Latent variable selection in structural equation models

Structural equation models (SEMs) are often formulated using a prespecified parametric structural equation. In many applications, however, the formulation of the structural equation is unknown, and its misspecification may lead to unreliable statistical inference. This paper develops a general SEM in which latent variables are linearly regressed on themselves, thereby avoiding the need to specify outcome/explanatory latent variables. A penalized likelihood method with a proper penalty function is proposed to simultaneously select latent variables and estimate the coefficient matrix in formulating the structural equation. Under some regularity conditions, we show the consistency and the oracle property of the proposed estimators. We also develop an expectation/conditional maximization (ECM) algorithm involving a minorization–maximization algorithm that facilitates the second M-step. Simulation studies are performed and a real data set is analyzed to illustrate the proposed methods.