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

  • John C. Chao


    (University of Maryland)

  • Norman R. Swanson


    (Rutgers University)

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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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
Date of revision:
Handle: RePEc:rut:rutres:200312
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  1. Harry H. Kelejian & Ingmar R. Prucha, 1999. "On the Asymptotic Distribution of the Moran I Test Statistic with Applications," Electronic Working Papers 99-002, University of Maryland, Department of Economics.
  2. Joshua D. Angrist & Alan B. Krueger, 1995. "Split Sample Instrumental Variables," NBER Technical Working Papers 0150, National Bureau of Economic Research, Inc.
  3. Douglas Staiger & James H. Stock, 1994. "Instrumental Variables Regression with Weak Instruments," NBER Technical Working Papers 0151, National Bureau of Economic Research, Inc.
  4. 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.
  5. Donald, Stephen G & Newey, Whitney K, 2001. "Choosing the Number of Instruments," Econometrica, Econometric Society, vol. 69(5), pages 1161-91, September.
  6. 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.
  7. 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.
  8. Joshua D. Angrist & Guido W. Imbens & Alan Krueger, 1995. "Jackknife Instrumental Variables Estimation," NBER Technical Working Papers 0172, National Bureau of Economic Research, Inc.
  9. Frank Kleibergen, 2002. "Pivotal Statistics for Testing Structural Parameters in Instrumental Variables Regression," Econometrica, Econometric Society, vol. 70(5), pages 1781-1803, September.
  10. Bekker, Paul A, 1994. "Alternative Approximations to the Distributions of Instrumental Variable Estimators," Econometrica, Econometric Society, vol. 62(3), pages 657-81, May.
  11. 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.
  12. John C. Chao & Norman R. Swanson, 2005. "Consistent Estimation with a Large Number of Weak Instruments," Econometrica, Econometric Society, vol. 73(5), pages 1673-1692, 09.
  13. 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.
  14. Fuller, Wayne A, 1977. "Some Properties of a Modification of the Limited Information Estimator," Econometrica, Econometric Society, vol. 45(4), pages 939-53, May.
  15. 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.
  16. 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.
  17. 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.
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