  The paper provides a parametrization of Vector Autoregression (VAR) that
enables one to look at the parameters associated with unit root dynamics and
those associated with stable dynamics separately. The task is achieved via a
novel factorization of the VAR polynomial that partitions the polynomial
spectrum into unit root and stable and zero roots via polynomial factors. The
proposed factorization adds to the literature of spectral factorization of
matrix polynomials. The main benefit is that using the parameterization,
actions could be taken to model the dynamics due to a particular class of
roots, e.g. unit roots or zero roots, without changing the properties of the
dynamics due to other roots. For example, using the parameterization one is
able to estimate cointegrating space with appropriate rank that maintains the
root structure of the original VAR processes or one can estimate a reduced rank
causal VAR process maintaining the constraints of causality. In essence, this
parameterization provides the practitioner an option to perform estimation of
VAR processes with constrained root structure (e.g., conintegrated VAR or
reduced rank VAR) such that the estimated model maintains the assumed root
structure.
