Before revision:  We prove Central Limit Theorems for the estimation error with an optimal rate and study the problems of optimal bandwidth and kernel selection. As in the case of a simple kernel estimator of spot volatility in the absence of microstructure noise, we show that the asymptotic variance of the pre-averaging/kernel estimator  is minimal for exponential or Laplace kernels, hence, justifying the need of working with unbounded kernels as proposed in this work.
After revision:  We prove Central Limit Theorems for the estimation error with an optimal rate and study the problems of optimal bandwidth and kernel functions. We show that the asymptotic variance of the pre-averaging/kernel estimator is minimal for exponential or Laplace kernels, hence, justifying the need of working with unbounded kernels as proposed in this work.