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

TL;DR

This paper introduces a heteroskedasticity-robust overidentifying restriction test suitable for high-dimensional linear instrumental variable models where covariates and instruments can exceed sample size, improving power and robustness.

Contribution

It develops a novel scale-invariant test that handles high-dimensional data, surpassing existing methods in power and robustness to heteroskedastic errors.

Findings

Test performs well in high-dimensional settings

Power-enhanced version detects extreme alternatives effectively

Empirical example demonstrates practical usefulness

Abstract

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.