A Heteroskedasticity-Robust Overidentifying Restriction Test with High-Dimensional Covariates

We propose a new overidentifying restriction test for linear instrumental variable models. The novelty of the proposed test is that it allows the number of covariates and/or instruments to be larger than the sample size and is robust to heteroskedastic errors. We show that the test has the desired theoretical properties under sparse high-dimensional models and is more powerful than existing overidentification tests. First, we introduce a test based on the maximum norm of multiple parameters that could be high-dimensional. The theoretical power based on the maximum norm is shown to be higher than that in the modified Cragg-Donald test (Koles\'{a}r, 2018), which is the only existing test allowing for large-dimensional covariates. Second, following the principle of power enhancement (Fan et al., 2015), we introduce the power-enhanced test, with an asymptotically zero component used to enhance the empirical power against some extreme alternatives with many locally invalid instruments. Focusing on hypothesis testing, we also provide a feasible estimator of endogenous effects for practitioners when instrument validity is not rejected. The simulation results show the superior performance of the proposed test, and the empirical power enhancement is clear. Finally, an empirical example of the trade and economic growth nexus demonstrates the usefulness of the proposed tests.