Quantile regression at quantile $\tau$ minimizes the following "pinball" loss function, $L_{\tau}$, and elicits conditional quantile $\tau$.
$$ l_{\tau}(y_i, \hat y_i) = \begin{cases} \tau\vert y_i - \hat y_i\vert, & y_i - \hat y_i \ge 0 \\ (1 - \tau)\vert y_i - \hat y_i\vert, & y_i - \hat y_i < 0 \end{cases}\\ L_{\tau}(y, \hat y) = \sum_{i=1}^n l_{\tau}(y_i, \hat y_i) $$
When $\tau = 0.5$, this loss function is absolute loss.
If we generalize squared loss the way that absolute loss generalizes to the pinball loss, denotes as $L^{^*}_{\tau}$ below, what is elicited?
$$ l^{^*}_{\tau}(y_i, \hat y_i) = \begin{cases} \tau\left( y_i - \hat y_i\right)^2, & y_i - \hat y_i \ge 0 \\ (1 - \tau)\left( y_i - \hat y_i\right)^2, & y_i - \hat y_i < 0 \end{cases}\\ L^{^*}_{\tau}(y, \hat y) = \sum_{i=1}^n l^{^*}_{\tau}(y_i, \hat y_i) $$
When $\tau = 0.5$, this is just the usual squared loss that elicits conditional means. When $\tau\ne 0.5$, I am not sure what such a loss function would elicit.
