# Interpretation of LAD under model misspecification

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.

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.