Inference with Many Weak Instruments and Heterogeneity

This paper considers inference in a linear instrumental variable regression model with many potentially weak instruments and treatment effect heterogeneity. I show that existing tests can be arbitrarily oversized in this setup. Then, I develop a valid procedure that is robust to weak instrument asymptotics and heterogeneous treatment effects. The procedure targets a JIVE estimand, calculates an LM statistic, and compares it with critical values from a normal distribution. To establish this procedure's validity, this paper shows that the LM statistic is asymptotically normal and a leave-three-out variance estimator is unbiased and consistent. The power of the LM test is also close to a power envelope in an empirical application.