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A Remark on Bimodality and Weak Instrumentation in Structural Equation Estimation

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Abstract

In a simple model composed of a structural equation and identity, the finite sample distribution of the 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 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.

Suggested Citation

  • Peter C. B. Phillips, 2005. "A Remark on Bimodality and Weak Instrumentation in Structural Equation Estimation," Cowles Foundation Discussion Papers 1540, Cowles Foundation for Research in Economics, Yale University.
    • Handle: RePEc:cwl:cwldpp:1540
    Note: CFP 1171.
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    1. Phillips, Peter C B, 1985. "The Exact Distribution of LIML: II," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 26(1), pages 21-36, February.
    2. Donald W.K. Andrews & James H. Stock, 2005. "Inference with Weak Instruments," Cowles Foundation Discussion Papers 1530, Cowles Foundation for Research in Economics, Yale University.
    3. Nelson, Charles R & Startz, Richard, 1990. "Some Further Results on the Exact Small Sample Properties of the Instrumental Variable Estimator," Econometrica, Econometric Society, vol. 58(4), pages 967-976, July.
    4. Phillips, P C B, 1986. "The Distribution of FIML in the Leading Case," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 27(1), pages 239-243, February.
    5. Phillips, P C B, 1980. "The Exact Distribution of Instrumental Variable Estimators in an Equation Containing n + 1 Endogenous Variables," Econometrica, Econometric Society, vol. 48(4), pages 861-878, May.
    6. Stock, James H & Wright, Jonathan H & Yogo, Motohiro, 2002. "A Survey of Weak Instruments and Weak Identification in Generalized Method of Moments," Journal of Business & Economic Statistics, American Statistical Association, vol. 20(4), pages 518-529, October.
    7. Hillier, Grant, 2006. "Yet More On The Exact Properties Of Iv Estimators," Econometric Theory, Cambridge University Press, vol. 22(5), pages 913-931, October.
    8. Forchini, G., 2006. "On The Bimodality Of The Exact Distribution Of The Tsls Estimator," Econometric Theory, Cambridge University Press, vol. 22(5), pages 932-946, October.
    9. Phillips, P.C.B., 1983. "Exact small sample theory in the simultaneous equations model," Handbook of Econometrics, in: Z. Griliches† & M. D. Intriligator (ed.), Handbook of Econometrics, edition 1, volume 1, chapter 8, pages 449-516, Elsevier.
    10. Phillips, P.C.B., 1989. "Partially Identified Econometric Models," Econometric Theory, Cambridge University Press, vol. 5(2), pages 181-240, August.
    11. Douglas Staiger & James H. Stock, 1997. "Instrumental Variables Regression with Weak Instruments," Econometrica, Econometric Society, vol. 65(3), pages 557-586, May.
    12. Woglom, Geoffrey, 2001. "More Results on the Exact Small Sample Properties of the Instrumental Variable Estimator," Econometrica, Econometric Society, vol. 69(5), pages 1381-1389, September.
    13. Maddala, G S & Jeong, Jinook, 1992. "On the Exact Small Sample Distribution of the Instrumental Variable Estimator," Econometrica, Econometric Society, vol. 60(1), pages 181-183, January.
    14. Jinyong Hahn & Jerry Hausman, 2003. "Weak Instruments: Diagnosis and Cures in Empirical Econometrics," American Economic Review, American Economic Association, vol. 93(2), pages 118-125, May.
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    Cited by:

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    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, 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.
    5. 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.
    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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    More about this item

    Keywords

    Attraction; Bimodality; Concentration parameter; Identity; Inverse normal; Point of compression; Structural Equation; Weak instrumentation;
    All these keywords.

    JEL classification:

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

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