# Publication

### Consistent Estimation with a Large Number of Weak Instruments

*Econometrica 73(5)*,

*1673-1692*, September 2005.

###### 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 = n^0.5. 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 (JIV E) 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 JIV E, are consistent provided that K_n^0.5/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 instrum