On the instrumental variable estimation with many weak and invalid instruments

We discuss the fundamental issue of identification in linear instrumental variable (IV) models with unknown IV validity. We revisit the popular majority and plurality rules and show that no identification condition can be "if and only if" in general. With the assumption of the "sparsest rule", which is equivalent to the plurality rule but becomes operational in computation algorithms, we investigate and prove the advantages of non-convex penalized approaches over other IV estimators based on two-step selections, in terms of selection consistency and accommodation for individually weak IVs. Furthermore, we propose a surrogate sparsest penalty that aligns with the identification condition and provides oracle sparse structure simultaneously. Desirable theoretical properties are derived for the proposed estimator with weaker IV strength conditions compared to the previous literature. Finite sample properties are demonstrated using simulations and the selection and estimation method is applied to an empirical study concerning the effect of trade on economic growth.

Testing Overidentifying Restrictions with High-Dimensional Data and Heteroskedasticity

This paper proposes a new test of overidentifying restrictions (called the Q test) with high-dimensional data. This test is based on estimation and inference for a quadratic form of high-dimensional parameters. It is shown to have the desired asymptotic size and power properties under heteroskedasticity, even if the number of instruments and covariates is larger than the sample size. Simulation results show that the new test performs favorably compared to existing alternative tests (Chao et al., 2014; Kolesar, 2018; Carrasco and Doukali, 2021) under the scenarios when those tests are feasible or not. An empirical example of the trade and economic growth nexus manifests the usefulness of the proposed test.