  This paper presents a novel estimator for a non-standard 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 underestimate 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 $p\times p$ 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.
