[Paper Review] Estimation of Causal Invertible VARMA Models

Anindya Roy, Tucker McElroy+1 more|arXiv (Cornell University)|Jun 18, 2014

Random Matrices and Applications34 references3 citations

TL;DR

This paper presents a novel reparameterization of multivariate causal invertible VARMA models using positive definite and orthogonal matrices to parameterize block Toeplitz matrices, enabling exact likelihood-based estimation under strict causality and invertibility constraints—previously intractable due to complex parameter space geometry. The key contribution is a mathematically rigorous characterization of Schur-stable matrix polynomials, ensuring stable estimation for forecasting and dynamical systems analysis.

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 a vector autoregression (VAR), where some moment based causal estimators are available. Even in the VAR case, the available likelihood based estimators are not causal. The maximum likelihood estimator based on the full likelihood that does not condition on the initial observations by definition satisfies the causal invertible constraints but optimization of the likelihood under the complex constraints is an intractable problem. The commonly used Bayesian procedure for VAR often has posterior mass outside the causal set because the priors are not constrained to the causal set of parameters. We provide an exact mathematical solution to this problem. An $m$-variate VARMA$(p, q)$ process contains $(p+ q) m^2 + \binom{m+1}{2}$ parameters, which must be constrained to a subset of Euclidean space in order to guarantee causality and invertibility. This space is implicitly described in this paper, through the device of parameterizing the entire space of block Toeplitz matrices in terms of positive definite matrices and orthogonal matrices. The parameterization has connection to Schur- stability of 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.

Motivation & Objective

• To address the lack of estimation procedures that maintain causality and invertibility in general VARMA models, which are critical for stable long-term forecasting.
• To resolve the intractability of optimizing the full likelihood under implicit constraints of causality and invertibility in multivariate time series models.
• To generalize classical Schur stability results to matrix polynomials, enabling a stable parameter space representation for VARMA processes.
• To develop a reparameterization that allows likelihood-based estimators to remain within the causal-invertible parameter space, overcoming limitations of standard MLE and Bayesian methods.
• To provide a computationally feasible framework for estimating VARMA models with guaranteed stability, particularly in macroeconomics and dynamical systems.

Proposed method

• Reparameterize the VARMA model using positive definite matrices and orthogonal matrices to represent the block Toeplitz covariance matrix structure.
• Use the Stein transformation and determinantal equations to link the coefficient matrix A to the covariance matrix, ensuring Schur stability.
• Characterize the parameter space via recursive formulas involving $ C_t = U(0) - \xi_t' \underline{U}_{t-1}^{-1} \xi_t $, ensuring positive definiteness at each step.
• Establish equivalence between Schur stability of the matrix polynomial $ A(z) $ and positive definiteness of the resulting covariance blocks $ C_t $.
• Construct a stable parameterization by ensuring $ \underline{U}_k $ is positive definite and $ C_k > 0 $, which implies $ |\lambda| < 1 $ for all eigenvalues of the companion matrix.
• Leverage block Toeplitz structure to embed the constraints of causality and invertibility directly into the parameter space, avoiding numerical optimization over complex manifolds.

Experimental results

Research questions

• RQ1 How can we parameterize the space of causal invertible VARMA models in a way that guarantees stability during estimation?
• RQ2 What mathematical conditions ensure that a VARMA model is both causal and invertible, and how can these be embedded into a likelihood-based estimation framework?
• RQ3 Can we generalize classical Schur stability results for scalar polynomials to matrix polynomials in multivariate time series models?
• RQ4 Why do standard likelihood-based and Bayesian estimators fail to preserve causality and invertibility, and how can this be corrected?
• RQ5 What is the role of block Toeplitz matrices and orthogonal transformations in constructing a stable, constrained parameter space for VARMA models?

Key findings

• The paper establishes a one-to-one correspondence between stable VARMA models and positive definite block Toeplitz matrices via a reparameterization that embeds causality and invertibility constraints.
• It proves that Schur stability of the matrix polynomial $ A(z) $ is equivalent to the positive definiteness of the sequence $ C_t $, derived recursively from the covariance structure.
• The authors generalize a classical result in dynamical systems by characterizing Schur-stable matrix polynomials through the positivity of the sequence $ C_0 > C_1 > \cdots > C_k > 0 $.
• The reparameterization ensures that maximum likelihood estimators based on the full likelihood remain within the causal-invertible set, solving a long-standing estimation challenge.
• The method enables likelihood-based inference with guaranteed stability, overcoming the limitations of conditional likelihood and unconstrained MLE in VARMA models.
• The framework supports improved forecasting and system identification in multivariate time series by ensuring long-run stability and interpretability.