The generalized method of moments (GMM) estimation technique is discussed for count data models with endogenous regressors. Count data models can be specified with additive or multiplicative errors. It is shown that, in general, a set of instruments is not orthogonal to both error types. Simultaneous equations with a dependent count variable often do not have a reduced form which is a simple function of the instruments. However, a simultaneous model with a count and a binary variable can only be logically consistent when the system is triangular. The GMM estimator is used in the estimation of a model explaining the number of visits to doctors, with as a possible endogenous regressor a self-reported binary health index. Further, a model is estimated, in stages, that includes latent health instead of the binary health index.