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

This paper proposes a nonparametric estimator of the spot volatility matrix with high-frequency data. Our newly proposed Positive Definite Fourier (PDF) estimator produces symmetric positive semi-definite estimates and is consistent with a suitable choice of the localizing kernel. The PDF estimator is based on a modification of the Fourier estimation method introduced by Malliavin and Mancino. The estimator has two parameters: the frequency N, which controls the biases due to the asynchronicity effect and the market microstructure noise effect; and the localization parameter M for the employed Gaussian kernel. The sensitivity of the PDF estimator to the choice of these two parameters is studied in a simulated environment. The accuracy and the ability of the estimator to produce positive semi-definite covariance matrices are evaluated by an extensive numerical analysis, against competing estimators present in the literature. The results of the simulations are confirmed under different scenarios, including the dimensionality of the problem, the asynchronicity of data, and several different specifications of the market microstructure noise. The computational time required by the estimator and the stability of estimation are also tested with empirical data.