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Shape-Enforcing Operators for Point and Interval Estimators

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Listed:
  • Xi Chen
  • Victor Chernozhukov
  • Iv'an Fern'andez-Val
  • Scott Kostyshak
  • Ye Luo

Abstract

A common problem in econometrics, statistics, and machine learning is to estimate and make inference on functions that satisfy shape restrictions. For example, distribution functions are nondecreasing and range between zero and one, height growth charts are nondecreasing in age, and production functions are nondecreasing and quasi-concave in input quantities. We propose a method to enforce these restrictions ex post on point and interval estimates of the target function by applying functional operators. If an operator satisfies certain properties that we make precise, the shape-enforced point estimates are closer to the target function than the original point estimates and the shape-enforced interval estimates have greater coverage and shorter length than the original interval estimates. We show that these properties hold for six different operators that cover commonly used shape restrictions in practice: range, convexity, monotonicity, monotone convexity, quasi-convexity, and monotone quasi-convexity. We illustrate the results with two empirical applications to the estimation of a height growth chart for infants in India and a production function for chemical firms in China.

Suggested Citation

  • Xi Chen & Victor Chernozhukov & Iv'an Fern'andez-Val & Scott Kostyshak & Ye Luo, 2018. "Shape-Enforcing Operators for Point and Interval Estimators," Papers 1809.01038, arXiv.org, revised Feb 2021.
    • Handle: RePEc:arx:papers:1809.01038
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    References listed on IDEAS

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    1. Groeneboom,Piet & Jongbloed,Geurt, 2014. "Nonparametric Estimation under Shape Constraints," Cambridge Books, Cambridge University Press, number 9780521864015, November.
    2. Beare, Brendan K. & Schmidt, Lawrence, 2011. "An Empirical Test of Pricing Kernel Monotonicity," University of California at San Diego, Economics Working Paper Series qt5572n8pc, Department of Economics, UC San Diego.
    3. Belloni, Alexandre & Chernozhukov, Victor & Chetverikov, Denis & Fernández-Val, Iván, 2019. "Conditional quantile processes based on series or many regressors," Journal of Econometrics, Elsevier, vol. 213(1), pages 4-29.
    4. Brendan K. Beare & Lawrence D. W. Schmidt, 2016. "An Empirical Test of Pricing Kernel Monotonicity," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 31(2), pages 338-356, March.
    5. Beare, Brendan K. & Moon, Jong-Myun, 2015. "Nonparametric Tests Of Density Ratio Ordering," Econometric Theory, Cambridge University Press, vol. 31(3), pages 471-492, June.
    6. Denis Chetverikov & Andres Santos & Azeem M. Shaikh, 2018. "The Econometrics of Shape Restrictions," Annual Review of Economics, Annual Reviews, vol. 10(1), pages 31-63, August.
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    Cited by:

    1. Guido W. Imbens, 2020. "Potential Outcome and Directed Acyclic Graph Approaches to Causality: Relevance for Empirical Practice in Economics," Journal of Economic Literature, American Economic Association, vol. 58(4), pages 1129-1179, December.
    2. Harold D. Chiang & Kengo Kato & Yuya Sasaki & Takuya Ura, 2021. "Linear programming approach to nonparametric inference under shape restrictions: with an application to regression kink designs," Papers 2102.06586, arXiv.org.
    3. Zheng Fang & Juwon Seo, 2021. "A Projection Framework for Testing Shape Restrictions That Form Convex Cones," Econometrica, Econometric Society, vol. 89(5), pages 2439-2458, September.
    4. Zheng Fang, 2021. "A Unifying Framework for Testing Shape Restrictions," Papers 2107.12494, arXiv.org, revised Aug 2021.

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