Symmetric Positive Semi-Definite Fourier Estimator of Spot Covariance Matrix with High Frequency Data

In this paper we propose a nonparametric estimator of the spot volatility matrix with high- frequency data. The newly proposed Positive Definite Fourier (PDF) estimator is proved to produce symmetric positive semi-definite estimates and to be consistent with a suitable choice of the localizing kernel. The PDF estimator relies on a modification of the Fourier estimation method introduced by Malliavin and Mancino, 2002. The estimator relies on two parameters: the frequency N , which is responsible for controlling both the biases due to the asynchronicity effect and the market microstructure noise effect, and the localization parameter M of the employed Gaussian kernel. The sensitivity of the estimator to the choice of the two parameters is studied in a simulated environment. The accuracy and the ability of the estimator to produce positive semi-definite covariance matrices is evaluated with an extensive numerical study, in comparison with the competing estimators present in the literature. The results of the simulation study are confirmed under many scenarios, that consider the dimensionality of the problem, the asynchronicity of data and the presence of several specifications of market microstructure noise.