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 de#ned 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 speci#cation condition. Showing that the correct speci#cation of the moment function is su#cient for the projection to exist is entirely new and not yet seen in the literature.