Machine LearningAdvanced Regression

Quantile Regression

LevelAdvanced

Duration75 mins

TopicAdvanced Regression

3 / 5

Conditional Quantiles

Modeling the Entire Conditional Distribution

Standard regression provides a single number for each input: the conditional mean $\mathbb{E}[Y \mid X = x]$. But this summary collapses an entire distribution into one point. What about the spread? The tails? The asymmetry?

Conditional quantile regression addresses these questions by estimating:

$$Q_\tau(Y \mid X = x) = \inf{y : P(Y \leq y \mid X = x) \geq \tau}$$

for any desired $\tau \in (0, 1)$. By fitting models for multiple quantile levels, we reconstruct the shape of the conditional distribution—not just its center.

Why This Matters:

Heteroscedasticity detection: If quantile spacing varies with $x$, the conditional variance changes
Distributional effects: A covariate might affect upper quantiles differently than lower quantiles
Risk assessment: Tail quantiles directly measure extreme outcomes
Inequality analysis: In economics, wage quantiles reveal how education affects different parts of the income distribution

What You Will Learn

By the end of this page, you will understand how to interpret quantile regression coefficients, detect heteroscedasticity through quantile spreads, visualize conditional distributions, and identify when covariates have quantile-specific effects.

The Conditional Quantile Function

Let's formalize the conditional quantile function and its relationship to the conditional CDF.

Definition (Conditional CDF):

For random variables $Y$ and $X$, the conditional cumulative distribution function is:

$$F_{Y|X}(y \mid x) = P(Y \leq y \mid X = x)$$

Definition (Conditional Quantile Function):

The conditional quantile function is the inverse:

$$Q_\tau(Y \mid X = x) = F_{Y|X}^{-1}(\tau \mid x) = \inf{y : F_{Y|X}(y \mid x) \geq \tau}$$

Key Properties:

Properties of Conditional Quantiles

•Probability interpretation: $P(Y \leq Q_\tau(Y | X = x) | X = x) = \tau$ (for continuous distributions)
•Monotonicity in τ: If $\tau_1 < \tau_2$, then $Q_{\tau_1}(Y | X = x) \leq Q_{\tau_2}(Y | X = x)$
•Full distribution characterization: The set ${Q_\tau(Y | X = x) : \tau \in (0,1)}$ completely determines $F_{Y|X}$
•Location-scale relationship: For $Y | X = x \sim \mu(x) + \sigma(x) \cdot \epsilon$, we have $Q_\tau(Y | X=x) = \mu(x) + \sigma(x) \cdot Q_\tau(\epsilon)$

Linear Conditional Quantile Model:

The most common specification assumes a linear model for each quantile:

$$Q_\tau(Y \mid X = x) = x^\top \beta(\tau)$$

where $\beta(\tau) \in \mathbb{R}^p$ is the coefficient vector specific to quantile $\tau$.

Critically, each quantile has its own coefficients. The effect of a covariate on the 90th percentile may differ substantially from its effect on the median or the 10th percentile.

Alternative: Location-Scale Model:

A more restrictive model assumes:

$$Y \mid X = X^\top \beta + (X^\top \gamma) \cdot \epsilon$$

where $\epsilon$ has a fixed distribution. Under this model:

$$Q_\tau(Y \mid X) = X^\top \beta + (X^\top \gamma) \cdot q_\tau^\epsilon$$

This implies $\beta_j(\tau) = \beta_j + \gamma_j \cdot q_\tau^\epsilon$—coefficients change linearly with $q_\tau^\epsilon$.

Model Flexibility

The unrestricted linear quantile model Q_τ(Y|X) = X'β(τ) is more flexible than location-scale models. It allows covariates to affect different quantiles in completely different ways—even with opposite signs. The data determine the pattern.

Interpreting Quantile Regression Coefficients

Interpreting quantile regression coefficients requires care—they mean something different than OLS coefficients.

OLS Coefficient Interpretation:

$$\mathbb{E}[Y \mid X = x] = x^\top \beta$$

$\beta_j$: The expected change in $Y$ for a one-unit increase in $x_j$, holding other covariates constant.

Quantile Regression Coefficient Interpretation:

$$Q_\tau(Y \mid X = x) = x^\top \beta(\tau)$$

$\beta_j(\tau)$: The change in the $\tau$-th conditional quantile of $Y$ for a one-unit increase in $x_j$, holding other covariates constant.

Example: Education and Wages

Suppose we model log wages with years of education as a predictor:

Quantile τ	β(τ) for Education
0.10	0.08
0.50	0.12
0.90	0.18

Interpretation:

At the 10th percentile of wages: one additional year of education increases wages by 8%
At the median: one additional year increases wages by 12%
At the 90th percentile: one additional year increases wages by 18%

Conclusion: Education has larger returns for high earners. The same variable has heterogeneous effects across the wage distribution.

When β(τ) is Constant Across τ

•Covariate shifts entire distribution uniformly
•Only the location changes, not the shape
•Homogeneous effects: same impact at all quantiles
•OLS coefficient tells the whole story
•Example: A fixed tax applied equally to all incomes

When β(τ) Varies With τ

•Covariate reshapes the distribution
•Location and/or scale change
•Heterogeneous effects: different impact by quantile
•OLS coefficient masks important variation
•Example: Exercise affects fitness differently for healthy vs. unfit individuals

Common Misinterpretation

β(0.9) does NOT describe the effect for 'high-y individuals.' It describes the effect on the 90th percentile of the conditional distribution at each x. The same individual could be at different conditional quantiles depending on their covariate values.

Detecting Heteroscedasticity Through Quantiles

One of the most powerful applications of quantile regression is detecting and characterizing heteroscedasticity—changing variance across covariate values.

The OLS Problem:

OLS assumes constant variance (homoscedasticity): $$\text{Var}(Y \mid X = x) = \sigma^2$$

When violated, OLS remains consistent but:

Standard errors are incorrect
Prediction intervals have wrong coverage
Efficiency is reduced

How Quantile Regression Reveals Heteroscedasticity:

If the conditional distribution's spread changes with $x$, the spacing between quantiles will change:

$$\text{IQR}(x) = Q_{0.75}(Y \mid X=x) - Q_{0.25}(Y \mid X=x)$$

If IQR varies with $x$, heteroscedasticity is present.

Geometric Interpretation:

In a quantile regression plot:

Parallel lines → Homoscedasticity (constant spacing)
Fan-shaped pattern → Heteroscedasticity (variance increases with $x$)
Funnel pattern → Heteroscedasticity (variance decreases with $x$)
Crossing lines → Complex distributional changes

heteroscedasticity_detection.py
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import numpy as np
import matplotlib.pyplot as plt
from sklearn.linear_model import QuantileRegressor
 
np.random.seed(42)
n = 500
 
# Scenario 1: Homoscedastic data
X_homo = np.random.uniform(0, 10, n)
y_homo = 2 * X_homo + 5 + np.random.normal(0, 2, n)
 
# Scenario 2: Heteroscedastic data (variance increases with X)
X_hetero = np.random.uniform(0, 10, n)
y_hetero = 2 * X_hetero + 5 + np.random.normal(0, 0.5 + 0.5 * X_hetero, n)
 
def fit_and_plot_quantiles(X, y, title, ax):
    """Fit quantile regressions and plot."""
    X = X.reshape(-1, 1)
    quantiles = [0.1, 0.25, 0.5, 0.75, 0.9]
    
    ax.scatter(X, y, alpha=0.3, s=20, c='gray')
    
    X_plot = np.linspace(X.min(), X.max(), 100).reshape(-1, 1)
    colors = plt.cm.RdYlBu(np.linspace(0.1, 0.9, len(quantiles)))
    
    for tau, color in zip(quantiles, colors):
        model = QuantileRegressor(quantile=tau, alpha=0, solver='highs')
        model.fit(X, y)
        y_pred = model.predict(X_plot)
        ax.plot(X_plot, y_pred, color=color, linewidth=2, label=f'τ = {tau}')
    
    ax.set_xlabel('X')
    ax.set_ylabel('y')
    ax.set_title(title)
    ax.legend(loc='upper left')
    ax.grid(True, alpha=0.3)
 
fig, axes = plt.subplots(1, 2, figsize=(14, 5))
 
fit_and_plot_quantiles(X_homo, y_homo, 'Homoscedastic: Parallel Lines', axes[0])
fit_and_plot_quantiles(X_hetero, y_hetero, 'Heteroscedastic: Fan Pattern', axes[1])
 
plt.tight_layout()
plt.show()
 
# Quantitative test: Compare IQR at different X values
print("Interquartile Range (IQR) Analysis:")
print("=" * 50)
 
for scenario, X, y in [("Homoscedastic", X_homo, y_homo), 
                        ("Heteroscedastic", X_hetero, y_hetero)]:
    X = X.reshape(-1, 1)
    
    # Fit Q25 and Q75
    model_25 = QuantileRegressor(quantile=0.25, alpha=0, solver='highs').fit(X, y)
    model_75 = QuantileRegressor(quantile=0.75, alpha=0, solver='highs').fit(X, y)
    
    # IQR at X=2 vs X=8
    iqr_low = model_75.predict([[2]]) - model_25.predict([[2]])
    iqr_high = model_75.predict([[8]]) - model_25.predict([[8]])
    
    print(f"
{scenario}:")
    print(f"  IQR at X=2: {iqr_low[0]:.2f}")
    print(f"  IQR at X=8: {iqr_high[0]:.2f}")
    print(f"  Ratio: {iqr_high[0]/iqr_low[0]:.2f}x")

Visual Diagnostic Power

Plotting quantile regression lines is often more informative than formal heteroscedasticity tests (Breusch-Pagan, White). The visual pattern immediately reveals not just whether variance changes, but how it changes and whether the effect is symmetric.

Distributional Effects: Beyond Location Shift

Quantile regression excels at capturing complex distributional effects—situations where covariates don't simply shift or scale the distribution but reshape it entirely.

Types of Distributional Effects:

1. Location Shift Only:

$Q_\tau(Y|X=x) = x\beta + q_\tau^0$ where $q_\tau^0$ is constant by $\tau$
All quantile lines are parallel
Covariate shifts entire distribution uniformly

2. Location-Scale Change:

$Q_\tau(Y|X=x) = x\beta + x\gamma \cdot q_\tau^0$
Quantile lines diverge (or converge) as $x$ changes
Covariate affects both center and spread

3. Skewness Change:

Upper and lower quantiles move asymmetrically
Distribution becomes more or less skewed with $x$
Example: Wages become more right-skewed at higher education levels

4. Complex Reshaping:

Arbitrary changes in distributional shape
Different covariates affecting different parts of the distribution
Example: A treatment helps weak performers, hurts strong performers

Patterns in Quantile Coefficients and Their Interpretations
Pattern in β(τ)	Interpretation	Example
Constant across τ	Pure location shift	Fixed dollar raise for all employees
Increases with τ	Positive compression (high quantiles affected more)	Percent-based raise; training with diminishing returns for low performers
Decreases with τ	Negative compression (low quantiles affected more)	Minimum wage increase; safety net policies
Positive for low τ, negative for high τ	Compression toward center	Regulations reducing inequality
Negative for low τ, positive for high τ	Expansion from center	Deregulation increasing variance
Non-monotonic (U-shape, etc.)	Complex distributional changes	Technology affecting middle-skilled workers negatively

distributional_effects.py
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import numpy as np
import matplotlib.pyplot as plt
from sklearn.linear_model import QuantileRegressor
 
np.random.seed(42)
n = 1000
 
# Create treatment indicator and baseline
treatment = np.random.binomial(1, 0.5, n)
baseline_ability = np.random.normal(0, 1, n)
 
# Treatment has HETEROGENEOUS effects:
# - Helps low-baseline individuals more  
# - Has minimal effect on high-baseline individuals
# This is a "compression" effect
 
treatment_effect = np.where(treatment == 1, 
                             3 - 1.5 * baseline_ability,  # Larger effect for low baseline
                             0)
 
y = 5 + 2 * baseline_ability + treatment_effect + np.random.normal(0, 1, n)
 
# Fit quantile regressions for treatment effect
X = np.column_stack([np.ones(n), treatment])
quantiles = [0.1, 0.25, 0.5, 0.75, 0.9]
 
treatment_effects = []
for tau in quantiles:
    model = QuantileRegressor(quantile=tau, alpha=0, solver='highs')
    model.fit(X, y)
    treatment_effects.append(model.coef_[1])  # Treatment coefficient
 
# Visualization
fig, axes = plt.subplots(1, 2, figsize=(14, 5))
 
# Plot 1: Treatment effect by quantile
ax1 = axes[0]
ax1.plot(quantiles, treatment_effects, 'o-', linewidth=2, markersize=10)
ax1.axhline(y=np.mean(treatment_effects), color='red', linestyle='--', 
            label=f'OLS estimate (approx.)')
ax1.set_xlabel('Quantile τ', fontsize=12)
ax1.set_ylabel('Treatment Effect β(τ)', fontsize=12)
ax1.set_title('Heterogeneous Treatment Effect Across Quantiles', fontsize=14)
ax1.legend()
ax1.grid(True, alpha=0.3)
 
# Add interpretation annotations
ax1.annotate('Larger effect
for lower quantiles
(struggling students)', 
             xy=(0.1, treatment_effects[0]), 
             xytext=(0.25, treatment_effects[0] + 0.5),
             fontsize=10, arrowprops=dict(arrowstyle='->', color='gray'))
 
# Plot 2: Distribution comparison
ax2 = axes[1]
control = y[treatment == 0]
treated = y[treatment == 1]
 
ax2.hist(control, bins=40, alpha=0.5, density=True, label='Control', color='blue')
ax2.hist(treated, bins=40, alpha=0.5, density=True, label='Treatment', color='green')
ax2.axvline(np.median(control), color='blue', linestyle='--', linewidth=2)
ax2.axvline(np.median(treated), color='green', linestyle='--', linewidth=2)
ax2.set_xlabel('Outcome y', fontsize=12)
ax2.set_ylabel('Density', fontsize=12)
ax2.set_title('Outcome Distributions by Treatment Status', fontsize=14)
ax2.legend()
ax2.grid(True, alpha=0.3)
 
plt.tight_layout()
plt.show()
 
print("Treatment Effect by Quantile:")
print("-" * 40)
for tau, effect in zip(quantiles, treatment_effects):
    print(f"τ = {tau}: β = {effect:.3f}")

Policy Relevance

Understanding distributional effects is crucial for policy analysis. A job training program with positive average effect might actually hurt high-ability workers while dramatically helping low-ability workers. Quantile regression reveals this heterogeneity invisible to OLS.

Quantile Coefficient Plots: The Essential Visualization

The quantile coefficient plot (also called the "quantile process plot") is the definitive visualization for quantile regression results.

Construction:

Fit quantile regression for a fine grid of $\tau$ values (e.g., 0.05, 0.10, ..., 0.95)
Extract coefficient $\hat{\beta}_j(\tau)$ for each covariate $j$ at each $\tau$
Plot $\hat{\beta}_j(\tau)$ vs. $\tau$ with confidence bands
Include OLS estimate as horizontal reference line

Interpretation Guide:

Reading Quantile Coefficient Plots

•Horizontal line pattern: Covariate has homogeneous effect across quantiles; OLS tells the whole story
•Upward slope: Covariate has larger effects at higher quantiles (e.g., education on wage inequality)
•Downward slope: Covariate has larger effects at lower quantiles (e.g., safety interventions help the vulnerable most)
•Confidence bands overlapping OLS: Statistical evidence consistent with homogeneous effects
•Confidence bands excluding OLS at some tau: Statistically significant heterogeneity
•Wide bands at extremes: Less precise estimation for tail quantiles (fewer observations)

quantile_coefficient_plot.py
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import numpy as np
import matplotlib.pyplot as plt
from sklearn.linear_model import QuantileRegressor, LinearRegression
from scipy import stats
 
def quantile_coefficient_plot(X, y, covariate_idx, covariate_name, 
                              quantiles=None, n_bootstrap=100, ax=None):
    """
    Create a quantile coefficient plot with confidence bands.
    
    Parameters:
    -----------
    X : np.ndarray
        Feature matrix (n_samples, n_features)
    y : np.ndarray
        Target variable
    covariate_idx : int
        Index of the covariate to plot
    covariate_name : str
        Name for plot label
    quantiles : list, optional
        Quantile levels to estimate
    n_bootstrap : int
        Number of bootstrap samples for confidence bands
    ax : matplotlib.axes, optional
        Axes to plot on
    """
    if quantiles is None:
        quantiles = np.arange(0.05, 0.96, 0.05)
    
    n = len(y)
    
    # Fit OLS for reference
    ols = LinearRegression().fit(X, y)
    ols_coef = ols.coef_[covariate_idx]
    
    # Fit quantile regressions
    qr_coefs = []
    for tau in quantiles:
        model = QuantileRegressor(quantile=tau, alpha=0, solver='highs')
        model.fit(X, y)
        qr_coefs.append(model.coef_[covariate_idx])
    
    qr_coefs = np.array(qr_coefs)
    
    # Bootstrap for confidence bands
    boot_coefs = np.zeros((n_bootstrap, len(quantiles)))
    
    for b in range(n_bootstrap):
        # Resample
        idx = np.random.choice(n, size=n, replace=True)
        X_boot, y_boot = X[idx], y[idx]
        
        for i, tau in enumerate(quantiles):
            try:
                model = QuantileRegressor(quantile=tau, alpha=0, solver='highs')
                model.fit(X_boot, y_boot)
                boot_coefs[b, i] = model.coef_[covariate_idx]
            except:
                boot_coefs[b, i] = np.nan
    
    # Compute confidence bands (2.5th and 97.5th percentiles)
    lower = np.nanpercentile(boot_coefs, 2.5, axis=0)
    upper = np.nanpercentile(boot_coefs, 97.5, axis=0)
    
    # Plot
    if ax is None:
        fig, ax = plt.subplots(figsize=(10, 6))
    
    ax.fill_between(quantiles, lower, upper, alpha=0.3, color='blue',
                    label='95% Confidence Band')
    ax.plot(quantiles, qr_coefs, 'b-', linewidth=2, label='Quantile Estimates')
    ax.axhline(y=ols_coef, color='red', linestyle='--', linewidth=2,
               label=f'OLS Estimate: {ols_coef:.3f}')
    ax.axhline(y=0, color='gray', linestyle=':', linewidth=1)
    
    ax.set_xlabel('Quantile τ', fontsize=12)
    ax.set_ylabel(f'β(τ) for {covariate_name}', fontsize=12)
    ax.set_title(f'Quantile Coefficients: {covariate_name}', fontsize=14)
    ax.legend(loc='best')
    ax.grid(True, alpha=0.3)
    
    return ax, qr_coefs, (lower, upper)
 
# Example: Simulated heterogeneous effect
np.random.seed(42)
n = 800
 
# X1: Constant effect across quantiles
# X2: Effect increases with quantile (heterogeneous)
X1 = np.random.normal(0, 1, n)
X2 = np.random.normal(0, 1, n)
 
# Generate y with heterogeneous effect for X2
epsilon = np.random.standard_t(df=5, size=n)  # Heavy-tailed errors
y = 3 + 2 * X1 + (1 + 0.5 * epsilon) * X2 + epsilon
 
X = np.column_stack([X1, X2])
 
# Create plots
fig, axes = plt.subplots(1, 2, figsize=(14, 5))
 
quantile_coefficient_plot(X, y, 0, 'X₁ (Homogeneous Effect)', ax=axes[0])
quantile_coefficient_plot(X, y, 1, 'X₂ (Heterogeneous Effect)', ax=axes[1])
 
plt.tight_layout()
plt.show()
 
print("X₁ has roughly constant effect across quantiles (homogeneous)")
print("X₂ shows increasing effect at higher quantiles (heterogeneous)")

Publication-Quality Plots

Quantile coefficient plots with confidence bands are the standard way to present quantile regression results in academic papers. They efficiently communicate effect heterogeneity, statistical significance, and comparison to OLS in a single elegant display.

Quantile Treatment Effects

In causal inference, Quantile Treatment Effects (QTE) provide richer information than the Average Treatment Effect (ATE).

Definition (Quantile Treatment Effect):

For treatment $D \in {0, 1}$ and outcome $Y$, the QTE at quantile $\tau$ is:

$$\text{QTE}(\tau) = Q_\tau(Y^1) - Q_\tau(Y^0)$$

where $Y^1$ and $Y^0$ are potential outcomes under treatment and control.

Interpretation:

QTE($\tau$) measures how the treatment shifts the $\tau$-th quantile of the outcome distribution.

Important Caveat:

The QTE compares the $\tau$-th quantiles of two distributions—not the same individuals. The person at the 90th percentile of $Y^1$ may not be the same as the person at the 90th percentile of $Y^0$.

When QTE Equals Individual Treatment Effect:

For QTE to equal the individual treatment effect for someone at quantile $\tau$, we need rank invariance—the assumption that individuals maintain their rank in the distribution regardless of treatment. This is strong but sometimes plausible.

Treatment Effect Concepts
Concept	Definition	Interpretation
ATE	E[Y¹ - Y⁰]	Average effect across entire population
QTE(τ)	Q_τ(Y¹) - Q_τ(Y⁰)	Effect on τ-th quantile of distributions
ATT	E[Y¹ - Y⁰ \| D=1]	Average effect on the treated
CATE	E[Y¹ - Y⁰ \| X=x]	Conditional average effect given covariates

Why QTE Matters:

Example: Minimum Wage Increase

Effect	Interpretation
ATE = $0.50/hr	Average wage increases modestly
QTE(0.10) = $1.50/hr	Large gains at the 10th percentile (low earners)
QTE(0.50) = $0.30/hr	Small median effect
QTE(0.90) = $0.00/hr	No effect on high earners

The ATE misses the distributional story: minimum wage disproportionately helps those at the bottom.

Estimation:

With random assignment and the linearity assumption: $$Q_\tau(Y \mid D) = \alpha(\tau) + \text{QTE}(\tau) \cdot D$$

Fitting quantile regression of $Y$ on treatment indicator $D$ gives QTE($\tau$) directly.

Rank Invariance Warning

Without rank invariance, QTE(τ) tells you about distributional shifts but not necessarily about individual treatment effects. The person at the 90th percentile when treated may have been at the 70th when untreated if treatment reshuffles ranks.

Handling Multiple Covariates

Real applications involve multiple covariates, each potentially having quantile-specific effects.

The Full Model:

$$Q_\tau(Y \mid X) = \beta_0(\tau) + \beta_1(\tau) X_1 + \cdots + \beta_p(\tau) X_p$$

Interpretation Challenges:

Dimensionality: With $K$ quantiles and $p$ covariates, you have $K \times p$ coefficients to interpret
Ceteris paribus: $\beta_j(\tau)$ is the effect of $X_j$ on $Q_\tau(Y|X)$ holding other $X$s constant
Multicollinearity: Affects quantile regression just like OLS—correlated features have unstable coefficients

Strategies for Summarization:

Managing Complexity with Many Covariates

•Focus on key quantiles: Report τ ∈ {0.1, 0.25, 0.5, 0.75, 0.9} rather than all quantiles
•Grid of coefficient plots: Create one quantile coefficient plot per covariate
•Interquartile range effects: Report β(0.75) - β(0.25) to summarize spread effects
•Coefficient tables: Traditional table with columns for different τ values
•Heat maps: Color-coded matrix of coefficients (covariates × quantiles)
•Test for homogeneity: Formal tests whether β(τ) varies significantly with τ

Testing for Quantile Coefficient Equality:

To test whether an effect is truly heterogeneous across quantiles, we can use:

Wald Test: $$H_0: \beta_j(\tau_1) = \beta_j(\tau_2) \quad \text{vs.} \quad H_1: \beta_j(\tau_1) eq \beta_j(\tau_2)$$

Procedure:

Estimate $\hat{\beta}_j(\tau_1)$ and $\hat{\beta}_j(\tau_2)$ with bootstrap standard errors
Compute $T = \frac{\hat{\beta}_j(\tau_1) - \hat{\beta}_j(\tau_2)}{\text{SE}(\hat{\beta}_j(\tau_1) - \hat{\beta}_j(\tau_2))}$
Compare to standard normal critical values

Joint Test (Khmaladze Transform):

More sophisticated tests consider the entire quantile process ${\beta_j(\tau) : \tau \in (0,1)}$ simultaneously, controlling for the multiple comparison problem.

Multiple Testing

When examining many covariates across many quantiles, some differences will appear significant by chance. Use appropriate corrections (Bonferroni, FDR) or joint testing procedures to avoid false discoveries.

Summary and Looking Ahead

We have explored how conditional quantiles reveal the full distributional relationship between covariates and outcomes.

Key Takeaways

•Conditional quantiles characterize the full distribution — Q_τ(Y|X) for all τ ∈ (0,1) completely specifies the conditional distribution.
•Each quantile has its own coefficients — β(τ) can vary arbitrarily with τ, revealing heterogeneous effects.
•Heteroscedasticity is visible — Fan-shaped quantile regression lines indicate changing variance.
•Distributional effects transcend location-scale — Covariates can compress, expand, or reshape distributions in complex ways.
•Quantile coefficient plots are essential — They efficiently communicate effect heterogeneity with uncertainty.
•QTE enriches causal inference — Treatment effects across the distribution provide richer policy insights than ATE alone.

What's Next:

In the next page, we'll turn to prediction intervals—using quantile regression to construct intervals that capture uncertainty in predictions. We'll see how pairs of quantile predictions (e.g., τ = 0.1 and τ = 0.9) form prediction intervals with controlled coverage.

Page Complete

You now understand how conditional quantiles reveal the full distributional relationship between features and outcomes. This goes far beyond mean regression, exposing heterogeneity, heteroscedasticity, and complex distributional effects. Next, we'll apply this understanding to construct reliable prediction intervals.

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Machine LearningAdvanced Regression

Quantile Regression

LevelAdvanced

Duration75 mins

TopicAdvanced Regression

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Conditional Quantiles

Modeling the Entire Conditional Distribution

Conditional quantile regression addresses these questions by estimating:

$$Q_\tau(Y \mid X = x) = \inf{y : P(Y \leq y \mid X = x) \geq \tau}$$

for any desired $\tau \in (0, 1)$. By fitting models for multiple quantile levels, we reconstruct the shape of the conditional distribution—not just its center.

Why This Matters:

Heteroscedasticity detection: If quantile spacing varies with $x$, the conditional variance changes
Distributional effects: A covariate might affect upper quantiles differently than lower quantiles
Risk assessment: Tail quantiles directly measure extreme outcomes
Inequality analysis: In economics, wage quantiles reveal how education affects different parts of the income distribution

What You Will Learn

The Conditional Quantile Function

Let's formalize the conditional quantile function and its relationship to the conditional CDF.

Definition (Conditional CDF):

For random variables $Y$ and $X$, the conditional cumulative distribution function is:

$$F_{Y|X}(y \mid x) = P(Y \leq y \mid X = x)$$

Definition (Conditional Quantile Function):

The conditional quantile function is the inverse:

$$Q_\tau(Y \mid X = x) = F_{Y|X}^{-1}(\tau \mid x) = \inf{y : F_{Y|X}(y \mid x) \geq \tau}$$

Key Properties:

Properties of Conditional Quantiles

•Probability interpretation: $P(Y \leq Q_\tau(Y | X = x) | X = x) = \tau$ (for continuous distributions)
•Monotonicity in τ: If $\tau_1 < \tau_2$, then $Q_{\tau_1}(Y | X = x) \leq Q_{\tau_2}(Y | X = x)$
•Full distribution characterization: The set ${Q_\tau(Y | X = x) : \tau \in (0,1)}$ completely determines $F_{Y|X}$
•Location-scale relationship: For $Y | X = x \sim \mu(x) + \sigma(x) \cdot \epsilon$, we have $Q_\tau(Y | X=x) = \mu(x) + \sigma(x) \cdot Q_\tau(\epsilon)$

Linear Conditional Quantile Model:

The most common specification assumes a linear model for each quantile:

$$Q_\tau(Y \mid X = x) = x^\top \beta(\tau)$$

where $\beta(\tau) \in \mathbb{R}^p$ is the coefficient vector specific to quantile $\tau$.

Critically, each quantile has its own coefficients. The effect of a covariate on the 90th percentile may differ substantially from its effect on the median or the 10th percentile.

Alternative: Location-Scale Model:

A more restrictive model assumes:

$$Y \mid X = X^\top \beta + (X^\top \gamma) \cdot \epsilon$$

where $\epsilon$ has a fixed distribution. Under this model:

$$Q_\tau(Y \mid X) = X^\top \beta + (X^\top \gamma) \cdot q_\tau^\epsilon$$

This implies $\beta_j(\tau) = \beta_j + \gamma_j \cdot q_\tau^\epsilon$—coefficients change linearly with $q_\tau^\epsilon$.

Model Flexibility

Interpreting Quantile Regression Coefficients

Interpreting quantile regression coefficients requires care—they mean something different than OLS coefficients.

OLS Coefficient Interpretation:

$$\mathbb{E}[Y \mid X = x] = x^\top \beta$$

$\beta_j$: The expected change in $Y$ for a one-unit increase in $x_j$, holding other covariates constant.

Quantile Regression Coefficient Interpretation:

$$Q_\tau(Y \mid X = x) = x^\top \beta(\tau)$$

$\beta_j(\tau)$: The change in the $\tau$-th conditional quantile of $Y$ for a one-unit increase in $x_j$, holding other covariates constant.

Example: Education and Wages

Suppose we model log wages with years of education as a predictor:

Quantile τ	β(τ) for Education
0.10	0.08
0.50	0.12
0.90	0.18

Interpretation:

At the 10th percentile of wages: one additional year of education increases wages by 8%
At the median: one additional year increases wages by 12%
At the 90th percentile: one additional year increases wages by 18%

Conclusion: Education has larger returns for high earners. The same variable has heterogeneous effects across the wage distribution.

When β(τ) is Constant Across τ

•Covariate shifts entire distribution uniformly
•Only the location changes, not the shape
•Homogeneous effects: same impact at all quantiles
•OLS coefficient tells the whole story
•Example: A fixed tax applied equally to all incomes

When β(τ) Varies With τ

•Covariate reshapes the distribution
•Location and/or scale change
•Heterogeneous effects: different impact by quantile
•OLS coefficient masks important variation
•Example: Exercise affects fitness differently for healthy vs. unfit individuals

Common Misinterpretation

Detecting Heteroscedasticity Through Quantiles

One of the most powerful applications of quantile regression is detecting and characterizing heteroscedasticity—changing variance across covariate values.

The OLS Problem:

OLS assumes constant variance (homoscedasticity): $$\text{Var}(Y \mid X = x) = \sigma^2$$

When violated, OLS remains consistent but:

Standard errors are incorrect
Prediction intervals have wrong coverage
Efficiency is reduced

How Quantile Regression Reveals Heteroscedasticity:

If the conditional distribution's spread changes with $x$, the spacing between quantiles will change:

$$\text{IQR}(x) = Q_{0.75}(Y \mid X=x) - Q_{0.25}(Y \mid X=x)$$

If IQR varies with $x$, heteroscedasticity is present.

Geometric Interpretation:

In a quantile regression plot:

Parallel lines → Homoscedasticity (constant spacing)
Fan-shaped pattern → Heteroscedasticity (variance increases with $x$)
Funnel pattern → Heteroscedasticity (variance decreases with $x$)
Crossing lines → Complex distributional changes

heteroscedasticity_detection.py
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import numpy as np
import matplotlib.pyplot as plt
from sklearn.linear_model import QuantileRegressor
 
np.random.seed(42)
n = 500
 
# Scenario 1: Homoscedastic data
X_homo = np.random.uniform(0, 10, n)
y_homo = 2 * X_homo + 5 + np.random.normal(0, 2, n)
 
# Scenario 2: Heteroscedastic data (variance increases with X)
X_hetero = np.random.uniform(0, 10, n)
y_hetero = 2 * X_hetero + 5 + np.random.normal(0, 0.5 + 0.5 * X_hetero, n)
 
def fit_and_plot_quantiles(X, y, title, ax):
    """Fit quantile regressions and plot."""
    X = X.reshape(-1, 1)
    quantiles = [0.1, 0.25, 0.5, 0.75, 0.9]
    
    ax.scatter(X, y, alpha=0.3, s=20, c='gray')
    
    X_plot = np.linspace(X.min(), X.max(), 100).reshape(-1, 1)
    colors = plt.cm.RdYlBu(np.linspace(0.1, 0.9, len(quantiles)))
    
    for tau, color in zip(quantiles, colors):
        model = QuantileRegressor(quantile=tau, alpha=0, solver='highs')
        model.fit(X, y)
        y_pred = model.predict(X_plot)
        ax.plot(X_plot, y_pred, color=color, linewidth=2, label=f'τ = {tau}')
    
    ax.set_xlabel('X')
    ax.set_ylabel('y')
    ax.set_title(title)
    ax.legend(loc='upper left')
    ax.grid(True, alpha=0.3)
 
fig, axes = plt.subplots(1, 2, figsize=(14, 5))
 
fit_and_plot_quantiles(X_homo, y_homo, 'Homoscedastic: Parallel Lines', axes[0])
fit_and_plot_quantiles(X_hetero, y_hetero, 'Heteroscedastic: Fan Pattern', axes[1])
 
plt.tight_layout()
plt.show()
 
# Quantitative test: Compare IQR at different X values
print("Interquartile Range (IQR) Analysis:")
print("=" * 50)
 
for scenario, X, y in [("Homoscedastic", X_homo, y_homo), 
                        ("Heteroscedastic", X_hetero, y_hetero)]:
    X = X.reshape(-1, 1)
    
    # Fit Q25 and Q75
    model_25 = QuantileRegressor(quantile=0.25, alpha=0, solver='highs').fit(X, y)
    model_75 = QuantileRegressor(quantile=0.75, alpha=0, solver='highs').fit(X, y)
    
    # IQR at X=2 vs X=8
    iqr_low = model_75.predict([[2]]) - model_25.predict([[2]])
    iqr_high = model_75.predict([[8]]) - model_25.predict([[8]])
    
    print(f"
{scenario}:")
    print(f"  IQR at X=2: {iqr_low[0]:.2f}")
    print(f"  IQR at X=8: {iqr_high[0]:.2f}")
    print(f"  Ratio: {iqr_high[0]/iqr_low[0]:.2f}x")

Visual Diagnostic Power

Distributional Effects: Beyond Location Shift

Quantile regression excels at capturing complex distributional effects—situations where covariates don't simply shift or scale the distribution but reshape it entirely.

Types of Distributional Effects:

1. Location Shift Only:

$Q_\tau(Y|X=x) = x\beta + q_\tau^0$ where $q_\tau^0$ is constant by $\tau$
All quantile lines are parallel
Covariate shifts entire distribution uniformly

2. Location-Scale Change:

$Q_\tau(Y|X=x) = x\beta + x\gamma \cdot q_\tau^0$
Quantile lines diverge (or converge) as $x$ changes
Covariate affects both center and spread

3. Skewness Change:

Upper and lower quantiles move asymmetrically
Distribution becomes more or less skewed with $x$
Example: Wages become more right-skewed at higher education levels

4. Complex Reshaping:

Arbitrary changes in distributional shape
Different covariates affecting different parts of the distribution
Example: A treatment helps weak performers, hurts strong performers

Patterns in Quantile Coefficients and Their Interpretations
Pattern in β(τ)	Interpretation	Example
Constant across τ	Pure location shift	Fixed dollar raise for all employees
Increases with τ	Positive compression (high quantiles affected more)	Percent-based raise; training with diminishing returns for low performers
Decreases with τ	Negative compression (low quantiles affected more)	Minimum wage increase; safety net policies
Positive for low τ, negative for high τ	Compression toward center	Regulations reducing inequality
Negative for low τ, positive for high τ	Expansion from center	Deregulation increasing variance
Non-monotonic (U-shape, etc.)	Complex distributional changes	Technology affecting middle-skilled workers negatively

distributional_effects.py
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import numpy as np
import matplotlib.pyplot as plt
from sklearn.linear_model import QuantileRegressor
 
np.random.seed(42)
n = 1000
 
# Create treatment indicator and baseline
treatment = np.random.binomial(1, 0.5, n)
baseline_ability = np.random.normal(0, 1, n)
 
# Treatment has HETEROGENEOUS effects:
# - Helps low-baseline individuals more  
# - Has minimal effect on high-baseline individuals
# This is a "compression" effect
 
treatment_effect = np.where(treatment == 1, 
                             3 - 1.5 * baseline_ability,  # Larger effect for low baseline
                             0)
 
y = 5 + 2 * baseline_ability + treatment_effect + np.random.normal(0, 1, n)
 
# Fit quantile regressions for treatment effect
X = np.column_stack([np.ones(n), treatment])
quantiles = [0.1, 0.25, 0.5, 0.75, 0.9]
 
treatment_effects = []
for tau in quantiles:
    model = QuantileRegressor(quantile=tau, alpha=0, solver='highs')
    model.fit(X, y)
    treatment_effects.append(model.coef_[1])  # Treatment coefficient
 
# Visualization
fig, axes = plt.subplots(1, 2, figsize=(14, 5))
 
# Plot 1: Treatment effect by quantile
ax1 = axes[0]
ax1.plot(quantiles, treatment_effects, 'o-', linewidth=2, markersize=10)
ax1.axhline(y=np.mean(treatment_effects), color='red', linestyle='--', 
            label=f'OLS estimate (approx.)')
ax1.set_xlabel('Quantile τ', fontsize=12)
ax1.set_ylabel('Treatment Effect β(τ)', fontsize=12)
ax1.set_title('Heterogeneous Treatment Effect Across Quantiles', fontsize=14)
ax1.legend()
ax1.grid(True, alpha=0.3)
 
# Add interpretation annotations
ax1.annotate('Larger effect
for lower quantiles
(struggling students)', 
             xy=(0.1, treatment_effects[0]), 
             xytext=(0.25, treatment_effects[0] + 0.5),
             fontsize=10, arrowprops=dict(arrowstyle='->', color='gray'))
 
# Plot 2: Distribution comparison
ax2 = axes[1]
control = y[treatment == 0]
treated = y[treatment == 1]
 
ax2.hist(control, bins=40, alpha=0.5, density=True, label='Control', color='blue')
ax2.hist(treated, bins=40, alpha=0.5, density=True, label='Treatment', color='green')
ax2.axvline(np.median(control), color='blue', linestyle='--', linewidth=2)
ax2.axvline(np.median(treated), color='green', linestyle='--', linewidth=2)
ax2.set_xlabel('Outcome y', fontsize=12)
ax2.set_ylabel('Density', fontsize=12)
ax2.set_title('Outcome Distributions by Treatment Status', fontsize=14)
ax2.legend()
ax2.grid(True, alpha=0.3)
 
plt.tight_layout()
plt.show()
 
print("Treatment Effect by Quantile:")
print("-" * 40)
for tau, effect in zip(quantiles, treatment_effects):
    print(f"τ = {tau}: β = {effect:.3f}")

Policy Relevance

Quantile Coefficient Plots: The Essential Visualization

The quantile coefficient plot (also called the "quantile process plot") is the definitive visualization for quantile regression results.

Construction:

Fit quantile regression for a fine grid of $\tau$ values (e.g., 0.05, 0.10, ..., 0.95)
Extract coefficient $\hat{\beta}_j(\tau)$ for each covariate $j$ at each $\tau$
Plot $\hat{\beta}_j(\tau)$ vs. $\tau$ with confidence bands
Include OLS estimate as horizontal reference line

Interpretation Guide:

Reading Quantile Coefficient Plots

•Horizontal line pattern: Covariate has homogeneous effect across quantiles; OLS tells the whole story
•Upward slope: Covariate has larger effects at higher quantiles (e.g., education on wage inequality)
•Downward slope: Covariate has larger effects at lower quantiles (e.g., safety interventions help the vulnerable most)
•Confidence bands overlapping OLS: Statistical evidence consistent with homogeneous effects
•Confidence bands excluding OLS at some tau: Statistically significant heterogeneity
•Wide bands at extremes: Less precise estimation for tail quantiles (fewer observations)

quantile_coefficient_plot.py
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import numpy as np
import matplotlib.pyplot as plt
from sklearn.linear_model import QuantileRegressor, LinearRegression
from scipy import stats
 
def quantile_coefficient_plot(X, y, covariate_idx, covariate_name, 
                              quantiles=None, n_bootstrap=100, ax=None):
    """
    Create a quantile coefficient plot with confidence bands.
    
    Parameters:
    -----------
    X : np.ndarray
        Feature matrix (n_samples, n_features)
    y : np.ndarray
        Target variable
    covariate_idx : int
        Index of the covariate to plot
    covariate_name : str
        Name for plot label
    quantiles : list, optional
        Quantile levels to estimate
    n_bootstrap : int
        Number of bootstrap samples for confidence bands
    ax : matplotlib.axes, optional
        Axes to plot on
    """
    if quantiles is None:
        quantiles = np.arange(0.05, 0.96, 0.05)
    
    n = len(y)
    
    # Fit OLS for reference
    ols = LinearRegression().fit(X, y)
    ols_coef = ols.coef_[covariate_idx]
    
    # Fit quantile regressions
    qr_coefs = []
    for tau in quantiles:
        model = QuantileRegressor(quantile=tau, alpha=0, solver='highs')
        model.fit(X, y)
        qr_coefs.append(model.coef_[covariate_idx])
    
    qr_coefs = np.array(qr_coefs)
    
    # Bootstrap for confidence bands
    boot_coefs = np.zeros((n_bootstrap, len(quantiles)))
    
    for b in range(n_bootstrap):
        # Resample
        idx = np.random.choice(n, size=n, replace=True)
        X_boot, y_boot = X[idx], y[idx]
        
        for i, tau in enumerate(quantiles):
            try:
                model = QuantileRegressor(quantile=tau, alpha=0, solver='highs')
                model.fit(X_boot, y_boot)
                boot_coefs[b, i] = model.coef_[covariate_idx]
            except:
                boot_coefs[b, i] = np.nan
    
    # Compute confidence bands (2.5th and 97.5th percentiles)
    lower = np.nanpercentile(boot_coefs, 2.5, axis=0)
    upper = np.nanpercentile(boot_coefs, 97.5, axis=0)
    
    # Plot
    if ax is None:
        fig, ax = plt.subplots(figsize=(10, 6))
    
    ax.fill_between(quantiles, lower, upper, alpha=0.3, color='blue',
                    label='95% Confidence Band')
    ax.plot(quantiles, qr_coefs, 'b-', linewidth=2, label='Quantile Estimates')
    ax.axhline(y=ols_coef, color='red', linestyle='--', linewidth=2,
               label=f'OLS Estimate: {ols_coef:.3f}')
    ax.axhline(y=0, color='gray', linestyle=':', linewidth=1)
    
    ax.set_xlabel('Quantile τ', fontsize=12)
    ax.set_ylabel(f'β(τ) for {covariate_name}', fontsize=12)
    ax.set_title(f'Quantile Coefficients: {covariate_name}', fontsize=14)
    ax.legend(loc='best')
    ax.grid(True, alpha=0.3)
    
    return ax, qr_coefs, (lower, upper)
 
# Example: Simulated heterogeneous effect
np.random.seed(42)
n = 800
 
# X1: Constant effect across quantiles
# X2: Effect increases with quantile (heterogeneous)
X1 = np.random.normal(0, 1, n)
X2 = np.random.normal(0, 1, n)
 
# Generate y with heterogeneous effect for X2
epsilon = np.random.standard_t(df=5, size=n)  # Heavy-tailed errors
y = 3 + 2 * X1 + (1 + 0.5 * epsilon) * X2 + epsilon
 
X = np.column_stack([X1, X2])
 
# Create plots
fig, axes = plt.subplots(1, 2, figsize=(14, 5))
 
quantile_coefficient_plot(X, y, 0, 'X₁ (Homogeneous Effect)', ax=axes[0])
quantile_coefficient_plot(X, y, 1, 'X₂ (Heterogeneous Effect)', ax=axes[1])
 
plt.tight_layout()
plt.show()
 
print("X₁ has roughly constant effect across quantiles (homogeneous)")
print("X₂ shows increasing effect at higher quantiles (heterogeneous)")

Publication-Quality Plots

Quantile Treatment Effects

In causal inference, Quantile Treatment Effects (QTE) provide richer information than the Average Treatment Effect (ATE).

Definition (Quantile Treatment Effect):

For treatment $D \in {0, 1}$ and outcome $Y$, the QTE at quantile $\tau$ is:

$$\text{QTE}(\tau) = Q_\tau(Y^1) - Q_\tau(Y^0)$$

where $Y^1$ and $Y^0$ are potential outcomes under treatment and control.

Interpretation:

QTE($\tau$) measures how the treatment shifts the $\tau$-th quantile of the outcome distribution.

Important Caveat:

When QTE Equals Individual Treatment Effect:

Treatment Effect Concepts
Concept	Definition	Interpretation
ATE	E[Y¹ - Y⁰]	Average effect across entire population
QTE(τ)	Q_τ(Y¹) - Q_τ(Y⁰)	Effect on τ-th quantile of distributions
ATT	E[Y¹ - Y⁰ \| D=1]	Average effect on the treated
CATE	E[Y¹ - Y⁰ \| X=x]	Conditional average effect given covariates

Why QTE Matters:

Example: Minimum Wage Increase

Effect	Interpretation
ATE = $0.50/hr	Average wage increases modestly
QTE(0.10) = $1.50/hr	Large gains at the 10th percentile (low earners)
QTE(0.50) = $0.30/hr	Small median effect
QTE(0.90) = $0.00/hr	No effect on high earners

The ATE misses the distributional story: minimum wage disproportionately helps those at the bottom.

Estimation:

With random assignment and the linearity assumption: $$Q_\tau(Y \mid D) = \alpha(\tau) + \text{QTE}(\tau) \cdot D$$

Fitting quantile regression of $Y$ on treatment indicator $D$ gives QTE($\tau$) directly.

Rank Invariance Warning

Handling Multiple Covariates

Real applications involve multiple covariates, each potentially having quantile-specific effects.

The Full Model:

$$Q_\tau(Y \mid X) = \beta_0(\tau) + \beta_1(\tau) X_1 + \cdots + \beta_p(\tau) X_p$$

Interpretation Challenges:

Dimensionality: With $K$ quantiles and $p$ covariates, you have $K \times p$ coefficients to interpret
Ceteris paribus: $\beta_j(\tau)$ is the effect of $X_j$ on $Q_\tau(Y|X)$ holding other $X$s constant
Multicollinearity: Affects quantile regression just like OLS—correlated features have unstable coefficients

Strategies for Summarization:

Managing Complexity with Many Covariates

•Focus on key quantiles: Report τ ∈ {0.1, 0.25, 0.5, 0.75, 0.9} rather than all quantiles
•Grid of coefficient plots: Create one quantile coefficient plot per covariate
•Interquartile range effects: Report β(0.75) - β(0.25) to summarize spread effects
•Coefficient tables: Traditional table with columns for different τ values
•Heat maps: Color-coded matrix of coefficients (covariates × quantiles)
•Test for homogeneity: Formal tests whether β(τ) varies significantly with τ

Testing for Quantile Coefficient Equality:

To test whether an effect is truly heterogeneous across quantiles, we can use:

Wald Test: $$H_0: \beta_j(\tau_1) = \beta_j(\tau_2) \quad \text{vs.} \quad H_1: \beta_j(\tau_1) eq \beta_j(\tau_2)$$

Procedure:

Estimate $\hat{\beta}_j(\tau_1)$ and $\hat{\beta}_j(\tau_2)$ with bootstrap standard errors
Compute $T = \frac{\hat{\beta}_j(\tau_1) - \hat{\beta}_j(\tau_2)}{\text{SE}(\hat{\beta}_j(\tau_1) - \hat{\beta}_j(\tau_2))}$
Compare to standard normal critical values

Joint Test (Khmaladze Transform):

More sophisticated tests consider the entire quantile process ${\beta_j(\tau) : \tau \in (0,1)}$ simultaneously, controlling for the multiple comparison problem.

Multiple Testing

Summary and Looking Ahead

We have explored how conditional quantiles reveal the full distributional relationship between covariates and outcomes.

Key Takeaways

•Conditional quantiles characterize the full distribution — Q_τ(Y|X) for all τ ∈ (0,1) completely specifies the conditional distribution.
•Each quantile has its own coefficients — β(τ) can vary arbitrarily with τ, revealing heterogeneous effects.
•Heteroscedasticity is visible — Fan-shaped quantile regression lines indicate changing variance.
•Distributional effects transcend location-scale — Covariates can compress, expand, or reshape distributions in complex ways.
•Quantile coefficient plots are essential — They efficiently communicate effect heterogeneity with uncertainty.
•QTE enriches causal inference — Treatment effects across the distribution provide richer policy insights than ATE alone.

What's Next:

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