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Instrument-residual estimator for multi-valued instruments under full monotonicity

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  • Kim, Bora
  • Lee, Myoung-jae

Abstract

In determining the effects of a binary treatment D on an outcome Y, a multi-valued instrumental variable (IV) Z=0,1,…,J often appears. Imbens and Angrist (1994, Econometrica) showed that the IV estimator (IVE) of Y on D using Z as an IV is consistent for a non-negatively weighted average of heterogeneous “complier” effects. However, Imbens and Angrist did not include covariates X. This paper generalizes their finding by explicitly allowing X to appear in the linear model for the IVE, and shows that the extra condition E(Z|X)=L(Z|X) is necessary for generalization, where L(Z|X)≡E(ZX′){E(XX′)}−1X is the linear projection. This paper therefore proposes an alternative IVE using Z−E(Z|X) as an IV, which is consistent for the same estimand without the restrictive extra condition. A simulation study demonstrates that the extra condition E(Z|X)=L(Z|X) is necessary for the usual IVE, but not for the alternative IVE proposed in this paper.

Suggested Citation

  • Kim, Bora & Lee, Myoung-jae, 2024. "Instrument-residual estimator for multi-valued instruments under full monotonicity," Statistics & Probability Letters, Elsevier, vol. 213(C).
    • Handle: RePEc:eee:stapro:v:213:y:2024:i:c:s0167715224001561
    DOI: 10.1016/j.spl.2024.110187
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    References listed on IDEAS

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    1. Victor Chernozhukov & Denis Chetverikov & Mert Demirer & Esther Duflo & Christian Hansen & Whitney Newey & James Robins, 2018. "Double/debiased machine learning for treatment and structural parameters," Econometrics Journal, Royal Economic Society, vol. 21(1), pages 1-68, February.
    2. Myoung-Jae Lee, 2018. "Simple least squares estimator for treatment effects using propensity score residuals," Biometrika, Biometrika Trust, vol. 105(1), pages 149-164.
    3. Victor Chernozhukov & Juan Carlos Escanciano & Hidehiko Ichimura & Whitney K. Newey & James M. Robins, 2022. "Locally Robust Semiparametric Estimation," Econometrica, Econometric Society, vol. 90(4), pages 1501-1535, July.
    4. Joshua D. Angrist & Jörn-Steffen Pischke, 2009. "Mostly Harmless Econometrics: An Empiricist's Companion," Economics Books, Princeton University Press, edition 1, number 8769.
    5. Imbens, Guido W & Angrist, Joshua D, 1994. "Identification and Estimation of Local Average Treatment Effects," Econometrica, Econometric Society, vol. 62(2), pages 467-475, March.
    6. Myoung‐jae Lee, 2021. "Instrument residual estimator for any response variable with endogenous binary treatment," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 83(3), pages 612-635, July.
    7. Joshua D. Angrist, 1998. "Estimating the Labor Market Impact of Voluntary Military Service Using Social Security Data on Military Applicants," Econometrica, Econometric Society, vol. 66(2), pages 249-288, March.
    8. Fan Li & Kari Lock Morgan & Alan M. Zaslavsky, 2018. "Balancing Covariates via Propensity Score Weighting," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 113(521), pages 390-400, January.
    9. Myoung-jae Lee & Chirok Han, 2024. "Ordinary least squares and instrumental-variables estimators for any outcome and heterogeneity," Stata Journal, StataCorp LP, vol. 24(1), pages 72-92, March.
    10. Jin‐young Choi & Myoung‐jae Lee, 2012. "Bounding endogenous regressor coefficients using moment inequalities and generalized instruments," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 66(2), pages 161-182, May.
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