This paper considers the estimation and testing of demand systems when the number of sample goods is smaller than the number of commodity choices available to consumers. In this case, the demand system is incomplete. The large majority of papers that appeared in the literature specifies and estimates a demand system in share format even when the system may be incomplete. The criterion for deciding whether a share format is admissible without loss of information is a test of the adding-up condition. This test, however, requires the estimation of a demand system in quantity format.