Bayesian estimation of panel data fractional response models with endogeneity: an application to standardized test rates

This paper develops new Bayesian estimation procedures for a nonlinear panel data model with a fractional dependent variable which is bounded between zero and one. Specifically, estimation is outlined for a panel data fractional probit model (PFP) and is presented in both a baseline case which assumes strict exogeneity as well as in a more complex simultaneous equation model which allows for potential endogeneity. We also present a new data augmentation technique which allows us to transform the likelihood function of a binary probit model into one which allows for a fractional response. In contrast to Frequentist estimation of the PFP models, Bayesian methods allow for direct evaluation of the joint likelihood of the fractional response, thereby leading to gains in estimation efficiency. Furthermore, transitioning from the strictly exogenous case to the case of endogeneity only requires slight adjustments and this allows for straightforward model specification tests. Combining data from the Florida Department of Education, National Center of Education Statistics, and the U.S. Census Bureau, the model is applied empirically in order to examine the relationship between school spending and student achievement among Florida elementary schools. The study finds that greater spending has a positive and statistically significant effect on student achievement. Though, this effect is roughly 50 % larger in the model which allows for endogenous spending. Specifically, a $1,000 increase in per-pupil spending is associated with an increase in standardized test pass rates ranging from 6.2 to 10.1 %.