I am familiarizing myself with quantile regression. I understand it is first and foremost an estimation method such as e.g. OLS. But I wonder about the probability distribution models for which quantile regression makes sense.
To use an analogy, the OLS estimator
$$
\hat\beta^{OLS}:=(X^\top X)^{-1}X^\top y
$$
is a minimum variance linear unbiased estimator when the true probability distribution model is
\begin{align}
y &= X\beta+\varepsilon, \\
\varepsilon &\sim d(0,\sigma^2)
\end{align}
or in other words, $y\mid X\sim d(X\beta,\sigma^2)$, where $d$ is some (unspecified) probability distribution.
Moreover, $\hat\beta^{OLS}$ is a maximum likelihood estimator when the probability model is
\begin{align}
y &= X\beta+\varepsilon, \\
\varepsilon &\sim N(0,\sigma^2)
\end{align}
or in other words, $y\mid X\sim N(X\beta,\sigma^2)$. So in some sense, the OLS estimator "naturally implies" the above probability distribution models.
Question 1: What could we say about the quantile estimator $$ \hat\beta^{QR}_{\tau}:= \arg\min_{\beta}\sum_{i=1}^{n}\rho_{\tau}(y_i-X_i\beta) $$ where $\rho_{\tau}$ is the quantile loss function (and the superscript $^{QR}$ stands for "quantile regression")? Does it "naturally imply" a probability distribution model for $y\mid X$?
Question 2: If we assumed particular true conditional quantiles
$$
\beta^{QR}_{\tau}:= \arg\min_{\beta}\mathbb{E}\left(\rho_{\tau}(y-X\beta)\right)
$$
(where $\beta_{\tau}$ may differ across different values of $\tau$) for a continuum of quantiles between 0 and 1, we would get an implicit conditional probability model for $y\mid X$.
But could the model be expresed explicitly in a nice way? If so, I would welcome a simple example.
P.S. In principle, the question covers more than just linear quantile regressions, but for practical purposes an answer addressing just the linear case would suffice.
