A Jackknife Lagrange Multiplier Test With Many Weak Instruments

This paper proposes a jackknife Lagrange multiplier (JLM) test for instrumental variable regression models, which is robust to (i) many instruments, where the number of instruments may increase proportionally with the sample size, (ii) arbitrarily weak instruments, and (iii) heteroskedastic errors. In contrast to Crudu, Mellace, and Sándor (2021, Econometric Theory 37, 281–310) and Mikusheva and Sun (2021, Review of Economic Studies 89, 2663–2686), who proposed jackknife Anderson–Rubin tests that are also robust to (i)–(iii), we modify a score statistic by jackknifing and construct its heteroskedasticity robust variance estimator. Compared to the Lagrange multiplier tests by Kleibergen (2002, Econometrica 70, 1781–1803) and Moreira (2001, Tests with Correct Size when Instruments Can Be Arbitrarily Weak, Working paper) and their modification for many instruments by Hansen, Hausman, and Newey (2008, Journal of Business & Economic Statistics 26, 398–422), our JLM test is robust to heteroskedastic errors and may circumvent a possible decrease in the power function. Simulation results illustrate the desirable size and power properties of the proposed method.