Treatment Effects Inference with High-Dimensional Instruments and Control Variables

Obtaining valid treatment effect inferences remains a challenging problem when dealing with numerous instruments and non-sparse control variables. In this paper, we propose a novel ridge regularization-based instrumental variables method for estimation and inference in the presence of both high-dimensional instrumental variables and high-dimensional control variables. These methods are applicable both with and without sparsity assumptions. To address the bias caused by high-dimensional instruments, we introduce a two-step procedure incorporating a data-splitting strategy. We establish statistical properties of the estimator, including consistency and asymptotic normality. Furthermore, we develop statistical inference procedures by providing a consistent estimator for the asymptotic variance of the estimator. The finite sample performance of the proposed method is evaluated through numerical simulations. Results indicate that the new estimator consistently outperforms existing sparsity-based approaches across various settings, offering valuable insights for more complex scenarios. Finally, we provide an empirical application estimating the causal effect of schooling on earnings by addressing potential endogeneity through the use of high-dimensional instrumental variables and high-dimensional covariates.