This paper develops asymptotic distribution theory for instrumental variable (IV) regression when the partial correlation between the instruments and a single endogenous variable is weak, modeled as local to zero. The authors provide asymptotic representations for various IV statistics, including the two-stage least squares (TSLS) and limited information maximum likelihood (LIML) estimators and their t-statistics. They find that these asymptotic distributions provide good approximations to sampling distributions even with only 20 observations per instrument. However, TSLS can be biased in large samples, while LIML is approximately median-unbiased in many cases. The theory offers guidelines for applied work, such as interpreting Angrist and Krueger's (1991) estimates of the returns to education. The results suggest that LIML and Anderson-Rubin (1949) confidence intervals are more reliable than conventional TSLS confidence intervals, especially when the first-stage F-statistic is small and the number of instruments is large. The paper also explores alternative estimators and test statistics, such as a modified estimator combining TSLS and OLS, and the Anderson-Rubin statistic for constructing confidence intervals.This paper develops asymptotic distribution theory for instrumental variable (IV) regression when the partial correlation between the instruments and a single endogenous variable is weak, modeled as local to zero. The authors provide asymptotic representations for various IV statistics, including the two-stage least squares (TSLS) and limited information maximum likelihood (LIML) estimators and their t-statistics. They find that these asymptotic distributions provide good approximations to sampling distributions even with only 20 observations per instrument. However, TSLS can be biased in large samples, while LIML is approximately median-unbiased in many cases. The theory offers guidelines for applied work, such as interpreting Angrist and Krueger's (1991) estimates of the returns to education. The results suggest that LIML and Anderson-Rubin (1949) confidence intervals are more reliable than conventional TSLS confidence intervals, especially when the first-stage F-statistic is small and the number of instruments is large. The paper also explores alternative estimators and test statistics, such as a modified estimator combining TSLS and OLS, and the Anderson-Rubin statistic for constructing confidence intervals.