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The Vector AutoRegressive Moving Average (VARMA) model is fundamental to thetheory of multivariate time series; however, identifiability issues have ledpractitioners to abandon it in favor of the simpler but more restrictive VectorAutoRegressive (VAR) model. We narrow this gap with a new optimization-basedapproach to VARMA identification built upon the principle of parsimony. Amongall equivalent data-generating models, we use convex optimization to seek theparameterization that is "simplest" in a certain sense. A user-specifiedstrongly convex penalty is used to measure model simplicity, and that samepenalty is then used to define an estimator that can be efficiently computed.We establish consistency of our estimators in a double-asymptotic regime. Ournon-asymptotic error bound analysis accommodates both model specification andparameter estimation steps, a feature that is crucial for studying large-scaleVARMA algorithms. Our analysis also provides new results on penalizedestimation of infinite-order VAR, and elastic net regression under a singularcovariance structure of regressors, which may be of independent interest. Weillustrate the advantage of our method over VAR alternatives on three real dataexamples.

Sparse Identification and Estimation of Large-Scale Vector AutoRegressive Moving Averages