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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 and 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. Utilizing data from the British Health and Lifestyle Survey 1991-1992, the GMM estimator is used in the estimation of a model explaining the number of visits to doctors, with a self-reported binary health index as a possible endogenous regressor. If this regressor is truly endogenous, one expects the pseudo-likelihood estimate of its coefficient to be biased upwards. Indeed, for the additive model, the estimated coefficient of the binary health index decreases in value when the possible endogeneity of this regressor is taken into account. Further indication of endogeneity is given by the fact that the overidentifying restrictions are rejected in the multiplicative model, but not in the additive model. Finally, a model that includes predicted latent health instead of the binary health index is estimated in stages.