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Multilevel Modeling with Correlated Effects

Published online by Cambridge University Press:  01 January 2025

Jee-Seon Kim*
Affiliation:
University of Wisconsin-Madison
Edward W. Frees
Affiliation:
University of Wisconsin-Madison
*
Requests for reprints should be sent to Jee-Seon Kim, Department of Educational Psychology, University of Wisconsin at Madison, 1025 Johnson Street, Madison, WI 53706, USA. E-mail: jeeseonkim@wisc.edu

Abstract

When there exist omitted effects, measurement error, and/or simultaneity in multilevel models, explanatory variables may be correlated with random components, and standard estimation methods do not provide consistent estimates of model parameters. This paper introduces estimators that are consistent under such conditions. By employing generalized method of moments (GMM) estimation techniques in multilevel modeling, the authors present a series of estimators along a robust to efficient continuum. This continuum depends on the assumptions that the analyst makes regarding the extent of the correlated effects. It is shown that the GMM approach provides an overarching framework that encompasses well-known estimators such as fixed and random effects estimators and also provides more options. These GMM estimators can be expressed as instrumental variable (IV) estimators which enhances their interpretability. Moreover, by exploiting the hierarchical structure of the data, the current technique does not require additional variables unlike traditional IV methods. Further, statistical tests are developed to compare the different estimators. A simulation study examines the finite sample properties of the estimators and tests and confirms the theoretical order of the estimators with respect to their robustness and efficiency. It further shows that not only are regression coefficients biased, but variance components may be severely underestimated in the presence of correlated effects. Empirical standard errors are employed as they are less sensitive to correlated effects when compared to model-based standard errors. An example using student achievement data shows that GMM estimators can be effectively used in a search for the most efficient among unbiased estimators.

Type
Theory and Methods
Copyright
Copyright © 2007 The Psychometric Society

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Footnotes

This research was supported by the National Academy of Education/Spencer Foundation and the National Science Foundation, grant number SES-0436274. We thank the editor, associate editor, and referees for detailed feedback that helped improve the paper.

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