Estimation of Generalized Linear Latent Variable Models

Generalized Linear Latent Variable Models (GLLVM), as defined in Bartholomew and Knott (1999) allow to model relationships between manifest and latent variables when the manifest variables are of various type, such as continuous or discrete. They extend structural equation modelling techniques which are very powerful modelling tools in the social sciences. However, because of the complexity of the log-likelihood function of GLLVM due to the fact that the latent variables are not directly observed, usually an approximation such as numerical integration is used to carry out estimation and inference. This can limit in a drastic way 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. It is based on a Laplace approximation of the likelihood function and can be computed even for models with a large number of variables. It is shown that the new estimator can be viewed as a M-estimator leading to readily available asymptotic properties and correct inference. A simulation study in various settings shows its excellent finnite sample properties, in particular when compared with a well established approach such as LISREL