Is it possible that quantile regression estimates coincides with OLS estimates? Let $Y_i$ and $X_i$ be random variables. Under the linear regression specification, we have
$Y_i = X_i \beta + \epsilon_i$, where $\mathbb{E}[\epsilon_i | X_i] = 0$. For this model, we may obtain the OLS estimate of $\beta$ by minimizing
$$\sum_{i=1}^{N}(y_i - x_ib)^2.$$
On the other hand, under the quantile regression specification, we have
$Q_{Y_i}(u|X_i) = X_i \beta(u)$ for $u \in (0,1)$, which is equivalent to
$Y_i = X_i \beta(u) + \epsilon_i(u)$ with the assumption that $P(\epsilon(u)<0 | X_i) = u $ almost surely. For this model, we may obtain the QR estimate of $\beta(u)$ by minimizing
$$\sum_{i=1}^{N}\rho_u(y_i - x_ib),$$
where $\rho_u(x) = x(u-I\{x<0\})$.
I'm wondering if there is any particular data structure such that OLS estimates will be the same as quantile regression estimates.

*

*In quantile regression you can change some of the values $y_i$ without changing the solution.


*In least squares regression a small change of a value $y_i$ will always have an effect on the end result.
Therefore the two functions for the estimates can never be the same.*
When you consider random situations of the values, e.g. the median and mean for a specific sample, then it can occur by some chance that the middle value is equal to the mean. An example is when you have a sample of a discrete distribution, then the sample mean may coincide with the sample median or with some other quantile.

*An exception is when you only have two data points, then the median and mean coincide. This is not a contradiction, in this case with two data points the first bulletpoint about quantile regression is not valid.
