Modeling of Low Rank Time Series

Rank-deficient stationary stochastic vector processes are present in many problems in network theory and dynamic factor analysis. In this article, we study the 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.