We provide a tractable characterization of the sharp identification region of the parameter vector θ in a broad class of incomplete econometric models. Models in this class have set-valued predictions that yield a convex set of conditional or unconditional moments for the observable model variables. In short, we call these models with convex moment predictions. Examples include static, simultaneous-move finite games of complete and incomplete information in the presence of multiple equilibria; best linear predictors with interval outcome and covariate data; and random utility models of multinomial choice in the presence of interval regressors data. Given a candidate value for θ, we establish that the convex set of moments yielded by the model predictions can be represented as the Aumann expectation of a properly defined random set. The sharp identification region of θ, denoted ΘI, can then be obtained as the set of minimizers of the distance from a properly specified vector of moments of random variables to this Aumann expectation. Algorithms in convex programming can be exploited to efficiently verify whether a candidate θ is in ΘI. We use examples analyzed in the literature to illustrate the gains in identification and computational tractability afforded by our method.