A Specification Error Test That Uses Instrumental Variables to Detect Latent Quadratic and Latent Interaction Effects

Abstract

The relations between the latent variables in structural equation models are typically assumed to be linear in form. This article aims to explain how a specification error test using instrumental variables (IVs) can be employed to detect unmodeled interactions between latent variables or quadratic effects of latent variables. An empirical example is presented, and the results of a simulation study are reported to evaluate the sensitivity and specificity of the test and compare it with the commonly employed chi-square model test. The results show that the proposed test can identify most unmodeled latent interactions or latent quadratic effects in moderate to large samples. Furthermore, its power is higher when the number of indicators used to define the latent variables is large. Altogether, this article shows how the IV-based test can be applied to structural equation models and that it is a valuable tool for researchers using structural equation models.

Keywords:

• diagnosing nonlinear effects
• instrumental variables
• latent interaction
• latent nonlinear effects
• latent quadratic effects
• structural equation model

Notes

¹ It should be mentioned that the test can also be used to check whether some or all of the predictors should be raised to arbitrary powers or whether the logarithm, the square root, and so omspaintn, of the predictors should be modeled.