Identifying Restrictions for Finite Parameter Continuous Time Models with Discrete Time Data

When a continuous time model is sampled only at equally spaced intervals, a priori restrictions on the parameters can provide natural identifying restrictions which serve to rule out otherwise observationally equivalent parameter values. Specifically, we consider identification of the parameter matrix in a linear system of first-order stochastic differential equations, a setting which is general enough to include many common continuous time models in economics and finance. We derive a new characterization of the identification problem under a fully general class of linear restrictions on the parameter matrix and establish conditions under which only floor(n/2) restrictions are sufficient for identification when only the discrete time process is observable. Restrictions of the required kind are typically implied by economic theory and include zero restrictions that arise when some variables are excluded from an equation. We also consider identification of the intensity matrix of a discretely-sampled finite Markov jump processes, a related special case where we show that only floor((n-1)/2) restrictions are required. We demonstrate our results by applying them to two example models from economics and finance: a continuous time regression model with three equations and a continuous-time model of credit rating dynamics.