Prediction of Latent Variables in a Mixture of Structural Equation Models, with an Application to the Discrepancy Between Survey and Register Data

We study the prediction of latent variables in a finite mixture of linear structural equation models. The latent variables can be viewed as well-defined variables measured with error or as theoretical constructs that cannot be measured objectively, but for which proxies are observed. The finite mixture component may serve different purposes: it can denote an unobserved segmentation in subpopulations such as market segments, or it can be used as a nonparametric way to estimate an unknown distribution. In the first interpretation, it forms an additional discrete latent variable in an otherwise continuous latent variable model. Different criteria can be employed to derive “optimal” predictors of the latent variables, leading to a taxonomy of possible predictors. We derive the theoretical properties of these predictors. Special attention ∗Meijer: RAND Corporation and University of Groningen; Rohwedder: RAND Corporation; Wansbeek (corresponding author): University of Amsterdam and University of Groningen, [email protected]. We would like to thank Jelmer Ypma and Arie Kapteyn for sharing their code with us and for discussions about their model, and Jos ten Berge and conference participants at Aston Business School, Birmingham, in particular Battista Severgnini, for stimulating discussions and comments on an earlier version of this paper. The National Institute on Aging supported the collection of the survey data and subsequent match to administrative records (grant R03AG21780).