  This paper proposes an overidentifying restriction test for high-dimensional
linear instrumental variable models. The novelty of the proposed test is that
it allows the number of covariates and instruments to be larger than the sample
size. The test is scale-invariant and is robust to heteroskedastic errors. To
construct the final test statistic, we first 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 higher than that in the modified
Cragg-Donald test (Koles\'{a}r, 2018), 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 power to detect some extreme
alternatives with many locally invalid instruments. Finally, an empirical
example of the trade and economic growth nexus demonstrates the usefulness of
the proposed test.
