The role of the exogenous randomness in the identification of conditional models

This paper contributes to the construction of a general theory for conditional models by making explicit the role of the exogenous randomness in the identification of conditional models. We start with a definition of identification in conditional models called weak identification, derived from the usual concept of identification in unconditional statistical models. A first theorem shows that such a natural definition coincides, under a separability condition, with that one repeatedly met in the statistical literature. The role of the exogenous randomness is next examined from two perspectives. Firstly by characterizing the probability that the realization of the exogenous variables provides the identification of the parameter of interest. We display a hierarchy of four levels of identification in conditional models and show that the weakest level may entertain rather peculiar properties, in particular about the role of the sample size. Secondly, we look at identification as a necessary condition for the existence of an estimator with suitable properties. Several theorems relate the different levels of identification with the existence of estimators enjoying properties of unbiasedness or consistency at corresponding levels. Several examples illustrate these properties and conclusions are drawn concerning some implications for model building.