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Consistent Estimation with a Large Number of Weak Instruments

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Author Info
John C. Chao () (University of Maryland, Robert H. Smith School of Business)
Norman Rasmus Swanson () (Rutgers, The State University of New Jersey, Douglass College)

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Abstract

This paper conducts a general analysis of the conditions under which consistent estimation can be achieved in instrumental variables regression when the available instruments are weak in the local-to-zero sense. More precisely, the approach adopted in this paper combines key features of the local-to-zero framework of Staiger and Stock (1997) and the many-instrument framework of Morimune (1983) and Bekker (1994) and generalizes both of these frameworks in the following ways. First, we consider a general local-to-zero framework which allows for an arbitrary degree of instrument weakness by modeling the first-stage coefficients as shrinking toward zero at an unspecified rate, say b_{n}^{-1}. Our local-to-zero setup, in fact, reduces to that of Staiger and Stock (1997) in the case where b_{n}= sqrt{n}. In addition, we examine a broad class of single-equation estimators which extends the well-known k-class to include, amongst others, the Jackknife Instrumental Variables Estimator (JIVE) of Angrist, Imbens, and Krueger (1999). Analysis of estimators within this extended class based on a pathwise asymptotic scheme, where the number of instruments K_{n} is allowed to grow as a function of the sample size, reveals that consistent estimation depends importantly on the relative magnitudes of r_{n}, the growth rate of the concentration parameter, and K_{n}. In particular, it is shown that members of the extended class which satisfy certain general condtions, such as LIML and JIVE, are consistent provided that sqrt{K_{n}}/r_{n}} --> 0, as n --> infinity. On the other hand, the two-stage least squares (2SLS) estimator is shown not to satisfy the needed conditions and is found to be consistent only if K_{n}/r_{n} --> 0, as n --> infinity. A main point of our paper is that the use of many instruments may be beneficial from a point estimation standpoint in empirical applications where the available instruments are weak but abundant, as it provides an extra source, by which the concentration parameter can grow, thus, allowing consistent estimation to be achievable, in certain cases, even in the presence of weak instruments. Our results, thus, add to the findings of Staiger and Stock (1997) who study a local-to-zero framework where K_{n} is held fixed and the concentration parameter does not diverge as sample size grows; in consequence, no single-equation estimator is found to be consistent under their setup.

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Paper provided by Yale School of Management in its series Yale School of Management Working Papers with number ysm374.

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Date of creation: 28 Jul 2004
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Handle: RePEc:ysm:somwrk:ysm374

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Related research
Keywords: Instrumental Variables; k-class Estimator; Local-to-zero Framework; Pathwise Asymptotics; Weak Instruments;

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Find related papers by JEL classification:
C31 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Cross-Sectional Models; Spatial Models; Treatment Effect Models
C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: General - - - Estimation

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References listed on IDEAS
Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
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    Other versions:
  4. 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. [Downloadable!] (restricted)
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    Other versions:
  7. Jinyong Hahn & Atsushi Inoue, 2002. "A Monte Carlo Comparison Of Various Asymptotic Approximations To The Distribution Of Instrumental Variables Estimators," Econometric Reviews, Taylor and Francis Journals, vol. 21(3), pages 309-336. [Downloadable!] (restricted)
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    Other versions:
  10. Frank Kleibergen, 2002. "Pivotal Statistics for Testing Structural Parameters in Instrumental Variables Regression," Econometrica, Econometric Society, vol. 70(5), pages 1781-1803, September. [Downloadable!] (restricted)
    Other versions:
  11. Peter C.B. Phillips, 1987. "Partially Identified Econometric Models," Cowles Foundation Discussion Papers 845R, Cowles Foundation, Yale University, revised Aug 1988. [Downloadable!]
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  16. Peter C. B. Phillips & Chirok Han, 2004. "GMM with Many Moment Conditions," Econometric Society 2004 Far Eastern Meetings 525, Econometric Society. [Downloadable!]
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    Other versions:
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Cited by:
(explanations, Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.)

  1. Giovanni Forchini, 2006. "The Asymptotic distribution of the LIML Estimator in a Partially Identified Structural Equation," Monash Econometrics and Business Statistics Working Papers 1/06, Monash University, Department of Econometrics and Business Statistics. [Downloadable!]
  2. Peter C. B. Phillips & Chirok Han, 2004. "GMM with Many Moment Conditions," Econometric Society 2004 Far Eastern Meetings 525, Econometric Society. [Downloadable!]
    Other versions:
  3. Kazuhiko Hayakawa, 2006. "Efficient GMM Estimation of Dynamic Panel Data Models Where Large Heterogeneity May Be Present," Hi-Stat Discussion Paper Series d05-130, Institute of Economic Research, Hitotsubashi University. [Downloadable!]
  4. Mehmet Caner, 2005. "Higher Order Expansions in GMM with Nearly Weak and Many Nearly Weak Instruments," Working Papers 209, University of Pittsburgh, Department of Economics, revised Jan 2005. [Downloadable!]
  5. Peter C.B. Phillips, 2003. "Vision and Influence in Econometrics: John Denis Sargan," Cowles Foundation Discussion Papers 1393, Cowles Foundation, Yale University. [Downloadable!]
    Other versions:
  6. Stanislav Anatolyev, 2007. "Inference about predictive ability when there are many predictors," Working Papers w0096, Center for Economic and Financial Research (CEFIR). [Downloadable!]
  7. John C. Chao & Norman R. Swanson, 2003. "Asymptotic Normality of Single-Equation Estimators for the Case with a Large Number of Weak Instruments," Departmental Working Papers 200312, Rutgers University, Department of Economics. [Downloadable!]
  8. Mathias D. Cattaneo & Richard K. Crump & Michael Jansson, 2007. "Optimal Inference for Instrumental Variables Regression with non-Gaussian Errors," CREATES Research Papers 2007-11, School of Economics and Management, University of Aarhus. [Downloadable!]
  9. 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. [Downloadable!]
  10. Antonio Ciccone & Giovanni Peri, 2004. "Long-Run Substitutability between More and Less Educated Workers: Evidence from U.S. States 1950-1990," Economics Working Papers 764, Department of Economics and Business, Universitat Pompeu Fabra. [Downloadable!]
    Other versions:
  11. 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. [Downloadable!]
  12. Cizek, P., 2009. "Generalized Methods of Trimmed Moments," Discussion Paper 2009-25, Tilburg University, Center for Economic Research. [Downloadable!]
  13. 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. [Downloadable!]
    Other versions:
  14. T. W. Anderson & Naoto Kunitomo & Yukitoshi Matsushita, 2008. "On the Asymptotic Optimality of the LIML Estimator with Possibly Many Instruments," CIRJE F-Series CIRJE-F-542, CIRJE, Faculty of Economics, University of Tokyo. [Downloadable!]
  15. 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. [Downloadable!] (restricted)
  16. Stanislav Anatolyev & Nikolay Gospodinov, 2008. "Specification Testing in Models with Many Instruments," Working Papers w0124, Center for Economic and Financial Research (CEFIR). [Downloadable!]
  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. [Downloadable!]
  18. Andreas Pick, 2007. "Financial contagion and tests using instrumental variables," DNB Working Papers 139, Netherlands Central Bank, Research Department. [Downloadable!]
  19. T. W. Anderson & Naoto Kunitomo & Yukitoshi Matsushita, 2006. "A New Light from Old Wisdoms : Alternative Estimation Methods of Simultaneous Equations with Possibly Many Instruments," CIRJE F-Series CIRJE-F-399, CIRJE, Faculty of Economics, University of Tokyo. [Downloadable!]
  20. John Chao & Norman Swanson, 2004. "Estimation and Testing Using Jackknife IV in Heteroskedastic Regressions With Many Weak Instruments," Departmental Working Papers 200420, Rutgers University, Department of Economics. [Downloadable!]
    Other versions:
  21. Mehmet Caner, 2006. "Near Exogeneity and Weak Identification in Generlized Empirical Likelihood estimators : Fixed and Many Moment Asymptotics," Working Papers 212, University of Pittsburgh, Department of Economics, revised Jan 2006. [Downloadable!]
    Other versions:
  22. Ciccone, Antonio & Peri, Giovanni, 2003. "Skills' Substitutability and Technological Progress: U.S. States 1950-1990," CESifo Working Paper Series CESifo Working Paper No. , CESifo Group Munich. [Downloadable!]
  23. T. W. Anderson & Naoto Kunitomo & Yukitoshi Matsushita, 2008. "On Finite Sample Properties of Alternative Estimators of Coefficients in a Structural Equation with Many Instruments," CIRJE F-Series CIRJE-F-577, CIRJE, Faculty of Economics, University of Tokyo. [Downloadable!]
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