Testing specifications in partial observability models : a Bayesian encompassing approach
A structural approach for modelling a statistical problem permits to introduce a contextual theory based in previous knowledge. This approach makes the parameters completely meaningful; but, in the intermediate steps, some unobservable characteristics are introduced because of their contextual meaning. When the model is completely specified, the marginalisation into the observed variables is operated in order to obtain a tatistical model.
The variables can be discrete or continuous both at the level of unobserved and at the level of observed or manifest variables. We are sometimes faced, especially in behavioural sciences, with ordinal variables; this is the case of the so-called Likert scales.
Therefore, an ordinal variable could be nterpreted as a discrete version of a latent concept (the discretization model). The normality of the latent variables simplifies the study of this model into the analysis of the structure of the covariance matrix of the "ideally" measured variables, but only a sub-parameter of these matrix can be identified and consistently estimated (i.e. the matrix of polychoric correlations). Consequently, two questions rise here: Is the normality of the latent variables testable? If not, what is the aspect of this hypothesis which could be testable?.
In the discretization model, we observe a loss of information with related to the information contained in the latent variables. In order to treat this situation we introduce the concept of partial observability through a (non bijective) measurable function of the latent variable. We explore this definition and verify that other models can be adjusted to this concept. The definition of partial observability permits us to distinguish between two cases depending on whether the involved function is or not depending on a Euclidean parameter. Once the partial observability is introduced, we expose a set of conditions for building a specification test at the level of latent variables. The test is built using the encompassing principle in a Bayesian framework.
More precisely, the problem treated in this thesis is: How to test, in a Bayesian framework, the multivariate normality of a latent vector when only a discretized version of that vector is observed. More generally, the problem can be extended to (or re-paraphrased in): How to test, in Bayesian framework, a parametric specification on latent variables against a nonparametric alternative when only a partial observation of these latent variables is available.
School:Université catholique de Louvain
Source Type:Master's Thesis
Keywords:bayesian statistics partial observability encompassing test
Date of Publication:10/04/2007