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We estimate the impact of extra health insurance coverage beyond a National Health System on the demand for several health services. Traditionally, the literature has tried to deal with the endogeneity of the private (extra) insurance decision by finding instrumental variables. Since a priori instrumental variables are hard to find we take a different approach. We focus on the most common health insurance plan in Portugal, ADSE, which is given to all civil servants and their dependents. We argue that this insurance is exogenous, i.e., not correlated with the beneficiaries' health status. This identifying assumption allows us to estimate the impact of having ADSE coverage on the demand for three different health services using a matching estimator technique. The health services used are number of visits, number of blood and urine tests, and the probability of visiting a dentist. Results show large positive effects of ADSE coverage for number of visits and tests among the young (18&-30 years old) but only the latter is statistically significantly different from zero. The effects represent 21.8% and 30% of the average number of visits and tests for the young. On the contrary, we find no evidence of moral hazard on the probability of visiting a dentist.