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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.

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In your question you claim: "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)". This is not true. A local linear regression is a nonparamteric estimator and it can approximate nonlinearities in a large class of functions. What you refer to is what you would get if you implemented a parametric approach (say you specify a linear model for both sides of the cutoff). A regression that forces linear functional forms on both sided of the cuttof is "globally" rather than locally linear.

Coming to your second question: the Kernel weighting is a way to capture the "locality" of the approach. Observations in your sample that are too far away from the (fixed) point x at which you want to predict the conditional mean E[Y|X] are not used (or if they are, they receive a very low weight), because they are not local enough.

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