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Abstract

This paper derives the limiting distributions of alternative jackknife IV (JIV) estimators and gives formulae for accompanying consistent standard errors in the presence of heteroskedasticity and many instruments. The asymptotic framework includes the many instrument sequence of Bekker (1994) and the many weak instrument sequence of Chao and Swanson (2005). We show that JIV estimators are asymptotically normal and that standard errors are consistent provided that \frac{\sqrt{K_{n}}}{n} \to \infty as n \to \infty, where K_n and r_n denote, respectively, the number of instruments and the concentration parameter. This is in contrast to the asymptotic behavior of such classical IV estimators as LIML, B2SLS, and 2SLS, all of which are inconsistent in the presence of heteroskedasticity, unless \frac{K_n}{r_n}\to 0. We also show that the rate of convergence and the form of the asymptotic covariance matrix of the JIV estimators will in general depend on the strength of the instruments as measured by the relative orders of magnitude of r_n and K_n.

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Paper provided by The Johns Hopkins University,Department of Economics in its series Economics Working Paper Archive with number 567.

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Date of creation: Oct 2010
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Handle: RePEc:jhu:papers:567

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Find related papers by JEL classification:

• C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
• C31 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Cross-Sectional Models; Spatial Models; Treatment Effect Models; Quantile Regressions; Social Interaction Models

This paper has been announced in the following NEP Reports:

• NEP-ALL-2011-01-30 (All new papers)
• NEP-ECM-2011-01-30 (Econometrics)

References

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1. James G. MacKinnon & Russell Davidson, 2006. "The case against JIVE," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 21(6), pages 827-833.
   
   • Russell Davidson & James G. MacKinnon, 2004. "The Case Against JIVE," Working Papers 1031, Queen's University, Department of Economics.
   • Russell Davidson & James MacKinnon, 2006. "The Case Against Jive," Departmental Working Papers 2004-02, McGill University, Department of Economics.
2. 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.
3. Douglas Staiger & James H. Stock, 1997. "Instrumental Variables Regression with Weak Instruments," Econometrica, Econometric Society, vol. 65(3), pages 557-586, May.
   
   • Douglas Staiger & James H. Stock, 1994. "Instrumental Variables Regression with Weak Instruments," NBER Technical Working Papers 0151, National Bureau of Economic Research, Inc.
4. Whitney Newey & Richard Smith, 2003. "Higher order properties of GMM and generalised empirical likelihood estimators," CeMMAP working papers CWP04/03, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
   
   • Whitney K. Newey & Richard J. Smith, 2004. "Higher Order Properties of Gmm and Generalized Empirical Likelihood Estimators," Econometrica, Econometric Society, vol. 72(1), pages 219-255, 01.
5. Frank Kleibergen, 2002. "Pivotal Statistics for Testing Structural Parameters in Instrumental Variables Regression," Econometrica, Econometric Society, vol. 70(5), pages 1781-1803, September.
6. Andrews, Donald W.K. & Stock, James H., 2007. "Testing with many weak instruments," Journal of Econometrics, Elsevier, vol. 138(1), pages 24-46, May.
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   • Angrist, J D & Imbens, G W & Krueger, A B, 1999. "Jackknife Instrumental Variables Estimation," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 14(1), pages 57-67, Jan.-Feb..
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   • Chirok Han & Peter C. B. Phillips, 2006. "GMM with Many Moment Conditions," Econometrica, Econometric Society, vol. 74(1), pages 147-192, 01.
   • Chirok Han & Peter C.B. Phillips, 2005. "GMM with Many Moment Conditions," Cowles Foundation Discussion Papers 1515, Cowles Foundation for Research in Economics, Yale University.
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10. 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.
    
    • John Chao & Norman Swanson, 2004. "Consistent Estimation with a Large Number of Weak Instruments," Departmental Working Papers 200421, Rutgers University, Department of Economics.
    • 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.
    • John C. Chao & Norman Rasmus Swanson, 2004. "Consistent Estimation with a Large Number of Weak Instruments," Yale School of Management Working Papers ysm374, Yale School of Management.
11. 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.
12. Blomquist, Soren & Dahlberg, Matz, 1999. "Small Sample Properties of LIML and Jackknife IV Estimators: Experiments with Weak Instruments," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 14(1), pages 69-88, Jan.-Feb..
13. Chuanming Gao & Kajal Lahiri, 2000. "A Comparison of Some Recent Bayesian and Classical Procedures for Simultaneous Equation Models with Weak Instruments," Econometric Society World Congress 2000 Contributed Papers 0230, Econometric Society.
    
    • Kajal Lahiri & Chuanming Gao, 2001. "A Comparison of Some Recent Bayesian and Classical Procedures for Simultaneous Equation Models with Weak Instruments," Discussion Papers 01-15, University at Albany, SUNY, Department of Economics.
14. Hahn, Jinyong, 2002. "Optimal Inference With Many Instruments," Econometric Theory, Cambridge University Press, vol. 18(01), pages 140-168, February.
15. Ackerberg, Daniel & Devereux, Paul J., 2008. "Improved JIVE Estimators for Overidentified Linear Models with and without Heteroskedasticity," CEPR Discussion Papers 6926, C.E.P.R. Discussion Papers.
    
    • Daniel A. Ackerberg & Paul J. Devereux, 2009. "Improved JIVE Estimators for Overidentified Linear Models with and without Heteroskedasticity," The Review of Economics and Statistics, MIT Press, vol. 91(2), pages 351-362, May.
    • Daniel A. Ackerberg & Paul J. Devereux, 2008. "Improved Jive Estimators for Overidentified Linear Models with and without Heteroskedasticity," Working Papers 200817, School Of Economics, University College Dublin.
16. James G. MacKinnon & Russell Davidson, 2006. "Reply to Ackerberg and Devereux and Blomquist and Dahlberg on 'The case against JIVE'," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 21(6), pages 843-844.
17. Whitney Newey & Frank Windmeijer, 2005. "GMM with many weak moment conditions," CeMMAP working papers CWP18/05, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
18. Hansen, Christian & Hausman, Jerry & Newey, Whitney, 2008. "Estimation With Many Instrumental Variables," Journal of Business & Economic Statistics, American Statistical Association, vol. 26, pages 398-422.
19. Marcelo J. Moreira, 2003. "A Conditional Likelihood Ratio Test for Structural Models," Econometrica, Econometric Society, vol. 71(4), pages 1027-1048, 07.
20. Bekker, Paul A, 1994. "Alternative Approximations to the Distributions of Instrumental Variable Estimators," Econometrica, Econometric Society, vol. 62(3), pages 657-81, May.
21. 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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Cited by:

1. Michal Kolesár & Raj Chetty & John N. Friedman & Edward L. Glaeser & Guido W. Imbens, 2011. "Identification and Inference with Many Invalid Instruments," NBER Working Papers 17519, National Bureau of Economic Research, Inc.
2. Matias D. Cattaneo & Michael Jansson & Whitney K. Newey, 2012. "Alternative Asymptotics and the Partially Linear Model with Many Regressors," CREATES Research Papers 2012-02, School of Economics and Management, University of Aarhus.
3. Brian P. Poi, 2006. "Jackknife instrumental variables estimation in Stata," Stata Journal, StataCorp LP, vol. 6(3), pages 364-376, September.

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