Instrumental Variables Regression with Weak Instruments

This paper develops asymptotic distribution theory for instrumental variables regression when the partial correlations between the instruments and the endogenous variables are weak, here modeled as local to zero. Asymptotic representation are provided for various statistics, including two-stage least squares and limited information maximum likelihood estimators, Wald statistics, and statistics testing overidentification and endogeneity. The asymptotic distributions provide good approximations to sampling distributions with ten-twenty observations per instrument. The theory suggests concrete guidelines for applied work, including using nonstandard methods for construction of confidence regions. These results are used to interpret J. D. Angrist and A. B. Krueger's (1991) estimates of the returns to education.