Models with Endogenous Regressors

This chapter examines various estimation and testing issues concerning models with endogenous regressors. The complexity of these issues increases as the number of potential unobserved heterogeneities increases with the dimension of the data. The chapter examines the properties of least squares type estimators, including theWithin estimator, under different specifications of the error components and different correlation assumptions with the regressors. The latter induces different types of endogeneity not studied previously. In terms of estimation, the chapter includes an extension to the well-known Hausman-Taylor estimator for models with multiple dimensions as well as method to accommodate cross sectional dependence by incorporating the common correlated effects estimator into the Hausman Talyor procedure. It also proposes a set of valid orthogonality conditions for purposes of implementing Generalised Method of Moments (GMM) estimators under these different specifications and endogeneity assumptions. A brief discussion on the selection of valid moments using machine learning techniques will also be presented. The theoretical results in this chapter identify consistent and efficient estimators for different specifications. These results allow an extension of the Hausman specification test to detect endogeneity in multi-dimensional panel data models. Other issues, such as mixed effects models, self-flow, incomplete data and higher dimensional models, will also be discussed.