Convex Estimation of Cointegrated VAR Models by a Nuclear Norm Penalty

Cointegrated Vector AutoRegressive (VAR) processes arise in the study of long run equilibrium relations of stochastic dynamical systems. In this paper we introduce a novel convex approach for the analysis of these type of processes. The idea relies on an error correction representation and amounts at solving a penalized empirical risk minimization problem. The latter finds a model from data by minimizing a trade-off between a quadratic error function and a nuclear norm penalty used as a proxy for the cointegrating rank. We elaborate on properties of the proposed convex program; we then propose an easily implementable and provably convergent algorithm based on FISTA. This algorithm can be conveniently used for computing the regularization path, i.e., the entire set of solutions associated to the trade-off parameter. We show how such path can be used to build an estimator for the cointegrating rank and illustrate the proposed ideas with experiments.