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Consistent Estimation with a Large Number of Weak Instruments

  • John C. Chao

    ()

    (University of Maryland, Robert H. Smith School of Business)

  • Norman Rasmus Swanson

    ()

    (Rutgers, The State University of New Jersey, Douglass College)

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
Contact details of provider: Web page: http://icf.som.yale.edu/

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  9. 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..
  10. In Choi & Peter C.B. Phillips, 1989. "Asymptotic and Finite Sample Distribution Theory for IV Estimators and Tests in Partially Identified Structural Equations," Cowles Foundation Discussion Papers 929, Cowles Foundation for Research in Economics, Yale University.
  11. 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.
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  13. Bekker, Paul A, 1994. "Alternative Approximations to the Distributions of Instrumental Variable Estimators," Econometrica, Econometric Society, vol. 62(3), pages 657-81, May.
  14. John Chao & Norman Swanson, 2004. "Consistent Estimation with a Large Number of Weak Instruments," Departmental Working Papers 200421, Rutgers University, Department of Economics.
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