Interpretation of LAD under model misspecification

Question

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.

Answer 1

It can be shown that LAD (and more generally quantile regression) minimizes a weighted expectation of the squared specification error, $(X_i \beta_\tau-Q_\tau(Y_i |X_i))^2$, where $Q_\tau(Y_i |X_i)$ is the $\tau$th quantile of the conditional distribution fucntion and $\beta_\tau$ is the estimate from quantile regression.

Theorems 1 and 2 of Angrist, Chernozhukov, and Fernandez-Val (2006) is the citation.