Bayesian Estimation of an Endogenous Bivariate Semiparametric Probit Model for Health Practitioner Utilisation in Australia

This paper presents Bayesian methodology for the estimation of a bivariate probit model with an endogenous effect and both parametric linear and flexible semiparametric exogenous effects. The model is prompted by an analysis of the utilisation of health services in Australia using data from the Australian National Health Survey. The semiparametric effects are modeled using an adaption of the recently developed approach of Lang and Brezger (2001), which these authors demonstrate is highly effective for smoothing in a generalised linear model framework. We extend their work to allow for the calculation of the probability of each effect being null, linear or strictly nonlinear. Variable selection from the linear exogenous variables is also undertaken using an approach similar to that suggested by Shively and Kohn (1999). It explores efficiently the large space of all possible permutations of the linear variables and provides an automatic means of robust identification of the linear coefficients. The entire model is estimated using a carefully constructed Markov chain Monte Carlo sampling scheme, which generates the endogenous coefficient and error correlation as a block. Analysis of the data suggests that need based variables do indeed drive utilisation of health services, while the uptake of private health insurance is driven by enabling variables. Strong nonlinear patterns are uncovered in the key exogenous variables, justifying the semiparametric analysis. A simulation from the design of the data provides reassurance of the reliability of the proce