Asymptotic Effect of Misspecification in the Random Part of the Multilevel Model

The authors examine the asymptotic effect of omitting a random coefficient in the multilevel model and derive expressions for the change in (a) the variance components estimator and (b) the estimated variance of the fixed effects estimator. They apply the method of moments, which yields a closed form expression for the omission effect. In practice, the model parameters are estimated by maximum likelihood; however, since the moment estimator and the maximum likelihood estimator are both consistent, the presented expression for the change in the variance components estimator asymptotically holds for the maximum likelihood estimator as well. The results are illustrated with an analysis of mathematics performance data.