Structural Equation Models That are Nonlinear in Latent Variables: A Least-Squares Estimator

Busemeyer and Jones (1983) and Kenny and Judd (1984) proposed methods to include interactions of latent variables in structural equation models (SEMs). Despite the value of these works, their methods are limited by the required distributional assumptions, by their complexity in implementation, and by the unknown distributions of the estimators. This paper provides a framework for analyzing SEMs (LISREL models) that include nonlinear functions of latent or a mix of latent and observed variables in their equations. It permits such nonlinear functions in equations that are part of latent variable models or measurement models. I estimate the coefficient parameters with a two-stage least squares estimator that is consistent and asymptotically normal with a known asymptotic covariance matrix. The observed random variables can come from nonnormal distributions. Several hypothetical cases and an empirical example illustrate the method.