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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.

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Bibliographic Info

Article provided by Cambridge University Press in its journal Econometric Theory.

Volume (Year): 22 (2006)
Issue (Month): 05 (October)
Pages: 947-960

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Handle: RePEc:cup:etheor:v:22:y:2006:i:05:p:947-960_06

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  1. Douglas Staiger & James H. Stock, 1997. "Instrumental Variables Regression with Weak Instruments," Econometrica, Econometric Society, vol. 65(3), pages 557-586, May.
  2. 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.
  3. Donald W.K. Andrews & James H. Stock, 2005. "Inference with Weak Instruments," NBER Technical Working Papers 0313, National Bureau of Economic Research, Inc.
  4. 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.
  5. 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.
  6. 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.
  7. 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 967-76, July.
  8. Hillier, Grant, 2006. "Yet More On The Exact Properties Of Iv Estimators," Econometric Theory, Cambridge University Press, vol. 22(05), pages 913-931, October.
  9. 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.
  10. 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.
  11. 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.
  12. 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.
  13. 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.
  14. 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.
  15. Peter C.B. Phillips, 1987. "Partially Identified Econometric Models," Cowles Foundation Discussion Papers 845R, Cowles Foundation for Research in Economics, Yale University, revised Aug 1988.
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Cited by:
  1. 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.
  2. Jan F. KIVIET, 2012. "Identification and Inference in a Simultaneous Equation Under Alternative Information Sets and Sampling Schemes," Economic Growth centre Working Paper Series 1207, Nanyang Technolgical University, School of Humanities and Social Sciences, Economic Growth centre.
  3. 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.
  4. Simon A. Broda & Raymond Kan, 2014. "On Distributions of Ratios," Tinbergen Institute Discussion Papers 13-211/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. 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.
  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. Simon A. Broda & Raymond Kan, 2013. "On Distributions of Ratios," UvA-Econometrics Working Papers 13-10, Universiteit van Amsterdam, Dept. of Econometrics.

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