Consistent Estimation with a Large Number of Weak Instruments

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  • 1 January 2004
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      • Published in RePEc
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 conce
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