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Posterior inference in curved exponential families under increasing dimensions

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  • Alexandre Belloni
  • Victor Chernozhukov

Abstract

In this paper, we study the large‐sample properties of the posterior‐based inference in the curved exponential family under increasing dimensions. The curved structure arises from the imposition of various restrictions on the model, such as moment restrictions, and plays a fundamental role in econometrics and others branches of data analysis. We establish conditions under which the posterior distribution is approximately normal, which in turn implies various good properties of estimation and inference procedures based on the posterior. In the process, we also revisit and improve upon previous results for the exponential family under increasing dimensions by making use of concentration of measure. We also discuss a variety of applications to high‐dimensional versions of classical econometric models, including the multinomial model with moment restrictions, seemingly unrelated regression equations, and single structural equation models. In our analysis, both the parameter dimensions and the number of moments are increasing with the sample size.

Suggested Citation

  • Alexandre Belloni & Victor Chernozhukov, 2014. "Posterior inference in curved exponential families under increasing dimensions," Econometrics Journal, Royal Economic Society, vol. 17(2), pages 75-100, June.
    • Handle: RePEc:wly:emjrnl:v:17:y:2014:i:2:p:s75-s100
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    File URL: http://hdl.handle.net/10.1111/ectj.12027
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    1. van Garderen, Kees Jan, 1997. "Curved Exponential Models in Econometrics," Econometric Theory, Cambridge University Press, vol. 13(6), pages 771-790, December.
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    4. Ghosal, Subhashis, 2000. "Asymptotic Normality of Posterior Distributions for Exponential Families when the Number of Parameters Tends to Infinity," Journal of Multivariate Analysis, Elsevier, vol. 74(1), pages 49-68, July.
    5. Heckman, James J, 1974. "Shadow Prices, Market Wages, and Labor Supply," Econometrica, Econometric Society, vol. 42(4), pages 679-694, July.
    6. Guido W. Imbens, 1997. "One-Step Estimators for Over-Identified Generalized Method of Moments Models," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 64(3), pages 359-383.
    7. Hansen, Lars Peter & Singleton, Kenneth J, 1982. "Generalized Instrumental Variables Estimation of Nonlinear Rational Expectations Models," Econometrica, Econometric Society, vol. 50(5), pages 1269-1286, September.
    8. Donald, Stephen G. & Imbens, Guido W. & Newey, Whitney K., 2003. "Empirical likelihood estimation and consistent tests with conditional moment restrictions," Journal of Econometrics, Elsevier, vol. 117(1), pages 55-93, November.
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    Cited by:

    1. Gallant, A. Ronald & Hong, Han & Leung, Michael P. & Li, Jessie, 2022. "Constrained estimation using penalization and MCMC," Journal of Econometrics, Elsevier, vol. 228(1), pages 85-106.

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