conditional mean and conditional median In Wooldridge's book (Page 452), it says


  When linear absolute deviation (LAD) methods are applied alongside OLS, thre are often reasons to think a priori that OLS and LAD will not produce similar slope estimates. (in fact, it is unlikely that the conditional mean and conditional median are both linear in $x_i$).


I understand that LAD estimates and OLS estimates are generally different, but the words in parentheses confuse me. Why they cannot be both linear? Could anyone give me an example?
In Alecos Papadopoulos's answer, both the conditional mean and the conditional median are linear in $X$. In the following example, the conditional mean is linear in $X$ while the conditional median is quadratic in $X$. This example is a constructed toy but it touches the heart of the issue.

Example: Suppose that $y = \beta_0 + \beta_1 x + u$ and $u = x^2 (w - 1)$, where $w \sim \chi_1^2$ and $x$ and $w$ are mutually independent. Then, as the R command qchisq(0.5,1) gives, $med(w) \doteq 0.455$ so that $med(u|x) = x^2 (-0.545)$, while $E(u|x) = 0$.  We thus have
\begin{equation}
  E(y|x) = \beta_0 + \beta_1 x, ~~\text{ while }~~ med(y|x) = \beta_0 + \beta_1 x - 0.545 x^2.
\end{equation}

Note: Asymmetry of the error distribution and the dependence of $u$ on $x$ are both important. If the error distribution is symmetric, then the conditional mean function and the conditional median function coincide. Also, if the distribution of $u$ is independent of $x$, then $med(u|x)$ is nonzero but constant and thus $med(y|x)$ is again linear in $x$.