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An Instrumental Variable Estimator for Mixed Indicators: Analytic Derivatives and Alternative Parameterizations

Published online by Cambridge University Press:  01 January 2025

Zachary F. Fisher Open the ORCID record for Zachary F. Fisher [Opens in a new window]  and
Kenneth A. Bollen
Zachary F. Fisher*
Affiliation:
University of North Carolina at Chapel Hill
Kenneth A. Bollen
Affiliation:
University of North Carolina at Chapel Hill
*
Correspondence should bemade to Zachary F. Fisher, University of North Carolina at Chapel Hill, Chapel Hill, USA. Email: fish.zachary@gmail.com
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Abstract

Methodological development of the model-implied instrumental variable (MIIV) estimation framework has proved fruitful over the last three decades. Major milestones include Bollen’s (Psychometrika 61(1):109–121, 1996) original development of the MIIV estimator and its robustness properties for continuous endogenous variable SEMs, the extension of the MIIV estimator to ordered categorical endogenous variables (Bollen and Maydeu-Olivares in Psychometrika 72(3):309, 2007), and the introduction of a generalized method of moments estimator (Bollen et al., in Psychometrika 79(1):20–50, 2014). This paper furthers these developments by making several unique contributions not present in the prior literature: (1) we use matrix calculus to derive the analytic derivatives of the PIV estimator, (2) we extend the PIV estimator to apply to any mixture of binary, ordinal, and continuous variables, (3) we generalize the PIV model to include intercepts and means, (4) we devise a method to input known threshold values for ordinal observed variables, and (5) we enable a general parameterization that permits the estimation of means, variances, and covariances of the underlying variables to use as input into a SEM analysis with PIV. An empirical example illustrates a mixture of continuous variables and ordinal variables with fixed thresholds. We also include a simulation study to compare the performance of this novel estimator to WLSMV.

Keywords

estimationlatent variablesordinal variablesdichotomous variablescontinuous variablesinstrumental variablestwo-stage least squares (2SLS)factor analysisstructural equation modeling
Type
Theory and Methods
Information
Psychometrika , Volume 85 , Issue 3 , September 2020 , pp. 660 - 683
Copyright
Copyright © 2020 The Psychometric Society

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