Model Identification in Bayesian Analysis of Static and Dynamic Factor Models
This thesis deals with the Bayesian estimation of static and dynamic models for exploratory factor analysis. Factor analysis serves as a tool for dimensional reduction. Its aim is to find latent variables, the common factors, which drive observable variables through factor loadings, where the dimensional reduction implies that the number of latent factors is much smaller than the number of observable variables. Due to the fact that the factors are latent and the loadings are unknown parameters, the factor model suffers from indeterminacies even if a clear distinction between the systematic and the idiosyncratic part is possible. The major model indeterminacy in exploratory factor analysis is the rotation problem. It depends on the applied estimation technique whether the rotation problem affects the estimation process or is merely of interest when interpreting the results. Contemporary Bayesian factor analysis typically uses the MCMC method of Gibbs sampling, where a common identification scheme is to impose a positive lower triangular structure on the loadings matrix to deal with the rotation problem. This identification scheme, however, leads to a behavior of the Gibbs sampler that depends on the ordering of the variables. This phenomenon has become known as the ordering problem. This thesis investigates the rotation problem by first analyzing the properties of the Gibbs sampler if no constraints are imposed to solve the rotation problem, called the unconstrained Gibbs sampler, and the properties of the corresponding posterior distribution. Next, an approach to solve the rotation problem is proposed, which postprocesses the output of the unconstrained Gibbs sampler. The capabilities of this ex-post approach are demonstrated in an analysis of a high- dimensional macroeconomic data set. Afterwards, it is shown how the proposed approach can be used to determine the number of factors in the model, to select the variables relevant for the factor structure, and to identify a sparse loadings pattern. Eventually, two model selection criteria are investigated and the approach is applied to analyze a high-dimensional data set of regional unemployment figures for Germany.