I notice that when I use the package rdrobust, I can easily replicate the resulting point estimate by using a local linear regression with the given bandwidth. Moreover, with robust standard errors in rdrobust and robust standard errors in the local linear regression, I get the same standard errors. However, when I cluster, the standard errors from the two methods are very different. Is this a bug? Or is there econometric reasoning why this would be the case?

If an example is required, I'd be happy to share mine, although I expect that this is a general result.