Estimation of VARMA under Causality and Invertibility Constraints

Abstract
We present a re-parameterization 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. Currently there are no procedures that can maintain the constraints in the estimated VARMA process, except in the special case of vector autoregression (VAR) where some moment based causal estimators are available. The constrained parameter space is implicitly described in this paper, through the device of parameterizing the entire space of block Toeplitz matrices. The parameterization has connection to Schur- stability of polynomials and associated Stein transformations that are often used in dynamical systems literature. We generalize the classical Stein transformation to provide a characterization of  general order Schur stable matrix polynomials.