Pre-averaged kernel estimators for the drift function of a diffusion process in the presence of microstructure noise
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We consider estimation of the drift function of a stationary diffusion process when we observe high-frequency data with microstructure noise over a long time interval. We propose to estimate the drift function at a point by a Nadaraya–Watson estimator that uses observations that have been pre-averaged to reduce the noise. We give conditions under which our estimator is consistent and asympotically normal. Its rate and asymptotic bias and variance are the same as those without microstructure noise. To use our method in data analysis, we propose a data-based cross-validation method to determine the bandwidth in the Nadaraya–Watson estimator. Via simulation, we study several methods of bandwidth choices, and compare our estimator to several existing estimators. In terms of mean squared error, our new estimator outperforms existing estimators.
KeywordsDiffusion process Nonparametric estimation Discrete observations High-frequency observations Microstructure noise Pre-averaging Drift estimation Nadaraya–Watson estimator
The research of W. Lee and N. Heckman was supported by the Natural Sciences and Engineering Research Council of Canada. We thank the referees for reading the paper carefully and for suggesting a number of improvements.
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