Estimation of Generalized Linear Latent Variable Models

Generalized Linear Latent Variable Models (GLLVM), as de ned in Bartholomew and Knott (1999) enable modelling of relationships between manifest and latent variables. They extend structural equation modelling techniques, which are powerful tools in the social sciences. However, because of the complexity of the log-likelihood function of a GLLVM, an approximation such as numerical integration must be used for inference. This can limit drastically the number of variables in the model and lead to biased estimators. In this paper, we propose a new estimator for the parameters of a GLLVM, based on a Laplace approximation to the likelihood function and which can be computed even for models with a large number of variables. The new estimator can be viewed as a M -estimator, leading to readily available asymptotic properties and correct inference. A simulation study shows its excellent nite sample properties, in particular when compared with a well established approach such as LISREL. A real data example on the measurement of wealth for the computation of multidimentional inequality is analysed to highlight the importance of the methodology. keywords CATEGORICAL VARIABLES; LAPLACE APPROXIMATION; M -ESTIMATORS; LISREL; PENALIZED QUASI-LIKELIHOOD; VARIMAX ROTATION ∗Department of Econometrics, University of Geneva, Blv. Pont d'Arve 40, CH-1211 Geneva, Switzerland ¶HEC, Faculty of Economic and Social Sciences and Faculty of Psychology and Education, University of Geneva, Blv. Pont d'Arve 40, CH-1211 Geneva, Switzerland §Partially supported by Swiss National Science Foundation, grant 610-057883.99