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Published online by Cambridge University Press:  13 May 2015

Ivana Komunjer*
University of California
Giuseppe Ragusa
Luiss University
*Address correspondence to Ivana Komunjer, Department of Economics, University of California, San Diego, 9500 Gilman Drive MC 0508, La Jolla, CA 92093-0508; e-mail:
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In this paper we propose primitive conditions under which a projection of a conditional density onto a set defined by conditional moment restrictions exists and is unique. Moreover, we provide an analytic expression of the obtained projection. The range of applications where conditional density projections are used is wide. The derived results are potentially useful in a variety of areas including: semiparametric efficient estimation and optimal testing in (conditional) moment models, Bayesian prior determination and inference in semiparametric models, density forecasting, and simulation-based econometric analysis.

Regarding existence, we propose three different combinations of assumptions that are all sufficient to show that the projection exists and is unique. The proposed conditions exhibit a clear trade off between restrictions put on the divergence between the conditional densities and on the moment function which defines the projection set. Depending on the nature of the application, the researcher can pick and choose which set of conditions to use. Our second set of results characterizes the projection. The expression for the projected density is new though not surprising given the previously obtained results for the unconditional case. The projection is characterized by the dual of the original projection problem. In establishing the strong duality, however, we work with a constraint qualification condition that is weaker than that used by Borwein and Lewis (1991a, 1992a, 1993 in their seminal work concerning the unconditional case.

Copyright © Cambridge University Press 2015 


We would like to thank the Co-Editor, Yuichi Kitamura, and two anonymous referees for their excellent comments and suggestions. We also thank the seminar participants at Rice University, Iowa State University, University of British Columbia, University of Pennsylvania, USC, University of Texas Austin, Rochester, and Joint Montréal Econometrics seminar for their feedback. Previous versions of this paper were circulated under the title “Existence and Uniqueness of Semiparametric Projections.



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