Estimation of a Model with Multiple Indicators and Multiple Causes of a Single Latent Variable

TLDR

The paper studies a latent variable model with multiple indicators and multiple causes, noting that its two key restrictions—rank‑one regression coefficients and a one‑factor residual covariance—are familiar to econometricians and psychometricians. The authors derive maximum‑likelihood estimators for the model, compute their asymptotic variance, and compare them with two alternative limited‑information estimators in terms of efficiency. The limited‑information estimators were found to be less efficient than the maximum‑likelihood estimators.

Abstract

Abstract We consider a model in which one observes multiple indicators and multiple causes of a single latent variable. In terms of the multivariate regression of the indicators on the causes, the model implies restrictions of two types: (i) the regression coefficient matrix has rank one, (ii) the residual variance-covariance matrix satisfies a factor analysis model with one common factor. The first type of restriction is familiar to econometricians and the second to psychometricians. We derive the maximum-likelihood estimators and their asymptotic variance-covariance matrix. Two alternative “limited information” estimators are also considered and compared with the maximum-likelihood estimators in terms of efficiency.