A varying-coefficient approach to estimating multi-level clustered data models

Most of the literature on clustered data models emphasizes two-level clustering, and within-cluster correlation. While multi-level clustered data models can arise in practice, analysis of multi-level clustered data models poses additional difficulties owing to the existence of error correlations both within and across the clusters. It is perhaps for this reason that existing approaches to multi-level clustered data models have been mostly parametric. The purpose of this paper is to develop a varying-coefficient nonparametric approach to the analysis of three-level clustered data models. Because the nonparametric functions are restricted only to some of the variables, this approach has the appeal of avoiding many of the curse of dimensionality problems commonly associated with other nonparametric methods. By applying an undersmoothing technique, taking into account the correlations within and across clusters, we develop an efficient two-stage local polynomial estimation procedure for the unknown coefficient functions. The large and finite sample properties of the resultant estimators are examined; in particular, we show that the resultant estimators are asymptotically normal, and exhibit considerably smaller asymptotic variability than the traditional local polynomial estimators that neglect the correlations within and among clusters. An application example is presented based on a data set extracted from the World Bank’s STARS database.