How the 2SLS/IV estimator can handle equality constraints in structural equation models: A system‐of‐equations approach

Abstract

Parameters in structural equation models are typically estimated using the maximum likelihood (ML) approach. Bollen (1996) proposed an alternative non-iterative, equation-by-equation estimator that uses instrumental variables. Although this two-stage least squares/instrumental variables (2SLS/IV) estimator has good statistical properties, one problem with its application is that parameter equality constraints cannot be imposed. This paper presents a mathematical solution to this problem that is based on an extension of the 2SLS/IV approach to a system of equations. We present an example in which our approach was used to examine strong longitudinal measurement invariance. We also investigated the new approach in a simulation study that compared it with ML in the examination of the equality of two latent regression coefficients and strong measurement invariance. Overall, the results show that the suggested approach is a useful extension of the original 2SLS/IV estimator and allows for the effective handling of equality constraints in structural equation models.

Estimators for longitudinal latent exposure models: examining measurement model assumptions

“…Given the also increased interest in longitudinal latent outcome models , it is important to evaluate how estimates of coefficients for observed predictors are influenced by incorrectly assuming time invariance in the measurement model for a longitudinal latent outcome . While MLE estimates will probably have the similar degree of bias as observed here, the robustness of the IV estimator will largely depend on whether time invariance constraints are imposed or not. Our study focused on continuous exposure surrogates, and we assumed multivariate normality to obtain ML estimates.…”

The social consequences and mechanisms of personality: How to analyse longitudinal data from individual, dyadic, round-robin and network designs

“…Growth models can be constructed within the structural equation modelling (SEM; see Little, 2013;Nestler, 2014a;Nestler, in press) and multilevel modelling (MLM; see Singer & Willett, 2003;Verbeke & Molenberghs, 2009) frameworks. There are notable advantages of each framework when studying within-person change (Stoel & Galindo-Garre, 2011).…”

Using Instrumental Variables to Estimate the Parameters in Unconditional and Conditional Second-Order Latent Growth Models

“…This approach also allows the user to obtain standard errors of these parameters, and a chi-square test statistic can be computed to test the overall model fit. For technical details see Bollen and Maydeu-Olivares (2007) and Nestler (2014).…”

“…The basic idea of this 2SLS/IV system-of-equations approach (see Nestler, 2014, for technical details) is to consider the single regressions derived from the original 2SLS/IV approach (e.g., the six equations given in Equation 16) as a system of regression equations that can be used to define appropriate system matrices; then, one can use these matrices to estimate the model parameters and the standard errors. Stated more simply, the indicator intercepts and factor loadings are not estimated in succession for each equation, but rather, they can be estimated in a single step for all equations at once.…”

“…However, when equality constraints are introduced, it allows these constrained parameters to be estimated consistently. Nestler (2014) found that the system-of-equations approach produced estimates that were as accurate as ML in the case of a CFA model that was estimated under the requirement of strong factorial invariance and in the case of a structural equation model in which the latent regression coefficients were constrained to be equal. These findings suggest that this estimator should produce accurate parameter estimates of the model parameters in an unconditional second-order LGM and a conditional second-order LGM, respectively, that are estimated under the restriction of strong factorial invariance.…”