Identifiability

Abstract: This paper provides an analysis of aggregate behavioural outcomes when individual utility exhibits social interaction effects. We study generalized logistic models of individual choice which incorporate terms reflecting the desire of individuals to conform to the behaviour of others in an environment of noncooperative decisionmaking. Laws of large numbers are generated in such environments. Multiplicity of equilibria in these models, which are equivalent to the existence of multiple self-consistent means for average choice behaviour, will exist when the social interactions exceed a particular threshold. Local stability of these multiple equilibria is also studied. The properties of the noncooperative economy are contrasted with the properties of an economy in which a social planner determines the set of individual choices. Finally, a likelihood function based on the theoretical model is given and conditions for the econometric identifiability of the model are established.

Abstract: The paper develops and explores tests, based on standard moment specifications, for the identifiability of parameters apparently estimable by instrumental variables. An asymptotic expansion under standard restrictive assumptions on the error distribution suggests a correction to the asymptotic distribution. A small sampling experiment indicates that the tests are of use.

Abstract: A statistical approach to a posteriori blockmodeling for digraphs and valued digraphs is proposed. The probability model assumes that the vertices of the digraph are partitioned into several unobserved (latent) classes and that the probability distribution of the relation between two vertices depends only on the classes to which they belong. A Bayesian estimator based on Gibbs sampling is proposed. The basic model is not identified, because class labels are arbitrary. The resulting identifiability problems are solved by restricting inference to the posterior distributions of invariant functions of the parameters and the vertex class membership. In addition, models are considered where class labels are identified by prior distributions for the class membership of some of the vertices. The model is illustrated by an example from the social networks literature (Kapferer's tailor shop).

Abstract: Summary. In a Bayesian analysis of finite mixture models, parameter estimation and clustering are sometimes less straightforward than might be expected. In particular, the common practice of estimating parameters by their posterior mean, and summarizing joint posterior distributions by marginal distributions, often leads to nonsensical answers. This is due to the so-called 'label switching' problem, which is caused by symmetry in the likelihood of the model parameters. A frequent response to this problem is to remove the symmetry by using artificial identifiability constraints. We demonstrate that this fails in general to solve the problem, and we describe an alternative class of approaches, relabelling algorithms, which arise from attempting to minimize the posterior expected loss under a class of loss functions. We describe in detail one particularly simple and general relabelling algorithm and illustrate its success in dealing with the label switching problem on two examples.