Generalised Cepstral Models for the Spectrum of Vector Time Series

The paper treats the modeling of stationary multivariate stochastic processes via a frequency domain model expressed in terms of cepstrum theory. The proposed model nests the vector exponential model of Holan et al. (2017) as a special case, and extends the generalised cepstral model of Proietti and Luati (2019) to the multivariate setting, answering a question raised by the last authors in their paper. Contemporarily, we extend the notion of generalised autocovariance function of Proietti and Luati (2015) to vector time series. Then we derive explicit matrix formulas connecting generalised cepstral and autocovariance matrices of the process, and prove the consistency and asymptotic properties of the Whittle likelihood estimators of model parameters. Asymptotic theory for the special case of the vector exponential model is a signi ficant addition to the paper of Holan et al. (2017). We also provide a mathematical machinery, based on matrix differentiation, and computational methods to derive our results, which differ signi ficantly from those employed in the univariate case. The utility of the proposed model is illustrated through Monte Carlo simulation from a bivariate process characterized by a high dynamic range, and an empirical application on time varying minimum variance hedge ratios through the second moments of future and spot prices in the corn commodity market.