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A Remark On Bimodality And Weak Instrumentation In Structural Equation Estimation

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  • Phillips, Peter C.B.

Abstract

In a simple model composed of a structural equation and identity, the finite-sample distribution of the instrumental variable/limited information maximum likelihood (IV/LIML) estimator is always bimodal, and this is most apparent when the concentration parameter is small. Weak instrumentation is the energy that feeds the secondary mode, and the coefficient in the structural identity provides a point of compression in the density that gives rise to it. The IV limit distribution can be normal, bimodal, or inverse normal depending on the behavior of the concentration parameter and the weakness of the instruments. The limit distribution of the ordinary least squares (OLS) estimator is normal in all cases and has a much faster rate of convergence under very weak instrumentation. The IV estimator is therefore more resistant to the attractive effect of the identity than OLS. Some of these limit results differ from conventional weak instrument asymptotics, including convergence to a constant in very weak instrument cases and limit distributions that are inverse normal.My thanks to Richard Smith and two referees for comments on an earlier version. Section 2 of the paper is based on lectures given to students over the 1970s and 1980s at Essex, Birmingham, and Yale. Partial support is acknowledged from a Kelly Fellowship at the University of Auckland School of Business and the NSF under Grant SES 04-142254.

Suggested Citation

  • Phillips, Peter C.B., 2006. "A Remark On Bimodality And Weak Instrumentation In Structural Equation Estimation," Econometric Theory, Cambridge University Press, vol. 22(5), pages 947-960, October.
    • Handle: RePEc:cup:etheor:v:22:y:2006:i:05:p:947-960_06
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    Cited by:

    1. Marcelo C. Medeiros & Eduardo Mendes & Les Oxley, 2014. "A Note on Nonlinear Cointegration, Misspecification, and Bimodality," Econometric Reviews, Taylor & Francis Journals, vol. 33(7), pages 713-731, October.
    2. Simon A. Broda & Raymond Kan, 2016. "On distributions of ratios," Biometrika, Biometrika Trust, vol. 103(1), pages 205-218.
    3. David Roodman & Jonathan Morduch, 2014. "The Impact of Microcredit on the Poor in Bangladesh: Revisiting the Evidence," Journal of Development Studies, Taylor & Francis Journals, vol. 50(4), pages 583-604, April.
    4. Jan F. Kiviet & Jerzy Niemczyk, 2014. "On the Limiting and Empirical Distributions of IV Estimators When Some of the Instruments are Actually Endogenous," Advances in Econometrics, in: Essays in Honor of Peter C. B. Phillips, volume 33, pages 425-490, Emerald Group Publishing Limited.
    5. Jan F. Kiviet, 2013. "Identification and inference in a simultaneous equation under alternative information sets and sampling schemes," Econometrics Journal, Royal Economic Society, vol. 16(1), pages 24-59, February.
    6. Phillips, Peter C.B. & Gao, Wayne Yuan, 2017. "Structural inference from reduced forms with many instruments," Journal of Econometrics, Elsevier, vol. 199(2), pages 96-116.
    7. Oliver Fabel & Razvan Pascalau, 2007. "Recruitment of Overeducated Personnel: Insider-Outsider Effects on Fair Employee Selection Practices," TWI Research Paper Series 18, Thurgauer Wirtschaftsinstitut, Universität Konstanz.
    8. Morduch, Jonathan & Ravi, Shamika & Bauchet, Jonathan, 2013. "Substitution Bias and External Validity: Why an Innovative Anti-poverty Program Showed no Net Impact," CEI Working Paper Series 2013-02, Center for Economic Institutions, Institute of Economic Research, Hitotsubashi University.
    9. Oliver Fabel & Razvan Pascalau, 2013. "Recruitment of Seemingly Overeducated Personnel: Insider--Outsider Effects on Fair Employee Selection Practices," International Journal of the Economics of Business, Taylor & Francis Journals, vol. 20(1), pages 57-82, February.

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    JEL classification:

    • C30 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - General

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