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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.

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File URL: http://cowles.econ.yale.edu/P/cd/d15a/d1540.pdf
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Paper provided by Cowles Foundation for Research in Economics, Yale University in its series Cowles Foundation Discussion Papers with number 1540.

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Length: 14 pages
Date of creation: Dec 2005
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Publication status: Published in Econometric Theory (2006), 22(5): 947-960
Handle: RePEc:cwl:cwldpp:1540

Note: CFP 1171.
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Postal: Cowles Foundation, Yale University, Box 208281, New Haven, CT 06520-8281 USA

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Keywords: Attraction; Bimodality; Concentration parameter; Identity; Inverse normal; Point of compression; Structural Equation; Weak instrumentation;

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References

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  1. Peter C.B. Phillips, 1982. "Exact Small Sample Theory in the Simultaneous Equations Model," Cowles Foundation Discussion Papers 621, Cowles Foundation for Research in Economics, Yale University.
  2. Giovanni Forchini, 2005. "On the Bimodality of the Exact Distribution of the TSLS Estimator," Monash Econometrics and Business Statistics Working Papers 14/05, Monash University, Department of Econometrics and Business Statistics.
  3. 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 518-29, October.
  4. Phillips, Peter C B, 1984. "The Exact Distribution of LIML: I," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 25(1), pages 249-61, February.
  5. Peter C.B. Phillips, 1985. "The Distribution of FIML in the Leading Case," Cowles Foundation Discussion Papers 739, Cowles Foundation for Research in Economics, Yale University.
  6. Jan F. Kiviet & Jerzy Niemczyk, 2006. "The Asymptotic and Finite Sample Distributions of OLS and Simple IV in Simultaneous Equations," Tinbergen Institute Discussion Papers 06-078/4, Tinbergen Institute.
  7. 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 861-78, May.
  8. Woglom, Geoffrey, 2001. "More Results on the Exact Small Sample Properties of the Instrumental Variable Estimator," Econometrica, Econometric Society, vol. 69(5), pages 1381-89, September.
  9. Phillips, P.C.B., 1989. "Partially Identified Econometric Models," Econometric Theory, Cambridge University Press, vol. 5(02), pages 181-240, August.
  10. Charles R. Nelson & Richard Startz, 1988. "Some Further Results on the Exact Small Sample Properties of the Instrumental Variable Estimator," NBER Technical Working Papers 0068, National Bureau of Economic Research, Inc.
  11. Douglas Staiger & James H. Stock, 1997. "Instrumental Variables Regression with Weak Instruments," Econometrica, Econometric Society, vol. 65(3), pages 557-586, May.
  12. Donald W.K. Andrews & James H. Stock, 2005. "Inference with Weak Instruments," NBER Technical Working Papers 0313, National Bureau of Economic Research, Inc.
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  14. Hillier, Grant, 2006. "Yet More On The Exact Properties Of Iv Estimators," Econometric Theory, Cambridge University Press, vol. 22(05), pages 913-931, October.
  15. Maddala, G S & Jeong, Jinook, 1992. "On the Exact Small Sample Distribution of the Instrumental Variable Estimator," Econometrica, Econometric Society, vol. 60(1), pages 181-83, January.
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Cited by:
  1. 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 S24-S59, 02.
  2. Jan F. KIVIET & Jerzy NIEMCZYK, 2013. "On the limiting and empirical distributions of IV estimators when some of the instruments are actually endogenous," Economic Growth centre Working Paper Series 1311, Nanyang Technolgical University, School of Humanities and Social Sciences, Economic Growth centre.
  3. Simon A. Broda & Raymond Kan, 2013. "On Distributions of Ratios," UvA-Econometrics Working Papers 13-10, Universiteit van Amsterdam, Dept. of Econometrics.
  4. Jan F. Kiviet, 2012. "Identification and Inference in a Simultaneous Equation under Alternative Information Sets and Sampling Schemes," Tinbergen Institute Discussion Papers 12-128/III, Tinbergen Institute.
  5. M.C. Medeiros & E. Mendes & Les Oxley, 2010. "A Note on Nonlinear Cointegration, Misspecification and Bimodality," Working Papers in Economics 10/01, University of Canterbury, Department of Economics and Finance.
  6. 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.
  7. Fabel, Oliver & Pascalau, Razvan, 2007. "Recruitment of Seemingly Overeducated Personnel: Insider-Outsider Effects on Fair Employee Selection Practices," MPRA Paper 7218, University Library of Munich, Germany.
  8. Simon A. Broda & Raymond Kan, 2014. "On Distributions of Ratios," Tinbergen Institute Discussion Papers 13-211/III, Tinbergen Institute.

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