Dynamic relations in sampled processes

Linear dynamical relations that may exist in continuous-time, or at some natural sampling rate, are not directly discernable at reduced observational sampling rates. Indeed, at reduced rates, matricial spectral densities of vectorial time series have maximal rank and thereby cannot be used to ascertain potential dynamic relations between their entries. This hitherto undeclared source of inaccuracies appears to plague off-the-shelf identification techniques seeking remedy in hypothetical observational noise. In this paper we explain the exact relation between stochastic models at different sampling rates and show how to construct stochastic models at the finest time scale that data allows. We then point out that the correct number of dynamical dependences can only be ascertained by considering stochastic models at this finest time scale, which in general is faster than the observational sampling rate.