Does local linear regression include a weighting Kernel?

#1
Hey,

I am applying a Regression Discontinuity Design (RDD) to estimate the effect of a policy change. In RDD I can apply the parametric approach (polynomial regression) and the non-parametric approach (local regression). The parametric approach is clear to me and also the basic idea of the non-parametric approach is rather straightforward. Where as the parametric approach uses every observation in the sample to model the outcome as a function of the rating variable and treatment status, the non-parametric approach estimates the functional form itself by using only part of the data. To my understanding the local linear regression allows the slope to differ on either side of the cut-point (but it is a straight, linear line on both sides). However, as I read more about the local linear regression, I came across approaches where a weighting Kernel is included in the local linear regression which makes the regression no longer linear (unless it is a Uniform Kernel, right?). So my question is now whether I should use a Kernel (and therefore include a weighting) in the local linear regression which basically makes it a locally weighted regression or whether I should stick to the linear approach?

I have also seen published articles that apply RDD and use for example a Epanechnikov Kernel for the estimation of the local linear regression (for example the paper by Bento et al. (2014)). In the Online Appendix Table E.13 of their paper they show the Local Linear Regression Discontinuity Estimates which were calculated by using an Epanechnikov Kernel. So, what determines on whether I should use a Kernel or not in the local linear regression?

I have read various papers about the topic, but could not really find an answer to that question.

Thank you very much for your help!
 
#2
Only kind of familiar with RDD, but wouldn't it be dependent on what you believe the underlying effect is (shapewise). I am also assuming you lose some interpretability with non-linear, but if it is appropriate it should probably be used. I may check out that paper tomorrow.
 
#3
That's what I thought as well. However, there is one source that states:
"The parametric approach tries to pick the right model to fit a given data set, while the non-parametric approach tries to pick the right data set to fit a given model"
"In the simplest terms, this strategy (local linear regression/non-parametric) views the estimation of treatment effects as local randomization and limits the analysis to observations that lie within the close vicinity of the cut-point (sometimes called a bandwidth), where the functional form is more likely to be close to linear. The main challenge here is selecting the right bandwidth."

Therefore, the we would always try to find a linear relation and simply adjust the bandwidth in order to get as close to a linear relation as possible.

What do you think?