Worked Analysis Of Wage Distribution Effects

Why the Estimand Matters in a Wage Analysis

A treatment can affect wages differently at different points of the wage distribution. A job-training program, for example, might raise wages most for workers who would otherwise earn less, or its apparent effect might vary because participants and nonparticipants have different characteristics. Quantile regression, distribution regression, and marginal quantile treatment effects offer related but distinct ways to describe such patterns.

The key distinction is between a conditional distribution, which compares wages among people with specified covariates, and a marginal distribution, which averages over a target population’s covariate distribution. A coefficient from a conditional quantile regression is not automatically an effect on the corresponding marginal quantile. The examples below make that difference explicit.

Definition: Let $Y(1)$ and $Y(0)$ denote a worker’s wage under treatment and control, and let $X$ denote observed baseline covariates. The conditional quantile treatment effect at quantile level $\tau$ and covariate value $x$ is $Q_{Y(1)\mid X}(\tau\mid x)-Q_{Y(0)\mid X}(\tau\mid x)$. The marginal quantile treatment effect for a target population is $Q_{Y(1)}(\tau)-Q_{Y(0)}(\tau)$, where each marginal distribution averages over that population’s distribution of $X$.

Here $Q_Y(\tau)=\inf\{y:F_Y(y)\geq\tau\}$ is the quantile defined as the smallest wage at which the CDF reaches or exceeds $\tau$. A marginal quantile treatment effect is a difference between two population quantiles. It does not generally equal the average of the conditional quantile effects, nor does it identify the effect for a particular worker.

Three Complementary Estimands

A conditional quantile regression specifies a quantile of wages given treatment and covariates. A simple specification allowing the treatment effect to vary with a binary covariate $X$ is

Q_{Y\mid D,X}(\tau\mid D,X) =\beta_0(\tau)+\beta_D(\tau)D+\beta_X(\tau)X+\beta_{DX}(\tau)DX.

Here $D=1$ indicates treatment and $D=0$ control. At $X=0$, the fitted conditional quantile contrast is $\beta_D(\tau)$; at $X=1$, it is $\beta_D(\tau)+\beta_{DX}(\tau)$. These are contrasts between conditional quantiles. They are not, without further steps, contrasts between marginal quantiles.

Distribution regression takes a different route. At each wage threshold $y$, it models the probability that wages are at or below that threshold:

F_{Y\mid D,X}(y\mid D,X) =P(Y\leq y\mid D,X) =\Lambda\!\left(a(y)+b(y)D+c(y)X+d(y)DX\right),

where $\Lambda(v)=1/(1+\exp(-v))$ is the logistic CDF. The coefficients can vary with $y$, so a separate regression is fitted at each threshold. The fitted probabilities form conditional CDF estimates. To recover a conditional quantile, find the wage threshold at which the estimated CDF reaches $\tau$. To obtain a marginal distribution for a target population, average fitted conditional CDFs over that population’s covariate values or covariate distribution.

For a target population with covariate distribution $F_X$, the marginal CDF under treatment level $d$ is

F_d(y)=\int F_{Y\mid D,X}(y\mid d,x)\,dF_X(x), \qquad Q_d(\tau)=\inf\{y:F_d(y)\geq\tau\}.

The marginal quantile treatment effect is $Q_1(\tau)-Q_0(\tau)$. This standardization compares treatment and control wage distributions over the same target distribution of baseline covariates, rather than allowing the covariate mix to differ between the two groups.

Worked Example: Conditional Median Wage Effects

Suppose a hypothetical randomized training study records hourly wages in dollars and a baseline indicator $X$, where $X=1$ marks a higher-education stratum. For illustration, the conditional wage distributions have the following probability masses. In the lower-education stratum, control wages are $10, $15, $20, and $30 with probabilities 0.20, 0.30, 0.30, and 0.20; treated wages are $12, $17, $22, and $32 with the same probabilities. In the higher-education stratum, control wages are $15, $20, $30, and $40 with probabilities 0.20, 0.30, 0.30, and 0.20; treated wages are $16, $21, $31, and $41 with those probabilities.

Use the inverse-CDF definition of the median. For the lower-education control group, the CDF reaches 0.50 at $15, so its conditional median is $15 per hour. The treated CDF reaches 0.50 at $17, so the conditional median contrast is $17-15=\$2$ per hour. For the higher-education stratum, the control median is $20 and the treated median is $21, giving $21-20=\$1$ per hour.

At $\tau=0.50$, an additive conditional quantile regression with the interaction has fitted coefficients $\beta_0=15$, $\beta_D=2$, $\beta_X=5$, and $\beta_{DX}=-1$, in dollars per hour. The fitted control median at $X=1$ is $15+5=\$20$; the treated median there is $15+2+5-1=\$21$. Thus, the treatment contrast is $2 per hour when $X=0$ and $2-1=\$1$ per hour when $X=1$. This interpretation concerns conditional medians, not the median effect for the combined population.

From Conditional CDFs to a Marginal Quantile Effect

The same hypothetical distributions show why the distinction matters. Suppose the target population is half in each education stratum. Its treatment and control CDFs are the averages of the two stratum-specific CDFs, not the CDFs in either stratum alone.

Worked Example: Standardizing the Wage Distribution

At the threshold $y=\$20$ per hour, the control CDF is 0.80 in the lower-education stratum and 0.50 in the higher-education stratum. The standardized control CDF is therefore

$$ F_0(20)=0.5(0.80)+0.5(0.50)=0.65. $$

The treatment CDFs at the same threshold are 0.50 and 0.20. Thus

$$ F_1(20)=0.5(0.50)+0.5(0.20)=0.35. $$

In this hypothetical population, 65% of control wages and 35% of treated wages are at or below $20 per hour. The lower treated CDF at this threshold is consistent with a shift toward higher wages; by itself, however, it does not give the size of a quantile effect.

For the marginal median, combine the two stratum-specific distributions using the same 50:50 weights. The control distribution places probabilities 0.10 at $10, 0.25 at $15, 0.30 at $20, 0.25 at $30, and 0.10 at $40 per hour. Its CDF is 0.35 at $15 and 0.65 at $20, so $Q_0(0.50)=\$20$. The treatment distribution places probabilities 0.10 at $12, 0.10 at $16, 0.15 at $17, 0.15 at $21, 0.15 at $22, 0.15 at $31, 0.10 at $32, and 0.10 at $41. Its CDF reaches 0.50 at $21, so $Q_1(0.50)=\$21$. The marginal median treatment effect is

Q_1(0.50)-Q_0(0.50)=21-20=\$1\text{ per hour}.

The conditional median effects were $2 and $1 per hour. Their 50:50 average is $1.50 per hour, which is not the marginal median effect of $1 per hour. Quantiles are nonlinear functionals of distributions: averaging conditional quantiles need not produce the quantile of the averaged distribution.

Estimating and Comparing Uncertainty

The preceding calculations use known probability masses for transparency. In an applied analysis, the conditional quantile coefficients, CDFs, and marginal quantiles are estimated from a sample. Their uncertainty depends on the estimator and the sampling or assignment design.

For conditional quantile regression, uncertainty concerns the fitted coefficient or a specified conditional contrast. Standard errors need to reflect the quantile-regression sampling distribution; density estimation near the conditional quantile can matter. For distribution regression, uncertainty concerns fitted CDF values or functions of them. Marginal quantile estimates add another step: fitted conditional CDFs are standardized, and the wage threshold at which the standardized CDF reaches $\tau$ is estimated. Uncertainty must account for all these steps. Resampling the original observations and refitting the entire procedure is one practical approach when its assumptions are appropriate.

Worked Example: A Marginal Median Estimate and Interval

Consider a separate, hypothetical randomized wage study with a sufficiently large sample and smoothly varying wage distributions near the median. After standardization to the same target covariate distribution, suppose the estimated control median is $20.00 per hour and the estimated treatment median is $21.00 per hour. The estimated marginal median effect is

\widehat{\Delta}(0.50)=\widehat{Q}_1(0.50)-\widehat{Q}_0(0.50) =21.00-20.00=\$1.00\text{ per hour}.

Suppose a bootstrap that resamples participants within assignment strata and refits both distribution regressions and the standardized quantiles gives $SE=0.62$ dollars per hour. Using a normal approximation, a 95% interval is

1.00\pm 1.96(0.62) =1.00\pm1.2152 =[-0.2152,\ 2.2152]\text{ dollars per hour}.

Rounded to three significant figures, the interval is $[-0.215,\ 2.22]$ dollars per hour. Under the repeated-sampling procedure and its regularity assumptions, this method is designed to cover the target marginal median effect in 95% of repeated samples. This interval includes zero, so the data do not establish a nonzero median effect at the corresponding two-sided 5% level. It does not mean there is a 95% probability that the fixed effect lies in this particular interval.

A conditional coefficient’s standard error cannot simply be reused as the standard error of a marginal quantile effect. The marginal estimate depends on the estimated conditional distributions, the covariate weights used in standardization, and the inversion that recovers each quantile. Inference should target the estimand actually reported. If a study reports effects over many quantile levels, simultaneous uncertainty bands may be needed for claims about the whole curve; pointwise intervals alone do not provide simultaneous coverage.

Assumptions and Interpretation in a Wage Study

A causal interpretation requires a design that identifies the treatment and control wage distributions. Under random assignment, treatment assignment is independent of the potential outcomes and baseline covariates, with positive probability of each assignment for the population being studied. Outcomes must be observed or handled using a valid approach to censoring and missingness. The analysis also requires a well-defined treatment and a consistent link between assigned treatment and the observed wage. If treatment was not randomized, identification instead requires appropriate assumptions about selection into treatment; regression adjustment alone does not guarantee a causal interpretation.

The models add their own requirements. A conditional quantile regression must represent the conditional quantiles adequately for the contrasts being interpreted. A distribution regression must estimate the conditional CDF adequately across relevant wage thresholds. Standardization requires a clearly defined target covariate distribution and sufficient observations to estimate the relevant conditional distributions across it. Quantile inference generally also needs regularity near the target quantile, such as a positive, well-behaved density there; ties, sparse tails, and quantile crossing can complicate estimation and inference.

Wage measurement deserves particular attention. If wages are only observed for employed workers, or are top-coded or otherwise censored, analyzing recorded wages as if they were complete uncensored outcomes can change the target and bias distributional estimates. The analyst must state which population and wage outcome are being studied and use methods appropriate to the observation process. A treatment-related change in employment can also make a wage comparison among observed workers answer a different question from a comparison for the full target population.

Common Mistake: Treating Conditional and Marginal Effects as Interchangeable

A tempting but incorrect argument is: “The treatment raises the conditional median by $2 in one stratum and $1 in the other, so the overall median effect is their average, $1.50.” That calculation averages conditional median contrasts. It does not calculate the difference between the marginal medians. The worked example instead builds each marginal CDF using the same covariate weights, finds each marginal median from its CDF, and then subtracts them; that gives $1 per hour.

Another interpretive error is to call a conditional quantile-regression coefficient “the treatment effect for workers at that wage rank.” The coefficient describes a contrast between conditional quantiles at fixed covariates. It does not, without additional assumptions, identify the effect on the same individuals who would occupy that rank under both treatment states. Always name the estimand, the target population, the wage units, and the assumptions supporting a causal reading.

Key takeaway: Conditional quantile regression compares quantiles within covariate values; distribution regression estimates conditional CDFs; marginal quantile effects compare quantiles after standardizing treatment-specific distributions to a common target population. These estimands can differ, so calculate and quantify uncertainty for the one that answers the research question.

Check Your Understanding

Answer each question using the definitions and calculations above.

In the conditional quantile model, what is the treatment contrast at $X=1$, and which coefficient accounts for its difference from the contrast at $X=0$?
Why does the standardized control CDF at $20 per hour equal 0.65 in the worked example?
The conditional median effects are $2 and $1 per hour. Why is their average not necessarily the marginal median effect?
What does the bootstrap standard error of 0.62 dollars per hour measure in the interval example?
Name one assumption required for a causal interpretation and one issue that could make recorded wages an invalid outcome for the intended target population.