Treatment Effects Inference with High-Dimensional Instruments and Control Variables

Obtaining valid treatment effect inference 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 remove the estimation bias, we introduce a two-step procedure employing a ridge regression coupled with data-splitting in the first step, and a ridge style projection matrix with a simple least squares regression in the second. We establish statistical properties of the estimator, including consistency and asymptotic normality. Furthermore, we develop practical statistical inference procedures by providing a consistent estimator for the asymptotic variance of the estimator. The finite sample performance of the proposed methods is evaluated through numerical simulations. Results indicate that the new estimator consistently outperforms existing sparsity-based approaches across various settings, offering valuable insights for complex scenarios. Finally, we provide an empirical application estimating the causal effect of schooling on earnings addressing potential endogeneity through the use of high-dimensional instrumental variables and high-dimensional covariates.