• Itô calculus;
  • sampling;
  • continuous–discrete state-space models;
  • nonlinear Kalman filtering;
  • Hermite orthogonal expansion;
  • numerical integration;
  • Monte Carlo simulation.

Stochastic differential equations (SDE) are used as dynamical models for cross-sectional discrete time measurements (panel data). Thus causal effects are formulated on a fundamental infinitesimal time scale. Cumulated causal effects over the measurement interval can be expressed in terms of fundamental effects which are independent of the chosen sampling intervals (e.g. weekly, monthly, annually). The nonlinear continuous–discrete filter is the key tool in deriving a recursive sequence of time and measurement updates. Several approximation methods including the extended Kalman filter (EKF), higher order nonlinear filters (HNF), the local linearization filter (LLF), the unscented Kalman filter (UKF), the Gauss–Hermite filter (GHF) and generalizations (GGHF), as well as simulated filters (functional integral filter FIF) are compared.