V. & Hansen, C. (2005). An IV Model of Quantile Treatment Effects.

EconometricaDOI: 10.1111/j.1468-0262.2005.00570.x

Develops an instrumental variable framework for quantile regression to address endogeneity. Proposes the inverse quantile regression (IQR) method that exploits moment conditions implied by the structural quantile model. Provides conditions under which quantile treatment effects are identified with endogenous treatments, extending quantile regression to credible causal inference settings.

Machado, J. A. F. & Santos Silva, J. M. C. (2019). Quantiles via Moments.

Journal of EconometricsDOI: 10.1016/j.jeconom.2019.04.009

Proposes a method for estimating conditional quantile functions in panel data with fixed effects. The method-of-moments quantile regression (MMQR) approach avoids the incidental parameters problem that plagues standard quantile regression with fixed effects. Enables quantile regression with individual fixed effects in short panels, filling a major gap in the panel data toolkit.

Application (2)

Angrist, J. D., Chernozhukov, V., & Fernandez-Val, I. (2006). Quantile Regression under Misspecification, with an Application to the U.S. Wage Structure.

EconometricaDOI: 10.1111/j.1468-0262.2006.00671.x

Examines the interpretation of quantile regression coefficients under misspecification and applies the framework to study the U.S. wage structure. Shows that quantile regression coefficients have a causal interpretation as weighted averages of quantile treatment effects under appropriate conditions. Demonstrates how returns to education vary across the wage distribution, with larger returns at higher quantiles.

Bitler, M. P., Gelbach, J. B., & Hoynes, H. W. (2006). What Mean Impacts Miss: Distributional Effects of Welfare Reform Experiments.

American Economic ReviewDOI: 10.1257/aer.96.4.988

A landmark application of quantile treatment effects to experimental data from the Connecticut Jobs First welfare reform program. Shows that the average treatment effect of the program masks dramatic heterogeneity: the program increased earnings for women at the bottom of the distribution but decreased earnings at the top. Demonstrates why distributional analysis is essential for evaluating social programs.

Tags

model-baseddistributionalheterogeneity