Simultaneous equations, error-in-variable models, and model integration in systems ecology

Numerous dynamic ecological models of varied time and spatial scales exist in systems ecology. In general, small-scale models are more accurate, more capable of reflecting tiny local variations in eco-processes, and more sensitive to the outside disturbances than large-scale models. On the other hand, large-scale models are more comprehensive, and usually describe the ecosystemʹs average properties. There has been increased interest in how to integrate accurate small-scale models with comprehensive large-scale models. The two-stage or three-stage least squares regression is the classic parameter estimation method for such purposes. In this study, a two-stage error-in-variable method is introduced to estimate the parameters for model integration. It is proved theoretically that when the restriction is exactly identifiable, the two-stage least squares regression and the two-stage error-in-variable model produce the same estimates. If the restriction is over identifiable, both methods have solutions, but the estimates are not necessarily identical. For under identifiable systems, the estimate from the error-in-variable model still exists, but the estimate from the two-stage least squares regression is not valid any more. An example is provided to demonstrate how to use the two-stage error-in-variable model in a step-by-step fashion.