Interpolating Exogenous Variables in Open Continuous Time Dynamic Models

As the exact discrete model induced by an open continuous time system depends on the continous time paths of the exogenous variables, these need to be interpolated for the purpose of estimation. We examine some recently suggested forms of interpolation and find them not to be completely satisfactory. Accordingly, we extend to arbitrary degree the interpolant proposed by Bergstrom (Economic Theory, 1986). The resulting exact discrete model has a VARMAX representation, the order of whose exogenous variable component is the degree of the interpolating polynominal. This allows an optimum interpolant systematically to be selected on the basis of the data.