Spectra of bivariate VAR(p) models

In this paper, we give a characterization of the range of spectral matrices, which are feasible for bivariate VAR(p) models. In addition to the marginal spectra and the cross-spectrum being ratios of trigonometric polynomials, as is also the case for VARMA(p,q) models, in the VAR(p  ) case the polynomials involved as numerators and denominators must fulfil further restrictions. We state these restrictions and we show that they are also sufficient for a spectral matrix to belong to a VAR. We demonstrate how these polynomials may be regained only from the marginal spectra and the coherency. This, in turn, is used to construct a visual goodness-of-fit criterion for fitting a VAR model, yielding also indications on the necessary order. The feasible phase lags between frequency-λλ-components of the two series are discussed in detail. Finally, we propose two methods for constructing VAR models with (partially) pre-specified spectral elements. Examples are provided illustrating the criterion, the phase-lag structure and the above methods.