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. We also show how our results could be used to investigate the structure of the latent low-rank stochastic process in a dynamic factor model.