Effect Specification, Identification, Estimation, and Inference in a Fractional Outcome Regression Model with an Endogenous Causal Variable

Abstract

Empirical economic research is primarily driven by the desire to offer scientific evidence that helps assess policy relevant cause-and-effect. The approach most often applied in pursuit of this objective involves regression modeling and estimation. In this dissertation, we focus on the specification, identification, estimation, and causal inference of a causal effect (CE) in the context of the fractional regression model (FRM) for which the support of the outcome variable of interest is restricted to the unit interval. Empirical applications of such models abound in health economics, health services research and health policy literatures. Examples from other disciplines include labor economics, development economics, political economics, commerce or finance. Various full information maximum likelihood and quasi-maximum likelihood regression estimators and nonlinear least squares approach have been proposed to account for the inherent nonlinearity in the FRM due to the unit interval support restriction (UISR) on the outcome variable. Additional nonlinearity is induced in the FRM when the presumed causal variable is subject to unobservable confounding (UC) [i.e., when the presumed causal variable is endogenous]. In such cases, the additional analytic and implementation effort required to account for both sources of nonlinearity (fractional outcome and UC) while avoiding UC bias (which precludes causal interpretability) can be daunting. We seek to develop and implement regression model specifications that account for the inherent nonlinearity implied by this restriction, as well as the nonlinearity that could be additionally imposed by the endogeneity of the presumed causal variable. We focus on the case where the presumed causal variable is continuous. We develop new models for FRM-based CE estimation that implement two-stage residual inclusion (2SRI) methods, as suggested by Terza et al. (2008). We assess the accuracy of our proposed new methods and compare them with extant 2SRI approaches using simulation study. An empirical application demonstrates the working of our proposed method.