  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.
