This dissertation consists of three chapters concerning the estimation and inference of multi-level models. The first chapter shows how various econometric methods can be extended to multi-level linear models. The second chapter is a natural extension of the first chapter which focuses on the estimation of nonlinear models with hierarchical structure. The third chapter provides a practical guidance on choosing between the one-step GEE estimator and two-step GEE estimator by conducting simulation studies.More specifically, the first chapter considers a three-level linear model with both nested and non-nested structures. The stylized example will be a model for the average score on some statewide test at the school level at various points in time; scores are observed at multiple points in time (level one) in multiple schools (level two) in various school districts (level three). In such a model, the schools are naturally nested within the districts but time is not nested within schools or districts. This chapter extends several important econometric methods of analysis for panel data models to multi-level models. In particular, this chapter generalizes the results of Hausman and Taylor and the subsequent literature to these multi-level models. This is a non-trivial extension because the three-level linear model now has more than one kind of heterogeneity and more than one kind of between regression. This chapter also generalizes the concept of testing the exogeneity assumptions by a simple variable addition test. It also provides a discussion about the GMM estimation without the assumption of no conditional heteroskedasticity. The second chapter considers the estimation of hierarchical nonlinear models. It shows how to extend the generalized estimating equations approach to models with more than two levels. When some covariates are correlated the unobserved heterogeneity, the GEE estimation method can be combined with a modified Chamberlain-Mundlak device to account for the unbalanced data structure. Another important contribution is that this chapter provides a systematic discussion about the misspecification issue of the working correlation matrix. When the working correlation matrix is misspecified, some structured working correlation matrices may not have a well-defined limit or their limiting matrix is not positive semi-definite, which will cause identification issue of the GEE estimator. When the conditional mean function associated with heterogeneity has a multiplicative form, a more efficient estimator is available, which extends the multivariate weighted nonlinear least squares estimation to multi-level models.The third chapter investigates the small sample performances of the one-step GEE estimator and two-step GEE estimator, which is similar to the comparison between the continuously updating GMM estimator and two-step GMM estimator. The simulation study shows that the these two estimators have similar finite sample performances when the efficient working correlation matrix is used. When the inefficient working correlation matrix is adopted, the two-step GEE estimator usually has a better finite sample performance in terms of the mean squared error. The simulation also shows the efficiency gains by relaxing the traditional generalized linear model (GLM) variance assumption. By allowing time-varying dispersion parameters, the efficiency of the GEE estimator can be further enhanced.
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