Partially sufficient statistics and identification in conditional models
Abstract: In this thesis, we give a general construction of a conditional model through embedding that concept into the concept of unconditional model. Formally, the conditional model is considered as a statistical model bearing on all the variables, i.e. on the "endogenous variables" Y and the conditioning, or "exogenous", variables Z such that j, the parameter characterizing the marginal distribution of Z, is a nuisance parameter that is identified and "well-separated” from q, the parameter of interest characterizing the Z-conditional distribution. Therefore, a family of marginal distributions on the exogenous variables and a family of “well specified” transitions of probabilities, playing a role of conditional probabilities in a global model, characterize a conditional model. Typically, but not always, j takes values in a "thick" subset F, of all the probability distributions of Z. From this construction, we analyze the identification of a conditional model in the framework of the identification of a function of the parameters in unconditional model. We propose a definition of identification in conditional models called weak identification, derived from the usual concept of identification in unconditional models. We show, under a separability condition, that weak identification may be considered as a generalization of definitions usually met in the statistical literature; in particular those in Manski (1988) and Matzkin (1993). However, an undesirable property of weak identification is shown, namely that under rather general conditions, the weak identification does not depend on the sample size. As an alternative, three other levels of identification are given, stressing the proper role of the randomness of the conditioning variables. Similar distinctions are also shown to be relevant for properties of estimators, such as unbiasedness or consistency. The relationships between these different levels of identification, unbiasedness and consistency are given.
Another aspect analyzed in this thesis is the concept of partial sufficiency. Our contribution to this area is to give some further properties of S-sufficiency. In particular, we establish the connection between S-sufficiency and the identification concept for unconditional models and also for conditional models with partially observable endogenous variables. We show that when we reduce the structural (latent) model by marginalizing w.r.t an S-sufficient statistic, we do not lose the identification of the parameter of interest in the statistical (reduced) model. Furthermore, we study the properties and the conditions of applicability of S-sufficiency, with a view to compare the properties of the standard concept of sufficiency and of S-sufficiency respectively.
As an application, we analyze the identification of the conditional binary response models from the semi-parametric point of view.
School:Université catholique de Louvain
Source Type:Master's Thesis
Keywords:conditional models partial sufficiency identification
Date of Publication:05/05/2003