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

Jiro Akahori, Reika Kambara, Nien-Lin Liu, Maria Elvira Mancino (), Tommaso Mariotti () and Yukie Yasuda
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Jiro Akahori: Department of Mathematical Sciences, Ritsumeikan University, Kusatsu 525-8577, Japan
Reika Kambara: Department of Mathematical Sciences, Ritsumeikan University, Kusatsu 525-8577, Japan
Nien-Lin Liu: Department of Business Economics, Tokyo University of Science, Tokyo 113-8654, Japan
Maria Elvira Mancino: Department of Economics and Management, University of Florence, Via delle Pandette 9, 50127 Firenze, Italy
Tommaso Mariotti: Department of Economics, Social Studies, Applied Mathematics and Statistics, University of Turin, Corso Unione Sovietica 218/bis, 10134 Torino, Italy
Yukie Yasuda: Department of Mathematical Sciences, Ritsumeikan University, Kusatsu 525-8577, Japan

Risks, 2025, vol. 13, issue 10, 1-30

Abstract: 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.

Keywords: nonparametric covariance estimator; risk management; factor analysis; Fourier analysis; positive semidefiniteness (search for similar items in EconPapers)
JEL-codes: C G0 G1 G2 G3 K2 M2 M4 (search for similar items in EconPapers)
Date: 2025
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