Hostname: page-component-669899f699-b58lm Total loading time: 0 Render date: 2025-04-26T05:41:34.576Z Has data issue: false hasContentIssue false
  • English
  • Français

Model-Implied Instrumental Variable—Generalized Method of Moments (MIIV-GMM) Estimators for Latent Variable Models

Published online by Cambridge University Press:  01 January 2025

Kenneth A. Bollen ,
Stanislav Kolenikov  and
Shawn Bauldry
Kenneth A. Bollen*
Affiliation:
Department of Sociology, University of North Carolina at Chapel Hill
Stanislav Kolenikov
Affiliation:
Abt SRBI
Shawn Bauldry
Affiliation:
Department of Sociology, University of Alabama at Birmingham
*
Requests for reprints should be sent to Kenneth A. Bollen, Department of Sociology, University of North Carolina at Chapel Hill, CB 3210 Hamilton, Chapel Hill, NC 27599-3210, USA. E-mail: bollen@unc.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

The common maximum likelihood (ML) estimator for structural equation models (SEMs) has optimal asymptotic properties under ideal conditions (e.g., correct structure, no excess kurtosis, etc.) that are rarely met in practice. This paper proposes model-implied instrumental variable – generalized method of moments (MIIV-GMM) estimators for latent variable SEMs that are more robust than ML to violations of both the model structure and distributional assumptions. Under less demanding assumptions, the MIIV-GMM estimators are consistent, asymptotically unbiased, asymptotically normal, and have an asymptotic covariance matrix. They are “distribution-free,” robust to heteroscedasticity, and have overidentification goodness-of-fit J-tests with asymptotic chi-square distributions. In addition, MIIV-GMM estimators are “scalable” in that they can estimate and test the full model or any subset of equations, and hence allow better pinpointing of those parts of the model that fit and do not fit the data. An empirical example illustrates MIIV-GMM estimators. Two simulation studies explore their finite sample properties and find that they perform well across a range of sample sizes.

Keywords

structural equation modelslatent variablesgeneralized method of momentsinstrumental variablesfactor analysis
Type
Original Paper
Information
Psychometrika , Volume 79 , Issue 1 , January 2014 , pp. 20 - 50
Copyright
Copyright © 2013 The Psychometric Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Anderson, J.C., Gerbing, D. (1984). The effect of sampling error on convergence, improper solutions, and goodness-of-fit indices for maximum likelihood confirmatory factor analysis. Psychometrika, 49, 155–173CrossRefGoogle Scholar
Anderson, T.W., Amemiya, Y. (1988). The asymptotic normal distribution of estimators in factor analysis under general conditions. The Annals of Statistics, 16, 759–771CrossRefGoogle Scholar
Angrist, J.D., Pischke, J. (2009). Mostly harmless econometrics: an empiricist’s companion, Princeton: Princeton University PressCrossRefGoogle Scholar
Bauldry, S., (forthcoming). miivfind: a program for identifying model-implied instrumental variables (MIIVs) for structural equation models in Stata. Stata Journal.Google Scholar
Bentler, P.M. (1982). Confirmatory factor analysis via noniterative estimation: a fast, inexpensive method. Journal of Marketing Research, 19, 417–424CrossRefGoogle Scholar
Bentler, P.M., Yuan, K. (1999). Structural equation modeling with small samples: test statistics. Multivariate Behavioral Research, 34, 181–197CrossRefGoogle ScholarPubMed
Bollen, K.A. (1989). Structural equations with latent variables, New York: WileyCrossRefGoogle Scholar
Bollen, K.A. (1996). An alternative Two Stage Least Squares (2SLS) estimator for latent variable equations. Psychometrika, 61, 109–121CrossRefGoogle Scholar
Bollen, K.A. (1996). A limited information estimator for LISREL models with and without heteroscedasticity. In Marcoulides, G.A., Schumacker, R.E. (Eds.), Advanced structural equation modeling, Mahwah: Erlbaum 227241Google Scholar
Bollen, K.A. (2001). Two-stage least squares and latent variable models: simultaneous estimation and robustness to misspecifications. In Cudeck, R., Toit, S.D., Sörbom, D. (Eds.), Structural equation modeling: present and future, a festschrift in honor of Karl Jöreskog, Lincolnwood: Scientific Software International 119138Google Scholar
Bollen, K.A. (2012). Instrumental variables in sociology and the social sciences. Annual Review of Sociology, 38, 37–72CrossRefGoogle Scholar
Bollen, K.A., Bauer, D.J. (2004). Automating the selection of model-implied instrumental variables. Sociological Methods & Research, 32, 425–452CrossRefGoogle Scholar
Bollen, K.A., Kirby, J.B., Curran, P.J., Paxton, P.M., Chen, F. (2007). Latent variable models under misspecification: two-stage least squares (2SLS) and maximum likelihood (ML) estimators. Sociological Methods & Research, 36, 48–86CrossRefGoogle Scholar
Bollen, K.A., Pearl, J. (2013). Eight myths about causality and structural equation models. In Morgan, S. (Eds.), Handbook of causal analysis for social research, New York: SpringerGoogle Scholar
Bollen, K.A., Stine, R. (1990). Direct and indirect effects: classical and bootstrap estimates of variability. Sociological Methodology, 20, 115–140CrossRefGoogle Scholar
Bollen, K.A., Stine, R. (1992). Bootstrapping goodness-of-fit measures in structural equation models. Sociological Methods & Research, 21, 205–229CrossRefGoogle Scholar
Boomsma, A., Hoogland, J.J. (2001). The robustness of LISREL modeling revisited. In Cudeck, R., Toit, S.D., Sörbom, D. (Eds.), Structural equation modeling: present and future, a festschrift in honor of Karl Jöreskog, Lincolnwood: Scientific Software International 139168Google Scholar
Browne, M.W. (1984). Asymptotically distribution-free methods for the analysis of the covariance structures. British Journal of Mathematical & Statistical Psychology, 37, 62–83CrossRefGoogle ScholarPubMed
Browne, M.W., Cudeck, R. (1993). Alternative ways of assessing model fit. In Bollen, K.A., Long, J.S. (Eds.), Testing structural equation models, Newbury Park: Sage 136162Google Scholar
Chausse, P., (2012). gmm: generalized method of moments and generalized empirical likelihood (R package). http://cran.r-project.org/web/packages/gmm/index.html.Google Scholar
Cragg, J.G. (1968). Some effects of incorrect specification on the small sample properties of several simultaneous equation estimators. International Economic Review, 9, 63–86CrossRefGoogle Scholar
Davidson, R., MacKinnon, J.G. (1993). Estimation and inference in econometrics, New York: Oxford University PressGoogle Scholar
Foster, E.M. (1997). Instrumental variables for logistic regression: an illustration. Social Science Research, 26, 487–504CrossRefGoogle Scholar
Glanville, J.L., Paxton, P. (2007). How do we learn to trust? A confirmatory tetrad analysis of the sources of generalized trust. Social Psychology Quarterly, 70, 230–242CrossRefGoogle Scholar
Godambe, V.P., Thompson, M. (1978). Some aspects of the theory of estimating equations. Journal of Statistical Planning and Inference, 2, 95–104CrossRefGoogle Scholar
Hall, A.R. (2005). Generalized method of moments, Oxford: Oxford University PressGoogle Scholar
Hägglund, G. (1982). Factor analysis by instrumental variables. Psychometrika, 47, 209–222CrossRefGoogle Scholar
Hansen, L.P. (1982). Large sample properties of generalized method of moments estimators. Econometrica, 50, 1029–1054CrossRefGoogle Scholar
Hu, L.T., Bentler, P.M., Kano, Y. (1992). Can test statistics in covariance structure analysis be trusted?. Psychological Bulletin, 112, 351–362CrossRefGoogle ScholarPubMed
Ihara, M., Kano, Y. (1986). A new estimator of the uniqueness in factor analysis. Psychometrika, 51, 563–566CrossRefGoogle Scholar
Jöreskog, K.G. (1969). A general approach to confirmatory maximum likelihood factor analysis. Psychometrika, 34, 183–202CrossRefGoogle Scholar
Jöreskog, K.G. (1973). A general method for estimating a linear structural equation system. In Goldberger, A.S., Duncan, O.D. (Eds.), Structural equation models in the social sciences, New York: Academic Press 85112Google Scholar
Jöreskog, K.G. (1977). Structural equation models in the social sciences: specification, estimation, and testing. In Krishnaiah, P.R. (Eds.), Applications of statistics, Amsterdam: North-Holland 265287Google Scholar
Jöreskog, K.G. (1983). Factor analysis as an error-in-variables model. In Wainer, H., Messick, S. (Eds.), Principles of Modern Psychological Measurement, Hillsdale: Erlbaum 185196Google Scholar
Kirby, J.B., Bollen, K.A. (2009). Using instrumental variable tests to evaluated model specification in latent variable structural equation models. Sociological Methodology, 39, 327–355CrossRefGoogle ScholarPubMed
Kolenikov, S. (2011). Biases of parameter estimates in misspecified structural equation models. Sociological Methodology, 41, 119–157CrossRefGoogle Scholar
Kolenikov, S., Bollen, K.A. (2012). Testing negative error variances: is a Heywood case a symptom of misspecification?. Sociological Methods & Research, 41, 124–167CrossRefGoogle Scholar
Lawley, D.N. (1940). The estimation of factor loadings by the method of maximum likelihood. Proceedings of the Royal Society of Edinburgh, 60, 64–82CrossRefGoogle Scholar
Madansky, A. (1964). Instrumental variables in factor analysis. Psychometrika, 29, 105–113CrossRefGoogle Scholar
Mardia, K.V. (1970). Measures of multivariate skewness and kurtosis with applications. Biometrika, 57, 519–530CrossRefGoogle Scholar
Mátyás, L. (1999). Generalized method of moments estimation, Cambridge: Cambridge University PressCrossRefGoogle Scholar
Micceri, T. (1989). The unicorn, the normal curve, and other improbable creatures. Psychological Bulletin, 1, 156–166CrossRefGoogle Scholar
Muthén, L.K., Muthén, B. (1998–2010). Mplus user’s guide, Los Angeles: Muthén & MuthénGoogle Scholar
Nevitt, J., Hancock, G.R. (2004). Evaluating small sample approaches for model test statistics in structural equation modeling. Multivariate Behavioral Research, 39, 439–478CrossRefGoogle Scholar
Newey, W.K., McFadden, D. (1986). Large sample estimation and hypothesis testing. In Engle, R.F., McFadden, D. (Eds.), Handbook of Econometrics, (1st ed.). Amsterdam: Elsevier 21112245Google Scholar
Paxton, P.M., Curran, P., Bollen, K.A., Kirby, J., Chen, F. (2001). Monte Carlo simulations in structural equation models. Structural Equation Modeling, 8, 287–312CrossRefGoogle Scholar
Pew Research Center (1998). Trust and citizen engagement in metropolitan Philadelphia: a case study, Washington: The Pew Research Center for the People and the PressGoogle Scholar
Sargan, J.D. (1958). The estimation of economic relationships using instrumental variables. Econometrica, 26, 393–415CrossRefGoogle Scholar
Satorra, A. (1990). Robustness issues in structural equation modeling: a review of recent developments. Quality and Quantity, 24, 367–386CrossRefGoogle Scholar
Satorra, A., Bentler, P.M. (1994). Corrections to test statistics and standard errors in covariance structure analysis. In von Eye, A., Clogg, C.C. (Eds.), Latent variable analysis, Thousand Oaks: Sage 399419Google Scholar
Searle, S.R. (1982). Matrix algebra useful for statistics, (1st ed.). New York: WileyGoogle Scholar
Skrondal, A., Hesketh, S.R. (2004). Generalized latent variable modeling, Boca Raton: Chapman & Hall/CRCCrossRefGoogle Scholar
Staiger, D., Stock, J.H. (1997). Instrumental variables regression with weak instruments. Econometrica, 65, 557–586CrossRefGoogle Scholar
StataCorp (2011). Stata statistical software: release 12, College Station: StataCorpGoogle Scholar
Stock, J.H., Yogo, M. (2005). Testing for weak instruments in linear IV regression. In Andrews, D.W.K. (Eds.), Identification and Inference for Econometric Models, New York: Cambridge University Press 80108CrossRefGoogle Scholar
Stock, J.H., Wright, J.H., Yogo, M. (2002). A survey of weak instruments and weak identification in generalized method of moments. Journal of Business & Economic Statistics, 20, 518–529CrossRefGoogle Scholar
van der Vaart, A.W. (1998). Asymptotic statistics, New York: WileyCrossRefGoogle Scholar
Wooldrige, J.M. (2010). Econometric analysis of cross section and panel data, Cambridge: MIT PressGoogle Scholar
Yuan, K., Hayashi, K. (2006). Standard errors in covariance structure models: asymptotic versus bootstrap. British Journal of Mathematical & Statistical Psychology, 59, 397–417CrossRefGoogle ScholarPubMed