Model identification and non-unique structure

Identification is an essential attribute of any model’s parameters, so we consider its three aspects of ‘uniqueness’, ‘correspondence to reality’ and ‘interpretability’. Observationally-equivalent overidentified models can co-exist, and are mutually encompassing in the population; correctly-identified models need not correspond to the underlying structure; and may be wrongly interpreted. That a given model is over-identified with all over-identifying restrictions valid (even asymptotically) is insufficient to demonstrate that it is a unique representation. Moreover, structure (as invariance under extended information) need not be identifiable. We consider the role of structural breaks to discriminate between such representations