Quantile Regression

How much do wages rise as work experience grows? Looking only at average salary tells you the overall relationship between experience and wages. But does the same relationship hold for both low-wage and high-wage workers?

The mean summarizes many people's wages in a single value. In data with large wage gaps, the low end of the wage distribution may barely change even as the mean rises. Conversely, only the high end might rise quickly.

Quantile Regression analyzes different points of the conditional distribution, such as the 25th percentile, the median, and the 75th percentile, instead of the mean. By comparing the regression coefficients at each point, it examines whether the relationship between the explanatory variable and the outcome is the same across the whole distribution.

In this article, using work experience and annual wages as an example, we look in turn at how to read quantile regression lines and result tables, how it differs from mean regression, and what to watch out for when reporting results.

Even among workers with the same 10 years of experience, wages vary widely. Ordinary linear regression estimates how the conditional mean wage changes as experience increases. Quantile regression examines how the lower, middle, and upper levels of the wage distribution change under the same conditions.

For example, τ=0.25 refers to the point at the bottom 25% of the conditional wage distribution of workers with the same experience. It does not mean placing an individual worker's wage exactly at the 25% rank.

In data with a wide range of outcomes, such as wage gaps, home prices, medical costs, and academic achievement, the mean alone can hardly explain every change. In particular, if the same explanatory variable is expected to relate differently to low and high outcome values, you can consider quantile regression.

Quantile regression is not a cure-all that replaces mean regression; it is a method to use alongside it when differences in the distribution are the research question.

Whereas least squares regression minimizes the sum of squared errors, quantile regression minimizes a loss function (check loss) that weights overprediction and underprediction differently depending on the quantile.

A quantile regression coefficient is the estimated change in the conditional quantile for a one-unit change in the explanatory variable, holding the other explanatory variables fixed.

Suppose we surveyed 150 hypothetical workers on their experience (0–20 years) and annual wages ($ thousands). This example uses data generated for illustration, not results from an actual labor market survey.

Looking at the spacing and slopes of the three regression lines together lets you explore how the wage distribution widens or changes as experience increases.

The results below were obtained by actually applying quantile regression to the example data. In the analysis, you vary τ and compare how the coefficient of the same explanatory variable changes.

The unit of the experience coefficient is the change in annual wage ($ thousands) per year of experience. The p-values in the table are for the hypothesis that the experience coefficient is 0 at each quantile.

In the example, the experience coefficient is larger at higher quantiles than at lower ones. However, whether the coefficients differ statistically between quantiles must be checked with a separate test of the difference or bootstrap confidence intervals.

If the experience coefficient at τ=0.25 is about 1.24, it is interpreted as a relationship in which, other things being equal, each additional year of experience raises the conditional 25th-percentile wage by about $1.24 thousand. At τ=0.75, the change is about $1.91 thousand.

Applying ordinary least squares (OLS) regression to the same example also gives a coefficient for the change in the mean wage. But a single mean regression line makes it hard to see how the spread of the outcome distribution changes with experience.

The mere fact that the results differ should not lead you to conclude that the effect at a particular quantile is significantly larger. You need to compare the estimation uncertainty as well.

In the graph above, the upper quantiles have steeper slopes, so the estimated gap between the upper and lower wage quantiles widens as experience increases. On this basis, you can write that the conditional distribution of wages is wider at higher levels of experience.

However, the graph only shows a statistical relationship within the observed range; it does not prove a causal relationship in which more experience actually creates wage gaps.

Quantile regression and polynomial regression can also be used together. To study whether the curved relationship differs by quantile, you can fit a quantile regression that includes a squared term.

Quantile regression does not require assuming normally distributed errors. But that does not mean it can be applied without any checks.

• Independence of observations: For repeated measurements of the same person or grouped data, take the cluster structure into account.
• Model form: Check that the relationship between the explanatory variables and the chosen conditional quantile is properly represented.
• Sample and extreme quantiles: Quantiles close to the extremes, such as 0.05 or 0.95, need enough observations.
• Missing values and outliers: Check for missing data and input errors, and keep wage units and ranges consistent.
• Crossing quantile lines: If you fit each quantile model separately, the estimated lines can cross each other.

The fact that the slope at the 0.25 quantile is smaller than at the 0.75 quantile is descriptively meaningful, but to judge whether that difference is more than chance, you need a confidence interval or test for the difference in coefficients. The fact that only one of the individual p-values is significant does not mean the two coefficients differ significantly.

Leaving out key variables such as occupation, education, and industry can distort the relationship with experience. Also, results about the conditional distribution in quantile regression should not be generalized to changes in the unconditional quantiles of the whole population.

When reporting results, it is good to state the quantiles used, the sample size, the units of the explanatory and outcome variables, the regression coefficients, the confidence intervals or significance levels, and the inference method.

The sentence above was written based on the results fitted to the example data. In real studies, present the confidence intervals and the standard error estimation method as well.

If you want to know not only the average change in a variable but also what differences appear at each location of the distribution, quantile regression is a useful complementary analysis.

It is important to know how much the average wage rises as experience increases, but looking at how that relationship differs at low and high wage levels lets you answer more specific research questions.

When reading the coefficients for each quantile, remember that they represent not the wage change of a specific individual or fixed group, but the corresponding quantile of the distribution conditional on the explanatory variables. It is good to check differences in coefficients between quantiles with separate inference.