# Log transformed outcome in quantile regression

I know in OLS, back transformation is not recommended so smearing estimators are often employed. As I understand it, this is not an issue in quantile regression -- you can simply exponentiate to back-transform. Can someone explain why this is the case, and how would you obtain the SEs and CIs when back-transforming? Thanks in advance.

## 1 Answer

The reason is that quantiles, in addition to the classical equivariance properties, enjoy a stronger property, called Equivariance to Monotonic Transformations. Let $h$ be a non-decreasing function, this property implies that for any $Y$ we have

$Q_{h(Y)}(\tau)=h(Q_{Y}(\tau))$

In words, the quantiles of the transformed random variable $h(Y)$ are the transformed quantiles of the original $Y$. For example, a conditional quantile of $\log(Y)$ is the $\log$ of the conditional quantile of $Y$:

$Q_{\log(Y)}(\tau)=\log(Q_{Y}(\tau))$

When one is dealing with skewed distributions this property is of particular importance as it allows log-transforming the outcome and back-transform the estimates.