VAR Based State-space Structures: Realization, Statistics and Spectral Analysis

The effective representation of multivariate time–series in terms of their structural indices is faced in this study, leading to the development of a corresponding scheme that is based in the interconnection between VAR and state–space models. To this, a specific state equation that retains a state matrix identical to the companion matrix of the original VAR polynomial is realized, uncovering many ”hidden” information of the initial process. In light of this realization, it is shown how closed form expressions for the Green function, the covariance generating function and the spectral density can be derived through the spectrum of the state matrix, thus allowing assessment and quantification (via the notion of dispersion analysis) of every structural mode. A Yule–Walker based estimate is also provided, which applies directly to the state equation. A structural system with two degrees of freedom and closely spaced modes serves as an application of the novel scheme, using Monte Carlo analysis.