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

TL;DR

This paper advances the Model-implied Instrumental Variable (MIIV) estimation framework by deriving analytic derivatives, extending to mixed variable types, and allowing for more flexible parameterizations, demonstrated through empirical and simulation studies.

Contribution

It introduces analytic derivatives for the PIV estimator, extends it to mixed variable types, and generalizes the model to include intercepts, means, and known thresholds.

Findings

The new estimator performs well in simulations compared to WLSMV.

It effectively handles mixed binary, ordinal, and continuous variables.

The empirical example demonstrates practical applicability.

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 (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 \& Maydeu-Olivares, 2007), and the introduction of a Generalized Method of Moments (GMM) estimator (Bollen, Kolenikov \& Bauldry, 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.