Generalized autocovariance matrices for multivariate time series

Abstract

The paper treats the modeling of stationary multivariate stochastic processes via frequency domain, and extends the notion of generalized autocovariance function, given by Proietti and Luati (2015) for univariate time series, to the multivariate setting. The generalized autocovariance matrices are defined for stationary multivariate stochastic processes as the Fourier transform of the power transformation of the spectral density matrix. Then we prove the consistency and derive the asymptotic distribution of frequency domain non-parametric estimators of the generalized autocovariance matrices, based on the power transformation of the periodogram matrix. Generalized autocovariance matrices are used to construct white noise hypothesis testing, to discriminate stochastic processes, and to introduce a generalized Yule–Walker estimator for the spectrum. A so-called λ–squared distance between two multivariate stochastic processes is also defined by using their generalized autocovariance matrices, and it serves for clustering time series and estimation by feature matching. Another use is in discriminant analysis.

Keywords:

• Stationary vector stochastic processes
• spectral density matrix
• Wold coefficients
• generalized autocovariance matrices
• spectral estimation
• hypothesis testing
• matching estimator
• discriminant analysis

JEL Classification:

• C22
• C52

Acknowledgements

This work is financially supported by a research grant (FAR 2021) of the University of Modena and Reggio E., Italy. We should like to thank the Editor in Chief of the journal, Professor Narayanaswamy Balakrishnan, and the two anonymous referees for their constructive comments and very useful suggestions and remarks which were most valuable for improvement of the final version of the paper.