...everyone likes the local Wald estimators for regression discontinuities because they're non-parametric. But it's precisely because they're non-parametric that you throw the local continuity at the cutoff assumption overboard. There is no guarante of local continuity at all. That's true of a lot of parametric versions that people like to do too.
This is not small potatoes. The only reason the regression discontinuity model is identified is the local continuity assumption, right?
I've never written a purely econometrics paper before. I don't know how it's really done. But the regression discontinuity literature - although relatively old - is new to economics. And this seems worth pinning down. It's easy enough to point out how the local continuity assumption is required to identify the model. I would think it would just be helpful to simulate a high variance data generating process with modest sample size and check on each method.
Parametric attempts at regression discontinuity models are not a silver bullet if the relationship is unusual or non-linear. But they're not sensitive to a few points around the cut-off the way the non-parametric estimates are. And it's precisely behavior around the cut-off that we have to be worried about.