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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

Information

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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References

Arminger, G., Küsters, U., Langeheine, R., & Rost, J. (1988). Latent trait models with indicators of mixed measurement level. Latent trait and latent class models, Boston, MA: Springer. 51– 73. CrossRefGoogle Scholar
Bock, D. R. (1972). Estimating item parameters and latent ability when responses are scored in two or more nominal categories. Psychometrika, 37 (1), 29– 51. CrossRefGoogle Scholar
Bollen, K. A. (1996). An alternative two stage least squares (2sls) estimator for latent variable equations. Psychometrika, 61 (1), 109– 121. CrossRefGoogle Scholar
Bollen, K. A. Kolenikov, S. Bauldry, S. (2014). Model-implied instrumental variable-generalized method of moments (MIIV-GMM) estimators for latent variable models. Psychometrika, 79 (1), 20– 50. CrossRefGoogle ScholarPubMed
Bollen, K. A., & Maydeu-Olivares, A. (2007). A polychoric instrumental variable (PIV) estimator for structural equation models with categorical variables. Psychometrika, 72 (3), 309– 326. CrossRefGoogle Scholar
Cragg, J. G., & Donald, S. G. (1993). Testing identifiability and specification in instrumental variable models. Econometric Theory, 9 (2), 222– 240. CrossRefGoogle Scholar
Cramér, H. (1999). Mathematical methods of statistics, Princeton: Princeton University Press. Google Scholar
Fisher, Z., Bollen, K., Gates, K., & Rönkkö, M. (2017). Miivsem: Model implied instrumental variable (miiv) estimation of structural equation models. r package version 0.5. 2.Google Scholar
Fisher, Z. F., Bollen, K. A., & Gates, K. M. (2019). A limited information estimator for dynamic factor models. Multivariate Behavioral Research, 54 (2), 246– 263. CrossRefGoogle ScholarPubMed
Hansen, L. P. (1982). Large sample properties of generalized method of moments estimators. Econometrica, 50 (4), 1029– 1054. CrossRefGoogle Scholar
Hayashi, F. (2011). Econometrics, Princeton: Princeton University Press. Google Scholar
Jin, S., & Cao, C. (2018). Selecting polychoric instrumental variables in confirmatory factor analysis: An alternative specification test and effects of instrumental variables. British Journal of Mathematical and Statistical Psychology, 71 (2), 387– 413. CrossRefGoogle ScholarPubMed
Jin, S., Luo, H., & Yang-Wallentin, F. (2016). A simulation study of polychoric instrumental variable estimation in structural equation models. Structural Equation Modeling, 23 (5), 680– 694. CrossRefGoogle Scholar
Jöreskog, K. G. (1994). On the estimation of polychoric correlations and their asymptotic covariance matrix. Psychometrika, 59 (3), 381– 389. CrossRefGoogle Scholar
Jöreskog, K. G. (2002). Structural Equation Modeling with Ordinal Variables using LISREL. Scientific Software International, Inc.Google Scholar
Katsikatsou, M., Moustaki, I., Yang-Wallentin, F., & Jöreskog, K. G. (2012). Pairwise likelihood estimation for factor analysis models with ordinal data. Computational Statistics & Data Analysis, 56 (12), 4243– 4258. CrossRefGoogle Scholar
Kirby, J. B., & Bollen, K. A. (2009). Using instrumental variable (IV) tests to evaluate model specification in latent variable structural equation models. Sociological Methodology, 39 (1), 327– 355. CrossRefGoogle ScholarPubMed
Lazarsfeld, P. F. (1950). The logical and mathematical foundation of latent structure analysis. Studies in Social Psychology in World War II Vol. IV : Measurement and Prediction, 4 362– 412. Google Scholar
Lee, S. -Y., Poon, W. -Y., & Bentler, P. M. (1995). A two-stage estimation of structural equation models with continuous and polytomous variables. British Journal of Mathematical and Statistical Psychology, 48 (2), 339– 358. CrossRefGoogle ScholarPubMed
Lord, F., Novick, M., & Birnbaum, A. (1968). Statistical theories of mental test scores, Oxford: Addison-Wesley. Google Scholar
Magnus, J. R. (1983). L-structured matrices and linear matrix equations. Linear and Multilinear Algebra, 14 (1), 67– 88. CrossRefGoogle Scholar
Magnus, J. R., & Neudecker, H. (1986). Symmetry, 0–1 matrices and Jacobians: A review. Econometric Theory, 2 (2), 157– 190. CrossRefGoogle Scholar
Monroe, S. (2018). Contributions to estimation of polychoric correlations. Multivariate Behavioral Research, 53 (2), 247– 266. CrossRefGoogle ScholarPubMed
Muthén, B. (1978). Contributions to factor analysis of dichotomous variables. Psychometrika, 43 (4), 551– 560. CrossRefGoogle Scholar
Muthén, B. (1984). A general structural equation model with dichotomous, ordered categorical, and continuous latent variable indicators. Psychometrika, 49 (1), 115– 132. CrossRefGoogle Scholar
Muthén, B. (1993). Latent trait models with indicators of mixed measurement level. Testing structural equation models, Thousand Oaks, CA: Sage. Google Scholar
Muthén, B., & Satorra, A. (1995). Technical aspects of Muthén’s liscomp approach to estimation of latent variable relations with a comprehensive measurement model. Psychometrika, 60 (4), 489– 503. CrossRefGoogle Scholar
Nel, D. G. (1980). On matrix differentiation in statistics. South African Statistical Journal, 14 (2), 137– 193. Google Scholar
Nestler, S. (2013). A Monte Carlo study comparing PIV, ULS and DWLS in the estimation of dichotomous confirmatory factor analysis. British Journal of Mathematical and Statistical Psychology, 66 (1), 127– 143. CrossRefGoogle Scholar
Olsson, U. (1979). Maximum likelihood estimation of the polychoric correlation coefficient. Psychometrika, 44 (4), 443– 460. CrossRefGoogle Scholar
Phillips, P. C. B. (1989). Partially identified econometric models. Econometric Theory, 5 (2), 181– 240. CrossRefGoogle Scholar
Rosseel, Y. (2012). lavaan: An R package for structural equation modeling. Journal of Statistical Software, 48 (2), 1– 36. CrossRefGoogle Scholar
Savalei, V., & Falk, C. F. (2014). Robust two-stage approach outperforms robust full information maximum likelihood with incomplete nonnormal data. Structural Equation Modeling, 21 (2), 280– 302. CrossRefGoogle Scholar
Shea, J. (1997). Instrument relevance in multivariate linear models: A simple measure. The Review of Economics and Statistics, 79 (2), 348– 352. CrossRefGoogle Scholar
Stock, J. H., Yogo, M., Stock, J. H., & Andrews, D. W. K. (2005). Testing for weak instruments in linear IV regression. Identification and inference for econometric models: Essays in honor of Thomas J. Rothenberg, Cambridge: Cambridge University Press. Google Scholar
Thurstone, L. (1925). A method of scaling psychological and educational tests. Journal of Educational Psychology, 16 (7), 433– 451. CrossRefGoogle Scholar
Yang-Wallentin, F., Jöreskog, K. G., & Luo, H. (2010). Confirmatory factor analysis of ordinal variables with misspecified models. Structural Equation Modeling, 17 (3), 392– 423. CrossRefGoogle Scholar
Yuan, K. -H., & Bentler, P. M. (2000). Three likelihood-based methods for mean and covariance structure analysis with nonnormal missing data. Sociological Methodology, 30 165– 200. CrossRefGoogle Scholar