Quantile Regression with Regression Discontinuity Sort of a methodological question:
If one has an exogenous binary treatment and a continuous outcome variable Y and wants to estimate quantile treatment effects by exploiting a (sharp) discontinuity in the treatment, what is the advantage of using non-parametric quantile RDD (specifically this: http://www.sciencedirect.com/science/article/pii/S0304407612000607 by Frandsen, Frolich and Melly (2010) but in general any non-parametric method) vs. the normal quantile regression (the one we would find in qreg in stata, for instance). 
It seems to me that since the treatment is exogenous and the discontinuity is "sharp", the normal qreg command would still give nice estimates of the quantile effects? Am I missing something here? The one advantage I see is that the non-parametric method estimates the entire distribution around the discontinuity which is more desirable but any other advantages? 
 A: I do not think you can use the regular qreg because it provides OLS estimations which might lead to biased results, which is in principle why RDDs are estimated through non-parametric methods such as the rdrobust command (references to Cattaneo et al, http://www-personal.umich.edu/~cattaneo/books/Cattaneo-Idrobo-Titiunik_2018_Cambridge-Part1.pdf).
I am having a similar issue finding a way to estimate a RDD with quantiles but I still have not find how.
A: It is actually the same problem as estimating the mean treatment effect in a regression discontinuity context - there is a tradeoff between bias and variance when you choose the bandwidth to perform the estimation.
Let's say your outcome variable Y is not linear in the running variable X. If you estimate the simple regession Y = aX + bT on your whole dataset, where T is the treatment variable, your estimate of the treatment effect will be biased by the curvature in the outcome. One possible solution is to use a polynomial of X, but if you do not know the exact functional form, this will still introduce bias (see Gelman and Imbens 2018). Another possible solution is to assume that the functional form is linear within a certain distance of the cutoff. So we can eliminate data far from the cutoff, but if we eliminate too much then the variance of our estimate gets higher. This results in a procedure to select a bandwidth that minimizes the tradeoff between bias and variance (see Imbens and Kalyanaraman 2011).
Non parametric methods use data-driven methods to select bandwidths and kernel weights that estimate the regression placing more weight on observations that are closer to the bandwidth. Most also allow the slope of the outcome to differ on either side of the cutoff.
Note on the previous answer - qreg does not use OLS - it is actually minimizing weighted absolute deviations.
