We present a reparameterization of vector autoregressive moving average (VARMA) models that allows estimation of parameters under the constraints of causality and invertibility. The parameter constraints associated with a causal invertible VARMA model are highly complex. An m-variate VARMA(p; q) process contains (p+q)m2 + m(m+1)/2 parameters, which must be constrained to a complicated subset of the Euclidean space in order to guarantee causality, invertibility. The main result of the paper is a bijection from the constrained set to the entire Euclidean space. The bijection helps in reparameterization of the VARMA process in terms of unconstrained parameters and thereby facilitates estimation via constrained optimization or via prior specification on the constrained space. Constrained estimators can have positive impact on how long-term forecasts are made based on VARMA models. Also, they are important for understanding the state of a linear dynamical system when the system is known to be stable. The proposed parameterization has connection to Schur- stability of matrix polynomials and the associated Stein transformation that are often used in dynamical systems literature. As an important by-product of our investigation, we generalize a classical result in dynamical systems to provide a characterization of Schur stable matrix polynomials.
This is joint work with Tucker McElroy and Peter Linton.