Constrained Parameterization of Reduced Rank and Co-integrated Vector Autoregression

Abstract

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.