This paper contributes in three dimensions to the literature on health care demand. First, it features the first application of a bivariate random effects estimator in a count data setting, to permit the efficient estimation of this type of model with panel data. Second, it provides an innovative test of adverse selection and confirms that high-risk individuals are more likely to acquire supplemental add-on insurance. Third, the estimations yield that in accordance with the theory of moral hazard, we observe a much lower frequency of doctor visits among the self-employed, and among mothers of small children.