  Rank-deficient stationary stochastic vector processes are present in many
problems in network theory and dynamic factor analysis. In this paper we study
hidden dynamical relations between the components of a discrete-time stochastic
vector process and investigate their properties with respect to stability and
causality. More specifically, we construct transfer functions with a full-rank
input process formed from selected components of the given vector process and
having a vector process of the remaining components as output. An important
question, which we answer in the negative, is whether it is always possible to
find such a deterministic relation that is stable. If it is unstable, there
must be feedback from output to input ensuring that stationarity is maintained.
This leads to connections to robust control. We also show how our results could
be used to investigate the structure of dynamic network models and the latent
low-rank stochastic process in a dynamic factor model.
