Deriving the Exact Discrete Analog of a Continuous Time System

We present a method of deriving the exact discrete model satisfied by equispaced data generated by a system of linear stochastic differential equations without implying the usual restrictions on observed discrete data that are capable of being rejected by a statistical test. The method involves integrating the solution of the continuous time model in state space form, and relies on a non-standard change in the order of three types of integration as a means of representing the exact discrete model as an asymptotically time-invariant VARMA model. Applying to the state space form of the model, the method is general, and delivers a parsimonious representation of the exact discrete model in any particular case. It is applied by way of example to the prototypical higher order model for mixed stock and flow data discussed by Bergstrom (1986, Econometric Theory 2, 350-373.