Suppose we have $n$ i.i.d. draws of $y_i$ and $x_i$ and consider a linear equation:
$$y_i =\beta_0 +\beta_1x_{1i}+...+\beta_kx_{ki}+u_i =x_i'\beta+u_i $$
A common assumption in OLS is $E[y|x]=x_i'\beta$. If this assumption fails, OLS still has a somewhat meaningful interpretation as the best linear approximation to the conditional expectation function. (With "best" being in a squared loss sense).
In LAD it is common to assume $Median[y_i|x_i]=x_i'\beta$. My question is, if the LAD assumption fails, is there a meaningful interpretation to the LAD estimates?
I tried fiddling around with the triangle inequality but made minimal progress, and am skeptical there is a meaningful interpretation.