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Asymptotic Normality of Single-Equation Estimators for the Case with a Large Number of Weak Instruments

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  • John C. Chao

    ()
    (University of Maryland)

  • Norman R. Swanson

    ()
    (Rutgers University)

Abstract

This paper analyzes conditions under which various single-equation estimators are asymptotically normal in a simultaneous equations framework with many weak instruments. In particular, our paper adds to the many instruments asymptotic normality literature, including papers by Morimune (1983), Bekker (1994), Angrist and Krueger (1995), Donald and Newey (2001), Hahn, Hausman, and Kuersteiner (2001), and Stock and Yogo (2003). We consider the case where instrument weakness is such that rn, the rate of growth of the concentration parameter, is slower than Kn, the growth rate of the number of instruments, but such that Kn^.5/rn --> 0 as n --> 1: In this case, the rate of convergence is shown to be rn/Kn^.5 . We also show that formulae for the asymptotic variances of various single-equation estimators are di®erent from those obtained under assumptions of stronger instruments, i.e., cases where rn is assumed to grow at the same rate or at a faster rate than Kn. An interesting finding of this paper is that, for the case we study here, both the LIML and the Fuller estimators can be shown to be asymptotically more e±cient than the B2SLS estimator not just for the case where the error distributions are assumed to be Gaussian but for all error distributions that lie within the elliptical family.

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Bibliographic Info

Paper provided by Rutgers University, Department of Economics in its series Departmental Working Papers with number 200312.

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Date of creation: 20 Oct 2003
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Handle: RePEc:rut:rutres:200312

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Related research

Keywords: CLT for bilinear forms; instrumental variables; k-class estimator; local-to-zero framework; pathwise asymptotics; weak instruments;

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References

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  1. H. Kelejian, Harry & Prucha, Ingmar R., 2001. "On the asymptotic distribution of the Moran I test statistic with applications," Journal of Econometrics, Elsevier, vol. 104(2), pages 219-257, September.
  2. Joshua D. Angrist & Guido W. Imbens & Alan Krueger, 1995. "Jackknife Instrumental Variables Estimation," NBER Technical Working Papers 0172, National Bureau of Economic Research, Inc.
  3. Frank Kleibergen, 2000. "Pivotal Statistics for Testing Structural Parameters in Instrumental Variables Regression," Tinbergen Institute Discussion Papers 00-055/4, Tinbergen Institute.
  4. Jiahui Wang & Eric Zivot, 1998. "Inference on Structural Parameters in Instrumental Variables Regression with Weak Instruments," Econometrica, Econometric Society, vol. 66(6), pages 1389-1404, November.
  5. Joshua D. Angrist & Alan B. Krueger, 1995. "Split Sample Instrumental Variables," NBER Technical Working Papers 0150, National Bureau of Economic Research, Inc.
  6. Angrist, Joshua D & Krueger, Alan B, 1995. "Split-Sample Instrumental Variables Estimates of the Return to Schooling," Journal of Business & Economic Statistics, American Statistical Association, vol. 13(2), pages 225-35, April.
  7. Choi, In & Phillips, Peter C. B., 1992. "Asymptotic and finite sample distribution theory for IV estimators and tests in partially identified structural equations," Journal of Econometrics, Elsevier, vol. 51(1-2), pages 113-150.
  8. Chao, John Chao & Norman R. Swanson, 2003. "Consistent Estimation with a Large Number of Weak Instruments," Cowles Foundation Discussion Papers 1417, Cowles Foundation for Research in Economics, Yale University.
  9. Koenker, Roger & Machado, Jose A. F., 1999. "GMM inference when the number of moment conditions is large," Journal of Econometrics, Elsevier, vol. 93(2), pages 327-344, December.
  10. Morimune, Kimio, 1983. "Approximate Distributions of k-Class Estimators When the Degree of Overidentifiability Is Large Compared with the Sample Size," Econometrica, Econometric Society, vol. 51(3), pages 821-41, May.
  11. James H. Stock & Motohiro Yogo, 2002. "Testing for Weak Instruments in Linear IV Regression," NBER Technical Working Papers 0284, National Bureau of Economic Research, Inc.
  12. Donald, Stephen G. & Whitney Newey, 1999. "Choosing the Number of Instruments," Working papers 99-05, Massachusetts Institute of Technology (MIT), Department of Economics.
  13. Douglas Staiger & James H. Stock, 1997. "Instrumental Variables Regression with Weak Instruments," Econometrica, Econometric Society, vol. 65(3), pages 557-586, May.
  14. Bekker, Paul A, 1994. "Alternative Approximations to the Distributions of Instrumental Variable Estimators," Econometrica, Econometric Society, vol. 62(3), pages 657-81, May.
  15. Fuller, Wayne A, 1977. "Some Properties of a Modification of the Limited Information Estimator," Econometrica, Econometric Society, vol. 45(4), pages 939-53, May.
  16. Peter C.B. Phillips, 1982. "Small Sample Distribution Theory in Econometric Models of Simultaneous Equations," Cowles Foundation Discussion Papers 617, Cowles Foundation for Research in Economics, Yale University.
  17. Phillips, Garry D A & Hale, C, 1977. "The Bias of Instrumental Variable Estimators of Simultaneous Equation Systems," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 18(1), pages 219-28, February.
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Citations

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Cited by:
  1. Peter C. B. Phillips & Chirok Han, 2004. "GMM with Many Moment Conditions," Econometric Society 2004 Far Eastern Meetings 525, Econometric Society.
  2. Norman R. Swanson & John C. Chao, 2004. "Estimation and Testing Using Jackknife IV in Heteroskedastic Regressions with Many Weak Instruments," Econometric Society 2004 Far Eastern Meetings 668, Econometric Society.
  3. Christian Hansen & Jerry Hausman & Whitney Newey, 2006. "Estimation with many instrumental variables," CeMMAP working papers CWP19/06, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
  4. D. S. Poskitt & C. L. Skeels, 2004. "Approximating the Distribution of the Instrumental Variables Estimator when the Concentration Parameter is Small," Monash Econometrics and Business Statistics Working Papers 19/04, Monash University, Department of Econometrics and Business Statistics.
  5. D. S. Poskitt & C. L. Skeels, 2005. "Small Concentration Asymptotics and Instrumental Variables Inference," Monash Econometrics and Business Statistics Working Papers 4/05, Monash University, Department of Econometrics and Business Statistics.

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