Generalized linear latent variables models: Estimation, inference and empirical analysis of financial data

Abstract

Generalized Linear Latent Variable Models (GLLVM), as defined in Bartholomew and Knott (1999), enable the modelling of relationships between manifest and latent variables. They extend the 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 estimation and inference. Depending on the choice of the approximation, the estimators can be biased, and/or their computation can be extremely time consuming. In this work, we propose a new estimator for the parameters of a GLLVM, the LAMLE, based on a Laplace approximation to the likelihood function and which can be computed even for models with a large number of variables. The LAMLE can be viewed as an M-estimator, leading to readily available asymptotic properties, correct inference and an Akaike Information Criterion as a model selection criterion. We introduce e new software called L-Cube that computes the LAMLE and propose new algorithms to increase the computation speed. In particular, we propose a sequence of three algortihm to find a good starting point for the optimization: a principal Component Analysis, a QR decomposition and a new rotation algorithm that we call LLD. We also propose specific algorithms for the computation of p-values and confidence intervals for the models parameters. A simulation study shows the excellent finite sample properties of the LAMLE, in particular when compared with well established approach such as LISREL. Like in stardard Factor Analysis, multiple solutions are possible for the estimators of a GLLVM. Here we show how constraints must be imposed to make a solution unique for independent latent variables as well as for correlated latent variables. Finally, two real datasets, one on forecasts on stock and bond markets made by brokers and one on the measurement of wealth for the computation of multidimensional inequality are analysed to highlight the importance of the methodology.