Kernel Estimation of Spot Volatility with Microstructure Noise Using Pre-Averaging
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We first revisit the problem of kernel estimation of spot volatility in a general continuous Itô semimartingale model in the absence of microstructure noise, and prove a Central Limit Theorem with optimal convergence rate, which is an extension of Figueroa and Li (2020) as we allow for a general two-sided kernel function. Next, to handle the microstructure noise of ultra high-frequency observations, we present a new type of pre-averaging/kernel estimator for spot volatility under the presence of additive microstructure noise. 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. Feasible implementation of the proposed estimators with optimal bandwidth is also developed. Monte Carlo experiments confirm the superior performance of the devised method.
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