Essays in the Econometric Theory of Panel and Multidimensional Data

This dissertation studies econometric models in the presence of unobserved heterogeneity when data is observed over multiple dimensions. Chapter~2 and 3 study this in the classic panel setting with two dimensions, which are usually individuals and time. Chapter~2 studies the setting where unobserved heterogeneity may enter non-linearly and nonseparably to the observed covariates. Established matrix completion methods and a group fixed-effect type estimator prove to approximate the model well. Chapter~3 studies the setting where unobserved heterogeneity enters linearly and separably, but is modelled as a generic functional transformation of unobserved characteristics. The factor model estimated with many factors approximates this form of unobserved heterogeneity well, and, like in Chapter~2, a group fixed-effects estimator also performs well in theory and in simulations. Chapter~4 studies this setting when three or more dimensions are observed in the data and restricts focus to the linear regression model. This chapter extends the notion of the group fixed-effects estimator to a nonparametric kernel style transformation that can be applied to any number of dimensions. The results in this chapter show that the current state-of-the-art factor model methods to approximate unobserved heterogeneity do not extend well to the setting with three or more dimensions. The results also show that the novel nonparametric kernel transformation proposed in this chapter control for unobserved heterogeneity sufficiently well to achieve the parametric rate of consistency under certain conditions