Simultaneous equations, error-in-variable models, and model integration insystems 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 sens itive to the outside disturbances than large-scale models. On the other han d, large-scale models are more comprehensive, and usually describe the ecos ystem's average properties. There has been increased interest in how to int egrate accurate small-scale models with comprehensive large-scale models. T he two-stage or three-stage least squares regression is the classic paramet er estimation method for such purposes. In this study, a two-stage error-in -variable method is introduced to estimate the parameters for model integra tion. It is proved theoretically that when the restriction is exactly ident ifiable, the two-stage least squares regression and the two-stage error-in- variable model produce the same estimates. If the restriction is over ident ifiable, 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 squa res 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 , (C) 2001 Elsevier Science B.V. All rights reserved.