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A practical guide to compact infinite dimensional parameter spaces

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  • Joachim Freyberger
  • Matthew A. Masten

Abstract

Compactness is a widely used assumption in econometrics. In this article, we gather and review general compactness results for many commonly used parameter spaces in nonparametric estimation, and we provide several new results. We consider three kinds of functions: (1) functions with bounded domains which satisfy standard norm bounds, (2) functions with bounded domains which do not satisfy standard norm bounds, and (3) functions with unbounded domains. In all three cases, we provide two kinds of results, compact embedding and closedness, which together allow one to show that parameter spaces defined by a ||·||s-norm bound are compact under a norm ||·||c. We illustrate how the choice of norms affects the parameter space, the strength of the conclusions, as well as other regularity conditions in two common settings: nonparametric mean regression and nonparametric instrumental variables estimation.

Suggested Citation

  • Joachim Freyberger & Matthew A. Masten, 2019. "A practical guide to compact infinite dimensional parameter spaces," Econometric Reviews, Taylor & Francis Journals, vol. 38(9), pages 979-1006, October.
    • Handle: RePEc:taf:emetrv:v:38:y:2019:i:9:p:979-1006
    DOI: 10.1080/07474938.2018.1514025
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    Cited by:

    1. Jun, Sung Jae & Zincenko, Federico, 2022. "Testing for risk aversion in first-price sealed-bid auctions," Journal of Econometrics, Elsevier, vol. 226(2), pages 295-320.
    2. Dalderop, Jeroen, 2023. "Semiparametric estimation of latent variable asset pricing models," Journal of Econometrics, Elsevier, vol. 236(1).
    3. Sukjin Han & Sungwon Lee, 2019. "Estimation in a generalization of bivariate probit models with dummy endogenous regressors," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 34(6), pages 994-1015, September.
    4. Isaac Loh, 2024. "Inference under partial identification with minimax test statistics," Papers 2401.13057, arXiv.org, revised Apr 2024.
    5. Yang Ning & Sida Peng & Jing Tao, 2020. "Doubly Robust Semiparametric Difference-in-Differences Estimators with High-Dimensional Data," Papers 2009.03151, arXiv.org.
    6. Ben Deaner, 2019. "Nonparametric Instrumental Variables Estimation Under Misspecification," Papers 1901.01241, arXiv.org, revised Dec 2022.
    7. An, Yonghong & Hong, Shengjie & Zhang, Daiqiang, 2023. "A structural analysis of simple contracts," Journal of Econometrics, Elsevier, vol. 236(2).
    8. Fan, Yanqin & Shi, Xuetao & Tao, Jing, 2023. "Partial identification and inference in moment models with incomplete data," Journal of Econometrics, Elsevier, vol. 235(2), pages 418-443.

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