Consistent Estimation with a Large Number of Weak Instruments

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 bn-1. Our local-to-zero setup, in fact, reduces to that of Staiger and Stock (1997) in the case where bn = /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 Kn  is allowed to grow as a function of the sample size, reveals that consistent estimation depends importantly on the relative magnitudes of rn, the growth rate of the concentration parameter, and Kn. 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{Kn/rn} → 0, as n → ∞. 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 Kn/rn → 0, as n → ∞. 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 Kn 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.

JEL Classification: C13, C31

Keywords: Instrumental variables, k-class estimator, Local-to-zero framework, Pathwise asymptotics, Weak instruments