A systems approach to the calibration of deterministic dynamic nonlinear simultaneous equation models with incomplete data

This paper develops an interactive three–stage systems approach for the calibration of the structural parameters and missing data within a deterministic, dynamic non–linear simultaneous equations model under arbitrary configurations of incomplete data. In Stage One, we minimize a quadratic loss function in the differences between the actual endogenous variables and the predicted solution values, relative to any feasible choice of the structural parameters. Missing exogenous variables and initial endogenous variables are treated as additional parameters to be calibrated; whereas missing current endogenous variables are treated by the missing data updating condition, in which the current solution values iteratively and sequentially replace those absent. Stage One may or may not lead to unique calibrations of the structural parameters—a fact that can be monitored a posteriori using singular value decompositions of the relevant Jacobian matrix. If not, there is an equivalence class of parameter values, all of which result in the same loss function value. If Stage Two is necessary, we attempt to exploit the non–linearity and simultaneity of the structural system to extract further information about the parameters from the same database, by minimizing the distance between the restricted and unrestricted reduced forms, while constraining the parameters also to lie within the Stage One equivalence class. This requires the use of higher–order numerical derivatives, and probably restricts its use in all but the simplest of cases to the next generation of supercomputers with massive numbers of parallel processors and much larger word–sizes. In Stage Three, various methods by which the original structural model can be simplified, given a non–unique Stage One calibration, are entertained.