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Decision Theory Applied to an Instrumental Variables Model

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  • Gary Chamberlain

Abstract

This paper applies some general concepts in decision theory to a simple instrumental variables model. There are two endogenous variables linked by a single structural equation; k of the exogenous variables are excluded from this structural equation and provide the instrumental variables (IV). The reduced-form distribution of the endogenous variables conditional on the exogenous variables corresponds to independent draws from a bivariate normal distribution with linear regression functions and a known covariance matrix. A canonical form of the model has parameter vector (rho, phi, omega), where phi is the parameter of interest and is normalized to be a point on the unit circle. The reduced-form coefficients on the instrumental variables are split into a scalar parameter rho and a parameter vector omega, which is normalized to be a point on the (k - 1)-dimensional unit sphere; rho measures the strength of the association between the endogenous variables and the instrumental variables, and omega is a measure of direction. A prior distribution is introduced for the IV model. The parameters phi, rho, and omega are treated as independent random variables. The distribution for phi is uniform on the unit circle; the distribution for omega is uniform on the unit sphere with dimension k-1. These choices arise from the solution of a minimax problem. The prior for rho is left general. It turns out that given any positive value for rho, the Bayes estimator of phi does not depend on rho; it equals the maximum-likelihood estimator. This Bayes estimator has constant risk; because it minimizes average risk with respect to a proper prior, it is minimax. Copyright The Econometric Society 2007.

Suggested Citation

  • Gary Chamberlain, 2007. "Decision Theory Applied to an Instrumental Variables Model," Econometrica, Econometric Society, vol. 75(3), pages 609-652, May.
  • Handle: RePEc:ecm:emetrp:v:75:y:2007:i:3:p:609-652
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    Cited by:

    1. Marcelo Moreira & Geert Ridder, 2019. "Efficiency loss of asymptotically efficient tests in an instrumental variables regression," CeMMAP working papers CWP03/19, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    2. Andrews, Isaiah, 2019. "On the structure of IV estimands," Journal of Econometrics, Elsevier, vol. 211(1), pages 294-307.
    3. Kolesár, Michal, 2018. "Minimum distance approach to inference with many instruments," Journal of Econometrics, Elsevier, vol. 204(1), pages 86-100.
    4. Marcelo J. Moreira & Mahrad Sharifvaghefi & Geert Ridder, 2017. "Optimal Invariant Tests in an Instrumental Variables Regression With Heteroskedastic and Autocorrelated Errors," Papers 1705.00231, arXiv.org, revised Aug 2021.
    5. Thomas A. Severini, 2023. "Integrated likelihood inference in multinomial distributions," METRON, Springer;Sapienza Università di Roma, vol. 81(2), pages 131-142, August.
    6. Keisuke Hirano & Jack R. Porter, 2015. "Location Properties of Point Estimators in Linear Instrumental Variables and Related Models," Econometric Reviews, Taylor & Francis Journals, vol. 34(6-10), pages 720-733, December.
    7. Kociecki, Andrzej, 2012. "Orbital Priors for Time-Series Models," MPRA Paper 42804, University Library of Munich, Germany.
    8. T. W. Anderson & Naoto Kunitomo & Yukitoshi Matsushita, 2009. "The Limited Information Maximum Likelihood Estimator as an Angle," CIRJE F-Series CIRJE-F-619, CIRJE, Faculty of Economics, University of Tokyo.
    9. Jann Spiess, 2017. "Bias Reduction in Instrumental Variable Estimation through First-Stage Shrinkage," Papers 1708.06443, arXiv.org, revised Oct 2017.
    10. Charles F. Manski, 2021. "Econometrics for Decision Making: Building Foundations Sketched by Haavelmo and Wald," Econometrica, Econometric Society, vol. 89(6), pages 2827-2853, November.
    11. Nascimento, Marcus Gerardus Lavagnole & Abanto-Valle, Carlos Antonio & Mendonça, Mario Jorge, 2019. "Multivariate Spatial IV Regression," Brazilian Review of Econometrics, Sociedade Brasileira de Econometria - SBE, vol. 38(2), January.
    12. Martin Emil Jakobsen & Jonas Peters, 2022. "Distributional robustness of K-class estimators and the PULSE [The colonial origins of comparative development: An empirical investigation]," The Econometrics Journal, Royal Economic Society, vol. 25(2), pages 404-432.
    13. Andrews, Donald W.K. & Moreira, Marcelo J. & Stock, James H., 2008. "Efficient two-sided nonsimilar invariant tests in IV regression with weak instruments," Journal of Econometrics, Elsevier, vol. 146(2), pages 241-254, October.
    14. Mills, Benjamin & Moreira, Marcelo J. & Vilela, Lucas P., 2014. "Tests based on t-statistics for IV regression with weak instruments," Journal of Econometrics, Elsevier, vol. 182(2), pages 351-363.
    15. Dong Jin Lee, 2020. "Optimal tests for parameter breaking process in conditional quantile models," The Japanese Economic Review, Springer, vol. 71(3), pages 479-510, July.
    16. Marc Hallin & Marcelo Moreira J. & Alexei Onatski, 2013. "Group Invariance, Likelihood Ratio Tests, and the Incidental Parameter Problem in a High-Dimensional Linear Model," Working Papers ECARES ECARES 2013-04, ULB -- Universite Libre de Bruxelles.
    17. Nascimento, Marcus Gerardus Lavagnole & Abanto-Valle, Carlos Antonio & Mendonça, Mario Jorge, 2018. "Multivariate Spatial IV Regression," Brazilian Review of Econometrics, Sociedade Brasileira de Econometria - SBE, vol. 38(2).

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