Conditional And Unconditional Quantile Effects

Two Different Quantile Questions

A conditional quantile asks where a specified fraction of outcomes falls among units with particular covariate values. A marginal, or unconditional, quantile asks where that fraction falls in the outcome distribution of the population as a whole. These are distinct estimands: the overall distribution depends both on the conditional outcome distributions and on how common different covariate values are.

For example, a regression of the conditional median on a binary group indicator compares the two groups’ medians. It does not, in general, describe how the population median would change if the proportion of units in each group changed. Recentered-influence-function (RIF) regression provides a way to summarize relationships between covariates and a marginal quantile, while making clear which distributional features are being summarized.

Definition: Let \(F_Y(y)=P(Y\leq y)\) be the marginal cumulative distribution function of an outcome \(Y\). Its \(\tau\)-quantile, for \(0<\tau<1\), is \(q_\tau=\inf\{y:F_Y(y)\geq\tau\}\). The conditional \(\tau\)-quantile at covariate value \(x\) is \(Q_\tau(Y\mid X=x)=\inf\{y:P(Y\leq y\mid X=x)\geq\tau\}\). The first is a feature of the overall outcome distribution; the second is a feature of the outcome distribution conditional on \(X=x\).

A conditional quantile regression models how \(Q_\tau(Y\mid X=x)\) varies with \(x\). A marginal quantile, by contrast, is calculated from \(F_Y(y)=\int F_{Y\mid X}(y\mid x)\,dP_X(x)\): it averages the conditional distributions using the population distribution \(P_X\) of covariates. Changing \(P_X\) can therefore change \(q_\tau\), even if the conditional distributions remain fixed.

Worked Examples: Conditional and Marginal Quantiles

Worked Example: Changing Group Composition

Suppose \(X\) is binary. Conditional on \(X=0\), \(Y\) is Uniform[0, 2]; conditional on \(X=1\), \(Y\) is Uniform[1, 3]. Let \(\pi=P(X=1)\). Find the conditional medians and the marginal median, and examine what happens as the group composition changes.

The conditional medians are the midpoints of their respective uniform distributions: \(Q_{0.5}(Y\mid X=0)=1\) and \(Q_{0.5}(Y\mid X=1)=2\). Their difference is 1 outcome unit.

For \(1\leq y\leq2\), the conditional CDFs are \(F_{Y\mid X=0}(y)=y/2\) and \(F_{Y\mid X=1}(y)=(y-1)/2\). Thus the marginal CDF in this range is

F_Y(y)=(1-\pi)\frac{y}{2}+\pi\frac{y-1}{2} =\frac{y-\pi}{2}.

Setting this equal to 0.5 gives \(q_{0.5}=1+\pi\), which lies in [1, 2] for \(0\leq\pi\leq1\). If \(\pi=0.5\), the population median is 1.5. If the share in group 1 rises from 0.5 to 0.6 while both conditional distributions stay the same, the marginal median rises from 1.5 to 1.6. The conditional medians have not changed. This is a composition change, not a change in either group’s conditional median.

The marginal median in this example is not the mean of the two conditional medians in every population: it depends on \(\pi\). More generally, a marginal quantile is not obtained by averaging conditional quantiles. The conditional distributions must first be combined into a marginal distribution, and then the desired quantile must be found.

Influence Function of a Marginal Quantile

The influence function describes the first-order change in a statistical functional when a small amount of probability mass is added to a distribution. For a marginal quantile, the familiar formula requires regularity. In particular, assume that \(F_Y\) is differentiable near \(q_\tau\), with a continuous density \(f_Y\) that is strictly positive at \(q_\tau\), and that \(F_Y(q_\tau)=\tau\). For a probability distribution \(G\), consider the mixture \(F_\varepsilon=(1-\varepsilon)F_Y+\varepsilon G\). To use the derivative formula below, also require \(G\) to be continuous at \(q_\tau\) and the mixture quantile to be locally differentiable at \(\varepsilon=0\).

Definition: Under those regularity conditions, the influence function of the \(\tau\)-quantile at an outcome value \(y\) is \(\operatorname{IF}(y;q_\tau,F_Y)=(\tau-\mathbf{1}\{y\leq q_\tau\})/f_Y(q_\tau)\). The recentered influence function is \(\operatorname{RIF}(y;q_\tau,F_Y)=q_\tau+\operatorname{IF}(y;q_\tau,F_Y)\).

Here \(\mathbf{1}\{y\leq q_\tau\}\) equals 1 when the condition holds and 0 otherwise. Under the stated conditions, \(P(Y\leq q_\tau)=\tau\), so the influence function has mean zero and the RIF has mean \(q_\tau\). The density in the denominator converts a small change in probability near the quantile into a change measured in the outcome’s units.

The derivative can be obtained directly from the defining quantile equation. Write \(q_\tau(\varepsilon)\) for the \(\tau\)-quantile of \(F_\varepsilon\). Under the smoothness conditions, \(F_\varepsilon(q_\tau(\varepsilon))=\tau\) near \(\varepsilon=0\). Differentiating at zero gives

0=G(q_\tau)-F_Y(q_\tau)+f_Y(q_\tau)q_\tau'(0) =G(q_\tau)-\tau+f_Y(q_\tau)q_\tau'(0).

Consequently,

q_\tau'(0)=\frac{\tau-G(q_\tau)}{f_Y(q_\tau)} =E_G\!\left[\operatorname{IF}(Y;q_\tau,F_Y)\right].

The last equality follows because \(G\) has no atom at \(q_\tau\), so \(E_G[\mathbf{1}\{Y\leq q_\tau\}]=G(q_\tau)\). This is a directional derivative along the specified mixture, subject to the conditions above; it is not a derivative formula valid for every possible \(G\).

Worked Example: Calculating RIF Values and a Regression Slope

Use the two-group uniform distributions from the first example, with \(P(X=0)=P(X=1)=0.5\), and focus on the marginal median. Calculate the RIF and the population linear regression of the RIF on an intercept and \(X\).

The marginal median is \(q_{0.5}=1.5\). In a neighborhood of 1.5, each conditional density is \(1/2\), so the marginal density is \(f_Y(1.5)=0.5\). The RIF is therefore

\operatorname{RIF}(Y;1.5,F_Y) =1.5+\frac{0.5-\mathbf{1}\{Y\leq1.5\}}{0.5}.

For \(Y\leq1.5\), this equals \(1.5+(0.5-1)/0.5=0.5\); for \(Y>1.5\), it equals \(1.5+(0.5-0)/0.5=2.5\). In group 0, \(P(Y\leq1.5\mid X=0)=1.5/2=0.75\). Thus the conditional mean RIF is \(0.75(0.5)+0.25(2.5)=1.0\). In group 1, \(P(Y\leq1.5\mid X=1)=(1.5-1)/2=0.25\), so the conditional mean RIF is \(0.25(0.5)+0.75(2.5)=2.0\).

With a binary regressor and an intercept, the population least-squares slope is the difference between the two conditional means. The RIF regression slope on \(X\) is therefore \(2.0-1.0=1.0\) outcome unit. In this particular example it agrees with \(dq_{0.5}/d\pi=1\), calculated in the first example. The agreement has a clear composition interpretation here; it does not establish that every RIF regression coefficient is a causal effect or equals every possible marginal quantile derivative.

RIF Regression: Definition and Interpretation

In practice, RIF regression constructs an outcome for each observation using the estimated marginal quantile and density, then regresses that outcome on covariates. Its population target is a linear projection, not necessarily a correctly specified conditional-mean model.

Definition: For a covariate vector \(X\), the population RIF regression coefficient \(\beta_\tau\) is the coefficient vector minimizing \(E[(\operatorname{RIF}(Y;q_\tau,F_Y)-X'\beta)^2]\), assuming the second moments exist and the projection is identified. Equivalently, it satisfies \(E[X\{\operatorname{RIF}(Y;q_\tau,F_Y)-X'\beta_\tau\}]=0\). An intercept should be included when the projection is intended to have one.

Because the RIF is built from the marginal distribution, this regression concerns the unconditional quantile rather than the conditional quantile at a fixed \(X\). A slope summarizes how the projected RIF changes with a covariate, holding the other included regressors fixed in the linear projection. Since RIF values are measured in outcome units, the slope is also expressed in outcome units per unit of that covariate.

One useful way to understand the connection to distribution changes is through the mixture derivative. If a population change alters the outcome distribution from \(F_Y\) toward \(G\), then, under the regularity conditions, the first-order change in \(q_\tau\) is the average influence function under \(G\). A RIF regression summarizes how this influence-function contribution varies with observed covariates. To turn a regression coefficient into the effect of a specific policy or covariate-distribution change requires specifying that change and establishing the relevant assumptions. A linear projection alone does not supply those assumptions.

The empirical procedure replaces \(q_\tau\) and \(f_Y(q_\tau)\) with estimates. Density estimation near the target quantile is especially consequential: when the density is small, the same change in probability corresponds to a larger change in the quantile. Inference must account for estimation of the quantile and density as well as the regression. Treating the constructed RIF as an ordinary observed outcome and automatically using standard least-squares standard errors can omit this additional uncertainty.

Why the Regularity Conditions Matter

Worked Example: A Point Mass at the Baseline Median

Let the baseline outcome be Uniform[0, 1], with \(\tau=0.5\), so \(q_{0.5}=0.5\) and \(f_Y(q_{0.5})=1\). Let \(G\) put all its probability mass at 0.5, and form \(F_\varepsilon=(1-\varepsilon)F_Y+\varepsilon G\) for \(\varepsilon>0\). Determine the mixture median and compare it with the unrestricted derivative formula.

Just below 0.5, the mixture CDF approaches \((1-\varepsilon)0.5\). At 0.5, the point mass in \(G\) makes the CDF jump to \((1-\varepsilon)0.5+\varepsilon=0.5+0.5\varepsilon\), which is at least 0.5. By the infimum definition of a quantile, the mixture median remains 0.5 for every \(\varepsilon>0\). Its right-hand derivative is therefore 0.

But \(G\) has an atom at the baseline quantile, and \(G(q_{0.5})=1\). Substitution into the formula that assumes continuity of \(G\) at the quantile would give \((0.5-1)/1=-0.5\), which is not the actual derivative. The assumptions fail, so that formula does not apply. This example shows why a quantile inequality alone cannot justify differentiating an equality for every perturbation direction.

Common Pitfalls

Equating conditional and marginal quantiles. A conditional quantile describes a distribution at a specified covariate value; a marginal quantile combines the full conditional distributions using the population covariate distribution.
Averaging conditional quantiles to obtain a marginal quantile. Quantiles are not generally linear. Combine the conditional CDFs first, then invert the marginal CDF.
Calling a RIF regression coefficient causal. The coefficient is a linear-projection summary. A causal interpretation requires a specified intervention and suitable identification assumptions.
Using the influence-function derivative without checking regularity. Positive density at the baseline quantile is not enough for every perturbation direction. An atom in \(G\) at \(q_\tau\) can invalidate the usual mixture derivative.
Ignoring the density factor. The marginal density at the quantile determines the scale of the quantile response; a small density can produce a large first-order change.

Key takeaway: Conditional quantiles describe outcomes at fixed covariates; marginal quantiles describe the overall outcome distribution and can change with its covariate composition. RIF regression links covariates to a marginal quantile through its influence function, but its interpretation depends on the projection, the distributional change of interest, and the regularity conditions.

Check Your Understanding

Answer each question by identifying the estimand and stating any assumptions needed.

In one sentence, distinguish \(Q_\tau(Y\mid X=x)\) from \(q_\tau\).
If conditional outcome distributions stay fixed but the population share in one group changes, can the marginal median change? Explain.
Why does the influence function for a marginal quantile include the density at the quantile?
What does the slope in a population RIF regression summarize, and why is it not automatically a causal effect?
What assumption about \(G\) fails when the perturbation places a point mass at the baseline quantile?