Common mistake 1 with Quantile regression

Confusing the quantile of y with a quantile of x. You are splitting the OUTCOME.

Common mistake 2 with Quantile regression

Reading crossing quantile lines as a finding. It usually means the model is misspecified.

Common mistake 3 with Quantile regression

Quoting a quantile coefficient as if it followed the same person over time. It compares different points of the distribution, not the same individual.

Common mistake 4 with Quantile regression

Estimating extreme quantiles - the 1st or the 99th - on a few hundred observations. The estimator is consistent there but converges slowly, and in small samples those coefficients are close to noise.

EconStatsCraft › Quantile regression

Quantile regression

OLS gives you the effect on the mean. Quantile regression gives you the effect at the 10th, 50th and 90th percentile - which is how you find out whether something matters more at the bottom of the distribution than the top.

What it needs

Dependent variable
Independent variables

The intuition

OLS answers "how does the AVERAGE of y move with x?" Quantile regression answers "how does the 10th percentile of y move with x?" - a different and often more policy-relevant question. If a training programme lifts the bottom of the wage distribution and does nothing at the top, the mean effect is a small positive number that describes nobody.

When to use it

When you suspect an effect differs across the distribution: a minimum wage matters at the bottom, a tax break at the top, and averaging them together hides both.

Where the estimator comes from

Minimise \sum_i \rho_\tau(y_i-x_i'\beta) where the check function \rho_\tau(u)=u\bigl(\tau-\mathbf{1}\{u<0\}\bigr) penalises over- and under-prediction asymmetrically. At \tau=0.5 the penalties are symmetric and you recover the median. The problem is piecewise linear and solved by linear programming, not calculus - there is no normal-equation analogue.

The model

Q_\tau(y\mid \mathbf{x})=\mathbf{x}^{\prime}\boldsymbol{\beta}_\tau,\qquad \hat{\boldsymbol{\beta}}_\tau=\arg\min_{\boldsymbol{\beta}}\sum_i \rho_\tau\!\left(y_i-\mathbf{x}_i^{\prime}\boldsymbol{\beta}\right),\quad \rho_\tau(u)=u\bigl(\tau-\mathbf{1}\{u<0\}\bigr)

What it assumes

The conditional quantile is linear in x.
Observations are independent.
It does NOT assume Normal or homoskedastic errors - that is its advantage.
There is enough data in the tail you are estimating. A 10th-percentile regression uses information concentrated near that quantile, so the effective sample is far smaller than the row count and the standard errors are correspondingly wide.

Statistical properties

Consistent and asymptotically Normal, with a variance that depends on the conditional density of the errors at the quantile.
Equivariant to monotone transformations: Q_τ(log y) = log Q_τ(y). The mean has no such property, so E[log y] ≠ log E[y].
Robust to outliers in y - a single extreme observation cannot drag the median fit the way it drags OLS.
Under homoskedasticity every quantile slope equals the OLS slope; differences ACROSS quantiles are precisely a form of heteroskedasticity.
Standard errors are less reliable in small samples; bootstrapping is the usual remedy.

How to read the output

Each column is a separate regression at that quantile.
A coefficient that rises across quantiles means the effect is stronger at the top of the distribution.
The median regression (τ = 0.5) is the robust alternative to OLS.
The plot is the key output - a sloping line means the mean effect was hiding heterogeneity.

A worked reading

β̂_educ = 0.0694 at τ = 0.10 and 0.0909 at τ = 0.90, against an OLS figure of 0.0786. The return to schooling is about a quarter lower at the bottom of the conditional wage distribution than at the top - 0.0694 is 76% of 0.0909 - so the single OLS number overstates what schooling pays low earners and understates what it pays high earners. The gap is real but moderate: this is a fanning-out of the distribution, not two different economies, and quoting it as such is the difference between a finding and a slogan.

Symptom → diagnosis → what to do

If you see this	It means	Do this
Quantile lines cross	Misspecification, or sparse data in the tails	Add flexibility, or restrict to the central quantiles
Coefficients rise steadily across τ	The effect is stronger at the top of the distribution	A real finding - report the plot, not just the median
All quantiles ≈ OLS	Homoskedastic, effects are uniform	OLS is a fair summary; say so
Wild estimates at τ = 0.05 or 0.95	Too few observations in the tail	Move inward; you need a lot of data for extreme quantiles

Common mistakes

Confusing the quantile of y with a quantile of x. You are splitting the OUTCOME.
Reading crossing quantile lines as a finding. It usually means the model is misspecified.
Quoting a quantile coefficient as if it followed the same person over time. It compares different points of the distribution, not the same individual.
Estimating extreme quantiles - the 1st or the 99th - on a few hundred observations. The estimator is consistent there but converges slowly, and in small samples those coefficients are close to noise.

Where to read this properly

Koenker & Bassett (1978) - the original paper
Koenker (2005), Quantile Regression
Angrist and Pischke (2009), Mostly Harmless Econometrics: An Empiricist's Companion, Princeton University Press, chapter 7

Related Regression

Multiple regression (OLS) · Logit / Probit · Tobit (censored) · Truncated regression · Multinomial logit · Ordered logit / probit · Poisson / Negative binomial

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Data & variables

Each example is simulated from a known model, so you can check whether the estimator recovers the truth - and nothing here is anyone's private data.

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Drag a variable into a slot on the right - or just click it.

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Statistic

Bands

How many

Cut points

Keep rows where

Value

and

Fill with

Value

Also mark which rows were filled

Rules

Base period = 100

Which part

Percent at each tail

Threshold

Survey weight optional

Direction

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Largest first

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Read this column as

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Quantile regression

What it needs

The intuition

When to use it

Where the estimator comes from

The model

What it assumes

Statistical properties

How to read the output

A worked reading

Symptom → diagnosis → what to do

Common mistakes

Where to read this properly

See also

Related Regression

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