Chapter 7 - Equivalence of non-nested models

Abstract

This chapter focuses on the equivalence of non-nested models. The concept of equivalence is a fundamental one in stochastic modeling. Equivalence of submodels need not be the result of underidentification of the original model, which also need not be equivalent to the submodels. In all cases of equivalence, the data cannot be used to distinguish between the equivalent models. The chapter formulates conditions for equivalence and thus, tries to answer the basic question as to when differently formulated models describe reality in the same way. It is shown that equivalent models need not be submodels of a common underidentified, that is, overparametrized model. The chapter gives conditions for equivalence that can also be applied in case of an identified common model. However, the regularity assumptions made in the chapter cannot be considered mild. There are relevant cases of equivalence where the regularity assumptions are not satisfied. The chapter considers, as a special case, systems of simultaneous equations; here the conditions that characterize equivalence allow for a computerized evaluation.