# Model interpretation resulted by quantile regression

I am very new to using the quantile regression (since I deal with heteroskedasticity). In my case, I estimate a quantile regression on three quantiles namely 0.10, 0.50 and 0.90 to model the effect of some explanatory variables (years of schooling, job experiences, education types, etc.) on worker's earnings.

I have two questions that may lead to further analysis:

1. Does the model yielded on each quantile represent a certain level of earning? (e.g., acts as a lower, medium or higher earning model)
2. Does quantile regression divide the total observations based on median/quartiles before performing the estimation algorithm?

Assuming Fajar Wisga Permana is talking about the traditional conditional quantile regression estimator (Koenker & Bassett, 1978), the answer is 'no' to both questions:

1. No. Since the traditional quantile regression (QR) estimator by Koenker & Bassett (1978) is a conditional QR estimator, it predicts quantile $$\tau$$ of outcome $$Y$$ conditional on all right hand side variables, $$X$$. Put differently, it models the conditional quantile function of $$Y$$ given $$X$$ for a specific quantile $$\tau$$, $$Q_\tau(Y|X)=X_i\beta_\tau$$, and, thus, concerns the distribution of $$Y$$ conditional on $$X$$, $$F_{Y|X}(y,x)$$. Depending on the values of the covariates, $$X$$, the predicted value of a higher quantile of $$Y$$, e.g., $$\tau=.90$$, can be lower than the predicted value of a lower quantile, e.g., $$\tau=.10$$. Note that, similarly, the Linear Regression Model (LRM), estimated via OLS or ML, models the conditional mean function. However, effects on the conditional mean aggregate to the unconditional mean in a straightforward manner and, thus, the regression coefficients from an LRM can be interpreted as effects on the unconditional mean in the population. In case of quantiles, we have to turn to unconditional quantile regression estimators, to learn about effects on or at quantiles of the unconditional distribution, $$F_{Y}(y)$$, or the unconditional distribution of $$Y$$ for different causal states of the world, $$F_{Y|D}(y,d)$$, where $$D$$ is the treatment variable of interest.

2. No. Just like Álvaro Méndez Civieta writes in his comment above, the traditional conditional QR estimator (Koenker & Bassett, 1978) uses all observations to arrive at its estimates. While the OLS estimator minimizes the sum of squared residuals, $$\sum_{i=1}^{N}(y_i-x_i\beta)$$, the QR estimator of Koenker & Bassett (1978) minimizes the sum of weighted absolute residuals, $$\sum_{i:y_i\geq x_i\beta_{\tau}}^{N}{\tau|y_i-x_i\beta_{\tau}|} +\sum_{i:y_i< x_i\beta_{\tau}}^{N}{(1-\tau)|y_i-x_i\beta_{\tau}|}$$

I address these and other questions surrounding quantile regression in a paper published in Child Development (Wenz, 2018). An ungated preprint is available at SoxArxiv.

## References

Koenker, R., & Bassett, G., Jr. (1978). Regression Quantiles. Econometrica, 46(1), 33–50. https://doi.org/10.2307/1913643

Wenz, S. E. (2018). What Quantile Regression Does and Doesn’t Do: A Commentary on Petscher & Logan (2014). Child Development. https://doi.org/10.1111/cdev.13141

This answer comes pretty late, but here it is:

1. When you build a quantile regression model for, lets say, the 0.9 quantile, what you get is the value of the 90% quantile of your response variable depending on your predictor variables. So yes, this model can act as a lower, medium and higher earning model

2. No, quantile regression takes into account all the observations of your dataset in order to get the estimations of the parameters.