An Introduction to Model Implied Instrumental Variables Using Two Stage Least Squares (MIIV-2SLS) in Structural Equation Models (SEMs)

Structural equation models (SEMs) are widely used to handle multiequation systems that involve latent variables, multiple indicators, and measurement error. Maximum likelihood (ML) and diagonally weighted least squares (DWLS) dominate the estimation of SEMs with continuous or categorical endogenous variables, respectively. When a model is correctly specified, ML and DWLS function well. But, in the face of incorrect structures or nonconvergence, their performance can seriously deteriorate. Model implied instrumental variable, two stage least squares (MIIV-2SLS) estimates and tests individual equations, is more robust to misspecifications, and is noniterative, thus avoiding nonconvergence. This article is an overview and tutorial on MIIV-2SLS. It reviews the six major steps in using MIIV-2SLS: (a) model specification; (b) model identification; (c) latent to observed (L2O) variable transformation; (d) finding MIIVs; (e) using 2SLS; and (f) tests of overidentified equations. Each step is illustrated using a running empirical example from Reisenzein’s (1986) randomized experiment on helping behavior. We also explain and illustrate the analytic conditions under which an equation estimated with MIIV-2SLS is robust to structural misspecifications. We include additional sections on MIIV approaches using a covariance matrix and mean vector as data input, conducting multilevel SEM, analyzing categorical endogenous variables, causal inference, and extensions and applications. Online supplemental material illustrates input code for all examples and simulations using the R package MIIVsem. Translational Abstract Theories in psychology hypothesize relationships between abstract variables that we can only imperfectly measure. To test these ideas requires models with latent variables to represent these abstract concepts and multiple measures to anchor the latent variables to those we can observe. Researchers routinely use latent variable structural equation models (SEMs) to test psychological theories and explanations, as well as to refine our measures. Current methods to estimate and test such models focus on the whole system under the assumption that the model is a fully accurate portrayal of reality. In the common situation of models as approximations to reality, these techniques are susceptible to spreading errors from one part of the system to another. This article describes an alternative approach, Model Implied instrumental variable, two stage least squares (MIIV-2SLS), with a focus on individual equations that better limits the spread of model misspecification errors that often occur. This didactic article describes the major steps to using MIIV-2SLS illustrated with an empirical example.