Multilevel Modeling with Correlated Effects
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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.
Keywordsgeneralized method of moments omitted variable bias model specification test robust estimation consistency empirical standard errors hierarchical linear models
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- Anderson, G.E., Jimerson, S.R., & Whipple, A.D. (2002). Grade retention: Achievement and mental health outcomes. National Association of School Psychologists. Google Scholar
- Diggle, P.J., Heagarty, P., Liang, K.-Y., & Zeger, S.L. (2002). Analysis of longitudinal data (2nd ed.). London: Oxford University Press. Google Scholar
- Fielding, A. (2004). The role of the Hausman test and whether higher level effects should be treated as random or fixed. Multilevel Modelling Newsletter, 16, 3–9. Google Scholar
- Frees, E.W. (2004). Longitudinal and panel data: Analysis and applications in the social sciences. Cambridge: Cambridge University Press. Google Scholar
- Gelman, A., Carlin, J.B., Stern, H.S., & Rubin, D.B. (2004). Bayesian data analysis. London: Chapman & Hall. Google Scholar
- Goldstein, H. (2003). Multilevel statistical models (3rd ed.). London: Oxford University Press. Google Scholar
- Hayashi, F. (2000). Econometrics. Princeton: Princeton University Press. Google Scholar
- Kim, J.-S., & Frees, E.W. (2005). Fixed effects estimation in multilevel models. University of Wisconsin working paper. Available at http://research.bus.wisc.edu/jfrees/.
- Little, R.J., & Rubin, D.B. (1987). Statistical analysis with missing data. New York: Wiley. Google Scholar
- Ludwig, J., & Bassi, L.J. (1999). The puzzling case of school resources and student achievement. Educational Evaluation and Policy Analysis, 21, 385–403. Google Scholar
- Rao, C.R. (1973). Linear statistical inference and its applications (2nd ed.). New York: Wiley. Google Scholar
- Raudenbush, S.W., & Bryk, A.S. (2002). Hierarchical linear models: Applications and data analysis methods (2nd ed.). Newbury Park: Sage. Google Scholar
- Rice, N., Jones, A., & Goldstein, H. (1997). Multilevel models where the random effects are correlated with the fixed predictors: A conditioned iterative generalised least squares estimator (CIGLS) (Centre for Health Economics Technical Paper 10). York: University of York. Google Scholar
- Skrondal, A., & Rabe-Hesketh, S. (2004). Generalized latent variable modeling: Multilevel, longitudinal and structural equation models. Boca Raton: Chapman & Hall/CRC. Google Scholar
- Snijders, T.A.B., & Berkhof, J. (2007). Diagnostic checks for multilevel models. In J. de Leeuw & I. Kreft (Eds.), Handbook of multilevel analysis (in press). New York: Springer. Google Scholar
- Snijders, T.A.B., & Bosker, R.J. (1999). Multilevel analysis: An introduction to basic and advanced multilevel modeling. London: Sage. Google Scholar
- Steele, F. (2003). Selection effects of source of contraceptive supply in an analysis of contraceptive discontinuation: Multilevel modelling when random effects are correlated with an explanatory variable. Journal of the Royal Statistical Society, Series A, 166, 407–424. Google Scholar
- Wooldridge, J.M. (2002). Econometric analysis of cross section and panel data. Cambridge: MIT Press. Google Scholar