A characterization of vector autoregressive processes with common cyclical features

Abstract

This paper presents necessary and sufficient conditions for the existence of common cyclical features in Vector Autoregressive (VAR) process integrated of order 0, 1, 2, where the common cyclical features correspond to serial correlation common features CS, commonality in the final equations CE and codependence CD. The results are based on polynomial rank factorizations of the reversed AR polynomial around the poles of its inverse. All processes with CS structures are found to present also CE structures and vice versa. The presence of CD structures, instead, implies the presence of both CS and CE structures, but not vice versa. Characterizations of the CS, CE, CD linear combinations are given in terms of linear subspaces defined in the polynomial rank factorizations.