Existence and Uniqueness of Semiparametric Projections

In this paper we propose primitive conditions under which a projec- tion of a conditional density onto a set dened by conditional moment restric- tions exists and is unique. Moreover, we provide an analytic expression of the obtained projection. Our rst result is to show the existence when the moment function is bounded. The result is as we would expect from the analogous results obtained in the unconditional case. Our second result relaxes the boundedness assumption and replaces it with a correct specication condition. Showing that the correct specication of the moment function is sucient for the projection to exist is entirely new and not yet seen in the literature.