INSTRUMENTAL VARIABLES REGRESSION WITH WEAK INSTRUMENTS

This paper develops asymptotic distribution theory for single-equation instrumental variables regression when the partial correlations betwe en the instruments and the endogenous variables are weak, here modeled as local to zero. Asymptotic representations are provided for various statistics, including mio-stage least squares (TSLS) and limited info rmation maximum likelihood (LIML) estimators, Wald statistics, and sta tistics testing overidentification and endogeneity. The asymptotic dis tributions are found to provide good approximations to sampling distri butions with 10-20 observations per instrument. The theory suggests co ncrete guidelines for applied work, including using nonstandard method s for construction of confidence regions. These results are used to in terpret Angrist and Krueger's (1991) estimates of the returns to educa tion: whereas TSLS estimates with many instruments approach the OLS es timate of 6%, the more reliable LIML estimates with fewer instruments fall between 8% and 10%, with a typical 95% confidence interval of (5% , 15%).