This paper presents a novel estimator for a nonstandard restriction to both symmetry and low rank in the context of high-dimensional cointegrated processes. Furthermore, we discuss rank estimation for high-dimensional cointegrated processes by restricted bootstrapping of the Gaussian innovations. We demonstrate that the classical rank test for cointegrated systems is prone to underestimating the true rank and demonstrate this effect in a 100-dimensional system. We also discuss the implications of this underestimation for such high-dimensional systems in general. Also, we define a linearized Kuramoto system and present a simulation study, where we infer the cointegration rank of the unrestricted
system and successively the underlying clustered network structure based on a graphical approach and a symmetrized low rank estimator of the couplings derived from a reparametrization of the likelihood under this unusual restriction.