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Existence And Characterization Of Conditional Density Projections

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  • Komunjer, Ivana
  • Ragusa, Giuseppe

Abstract

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.

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  • Komunjer, Ivana & Ragusa, Giuseppe, 2016. "Existence And Characterization Of Conditional Density Projections," Econometric Theory, Cambridge University Press, vol. 32(4), pages 947-987, August.
    • Handle: RePEc:cup:etheor:v:32:y:2016:i:04:p:947-987_00
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    2. Timothy Christensen & Benjamin Connault, 2019. "Counterfactual Sensitivity and Robustness," Papers 1904.00989, arXiv.org, revised May 2022.
    3. Qiu, Chen & Otsu, Taisuke, 2022. "Information theoretic approach to high dimensional multiplicative models: stochastic discount factor and treatment effect," LSE Research Online Documents on Economics 110494, London School of Economics and Political Science, LSE Library.
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    5. Andreas Tryphonides, 2018. "Tilting Approximate Models," Papers 1805.10869, arXiv.org, revised Mar 2024.
    6. Chen Qiu & Taisuke Otsu, 2022. "Information theoretic approach to high‐dimensional multiplicative models: Stochastic discount factor and treatment effect," Quantitative Economics, Econometric Society, vol. 13(1), pages 63-94, January.
    7. Christis Katsouris, 2023. "Optimal Estimation Methodologies for Panel Data Regression Models," Papers 2311.03471, arXiv.org, revised Nov 2023.
    8. Naoya Sueishi, 2022. "A Misuse of Specification Tests," Papers 2211.11915, arXiv.org.
    9. Tryphonides, Andreas, 2017. "Conditional moment restrictions and the role of density information in estimated structural models," SFB 649 Discussion Papers 2017-016, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.
    10. Timothy Christensen & Benjamin Connault, 2023. "Counterfactual Sensitivity and Robustness," Econometrica, Econometric Society, vol. 91(1), pages 263-298, January.

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