When does quantile regression produce biased coefficients (if ever)? It is easy to show using matrix algebra when least squares will produce bias.
\begin{equation}
\begin{split}
\text{E}[B]& = \text{E}[(X'X)^{-1}]\times\text{E}[X'Y] \\
& = \text{E}[(X'X)^{-1}]\times\text{E}[X'(XB +\epsilon)]  \\
& = \text{E}[(X'X)^{-1}]\times\text{E}[X'XB] + \text{E}[(X'X)^{-1}]\times\text{E}[X'\epsilon] \\
& = B + \text{E}[(X'X)^{-1}]\times\text{E}[X'\epsilon]  \\
\end{split}
\end{equation}
Bias is introduced in the last term when there is correlation between X and e.
My question is, how do we know when $\text{E}[B]$ for the LAD estimator will be biased?   Will bias arise when there is correlation between $X$ and $\epsilon$, as in least squares? Or is quantile regression robust to correlation between $X$ and $\epsilon$? I'm guessing it can't be demonstrated using matrix algebra because the LAD estimator is the following:
\begin{align}
B(\tau) = \operatorname{argmin} \text{E}[\rho(Y_i - X'B)]
\end{align}
and the LAD estimator is not calculated using linear algebra (?). If that's right, then how can we demonstrate when quantile regression will produce bias? Use a monte carlo simulation?
 A: If you have a model 
$$ 
Y_i = X_i B(\tau) + \epsilon_i(\tau)
$$
then a sufficient condition for $\tau$-quantile regression to give an unbiased estimate of $B(\tau)$ is that the $\tau$-th quantile of $\epsilon(\tau)$ conditional on $X$ is zero. This follows from the fact that (i) the sample quantile regression objective function, $\mathbb{E}_n[ \rho_\tau(Y-X'\beta)]$ converges uniformly to $\mathrm{E}[\rho_\tau(Y-X'\beta)]$ and (ii) $\mathrm{E}[\rho_\tau(Y-X'\beta)]$ is "uniquely" (actually something slightly stronger is needed) minimized at $B(\tau)$. (i) will be true under standard regularity conditions. (ii) is true because $\mathrm{E}[\rho_\tau(Y-X'\beta)]$ is convex as a function of $\beta$ and its first order condition can be written
$$
\tau \mathrm{E}[ P(Y - X\beta>0|X) X] = (1-\tau) \mathrm{E}[ P(Y - X\beta<0|X) X]
$$
which is satisfied at $\beta = B(\tau)$ if $Q_\tau(\epsilon(\tau)|X) = 0$. You can also see from this that $Q_\tau(\epsilon(\tau)|X) = 0$ is stronger than needed, but there doesn't seem to be an easy to interpret weaker condition. 
None of the above tells you what the bias in $\hat{B}(\tau)$ would be if $Q_\tau(\epsilon(\tau)|X) \neq 0$. I don't know of a general expression for the bias like you can get for OLS, but you can get some nice results in a few cases. For example, Angrist, Chernozhukov, and Fernandez-Val (2006) give an omitted variables bias formula for quantile regression. If your model satisfies the conditions above, and $X = (X_1, X_2)$, and then you estimate a quantile regression leaving out $X_2$, then the expectation of your estimated coefficient on $X_1$ is
$$
\beta_1(\tau) + \mathrm{E}[w_\tau(X) X_1'X_1]^{-1} \mathrm{E}[w_\tau(X)X_1' (X_2' \beta_2(\tau))]
$$
where $w_\tau(X)$ are some weights that depend on $X$, $\tau$, and the distribution of $\epsilon$. 
