Conditional Quantiles as Econometric Estimands

Why Conditional Quantiles?

An outcome’s average does not describe every economically relevant part of its distribution. For example, the median duration of a process may differ across covariate groups even when their means are similar, and changes near the upper tail may matter more than changes near the center. A conditional quantile describes a selected point in the outcome distribution for units with specified covariates.

The key is to be precise about what is being conditioned on and which population distribution is the target. A conditional quantile is not an individual’s outcome, and it is not automatically a causal effect. It is a feature of the distribution of an outcome among units with a given covariate value.

Definition: Let \(Y\) be a real-valued outcome and \(X\) a covariate vector. For a covariate value \(x\) and quantile index \(0<\tau<1\), define the conditional distribution function \(F_{Y\mid X}(y\mid x)=P(Y\leq y\mid X=x)\). Using the lower-quantile convention, the conditional \(\tau\)-quantile is \(Q_\tau(Y\mid X=x)=\inf\{y:F_{Y\mid X}(y\mid x)\geq\tau\}\). As \(x\) varies, \(Q_\tau(Y\mid X=x)\) is the conditional quantile function at level \(\tau\).

This definition applies to continuous and discrete outcomes. For a discrete outcome, the cumulative distribution can jump over \(\tau\), so the quantile is generally a value at which the cumulative probability reaches or exceeds \(\tau\), not necessarily a value where it equals \(\tau\). The lower-quantile convention selects the smallest such value. Conditional distributions, and therefore conditional quantiles, are understood for covariate values in the relevant support; a value \(x\) with no population support does not have an empirically identified conditional distribution without additional modeling assumptions.

Quantiles From a Conditional Distribution

For a fixed \(x\), write \(F(y)=F_{Y\mid X}(y\mid x)\) and \(q=Q_\tau(Y\mid X=x)\). A useful characterization is \(F(q-)\leq\tau\leq F(q)\), where \(F(q-)\) is the probability accumulated strictly below \(q\). In a discrete distribution, the mass at \(q\) may be what makes the cumulative probability cross \(\tau\).

Justification: The CDF is nondecreasing. By definition, \(q\) is the infimum of values where \(F(y)\geq\tau\). Thus values strictly below \(q\) have cumulative probability below \(\tau\), so \(F(q-)\leq\tau\). Right-continuity of a CDF ensures that the cumulative probability at the lower quantile reaches or exceeds \(\tau\), so \(\tau\leq F(q)\). If the distribution has a jump at \(q\), the two bounds can be strict on either side.

Worked Example: Finding Conditional Quantiles With Ties

Suppose a hypothetical service-duration outcome \(Y\), measured in days, has the following conditional distribution for units with \(X=0\): \(P(Y=2\mid X=0)=0.20\), \(P(Y=5\mid X=0)=0.50\), and \(P(Y=9\mid X=0)=0.30\). Find the conditional 25th, 70th, and 80th percentiles.

The conditional CDF is \(0\) below 2, \(0.20\) from 2 up to but not including 5, \(0.70\) from 5 up to but not including 9, and \(1\) at and above 9.

For \(\tau=0.25\), the CDF at 2 is \(0.20<0.25\), while the CDF at 5 is \(0.20+0.50=0.70\geq0.25\). Therefore \(Q_{0.25}(Y\mid X=0)=5\) days. For \(\tau=0.70\), the CDF first reaches 0.70 at 5, so \(Q_{0.70}(Y\mid X=0)=5\) days. For \(\tau=0.80\), the CDF at 5 is \(0.70<0.80\), and at 9 it is \(0.70+0.30=1.00\geq0.80\), so \(Q_{0.80}(Y\mid X=0)=9\) days.

The conditional mean is \(2(0.20)+5(0.50)+9(0.30)=0.40+2.50+2.70=5.60\) days. It is not the same object as the conditional median, which here is 5 days. Each number summarizes a different feature of this hypothetical conditional distribution.

The Covariate-Specific Estimand

An estimand is the population quantity an analysis aims to describe or estimate. At a fixed quantile index \(\tau\), the covariate-specific estimand is \(Q_\tau(Y\mid X=x)\), evaluated at a specified \(x\). It depends on the conditional distribution in the target population, not just on the outcome values observed in a particular sample.

The conditioning statement matters. For example, \(Q_{0.5}(Y\mid X=x)\) is the median among units with covariates \(x\); it is not the median for the whole population. Likewise, changing the target population can change the conditional distribution and the estimand. In applications involving duration, one must also be clear about which duration is the outcome and how censoring affects what is observed. Defining a target conditional quantile does not, by itself, solve the problem of recovering it from censored observations.

The Linear Conditional Quantile Model

A linear conditional quantile model restricts the conditional quantile function to be linear in specified regressors. Let \(Z(X)\) be a vector of regressors constructed from \(X\), often including an intercept. At a particular quantile index \(\tau\), the model is

Q_\tau(Y\mid X=x)=Z(x)'\beta_\tau.

Here \(\beta_\tau\) is the coefficient vector for the chosen quantile index. This equation is a model for a conditional quantile, not for the conditional mean. The coefficients can differ across quantile levels, and the equation asserts a linear specification for the selected conditional quantile over the covariate values to which it is applied. It does not assert that the entire conditional distribution is linear, or that each person’s outcome lies on the fitted line.

Interpretation: In a linear conditional quantile model, a coefficient describes how the modeled conditional \(\tau\)-quantile changes with a regressor, holding the other included regressors fixed. Its units are outcome units per unit of that regressor. The interpretation is conditional and distributional; it is not automatically a causal effect or a change in the conditional mean.

Worked Example: Interpreting a Linear Conditional Median

Suppose a hypothetical model for service duration in days is \(Q_{0.50}(Y\mid \text{age},\text{remote})=12+2(\text{age})-3(\text{remote})\), where age is measured in years and remote equals 1 for remote service and 0 otherwise. Use the model to calculate the fitted conditional median for a 50-year-old receiving remote service, and interpret the coefficients.

For this covariate profile, substitute age \(=50\) and remote \(=1\): \(12+2(50)-3(1)=12+100-3=109\) days. The model’s conditional median duration for this profile is 109 days.

Holding remote status fixed, a one-year increase in age corresponds to a 2-day increase in the modeled conditional median. Holding age fixed, remote service corresponds to a conditional median 3 days lower than non-remote service in this model. These are statements about the fitted conditional median. They do not establish that changing a person’s service mode would cause a 3-day reduction. Also, the intercept is the fitted median at age zero and remote \(=0\); if age zero is outside the population support, that intercept need not have a useful substantive interpretation.

A linear specification is a substantive modeling restriction, not a consequence of defining quantiles. If the true conditional quantile function is curved in a covariate, a straight-line specification may summarize it inadequately. Interactions, transformations, or a different functional form may be needed to represent the conditional quantile more plausibly. These choices concern the model for the estimand; they do not change the definition of a conditional quantile.

Conditional Quantiles Are Not Unconditional Quantiles

An unconditional quantile is calculated from the marginal distribution of \(Y\), after averaging over the population distribution of \(X\). In general, it is not obtained by averaging conditional quantiles. A mixture of conditional distributions can have a quantile that differs from the quantile in every subgroup.

Worked Example: A Conditional Median and a Population Median

Consider a hypothetical population with two equally common covariate groups, \(A\) and \(B\), so \(P(A)=P(B)=0.50\). The outcome \(Y\) is measured in units. In group \(A\), the probabilities of \(Y=0,40,60,100\) are respectively \(0.40,0.40,0.20,0\). In group \(B\), they are respectively \(0,0,0.20,0.80\). Find both group medians and the marginal population median.

In group \(A\), the cumulative probability at 0 is 0.40 and at 40 is \(0.40+0.40=0.80\). The conditional median is therefore \(Q_{0.50}(Y\mid A)=40\). In group \(B\), the cumulative probability at 60 is 0.20 and at 100 is \(0.20+0.80=1.00\), so \(Q_{0.50}(Y\mid B)=100\).

To obtain the marginal distribution, multiply each group-specific probability by the group’s population share and add. Thus \(P(Y=0)=0.50(0.40)+0.50(0)=0.20\); \(P(Y=40)=0.50(0.40)+0.50(0)=0.20\); \(P(Y=60)=0.50(0.20)+0.50(0.20)=0.20\); and \(P(Y=100)=0.50(0)+0.50(0.80)=0.40\). These probabilities sum to \(0.20+0.20+0.20+0.40=1.00\).

The marginal cumulative probabilities at 0, 40, 60, and 100 are 0.20, 0.40, 0.60, and 1.00. Hence the population median is 60. It is neither group median, and it is not the average of them: \(0.50(40)+0.50(100)=70\). This demonstrates why a conditional quantile and a marginal quantile answer different questions.

Conditional Quantiles Are Not Individual Treatment Effects

A conditional quantile describes the distribution of observed outcomes among units with specified covariates. An individual treatment effect instead refers to a unit-level contrast between potential outcomes under two treatment states, commonly written \(Y_i(1)-Y_i(0)\). The conditional quantile function alone does not identify the joint pairing of \(Y_i(1)\) and \(Y_i(0)\) for each unit.

Even if treatment-specific conditional outcome distributions were known, a difference between their quantiles would compare two distributional summaries. It would not, without further assumptions, equal a quantile of individual treatment effects. The reason is that separate marginal distributions do not reveal which potential outcome under one state belongs to the same unit as which potential outcome under the other state. Distinguishing these objects prevents a distributional comparison from being misdescribed as a statement about each individual.

Common Mistakes

Calling a conditional quantile a mean. A conditional median is a 50th percentile, not the expected outcome. The two can differ, as the 5-day median and 5.60-day mean in the first example show.
Treating a subgroup quantile as a population quantile. Conditioning on \(X=x\) defines a subgroup distribution; the marginal distribution mixes across covariate values.
Assuming the quantile must have CDF value exactly \(\tau\). With atoms or ties, the CDF can jump over \(\tau\). Under the lower-quantile convention, use the first value where the CDF reaches or exceeds \(\tau\).
Reading a linear quantile coefficient as causal or individual-level. The coefficient describes a modeled conditional quantile contrast, with other included regressors held fixed. Causal language and claims about individual treatment effects require additional identification assumptions.

Key takeaway: A conditional quantile is a covariate-specific feature of an outcome distribution. The linear conditional quantile model imposes a linear form on that feature at a chosen quantile level; it does not turn the quantile into a mean, an unconditional summary, or an individual effect.

Check Your Understanding

Use the definitions and distinctions in this tutorial to answer the following.

If \(F_{Y\mid X}(4\mid x)=0.35\) and \(F_{Y\mid X}(7\mid x)=0.75\), what is \(Q_{0.50}(Y\mid X=x)\), assuming no outcome values between 4 and 7 have positive probability?
In a linear conditional quantile model, what does a slope coefficient describe, and why is it not automatically a mean effect?
Why can the marginal median differ from the medians in each of two covariate groups?
Explain why the difference between two conditional outcome quantiles is not generally a quantile of individual treatment effects.
For a discrete conditional distribution, why can \(F_{Y\mid X}(q\mid x)\) be greater than \(\tau\) at \(q=Q_\tau(Y\mid X=x)\)?